New Wave Maths - Teachers Guides: Level D - Ages 8-9 by Teacher Superstore

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New Wave Maths Teachers Guide – D Published by R.I.C Publications® PO Box 332, Greenwood Western Australia 6924

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Robert Dayman 2003

RIC-1087 ISBN 978-1-86311-708-1 Copyright Notice No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying or recording, or by any information storage or retrieval system without written permission from the publisher.

Foreword The New Wave Maths Teachers Guide has been written to both supplement and support the New Wave Maths Workbook series based on the Western Australian Mathematics Student Outcome Statements.The New Wave Maths Teachers Guide provides a summary of three documents that are at the forefront of mathematical teaching and learning:

• Curriculum Frameworks;

• Student Outcome Statements; and

• National Outcome Statements.

Between the New Wave Maths Teachers Guide and the New Wave Maths Workbook, there is a comprehensive coverage of activities to assist the development of the students' mathematical concepts. However, student progress is very much in the hands of the teacher, his or her style of teaching and the provision made for each individual to ensure complete mastery of concepts is gained.

r o e t s Bo r e p ok u S This series caters for:

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Assessment followed in this series is consistent with the approach outlined within the appropriate section in the Curriculum Framework document.

• sharing ideas through discussion;

• school–home partnerships through parent information sheets;

• mixed ability groups through the use of challenge activities; and

• the use of concrete materials where required by teachers and students.

R.I.C. Publications has a recommended range of blackline masters that, together with New Wave Maths, will ensure a thorough coverage of the mathematics outcomes and further develop the students' mathematical competency at this level.

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The author and publisher wish to acknowledge the Education Department of Western Australia for its permission to reproduce selected information contained within this document.

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References Learning Mathematics Handbook: Pre-primary to Stage Seven Mathematics Syllabus, Curriculum Programs Branch, Ministry of Education, Perth, WA – 1989 Learning Mathematics Pre-Primary to Stage Seven, Curriculum Programs Branch, Ministry of Education, Perth, WA – 1989 Curriculum Framework, Curriculum Council of Western Australia, Perth, WA – 1998 A National Statement in Mathematics for Australian Schools, The Australian Education Council and Curriculum Corporation, Australian Education Council, Carlton, Vic. – 1991 Mathematics – Student Outcome Statements, Education Department of Western Australia, 1998

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New Wave Maths Book D – Teachers Guide • i •

Contents Introduction..........................................................................................................................................................1 Appreciating Mathematics............................................................................................................................2 Learning Environment.....................................................................................................................................3 Language and Mathematics..........................................................................................................................4 Mixed Abilities.....................................................................................................................................................4 General Content Outline................................................................................................................. 5 – 11 Technology......................................................................................................................................................... 12 Assessment........................................................................................................................................................ 13 Cross-curriculum Linkages........................................................................................................................ 14

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How to Use the Teachers Notes.......................................................................................................... 16 Materials List...................................................................................................................................................... 17 Overview of Activities Term One—Units 1 – 10..................................................................................................................... 18 Term Two—Units 11 – 20................................................................................................................... 19 Term Three—Units 21 – 30................................................................................................................ 20 Term Four—Units 31 – 40.................................................................................................................. 21 Lesson Notes, Consolidation and Answers Term One—Units 1 – 10..........................................................................................................22 – 61 Term Two—Units 11 – 20.....................................................................................................62 – 101 Term Three—Units 21 – 30...............................................................................................102 – 141 Term Four—Units 31 – 40.................................................................................................142 – 181

Additional Activities

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Teachers Notes and Answers

Space Activities...............................................................................................................................184 – 185 Measurement Activities..............................................................................................................186 – 187 Number Activities....................................................................................................................................... 188

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Reference to Student Outcomes....................................................................................................... 190 Record Sheets – Blank.................................................................................................................191 – 195 Proforma – Blank.......................................................................................................................................... 196

Photocopiable Resources

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Grid Paper.........................................................................................................................................198 – 201 Number Charts and Cards......................................................................................................202 – 204 Place Value Charts.........................................................................................................................205 – 207 Number Lines and Fraction Chart ....................................................................................208 – 209 Spinners..............................................................................................................................................210 – 211 Calendar – Any year..................................................................................................................................... 212 Clocks................................................................................................................................................................. 213 Money................................................................................................................................................................ 214 Bingo Cards.......................................................................................................................................215 – 218 3-D Model Attribute Table..................................................................................................................... 219 Venn diagrams – Blank............................................................................................................................... 220 Carroll diagram............................................................................................................................................. 221 3-D Shapes...................................................................................................................................................... 222 Tangrams.............................................................................................................................................223 – 226 Nets.......................................................................................................................................................227 – 233 Paper Circles.................................................................................................................................................. 234 Graphs and Table – Blank...........................................................................................................235 – 236

o c . che e r o t r s super Parent Information Sheets

Expectations of Knowledge of Basic Facts.................................................................................... 238 Primary School Mathematics..................................................................................................239 – 240 Problem-solving Strategies..................................................................................................................... 241 Concrete to Mental................................................................................................................................... 242 Mathematical Learning Areas................................................................................................................ 243 Homework Policy........................................................................................................................................ 244 • ii • New Wave Maths Book D – Teachers Guide

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Introduction Mathematics provides methods of representing patterns, relationships and logic and developing mathematical knowledge. Students should be encouraged to speculate, observe and investigate, to explore and solve problems in mathematics in real-life situations. Mathematics is important to people in providing tools which can be used at the personal, civic and vocational level. A National Statement on Mathematics for Australian Schools, 1990 (pages 11 – 14) lists the following goals for school mathematics: 1. Students should develop confidence and competence in dealing with commonly occurring situations. 2. Students should develop positive attitudes towards their involvement in mathematics.

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3. Students should develop their capacity to use mathematics in solving problems individually and collaboratively. 4. Students should learn to communicate mathematically. 6. Students should exercise the processes through which mathematics develops.

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5. Students should learn techniques and tools which reflect modern mathematics. A National Statement in Mathematics for Australian Schools, 1990 (page 15) continues in goal identification by determining, that as a result of learning mathematics in school, all students should: 1. realise that mathematics is relevant to them personally and to their community; 2. gain pleasure from mathematics and appreciate its fascination and power; 3. realise that mathematics is an activity requiring the observation, representation and application of patterns;

(a) conduct everyday affairs such as money exchanges, planning and organising events, and measuring;

(b) make individual and collaborative decisions at the personal, civic and vocational levels; and

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4. acquire the mathematical knowledge, ways of thinking and confidence to use mathematics to:

5. develop skills in presenting and interpreting mathematical arguments; 6. possess sufficient command of mathematical expressions, representations and technology to:

(a) interpret information (for example, from a court case or media report) in which mathematics is used;

(b) continue to learn mathematics independently and collaboratively; and

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(a) that mathematics is a dynamic field with its roots in many cultures; and

(b) its relationship to social and technological changes.

New Wave Maths Book D – Teachers Guide • 1 •

Appreciating Mathematics The following attitudes are seen as fundamental to the acquisition of processes and content and should be the focus of mathematical development.The attitudes are listed in Learning Mathematics Pre-Primary to Stage Seven Mathematics Syllabus Handbook (pages 6 – 7) as: 1. an awareness of the relevance of mathematics to life; 2. an ability to enjoy mathematical games and pursuits; 3. having pride in their skills and abilities; 4. being confident of their ability to experiment and solve problems; and 5. a willingness to express ideas and hypotheses. These are summarised as part of the Appreciating Mathematics substrand found in The Curriculum Framework 1998 (page 180):

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The development of positive attitudes towards mathematics is an important goal. This may be done by:

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1. Show a disposition to use mathematics to assist with understanding new situations, solving problems and making decisions, showing initiative, flexibility and persistence when working mathematically and a positive attitude to their own continued involvement in learning and doing mathematics.

1. providing mathematical experiences relevant to the students' world;

2. providing students with mathematical opportunities to gain personal enjoyment and satisfaction; 3. providing activities which construct conceptual understanding through manipulation of materials and time to reflect on the activities; 4. allowing free discussion of mathematical experiences;

5. providing mathematical activities which are appropriate to the students' levels of development; 6. recognising that students require differing amounts of time to complete tasks as they explore problems and ideas in a variety of ways; 7. assessment that reflects the teaching methods used; and

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8. modelling positive attitudes towards mathematics.

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Learning Environment Much has been learnt about how students learn mathematics and the classroom conditions required to support that learning. The teaching of mathematics requires a suppor tive, stimulating, varied and rich mathematical learning environment that reflects the diversity of Australian society.There should be a wide range of resources that includes collected and commercial products.The classroom learning environment should encourage practical activity, the use of appropriate technology and discussion. Mathematics lessons should extend beyond a ‘chalk and talk’ or ‘textbook, pencil and paper’ subject. The Curriculum Framework, 1998 (pages 206 – 209) highlights the following perspectives on learning mathematics:

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• Opportunity to learn

Learning experiences should enable students to engage with, observe and practise the actual ideas, processes, products and values which are expected of them. • Connection and challenge

Learning experiences should connect with students’ existing knowledge, skills and values while extending and challenging their current ways of thinking and acting.

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• Action and reflection

Learning experiences should be meaningful and encourage both action and reflection on the part of the learner.

• Motivation and purpose

Learning experiences should be motivating and their purpose clear to the student.

• Inclusivity and difference

Learning experiences should respect and accommodate differences between learners.

• Independence and collaboration

Learning experiences should encourage students to learn both from, and with, others as well as independently.

• Supportive environment

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The school and classroom setting should be safe and conducive to effective learning. These perspectives have several implications for teaching. They are listed as:

• a supportive environment for learning;

• appropriate mathematical challenge is provided; and

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• fostering processes which enhance learning.

The teaching of mathematics is not definitive in approach or style but rather is influenced by the mathematical concept being taught, and the abilities, experiences and attitudes of the students. Enhanced mathematical learning is likely to occur when activities are provided which build upon and respect students’ experiences, and which the learner regards as purposeful and interesting. Feedback is critical to enhanced learning. Students need to believe that mathematics makes sense; therefore, clear and logical feedback on errors or inconsistencies is required. Students should be encouraged to take risks in a challenging environment to extend their knowledge. Challenges need to be achievable as success is critical in building positive attitudes towards mathematics. Success on easy or rote tasks does not enhance mathematical learning.

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New Wave Maths Book D – Teachers Guide • 3 •

Language and Mathematics Developing appropriate language is important to the growth of a student’s conceptual understanding. Teachers need to be aware of the natural language used by students and respond appropriately to it. To assist in developing an understanding of mathematical ideas, students need to represent their knowledge in spoken and written words; with concrete materials; pictures; diagrams and graphs; and symbols. The use and development of appropriate language should also enhance mathematical learning. The use of appropriate language helps in working through and clarifying ideas.

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Mathematical ideas are more likely to be developed when they are clearly labelled when discussed by students. Regular, clear and explicit use of mathematical expressions by the teacher is essential. Students should be encouraged to develop their knowledge and understanding of mathematical expressions by being encouraged to describe orally or in writing the situations in which they are involved.

Mixed Abilities© R. I . C.Publ i cat i ons

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Teachers need to be aware of the individual differences of all children and provide learning experiences which develop a level of success and independence for each child.To do this, teachers plan lessons that build on current knowledge and allow progress and success at the students’ own rate. New concepts should be introduced in simple form leading to the complex by using concrete materials and relevant examples. Where possible, use group work to allow for content language and ability differences. Keep parents well informed of their child’s progress and work with them to aid students in reaching their potential. Above all, provide a positive, receptive learning environment, acknowledging various differences. Students with special needs can be catered for by ensuring that fundamental concepts are understood before proceeding with dependent concepts. The identification of the initial point of difficulty must be made and the concept then developed from this stage. Instructions need to be given slowly, simply and clearly and then checked for understanding.

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New Wave Maths allows individuals to achieve at their own rate by providing a number of similar activities.The series may also be used at differing stages of students’ development so the workbook chosen is level-appropriate rather than Year-level specific, because each book is sequentially developmental with both the previous and following book. By allowing students to work to their capacity on activities, teachers are also able to provide the learning opportunities for individual students to perform at their optimum level.

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General Content Outline Goals and Guidelines After completing and understanding Year 3 well, students should then move onto Year 4. In this stage, students now begin to think abstractly rather than relying on visual perception or concrete experiences, although these aids will enhance the learning of new mathematical concepts. With the increased ability to think abstractly there is an improved capacity to think hypothetically and reason logically. Students value mathematics the more the learning experiences provided recognise their interests.The development of an ownership of their own problems and the solutions will occur if the problems attract and involve the students.

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The teaching of processes is necessary to develop independent problem solvers.Therefore, for students to acquire concepts, skills and factual knowledge, opportunities need to be provided in settings that foster positive attitudes to mathematics.The Curriculum Programs Branch, Ministry of Education, 1989, publication Learning Mathematics: Pre-Primary to Stage Seven Mathematics Syllabus Handbook (page 4) lists the following processes as part of the learning of mathematics. These processes are not tied to one particular aspect of content but are used across a range of areas:

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Students should be encouraged to persist with problems and ask questions. They are also more able to think of concepts as mathematical objects in their own right. With teaching emphasising the investigation of mathematical ideas and relationships, students should also be learning to make speculations and test them by thinking hypothetically and reasoning logically.

1. comprehension of mathematical information given in oral and/or written forms;

4. selection of appropriate operations to solve problems; 5. reflection in actions to formulate ideas;

6. expression of mathematical ideas in words, pictures and symbols; 7. construction of lists, tables and graphs;

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9. identification of patterns and relationships; 10. classification, ordering and comparing; 11. analysis and interpretation of information; 12. formulation of hypotheses; and

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8. estimation of number and measurement activities;

o c . che e r o t r s super 13. justification of conclusions and inferences.

Understanding, skills and knowledge relationships make up the content that builds up conceptual structures. In the New Wave Maths series the following areas of mathematical content are included:

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1. Working Mathematically – develops mathematical thinking processes through conceptualising, investigating, applying and verifying and reasoning mathematically.

2. Space – describes and analyses the features of objects, environments and movements through location, shape, transformations and geometric reasoning.

3. Measurement – using direct and indirect measurement and estimation skills in length, area, mass, volume and capacity and time.

4. Chance and Data – using knowledge of chance and data processes to collect and organise data, summarise and represent data, interpret data and understand chance.

New Wave Maths Book D – Teachers Guide • 5 •

5. Number – using operations, number concepts and relationships in the number system to calculate, reason about number patterns and understand numbers and operations.

The Curriculum Framework, 1998 (pages 183 – 193) identifies seven clusters of outcomes, some of these being:

Appreciating Mathematics Students appreciate mathematics through using it to assist with understanding new situations, solving problems and decision making, and show a positive attitude in learning and doing mathematics. They should also recognise mathematical origins from a range of cultures, its significance in reflecting social and historical contents and understand its significance in explaining and influencing aspects of our lives.

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Students should not wait to be told but rather be actively involved in calling on a range of problem-solving techniques, personal and collaborative management strategies and appropriate technology to find solutions to practical problems. To do this students need to choose mathematical ideas and tools to fit the constraints of a practical situation.They need to interpret and make sense of the results within the content then evaluate the work done to determine the appropriateness of the methods used. Much of the work done will involve investigation, generalisation and reasoning about patterns in number, space and data and justification of conclusions reached.

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Working Mathematically

Problems in the New Wave Maths series relate to the students' immediate physical and social world. Problems are aimed at attracting and involving children so they develop an ownership over them and their solutions. Children should be encouraged to persist with problems and checking their mathematical work. Children are encouraged to make speculations and test them under a range of circumstances.

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The classroom teacher has an important role in the development of processes used in problem-solving.Through guidance, discussion and experimentation, students are able to adopt different strategies to solve problems and appreciate that there is more than one approach to a solution. The following broad strategies may be of assistance in helping students solve non-routine problems: 1. Understand the problem – rewording, breaking into smaller parts may assist. 2. Prepare a plan to solve the problem – working from the known to the unknown, draw diagrams, tables, charts to assist.

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3. Carry out the plan – using different strategies as appropriate.

4. Review final solution to check and discuss its reliability and validity.

By asking questions of the student, providing hints (without providing answers), having students suggest strategies, guiding discussion and comparison of strategies used and providing extension to the original problem, the teacher helps the students develop processes which allow generalisation to a variety of other situations. It is the teacher’s responsibility to provide experiences which contribute to the construction of each student’s mathematical understanding. Each student is an individual with different experiences and knowledge.The teacher should recognise that because of this the student may interpret the teaching in a different way. In accepting the individuality of each student, teachers should also accept that students will interpret teaching in different ways and need new content to be presented by easily understood, believable methods and for that content to be seen as more useful than knowledge already held. Knowledge that students already hold is important to later learning and should be used as the basis for subsequent teaching through learning activities which are relevant to the students’ environment. Encouragement of discussion within the class

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allows for reflection on experiences and understanding. Where students lack the skills required to complete a task satisfactorily, more effective alternative methods that nurture their understanding need to be used. Students move through a number of phases as they develop understanding. Students manipulate the materials and work through activities guided by open-ended questioning and discussion. The activities are explored by the students using the processes as listed in Learning Mathematics Pre-Primary to Stage Seven Mathematics Syllabus Handbook, 1989 (pages 16 – 18): 1. observing and identifying; 2. comparing, ordering and classifying; 3. making patterns and arrangements;

r o e t s Bo r e p ok u S 4. constructing models;

5. estimating and measuring; 6. recording and calculating;

8. discussing what they are doing.

Following this the students express, represent and interpret their workings by: 1. discussing findings and interpretations; 2. identifying patterns and relationships; 3. using symbols and words; 4. drawing pictures, diagrams and graphs; 5. constructing models; 6. translating between relationships;

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7. inferring, predicting and hypothesising; and

9. interpreting results; and

10. communicating findings.

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Then follows a period of consolidation of understanding through further activities that embody the mathematical idea. Students should apply and extend their understanding through work in familiar, and then more novel, contexts.

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New Wave Maths Book D – Teachers Guide • 7 •

Number Students need to read, write, say, interpret and use numbers, understand the meaning, order and relative magnitude of numbers, including whole numbers, decimals, fractions, percentages and negative numbers. Students will be able to carry out the four operations, identify which operation is required in situations where there are no obvious verbal clues and understand the meaning of addition, subtraction, multiplication and division.

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Number work in the New Wave Maths series develops place value concepts to cater for understanding of large whole numbers and decimal numbers. Estimation skills should be considered in mental, written and electronic computational algorithms. In particular, estimation should be used to alert students to possible errors in their computations. Errors should be identified at their source.

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Students should be able to use mental, written and calculator computations in each operation as required. Written operations are to be seen as a backup to mental computations unable to be effected solely mentally. Calculators and computers should be used to work out repetitive, complex or lengthy calculations. An integral component of number work is the ability to estimate and approximate.

Basic facts should be known to the extent of automatic response. If knowledge is not to this level, then memorisation of basic facts should be enhanced through use of concrete materials, diagrams and calculators. Mental computational skills should continue to be developed. Calculators and pencil and paper calculations should be used as a back up to calculations that cannot be done completely in the head.

Recommended Progression for Algorithms

• Use concrete materials to manipulate and arrange objects with either oral or written answer in addition and subtraction.

• Counting equivalent sets by two, threes, fours and fives up to 20.

• Sharing objects in practical situations.

Year 2 • Using basic facts to 9 + 9 = 18 and adding three numbers each less than 6. It is recommended that concrete materials are used.

• Symbol ‘x’ is introduced to assist with grouping. Use of language to support activities – ‘lots of ’, ‘sets of ’ or ‘groups of ’ to 20 or 30.

• Division experience is through sorting, sharing and grouping activities using concrete materials.

• Introduction to open number sentences. For example:

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4–1=

3 lots of 4 =

Year 3

• Activities without regrouping may be done without concrete materials; for example: 4 +5

17 +2

21 23 + 14

372 + 416

• All subtraction working out must start with top line number; for example, 9 take 7 equals 2. 36 – 3

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43 + 24

54 – 22

469 – 217

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• Addition and subtraction requiring regrouping should be done with the assistance of concrete materials, particularly Base 10 MAB; for example:

18 + 19 =

21 + 14 + 37 =

256 + 48 =

329 + 257 =

76 – 25 =

100 – 60 =

700 – 300 =

638 – 73 =

• Use Base 10 MAB and other concrete materials for multiplication; for example:

40 x 2 =

42 =

34 x 2 =

200 =

6 x 100 =

x 40

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30 x

= 90

• Division is set out as shown in the examples below. 6 4 24

24 ÷ 4 = 6

Year 4

• Addition and subtraction with regrouping and up to two decimal places; emphasise use of linear measure and money.

• Written multiplication of sums as shown by these examples. 30 x 6

54 x 2

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x 21

18 x 3

• Initially using basic facts in division, such as: 5 5 25

8r1 6 49

• Later with dividend less than 100 and divisor up to 10, such as: 3 96

5 76

Year 5 • Addition and subtraction examples extended to three decimal places with regrouping.

• Addition and subtraction of fractions with like denominators is introduced. Emphasis on concrete support. 1 + 1 = 4 4

2 + 3 = 8 8

3 – 1 = 4 4

7 – 3 = 10 10

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1 + 1 = 1 3 3

2 – 1 = 3 3

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• Multiplication of whole numbers to two digits by two digits; for example: 463 x 6

• Also, multiplication of a number with up to two decimal places by a whole number.

• Division extends to examples such as: 4 753

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34 x 23

40 720

3 60.24

50 267

New Wave Maths Book D – Teachers Guide • 9 •

Measurement Students use direct and indirect measurement and estimation skills in length, area, mass, volume and capacity, time and angles. Measurement work needs to be practical, concrete and relevant and students encouraged to make sensible choices as to which units to use. Estimation skills should be continuously developed. They are now conservers of mass and area and still require concrete experiences to assist in mathematical learning. Pictorial and symbolic representation is used more. In the New Wave Maths series, measurement activities are focused on developing an awareness of particular attributes of objects and events including length, area, mass, volume and capacity and time. Activities extend from early notions of more, less or equal. A great deal of early work involves direct comparison of quantities. Learning to estimate is given due emphasis.

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Students are able to use and understand the language of chance and from this make a statement about the likelihood an event will occur.

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Chance and Data

Students are to be able to plan and undertake data collection and then to organise, summarise and represent data for effective and valid interpretation and communication. Students are able to locate data that has been published, interpret, analyse and draw conclusions from this data taking into account data collection techniques and chance processes involved. In the New Wave Maths series, students are directed to make sensible judgments about the quality of the data and then to make a decision and draw inferences from the data.

Chance and data work should focus on collecting, representing and interpreting data. Data collection activities should lead to classification, organisation, summarising and displaying in a variety of ways. In the New Wave Maths series, students are introduced to activities that include an element of unpredictably and refine their use of some of the everyday language of chance.

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Classification skills are developed through a variety of activities. Where practical, students are asked to record and represent data. Students are directed to construct graphs or represent data in a format that is logical and easy to read.

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Space Students are to recognise shapes as well as visualise, draw and model shapes, locations and arrangements and predict and show the effect of transformations on them. Using their knowledge of shapes, transformations and arrangements, students are able to solve problems and justify solutions. Space activities should emphasise the investigation of the features of objects in the environment, including their shape and the effect on them of changes in shape, size and position, and include symmetry and tessellations. The features of objects should be emphasised in space activities. Relationships between three-dimensional shapes and two-dimensional shapes are represented by nets, diagrams and scale models. Sorting and classifying of shapes continues. Angles and directions are related to compass directions.

r o e t s Bo r e p ok u S Pre-Algebra

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The New Wave Maths series develops space exploration of the students' own environment and objects within it. By manipulating materials in a variety of ways students learn to observe and describe them in everyday language. Estimation and measuring skills using standard units should be completed.

Work in algebra is based on patterns in space and number strands. Relationships between two quantities should be noted when one of the quantities is varied. Where possible, relationship graphs should be used to explain relationships. Students should be finding ways to explain generalisations in these early stages of development of algebra. There is little algebra covered in the New Wave Maths series; however, teachers should be aware of this outcome, particularly for talented students who may recognise and describe the nature of variation in situations and are able to read, write and understand the meaning of symbolic expressions. They may also write equations and inequalities to describe situations.

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Number patterns are covered in much of the number work, which in turn leads to the development of algebra.

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New Wave Maths Book D – Teachers Guide • 11 •

Technology Calculators are an important technological resource in the teaching and learning of mathematics. The calculator should be used as both an instructional aid and as a computational tool. With the advent of cheaper and more sophisticated calculators there comes a natural deemphasis on written calculations.There is, as a consequence, a reduction in the complexity of written computation work but a clear emphasis on the use of concrete material to improve understanding of concepts to be developed through the New Wave Maths series. Greater emphasis is placed on quick and accurate mental computation. Students’ expected level of written computational skill is to a two-digit by two-digit multiplication, addition or subtraction sum, and a single divisor into a two-digit number for division.

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Mental calculations and calculator use need to be developed as these form the basis of most computational needs of adults in real-life situations. It is strongly recommended that all students use calculators at all Year levels (K–12). The Learning Mathematics Handbook Pre-Primary to Stage Seven Mathematics Syllabus, 1989 (pages 30 – 31) details where calculators can be used as an instructional aid to:

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An emphasis on knowledge of basic addition and multiplication facts and relationships, place value understanding, estimation, checking of results and confidence in applying appropriate calculations is essential.

• assist in the development of mathematical content and processes; for example, place value, multiplication as repeated addition and the learning of basic facts;

• provide immediate feedback on a student’s own calculation so errors and misunderstandings can be remedied; and

• improve attitudes towards mathematics through its effective use.

• enable attention to focus on mathematical processes by allowing calculations to be done swiftly and accurately by all children; for example, in problem-solving or investigative activities;

• enable rules or patterns to be discovered and investigated, by generating many examples in a short time;

• encourage students to employ a wider range of strategies to solve problems; and

• allow students to use data drawn from real life, rather than artificial numbers chosen to make the computation easier; for example, in exploring distances or costs of shopping.

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As a computational aid, the calculator can:

Computers also have their place in the mathematical learning environment and should be accorded appropriate time. Computers may be used for ‘number crunching’ and data analysis; as a simulation device; for graphics and symbol manipulation; and for running spreadsheets.

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Teachers need to select software which is sufficiently flexible and open-ended to allow students to develop their own ideas and use their initiative. The computer can be used in problem-solving, investigations, modelling, strategy games, refining ideas, concept development, skill development and gaining factual knowledge. There is still a place for textbooks in the teaching and learning of mathematics. However, emphasis must be placed on the need to use a variety of print materials. No single text is likely to cater for the interests of all students or cover the mathematics curriculum in full. The New Wave Maths series provides a solid foundation and allows teachers the opportunity to add their own ideas and activities to suit their individual class and students.

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Assessment Assessment is a critical component of the teaching program and is outlined in The Curriculum Framework, 1998 (pages 210 – 212) by these points:

• Valid

Assessment should provide valid information on the actual ideas, processes, products and values which are expected of students.

• Educative

Assessment should make a positive contribution to students' learning.

• Explicit

Assessment criteria should be explicit so that the basis for judgments is clear and public.

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• Fair

Assessment should be demonstrably fair to all students and not discriminate on grounds that are irrelevant to the achievement of the outcome. • Comprehensive

Judgments on student progress should be based on multiple kinds and sources of evidence.

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Assessment is a crucial aspect of the mathematics learning process. Assessment provides feedback on individual development to the student, teachers and parents. It provides the information for future teaching. All the outcomes of the school mathematics curriculum should be reflected in the assessment process. All assessments should be demonstrably fair, valid and reliable. The fairness of mathematical testing is brought into question by the practice of using one form of test only. Individual students respond to different environments in different ways; therefore the use of a single assessment tool, such as a pencil and paper test, may be valid and reliable but not fair, as the individual may respond better to short-answer questions, extended response questions or other forms of assessment. Hence, using nonrepresentative sampling of the mathematics curriculum outcomes or narrow sampling methods of assessment may be unfair to many students.

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It is clearly understood now that conventional forms of tests (pencil and paper) cannot address all areas of the mathematics curriculum; therefore, additional, not alternative, methods of assessment must be developed. Such methods include: teacher observation and questioning; structured interviews with students; paper and pencil tests; oral tests; practical skill tests; work- or project-based assessment; collected samples of students’ independent work; individual homework assignments; group reports; anecdotal records; self-assessment; and peer assessment.

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It is recommended that students’ mathematics be assessed using the Student Outcome Statements. Commercially prepared assessment packages are available from R.I.C. Publications® as follows: Maths Assessment Level 1 (RIC-0028) Maths Assessment Level 2 (RIC-0029) Maths Assessment Level 3 (RIC-0030) Maths Assessment Level 4 (RIC-0087) Where possible, links to these pages have been included in the teachers notes, pages 22 – 181. New Wave Maths is not a stand-alone assessment document. Activities may be assessed based on Student Outcome Statements. Teachers will need to be familiar with these to make the appropriate assessments. All activities may be assessed in this way. It is suggested that a random sample of activities only is assessed using Student Outcome Statements to determine progress.

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New Wave Maths Book D – Teachers Guide • 13 •

Cross-curriculum Linkages The learning and application of mathematics occurs across all curriculum areas. Literacy skills are developed in the English learning area where language foundations are provided that are essential for the learning of mathematics. Mathematics also provides for the development of language skills.Together, English and mathematics provide the information skills used in activities such as reading the newspaper, information text such as a telephone directory, and preparing and presenting reports. Spatial and measurement tasks are interwoven in many art activities which may in themselves provide alternative stimulus for the learning of mathematical skills. Data collection and interpretation skills as well as measuring activities are a part of both The Society and Environment and Health and Physical Education areas.

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Science provides for a variety of measurement activities with particular emphasis on the measurement component.

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Design activities and spatial knowledge development are a practical component of the Technology and Enterprise learning area. Activities in this learning area provide a wider diversity of learning opportunities than those provided from the basic mathematics syllabus.

The cultural significance of mathematics, its origins and different developments may be explored in the Languages Other than English and Society and Environment learning areas.

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• 14 • New Wave Maths Book D – Teachers Guide

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Teachers Notes and Answers

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Contents

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New Wave Maths Book D – Teachers Guide • 15 •

How to Use the Teachers Notes Unit and student page shown here as a quick reference to the equivalent page in the student workbook.

Indicators from the Student Outcome Statements have been included as a quick guide. These are directly related to the main activity only.

Outcomes relevant to all activities on the student workbook page have been listed as a ready reference.

A space for you to record notes relevant to the lesson has been provided. This space could be used for any purpose. Some suggestions: • record any improvements you made to the lesson; • record any problems you or your students experienced during the lesson; • record individual student's progress or development; • add any ideas for extension or remediation of the lesson; or • include any interesting facts or ideas you came across which were relevant to the lesson.

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Skills relevant to the main activity have been listed.

Resources have been listed to aid organisation before the lesson.

Language terms relevant to the workbook page have been listed here. It is preferred these words be introduced before beginning the activity to ensure students have a clear understanding of the terminology used in the activities.

The student workbook page is broken into distinct sections. These are each discussed in detail in this section of the teachers notes. The section is stated, followed by the relevant outcome in brackets. Then bullet points are used to guide you through the activity.

The great thing is that once this information is recorded, when you come to teach the lesson again, these notes will refresh your memory and enhance the smooth running of the lesson.

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This section is a guide only and you are more than welcome to take from it what you choose, modify it or add your own touches.

Answers have been provided to assist teachers in marking students' work. Some answers do require a teacher check as they are dependent on the classroom environment and the students in your class. Where possible, all answers are given. The answers for the Challenge activities are generally an example of one possible solution, as many solutions are often possible.

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Where possible, links to a relevant assessment activity in the R.I.C. Publications® Maths Assessment Level 3 document have been provided.

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Suggested activities for consolidation of the main activity on the workbook page have been provided as a guide only. Feel free to use, modify, extend or disregard these as you feel necessary. • 16 • New Wave Maths Book D – Teachers Guide

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Materials List The following list of suggested materials is a guide only. It is not suggested that they must be purchased or are the only items that may be used. If compiling a set of materials that will both supplement and compliment the teaching program, the following items will assist. Some items are required to complete the workbook activities. These are listed in more detail on the relevant page in the teachers notes.

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• Denotes items produced in New Wave Maths Teachers Guide as a blackline master which are available on pages 198 to 236. Teachers may photocopy and use them with their class(es). – A4 and A3 adhesive tape spinners • pages 210 – 211 height measuring stick aerosol caps square tiles hoops analog clock/watch stopwatch ice-cream containers atlas/street directory straws insect – model or picture attribute blocks string interlocking cubes balance sundials jellybeans balloons tangrams • pages 223 – 226 kitchen scales balls (various kinds) tape measure large circles • page 234 Base 10 MAB tessellating sets lead pencil bin timer leaves bottle tops toothpicks or equivalent light card – coloured or plain bucket trundle wheel – A4 and A3 bundles of 10s and 1s wire magazines calculator wool markers (cones) calendar • page 212 1-cm grid paper • page 199 measuring containers (mL/L) candles 2-cm grid paper • page 200 metre rule cardboard strips 1- and 2-cm cubes mirror/mira cardboard tubes modelling clay chalk money (coins/notes) • page 214 class members nails clock – clock faces • page 213 nets • pages 227 – 233 clock stamp newspapers coloured counters number cards • page 204 coloured pencils number charts • page 202 – 203 coloured rods number lines • page 208 compass number squares containers – various shapes and sizes objects for weighing activities cotton overhead of a Carroll diagram • page 221 cups overhead projector dice – 6- and 10-sided overhead of a Venn diagram • page 220 digital clock/watch packs of playing cards dominoes paper clips dot paper • page 198 paper – coloured or plain drawing pins – A4 and A3 eggtimer pattern blocks elastic bands pegboards erasers pegs felt-tip pens pipe-cleaners fishing line place value charts • pages 205 – 207 flashcards 0 – 9 plastic polygons flashcards – number names playing cards fraction cake polyominoes fraction/decimal number line • page 209 popsticks fraction grid • page 209 protractor fraction squares reading or library books geoboards ruler geometric blocks scissors geostrips shapes – 2-D glue shapes – 3-D graphs • pages 235 – 236 skipping rope gumnuts small plastic bottles heavy card – coloured or plain pages ®

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New Wave Maths Book D – Teachers Guide • 17 •

Term One Week Unit Outcomes

Page

1 1 S3.1—Follow compass directions to move between places.

N3.1a—Identify place value of three-digit numbers.

M3.2, C&D3.3—Use uniform units to measure and compare the area of an object. Record and display data in a pictograph.

2 N3.1a—Use Base 10 MAB to represent two- and three-digit numbers.

C&D3.3—Collect, record and display data.

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N3.1a—Enter two- and three-digit numbers in a place value chart.

3 3 S3.1—Draw items on a grid according to coordinates. Identify half turns clockwise and anticlockwise and full turns.

M3.2—Use a ruler to measure lines to the nearest centimetre. 4

4 S3.3—Identify and continue patterns based on colour, shape, symmetry and systematic movement.

N3.4, M3.2—Identify and complete number patterns. Use a balance scale to compare the mass of objects.

5 S3.3—Identify tessellating shapes.

N3.3—Reorder addends to make addition easier.

N3.4—Represent square number patterns in arrays on a grid. 6

N3.1a—Rearrange order of numbers. Identify place value in three-digit numbers. Convert measures to metres only. 5

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N3.4, N3.1a—Identify and follow rules to complete number sequences. Count in tens from various starting numbers.

6 WM3.2, WM3.3, M3.3—Work mathematically to estimate the time taken to complete an event.

15 16 17

N3.3, N3.1a—Reorder addends to make addition easier. Order measurements represented in decimals.

7 S3.2, S3.3, S3.4—Show and match congruent parts on 2-D and 3-D objects.

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N3.3, N3.1a—Reorder addends to make addition easier. Identify the higher and lower three-digit number.

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C&D3.1, C&D3.2, C&D3.3, C&D3.4—Collect, record and analyse data. Rate data from most likely to least likely.

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M3.2, WM3.1—Construct shapes with the same volume, using cubes. Compare the string length and swings per minute of pendulums.

8 8 N3.3, C&D3.2, C&D3.3, C&D3.4—Complete a speed test of multiplication and division basic facts. Record and graph results.

N3.4, C&D3.2, C&D3.4—Identify and describe number patterns in shapes. Record and analyse data.

N3.1a, N3.3—Complete number sentences using =, ≠, not true, less than or greater than. Complete number sentences involving multiplication of basic facts.

9 S.3.1, M3.4b—Draw a bird’s-eye view of a detailed plan.

N3.3, N3.4—Complete number sentences involving the multiplication of single-digit numbers by 100.

M3.4a, M3.2, C&D3.3—Measure the area and perimeter of shapes.

10 N3.1a, M3.2—Round three-digit numbers to the nearest ten. Read the time on analog clocks.

C&D3.1—Decide if an event is unlikely, likely or certain.

N3.1a—Read and write numbers in numerals and words.

• 18 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Term Two Week Unit

Outcomes

Page

1 11 S3.2, S3.4—Write descriptions for solid three-dimensional shapes.

S3.2—Construct models following a plan, using 2-cm cubes.

M3.1—Choose appropriate units of measure and measuring devices.

12 N3.1a—Rearrange digits to make different numbers.

C&D3.3, C&D3.4—Complete tree and arrow diagrams.

N3.1b—Identify the correct diagram to shade and show given fractions.

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13 S3.2—Sketch 3-D shapes from models.

37 38

M3.2, M3.4a, WM3.3—Work mathematically to measure the circumference of a hoop.

14 N3.1b—Identify equivalent fractions in diagrams.

M3.2, C&D3.2, C&D3.3—Measure and compare the length and mass of objects with uniform units. Record and summarise data by tallying. N3.1a, N3.3, N3.2—Use rounding techniques to estimate answers. Solve division word problems. 5

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N3.3, N3.1a—Estimate and check multiplication of whole numbers by 10. Trade with Base 10 MAB.

15 M3.2, M3.4a—Double the dimension of a shape to compare the height, width and area.

N3.3, N3.1a—Use a variety of methods to add two-digit numbers. Reorder a four-digit number to make new numbers.

M3.2—Read and show the time on analog clocks to the nearest five minutes.

40 41 42 43 44 45

6 16 N3.1a, N3.3, N3.2—Use rounding techniques to estimate answers to multiplication problems. Solve division word problems.

S3.2, WM3.2—Construct and draw 3-D models given set criteria.

N3.1a, N3.3—Use rounding techniques to estimate answers to subtraction problems.

17 WM3.2, S3.3—Pose, ask and contribute mathematical questions prompted by a specific stimulus.

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N3.2—Identify prime and composite numbers. M3.2, C&D3.2—Interpret a calendar. List events in order. 8

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18 N3.1a, M3.2—Convert cents into dollars and centimetres to metres. Measure lines in centimetres.

49 50 51 52

C&D3.1, C&D3.2, C&D3.3, C&D3.4—Investigate chance events and tally, graph and summarise results.

N3.4, N3.2, N3.3—Build 3-D models according to a given arrangement. Solve multiplication problems.

19 S3.3—Identify symmetry in 2-D shapes.

N3.2, N3.3, N3.4—Estimate, then use a calculator to solve repeated addition and subtraction problems.

M2.2—Measure and order lines in centimetres. Measure the area of an irregular shape.

10 20 N3.3, WM3.4—Estimate, then use a calculator to solve addition problems. Discuss estimation techniques.

C&D3.3, N3.2, S3.4—Categorise and represent data in a Venn diagram.

N3.1b—Show specified unit fractions in collections of objects.

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New Wave Maths Book D – Teachers Guide • 19 •

Term Three Week Unit Outcomes

Page 61

N3.1a—Calculate amounts of money. Show amounts of money in different ways.

M3.2, WM3.3—Devise a method for measuring height without direct measurement.

2 22 N3.4—Follow rules to continue and describe number patterns on a 1 – 100 grid. Generate and describe own patterns.

C&D3.1, C&D3.2, C&D3.3, C&D3.4—Investigate and record the results of chance activities.

N3.1a, N3.3—Demonstrate understanding of place value to complete multiplication and division number chains.

3 23 S3.3—Copy a shape with its dimensions halved and then doubled.

N3.1a—Use the correct decimal notation to record amounts of change. Show amounts of money using the fewest number of notes and coins.

S3.2—Identify side, top and front views of 3-D models.

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1 21 S3.3—Complete symmetrical drawings.

4 24 N3.3, C&D3.2, C&D3.3, C&D3.4—Complete a speed test of addition and subtraction basic facts. Record and graph results.

C&D3.3, C&D3.4—Collect, record and analyse data in a Carroll diagram.

N3.2, N3.3—Double single- and double-digit numbers.

5 25 S3.3, WM3.4—Investigate, describe and draw systematic movements of shapes. N3.3, N3.4—Identify, complete and describe a multiplication number pattern.

M3.2, C&D3.3, C&D3.4—Measure, record and graph wrist perimeters.

C&D3.2, C&D3.3—Survey, tally and analyse data.

N3.1.b—Show equivalent fractions on diagrams.

7 27 S3.3, M3.2—Reduce and enlarge shapes on grid paper. Measure and compare height, width and area.

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N3.1b—Order fractions from smallest to largest.

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6 26 N3.4—Identify, complete and describe number patterns.

WM3.1, WM3.2, C&D3.2—Work mathematically to describe how chance is built into some familiar games.

8 28 N3.1a, N3.3—Use rounding techniques to estimate answers to addition problems.

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C&D3.1, C&D3.2, C&D3.3, C&D3.4—Investigate and record the results of chance activities.

N3.1b—Identify fractional equivalents. Identify the greater fraction in a pair.

9 29 S3.2, S3.4—Draw and compare a variety of four-sided shapes.

N3.1a, N3.3—Use rounding techniques to estimate the difference between measurements. Solve, or check with a calculator.

M3.2, N3.1a, M3.4a—Measure objects in centimetres and metres. Measure the perimeter of regular polygons.

10 30 N3.3—Halve single- and double-digit numbers. Identify odd and even numbers. Multiply single-digit numbers by 100.

C&D3.2, C&D3.3, C&D3.4, N3.2—Collect, record and analyse data.

N3.2, N3.4, N3.1a—Identify multiples on a grid. Order numbers from smallest to largest.

• 20 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Term Four Week Unit

Outcomes

Page 91

N3.3—Solve multiplication basic facts.

M3.2, M3.3—Estimate, measure and compare the mass of cupfuls of different objects.

2 32 S3.3, C&D3.3—Investigate symmetry in upper case letters and record in a Venn diagram.

C&D3.1, C&D3.2, C&D3.3, C&D3.4—Investigate and record the results of chance activities.

N3.3, N3.2, N3.1a—Divide amounts of money. Trade with Base 10 MAB.

3 33 S3.3—Describe systematic movements of shapes.

N3.1a—Round three-digit numbers to the nearest ten or hundred.

M3.2, M3.1—Measure the mass of objects. Record the number of objects needed to balance with another.

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4 34 N3.3, M3.2—Solve multiplication word problems. Measure the area of squares. C&D3.1, C&D3.3—Describe and order situations from least to most likely. Collect and record data on a graph. S3.2, S3.4—Investigate vertices, edges and faces of 3-D shapes.

100

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1 31 S3.1, M3.4b—Draw a bird’s-eye view of a familiar location.

101 102

5 35 S3.2, S3.3—Investigate symmetry in cross-sections of 3-D shapes.

103

N3.1a—Write whole numbers in words. Count up and down by 10 and 100.

104

M3.2, M3.3—Measure distance using a trundle wheel. Estimate and measure body measurements.

105

6 36 WM3.2, WM3.4, N3.1a, N3.2, N3.3—Work mathematically to discuss and explain the best way to solve a problem.

106

C&D3.3, C&D3.4, WM3.2—Interpret a tree diagram.

107

N3.1a, N3.3—Read and say whole numbers in the thousands. Regroup notes and coins. Estimate and solve word problems.

108

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N3.4—Describe and create number patterns made with a calculator. M3.2, C&D3.3—Measure and record time intervals.

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8 38 N3.4—Follow instructions to find number patterns in tables.

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7 37 S3.1, M3.4b—Draw a map of a specific route.

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C&D3.2, C&D3.3—Use tallies to record data in straightforward tables.

110 111 112 113

N3.3, N3.1a—Use conventional algorithms to solve word problems.

114

9 39 S3.1—Show a sense of proximity when placing items on a plan.

115

N3.1a—Read and write whole numbers into the thousands.

116

M3.2, C&D3.3—Time events using a stopwatch. Record data and display in a bar graph.

117

10 40 N3.2—Match the correct algorithm to a word problem. Write a word problem to match an algorithm.

118

M3.3, WM3.4—Estimate and measure weight of objects. Evaluate and improve estimates with successive objects.

119

N3.2—Understand that multiplication is repeated addition.

120

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New Wave Maths Book D – Teachers Guide • 21 •

Unit 1–1

Student page 1

Outcomes

Indicators

N3.3, S3.1

The student is able to: • use directional language associated with quarter and half turns (such as north, south, east, west, right angle, quarter turn, right, left) to describe a route.

Skills • using a compass • measuring • recording • comparing • planning

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • ruler • 1-cm grid paper (see page 199) • compass

Language • direction • north • south • east • west • distance • centimetres • length • path

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Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.1) Warm up

• Discuss with the students the path they take coming to school. Ask: – Do you always travel the same way to school? – Do other children in your street take the same pathway to school as you do? – How many different ways could you come to school? • Introduce the concept of a compass and discuss the directions—north, south, east and west. Also discuss the different situations in which a compass could be used. Write the students’ suggestions on the board. • Have a compass available for students to view and use. Allow them the time and opportunity to become familiar with its function.

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• The focus for this unit is multiplication tables and addition of two numbers each less than 10.

What to do

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• Read through Exercises 3 and 4 with the students to ensure a clear understanding of the task. • Students complete the exercises, marking the pathway, then measuring its length and recording their answers. • Students move into small groups to compare their pathways and length. Students whose pathway differs should be encouraged to take another look at the directions provided. • For Exercise 5, suggest to the students to draw their 20-cm pathway on a sheet of grid paper before attempting to write the instructions. In this way, they already have the answer for checking purposes, making the process a lot easier to follow.

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• Discuss with the students what they think the question is asking. Is it asking to measure the length or width of the foot, or, is it asking to measure the area? • Ask students how they might be able to work out the area of their foot. • Allow them to work out the size of their foot using their own method. • Students can move into small groups to discuss how they solved the problem and compare their foot sizes. • As an extension activity, students can order their feet from smallest to largest. For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 6 – 7. • 22 • New Wave Maths Book D – Teachers Guide

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Unit 1–2

Student page 2

Outcomes

Indicators

N3.3, N3.1a

The student is able to: • produce and use standard partitions (that is, into hundreds, tens and units) of two- and threedigit numbers.

Skills • discussing • recording • reading numbers • defining values

Resources

Language

• calculator • Base 10 MAB

• place value • hundreds • tens • ones • value • digit

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Memory Masters (N3.3)

Notes

Number (N3.3)

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• The focus for this unit is multiplication tables and addition of two numbers each less than 10.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a) Warm up

• Review the relationship between numbers; i.e. the position of the number decides its value. – How does the way we order the numbers affect the number? – Would it matter where we position numbers?

• Discuss the meaning of ‘place value’ with the students. (The position of each number within a number tells the value of each number.) Direct students to look at the first example in Exercise 3. Three hundred and sixty-four—the three is in the hundreds position, so its value is 300. The six is in the tens position, so its value is 60 and the 4 is in the ones position, so its value is 4. When vocalising numbers such as ‘three hundred and sixty-four’, the place value is made clear.

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• Work through the second example together. In the number 817, what is the value of the eight? What is the value of the one? What is the value of the seven? Students write the numbers in the correct boxes in the table. • Students then complete the table. Some students may need more explicit instruction and the use of MABs may help them to visualise the value of each number. • Discuss with the students what Exercise 4 is asking. Is it asking to state the value or the position of each number? (It is asking for the position.) They will need to write the names of the position of each number. Therefore, their answers should be either hundreds, tens or ones. Study the first example which has been done for them. Work through the second example as a group and then ask the students to complete the rest of the exercise on their own. • Discuss with the students what Exercise 5 is asking. Is it asking to state the value or the position of each number? (It is asking for the value.) They will need to write the value of each number according to its position. Study the first example which has been done for them. Work through the second example as a group and then ask the students to complete the rest of the exercise.

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• Students may find this task easier to complete with the use of a calculator. It will also help develop the students’ calculator skills. • Discuss the use of +, –, x and ÷ signs. In what situations would we use each sign? • Allow students the opportunity to explore possible solutions. Students will have different approaches and follow different pathways to find a solution.The method used by each student should be praised if it reaches the required result.This is part of developing the skills for working mathematically.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 36 – 37. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 23 •

Unit 1–3

Student page 3

Indicators

Outcomes N3.3, M3.2, C&D3.3

The student is able to: • directly compare the areas of two regions by superimposing one on the other and making visual adjustments for non-overlapping parts or by cutting and rearranging the pieces of one to fit over the other.

Skills • finding area • recording • graphing

• use a uniform unit to compare the areas of two regions where the units are reasonably small relative to the shape.

Resources • calculator • Base 10 MAB • counters • square tiles (use pattern blocks) • triangles (use pattern blocks) • leaves

Language • cover • outline • pictograph • number

r o e t s Bo Notes r e p ok u S • display data in pictographs where each symbol represents more than one unit.

Memory Masters (N3.3) Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results. • Note: Additional teacher instruction may be required as students attempt regrouping with addition for the first time.

Main Activity (M3.2, C&D3.3) Warm up

• Inform students they will be using a variety of objects to measure their hand outline. One of these will be leaves. Take students outside to collect a number of fallen leaves and sort into similar sizes. (This could be done immediately prior to the lesson or on another day.)

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• Ask students to trace around their hand on a separate sheet of paper. • Ask students to find one object or item, other than their hand, that will cover the outline of their hand. • Distribute counters, square tiles, triangles and the leaves students have collected that can be used to cover the hand outline. • Read through the instructions for the activity, then ask the students to use the counters to cover their hand outline and record how many counters they used. • Repeat using square tiles, triangles and leaves. • Using the records for each of the items used to cover the hand outline, discuss how to complete the pictograph with the students. • Remind students that one or more symbols can be used to represent the counters, squares, triangles or leaves. • If students are unable to fit the required symbols in the space provided as a single line, ask them how else they might show their data collection. • Answers may include—use two or more columns, let one symbol show two or more counters, triangles, squares or leaves. • Remind students they must use the same ratio of items used to symbols drawn when completing the pictograph.

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• The focus for this unit is multiplication tables and addition of two numbers each less than 10.

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Challenge • Ask students to explain how they can tell whether large numbers are odd or even. Examples: 6847 3974 5492 8931 2530 • Students may be directed to work with smaller numbers; e.g. between 10 and 50 to see if they can discover the clue. • Students should discover they only need to view the digit in the ones place to verify the answer.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 114 – 115. • 24 • New Wave Maths Book D – Teachers Guide

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Unit 1—Answers

Student pages 1 – 3 Unit 1–1

Unit 1–2

Memory Masters 1. (a) 4 (b) 12 (c) 4 (d) 0 (e) 8 (f) 10 (g) 13 (h) 6 (i) 6 (j) 13 Number 2. (a) 56 (b) 69 (c) 98 (d) 97 (e) 69 (f) 88 Main Activity 3.

Memory Masters 1. (a) 12 (b) 7 (c) 0 (d) 1 (e) 9 (f) 8 (g) 6 (h) 9 (i) 11 (j) 10 Number 2. (a) 157 (b) 148 (c) 146 (d) 127 (e) 145 (f) 178 Main Activity 3. 817 265 734 489 972 364

300 60 4

800 10 7

200 60 5

700 30 4

400 80 9

900 70 2

4. (a) tens (d) hundreds (g) hundreds (b) ones (e) tens (h) ones (c) ones (f) hundreds (i) tens 5. (a) 200 (d) 300 (g) 60 (b) 3 (e) 800 (h) 30 (c) 6 (f) 5 (i) 30 Challenge Answers will vary; possible solutions working left to right: 4÷4=1 4+4–4=4 4 x 4 + 4 + 4 + 4 + 4 + 4 + 4 ÷ 4 = 10

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Place Value

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4. 15 cm 5. Teacher check Challenge Answers will vary

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• Use 1-cm grid paper (see page 199) to plan and give compass directions to follow between two places.

Consolidation 1–2

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Memory Masters 1. (a) 2 (b) 14 (c) 0 (d) 16 (e) 6 (f) 15 (g) 7 (h) 2 (i) 11 (j) 14 Number 2. (a) 63 (b) 66 (c) 82 (d) 91 (e) 81 (f) 72 Main Activity 3. Teacher check 4. Teacher check Challenge Viewing the digit in the ones place will verify if a number is odd or even.

• Provide further opportunities to practise identifying the place and value of digits in randomly selected numbers.

Consolidation 1–3

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• Repeat the activity using an irregular classroom object as the item to measure.

New Wave Maths Book D – Teachers Guide • 25 •

Unit 2–1

Student page 4

Outcomes

Indicators

N3.3, N3.1a

The student is able to: • read and write any whole number into the thousands. • count up and down in tens from any starting number.

Skills • counting • writing numbers

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • paper • 0 – 9 flashcards (see page 204)

Language • thousands • hundreds • tens • ones

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Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a) Warm up

• Revise what different pieces of Base 10 MABs represent. Display each piece, and ask for the students to give you the answers. • Provide each student (or pair) with a number of Base 10 MABs. – Ask the students to put out 2 longs and 3 ones—what number does this make? – Ask the students to put out 1 flat and 4 ones—what number does this make? – Ask the students to put out 1 flat, 2 longs and 2 ones—what number does this make? • Continue with other examples, until students have developed the concept that the Base 10 MABs can be used to represent numbers.

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Teac he r

• The focus for this unit is multiplication tables and addition of two numbers each less than 10.

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• Direct students to complete the examples on the page. Some students may need to physically reproduce the pictures with their blocks. • Once students have completed the examples on the page, direct them into pairs. • Within their pairs, take turns at putting out various pieces of Base 10 MABs. Draw the blocks and write the numbers represented.

Challenge

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What to do

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• Distribute 0 – 9 flashcards to the students. • Ask the students to select two number cards to make a new number. For example, 3 and 4. Students could either present these cards as 34 or 43. • Students may find it easier to work with number cards, as they can be rearranged accordingly until the students are satisfied with each solution.These can then be recorded on the page. • Ask students to complete the first example. As a class, discuss the possible numbers they made from 6, 3 and 8. This provides less able students with an insight how to solve the problem. • Allow the students time to complete the remainder of the activity.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 36 – 37. • 26 • New Wave Maths Book D – Teachers Guide

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Unit 2–2

Student page 5

Outcomes N3.3, C&D3.3

Skills • measuring • tallying • graphing

Indicators

Resources

Language

The student is able to: • display data in pictographs where each symbol represents more than one unit. • display frequency data in (vertical and horizontal) bar graphs where one axis shows the whole numbers.

• calculator • Base 10 MAB • plastic aerosol caps • ice-cream container • small plastic bottles • plastic cups • bin • basketballs • soccer balls • netballs • footballs

• fill • containers • results • bar graph • pictograph • record

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Notes

Teac he r

Memory Masters (N3.3)

• The focus for this unit is multiplication tables and addition of two numbers each less than 10.

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Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.3) Warm up

• Arrange the class into small groups and move to a sandpit. • Provide each group with a plastic aerosol cap, ice-cream container, small plastic bottle and a plastic cup. • Allow time for students to fill containers freely.

Challenge

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• Direct students to use the cap to fill the ice-cream container, recording the number used with tally marks. Write the total in the space provided. • Repeat using the plastic bottle then the cup. • Draw a bar graph or a pictograph to show the results. • Provide a bin, basketballs, soccer balls, footballs and netballs. • In turn, fill the bin with each type of ball. • Complete the graph as each ball is placed in the bin, or record the total of each type of ball used before completing the graph.

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• Students create a two-digit number where the two digits are one digit apart; e.g. 65. Reverse the digits and find the difference. • Repeat for other two-digit numbers. What do you notice?

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 118 – 119. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 27 •

Unit 2–3

Student page 6

Indicators

Outcomes N3.3, N3.1a

The student is able to: • produce and use standard partitions (that is, into hundreds, tens and units) of two- and threedigit numbers.

Skills • recording place value

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • place value charts (see page 205) • tape measure

Language • place value • ones • tens • hundreds • calculator

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Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a) Warm up

• Divide the class into small groups. Distribute place value charts to each group. (It is recommended that the charts be laminated for durability and reuse.) • Ask students to point to the columns as they are called out—ones, tens and hundreds. • Ask students to write the digits for the following numbers in the correct place value column—824, 6, 95, 420. • Ask the students to take out their calculators and enter the number 5. • Ask students to multiply 5 by 10. What number is obtained? • Repeat this process with 8, 4, 15 and 26.

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Teac he r

• The focus for this unit is division and subtraction of basic facts.

What to do

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• Read the instructions in the workbook and direct the students to complete the activity. • When students have completed the activity, ask them to examine their place value chart then describe any patterns they may see on the chart. • Students can share their discoveries with the class.

Challenge

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• Students are to measure their height and arm span. Working in pairs or small groups may help. • Compare the two measurements, then determine whether the two measures are or are not the same as or very close to. • Expressions of mathematical logic should be the key to the explanations.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 36 – 37. • 28 • New Wave Maths Book D – Teachers Guide

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Unit 2—Answers

Student pages 4 – 6 Unit 2–2

Unit 2–1

Memory Masters 1. (a) 20 (b) 5 (c) 18 (d) 25 (e) 0 (f) 3 (g) 12 (h) 12 (i) 10 (j) 5 Number 2. (a) 23 (b) 21 (c) 31 (d) 14 (e) 64 (f) 21 Main Activity 3. Teacher check 4. Teacher check Challenge The difference will always be nine.

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Memory Masters 1. (a) 15 (b) 9 (c) 0 (d) 6 (e) 16 (f) 8 (g) 4 (h) 13 (i) 12 (j) 16 Number 2. (a) 54 (b) 33 (c) 31 (d) 34 (e) 14 (f) 22 Main Activity 3. (a) 46 (b) 13 (c) 24 (d) 51 (e) 242 (f) 136 (g) 164 Challenge 6, 3, 8 – 863, 836, 683, 638, 386, 368 7, 0, 2 – 720, 702, 270, 207, (0)72, (0)27 9, 5, 1 – 951, 915, 591, 519, 195, 159

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• Work in pairs. One student chooses a two- or three-digit number.The other makes up that number using Base 10 MABs. Check if correct by working together.

Consolidation 2–2

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Memory Masters 1. (a) 2 (b) 8 (c) 3 (d) 1 (e) 5 (f) 4 (g) 3 (h) 5 (i) 7 (j) 0 Number 2. (a) 410 (b) 450 (c) 660 (d) 330 (e) 420 (f) 540 Main Activity Hundreds Tens Ones 3. 1 3 13 1 3 0 10 x 13

• If students used a pictograph to show the results of Exercise 3, ask them to show the results as a bar graph or vice versa.

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18 10 x 18

1 8

8 0

24 10 x 24

2 4

36 10 x 36

3 6

6 0

48 10 x 48

4 8

8 0

4 0

Consolidation 2–3

• Repeat the activity at a later date. Students can work in pairs, choosing two-digit numbers at random, entering them on a place value chart and multiplying the number by 10. Calculators can be used if necessary.

5 6 56 5 6 0 10 x 56 moving one place left; increasing by the power of 10 Challenge True. Teacher check student explanations. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 29 •

Unit 3–1

Student page 7

Outcomes

Indicators

N3.3, S3.1

The student is able to: • order and show a sense of the proximity of things in locating key features on maps. • attend to the order and proximity of things in giving directions.

Skills • locating points • plotting items • following directions

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • atlases • street directories • coloured pencils

Language • coordinates • location • clockwise • anticlockwise • half turn • full turn

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Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.1) Warm up

• Discuss with the students the use of coordinate grid points to give a location. – Where would you use coordinate grid points to help find something? (Elicit from students the use of an atlas or street directory.) – How does using coordinate points help find something? – Are there any games you know of that use grid coordinates? (Elicit from students the game of ‘battleship’.)

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• The focus for this unit is division and subtraction of basic facts.

• Introduce the use of the vertical and horizontal axis. Remind students that the horizontal axis is read from first.

• Follow the same procedure using an atlas to find his or her town or city.

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What to do

• Direct students’ attention to Exercise 3. Under the location column, grid coordinates are given with the horizontal axis first. As a group, locate (B, 1). Ask the students to use their red pencil to draw a star in this box.

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• Ask students to use the street directory to find their street. Using the index, write the coordinates given for his or her street, then go to the correct page.The directory gives the location by providing the horizontal axis first then the vertical axis second. Once the students have found their street, they could find the location of the school.

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• Students can check the work of the person next to them to make sure they have drawn the star in the correct location. • Direct the students to complete the remainder of the task on their own.

• Discuss with the students the direction hands move on a clock. (They move clockwise.) Ask the students to turn in a clockwise direction, then in an anticlockwise direction. Demonstrate a half-turn clockwise and half-turn anticlockwise. • Ask students to complete Exercise 4, keeping in mind the direction hands move on a clock. • Ask students: ‘Does it matter if you do a full turn clockwise or anticlockwise?’.

Challenge • Discuss with the students a scenario of getting from their classroom to the principal’s office. Ask them to think about which pathway they would take. • In small groups, discuss the pathways each student would take. Students may find they have different ways to get to the principal’s office. • Explaining their pathway to their peers will clarify the directions they need to use in their written description. • Students can swap their description with partners. How clear was the description?

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 8 – 9. • 30 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 3–2

Student page 8

Outcomes

Indicators

N3.3, N3.4, N3.1a

The student is able to: • identify the starting number and the constant multiplier needed to generate a number sequence. • follow a rule to generate a number sequence based on an addition or subtraction strategy. • count up and down in tens from any starting number.

Skills • completing patterns • counting

Resources

Language

• Base 10 MAB • calculator

• patterns

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Notes

Memory Masters (N3.3)

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• The focus for this unit is division and subtraction of basic facts.

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.4, N3.1a) Warm up

• Ask the class to count to 20 by ones. • Ask the class to count to 20 by twos. • Ask the class to count to 20 by fours. • Ask the class to count to 20 by fives. • Ask for an explanation of what they have just been doing. If you get an answer of calling out number patterns, continue—if not, explain to the class that each of their counting exercises was, in fact, a number pattern.

What to do

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Challenge

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• Direct the students to look at the number patterns in their workbook. • From the information shown, ask the students how they can discover the pattern. (Some examples are more obvious than others.) • Work with the class on the first pattern, asking students what the pattern is. For those having difficulty, ask them to look at the first two numbers of the pattern and see if they can think of two missing numbers to fill the space to 10. • Encourage the examination of the relationship between numbers to determine each pattern. • Students complete Exercise 3 at their own pace. • Exercise 4 may be read through with the class and should be able to be completed fairly readily. If not, suggest students use their calculator to assist, by adding 10 to the last number.

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• Suggest that students use the information they have gained from the main activity to develop their own number pattern. • Students should check that the pattern is consistent with their rule, then ask one of their classmates to try and complete the pattern.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 70 – 71. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 31 •

Unit 3–3

Student page 9

Indicators

Outcomes

The student is able to: • use a uniform unit of length carefully to make their own graduated scale to measure things in their environment.

N3.3, M3.2

Skills • measuring • estimating • recording • using a ruler

Teac he r

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • ruler • string

Language • measure • nearest • centimetre

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• The focus for this unit is division and subtraction of basic facts.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2) Warm up

• Ask students to take out their ruler. • Ask students to point to the mark on their ruler that will be used as the starting mark to measure a line. Ensure all are pointing to the 0 mark. • Remind students not to start from the end of the ruler unless that is also the 0 point.

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• Ask all students to place their ruler on the first line 3(a) in their workbook. Check to see that all are at the correct measuring point. • Tell the students to measure the line and write the length in the space provided. • Direct students to measure lines (b) and (c). • Ask students how they will measure the lines in Exercise 4. • Students could use string to measure the lines, then use a ruler to measure the length of the string. • If using just their ruler, students may need reminding to return to the 0 point for each change in direction and add the total measures, or ensure that the end point of each line is the starting measure mark for the change in direction.

Challenge

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What to do

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Number (N3.3)

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• Students provide their own thoughts for this challenge. Encourage creative but reasonable answers. • Share ideas with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 78 – 79. • 32 • New Wave Maths Book D – Teachers Guide

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Unit 3—Answers

Student pages 7 – 9 Unit 3–1

Unit 3–2

Memory Masters 1. (a) 4 (b) 5 (c) 1 (d) 3 (e) 1 (f) 1 (g) 5 (h) 4 (i) 2 (j) 5 Number 2. (a) 48 (b) 63 (c) 36 (d) 72 (e) 56 (f) 81 Main Activity 3. 7

Memory Masters 1. (a) 5 (b) 3 (c) 4 (d) 2 (e) 7 (f) 7 (g) 2 (h) 0 (i) 8 (j) 4 Number 2. (a) 8 (b) 6 (c) 6 (d) 9 (e) 8 (f) 6 Main Activity 3. (a) 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 (b) 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 (c) 8, 12, 16, 20, 24, 28, 32, 36, 40 (d) 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 (e) 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 (f) 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 (g) 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 (h) 7, 14, 21, 28, 35, 42, 48, 56, 63, 70 4. (a) 6, 16, 26, 36, 46, 56, 66, 76, 86, 96 (b) 3, 13, 23, 33, 43, 53, 63, 73, 83, 93 (c) 7, 17, 27, 37, 47, 57, 67, 77, 87, 97 (d) 4, 14, 24, 34, 44, 54, 64, 74, 84, 94 (e) 12, 22, 32, 42, 52, 62, 72, 82, 92, 102 (f) 21, 31, 41, 51, 61, 71, 81, 91, 101, 111 Challenge Teacher check

6 5 4

2 1

Challenge Teacher check

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• Add to the list of items to locate in Exercise 3. Provide a grid coordinate not yet used and complete the activity. (This could be done in pairs.)

Consolidation 3–2

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Memory Masters 1. (a) 3 (b) 2 (c) 9 (d) 7 (e) 3 (f) 5 (g) 3 (h) 3 (i) 9 (j) 2 Number 2. (a) 16 (b) 15 (c) 12 (d) 25 (e) 15 (f) 3 Main Activity 3. (a) 14 cm (b) 7 cm (c) 12 cm 4. (a) 15 cm (b) 16 cm (c) 31 cm Challenge Teacher check

• Provide further opportunities for students to work independently or in pairs to make their own number patterns and test on a partner.

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Consolidation 3–3

• Practise measuring items shorter than a ruler’s length, ensuring the 0 mark is the starting point.

New Wave Maths Book D – Teachers Guide • 33 •

Unit 4–1

Student page 10

Outcomes

Indicators

N3.3, S3.3

The student is able to: • identify repetitions of component parts in symmetrical objects/ arrangements and demonstrate by moving one component over another. • use multiple copies of figures to create patterns based on systematic movements of the shape and informally describe the movement used.

Skills • identifying patterns • completing patterns

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • pattern blocks

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.3)

Notes

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• patterns • semicircles • rectangles • triangles • circles • squares

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• The focus for this unit is division of basic facts and subtraction of two numbers without regrouping.

Warm up

Language

• Allow the students time to have directed free play with pattern blocks. Encourage them to use colour and shape to create designs and patterns. Ask them to make a pattern and have a classmate continue it. Direct students to make a variety of patterns, including only using colour, only using shape or using the direction of the shape. Ask students: – What is a pattern? – Where would you see patterns in the environment? – How do you make a pattern?

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• Direct students to complete the patterns on the page. Attention should be paid to the variety of patterns. Some patterns have been made using the same shape which has been flipped, rotated or turned. Some patterns use various shapes, while others use shape, direction and colour to form a pattern. • Some students may need to use concrete materials to recreate the patterns on the page before they can complete them. As they become more confident, this technique should eventually lead them to completing the patterns directly on the page.

Challenge

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What to do

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• Students may find this task easier to complete with the use of a calculator. It will also help develop the students’ calculator skills. • Discuss the use of +, –, x and ÷ signs. In what situations would we use each sign? • Allow students the opportunity to explore possible solutions. Students will have different approaches and follow different pathways to find a solution. The method used by each student should be praised if it reaches the required result. This is part of developing the skills for working mathematically.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 24 – 25. • 34 • New Wave Maths Book D – Teachers Guide

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Unit 4–2

Student page 11

Outcomes N3.3, N3.4, M3.2

Skills • describing patterns • recording information • measuring • balancing

Indicators

Resources

Language

The student is able to: • identify the starting number and the constant multiplier needed to generate a number sequence. • fill in number sequences involving addition or subtraction by a constant amount. • choose to make numerical measurements of objects to order the objects.

• Base 10 MAB • calculator • model or picture of an insect • balance scale • objects of various sizes and weights

• subtract • table • patterns • seesaw balance • mass • weighings

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Notes

Memory Masters (N3.3)

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Teac he r

• The focus for this unit is multiplication tables and addition of two numbers each less than 10.

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.4, M3.2) Warm up

• A science lesson on insect body parts may be a good introduction to this activity. • Display a model or a picture of an insect to the class. • Ask students how many legs and how many body parts an insect has. • If I have two insects, how many legs and body parts now? • If I have three insects, how many legs and body parts this time?

What to do

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Challenge

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• Direct students to their workbook. • Complete recording the information from the warm up on the chart. • Continue for four to seven insects recording the number of legs and body parts. • Ask students if they can see any patterns in the two lists of numbers. • Students may give doubling across the columns as well as counting by three (multiples of 3) and by six (multiples of 6). • For Exercise 4 organise the class into small groups and distribute a number of objects of various sizes and weights and a balance scale. • Ask students to find objects that balance with each other and groups of objects that balance with one object. • Record findings each time on the table provided.

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• Students will need to use their imagination for this activity unless you are able to provide objects of the same size and mass with one lighter. • Students should record their working so they remember what they did and how they found their answer. • Refer to the answers on page 37 for the quickest method. Some students may have this as the answer, while others may have a correct, but longer method.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 72 – 73 and 96 – 97. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 35 •

Unit 4–3

Student page 12

Indicators

Resources

Language

The student is able to: • read and write any whole number into the thousands. • produce and use standard partitions of two and three digit numbers. • use the decimal point in representing quantities and money.

• Base 10 MAB • calculator • 0 – 9 number cards (see page 204) • place value charts (see pages 205 – 206)

• rearrange • smallest • largest • digit • decimal place • tenths • measures • metres • centimetres • difference

Outcomes N3.3, N3.1a

Skills • rearranging numbers • ordering • recognising place value • converting measures

Memory Masters (N3.3)

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Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a) Warm up

• Display the numbers 0 – 9 on separate cards. Ask the students to select two numbers to make a new number. For example; 2 and 1. Students could present these cards as either 12 or 21. — Does the way we order the numbers make a difference to the number? — How many different ways could you arrange the numbers? • Repeat the activity with students choosing three numbers. Follow the same procedure and questioning process as above. Students will have more options as they have more numbers to rearrange.

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Teac he r

• The focus for this unit is multiplication tables and addition of two numbers each less than 10.

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• Students can now complete Exercise 3 independently. Give assistance if required. • Use a place value chart to show the ones, tens and hundreds places. Students can refer to this if necessary when completing Exercise 4. • Ask students how many centimetres in a metre. (100) • Ask: If I have 24 centimetres, how many metres do I have? (0.24 m) • Ask where the 2 and the 4 would be written on a place value chart. • Ask: If I have 1 metre 24 centimetres, how many metres do I have? (1.24 m) • Ask where the 1, 2 and 4 would be written on a place value chart. • If students readily point to ones, tenths and hundredths, allow them to complete Exercise 5. If not, repeat this process several times using 50 cm, 1 metre 50 cm, 29 cm, 1 metre 29 cm, 92 cm and 1 metre 92 cm as possible examples before asking students to complete Exercise 5.

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What to do

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Challenge • Students will need to apply their knowledge of place value to make the largest and the smallest number they can from the four digits. • Students will then need to determine how they can find the difference. • Leave this activity as an individual exercise with no teacher input unless students are having difficulty. A calculator could be suggested to assist in working out the difference as the problem requires regrouping to find the answer. (Many students may think to do this anyway.) For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 36 – 37. • 36 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 4—Answers

Student pages 10 – 12 Unit 4–2

Memory Masters 1. (a) 8 (b) 4 (c) 2 (d) 5 (e) 2 (f) 9 (g) 5 (h) 5 (i) 8 (j) 6 Number 2. (a) 5 (b) 7 (c) 6 (d) 7 (e) 6 (f) 8 Main Activity 3.

Memory Masters 1. (a) 18 (b) 24 (c) 24 (d) 0 (e) 3 (f) 14 (g) 6 (h) 9 (i) 4 (j) 9 Number 2. (a) 8c (b) $8 (c) $7 (d) 4c (e) 6c (f) 4c Main Activity 3. Multiples of 3; multiples Number of … insects body parts legs of 6; doubling multiples 1 3 6 of 3 gives multiples of 6. 2 6 12 3 9 18 4 12 24 5 15 30 6 18 36 7 21 42 4. Teacher check Challenge The quickest method is: • Weigh two coins on each side of a balance scale. If one side is heavier, then the lighter coin is on the opposite side. Weigh those two coins to find out which one is lighter. • If the coins balanced on the first weighing, then the lighter coin must be the spare coin.

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Challenge Answers will vary; possible solutions working left to right: 7x7+7÷7=8 7+7÷7=2 7+7+7+7+7÷7=5

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Unit 4–1

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• Play games with patterns. Students work in pairs using a coin. One student is ‘heads’ and the other is ‘tails’. Each toss of the coin decides which student can contribute to the pattern next. Between the two of them, they must work toward making a distinct pattern.

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Memory Masters 1. (a) 0 (b) 18 (c) 32 (d) 2 (e) 27 (f) 11 (g) 9 (h) 7 (i) 0 (j) 5 Number 2. (a) 9 (b) 6 (c) 9 (d) 8 (e) 8 (f) 5 Main Activity 3. (a) 489, 984 (e) 568, 865 (b) 239, 932 (f) 255, 552 (c) 347, 743 (g) 256, 652 (d) 167, 761 (h) 257, 752 4. (a) 4 6 5 (d) 6 2 7 (g) 8 6 4 (b) 7 8 3 (e) 6 5 9 (h) 2 5 7 (c) 1 4 8 (f) 2 9 9 (i) 3 4 6 5. (a) 3.4 m (f) 5.8 m (b) 9.5 m (g) 6.1 m (c) 4.56 m (h) 7.28 m (d) 1.16 m (i) 2.81 m (e) 8.32 m (j) 4.73 m Challenge 8732 – 2378 6354

Consolidation 4–2

• Repeat Exercise 3 using a spider’s body parts (two) and legs (eight). • Estimate first, then check the mass of two objects to see if they balance or not.

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Consolidation 4–3

• Provide further opportunities to write measures as metres only. When students are adept at changing metres and centimetres to metres only, introduce changing centimetres to metres only. This will require a ‘0’ in the ones place; for example, 42 cm = 0.42 m.

New Wave Maths Book D – Teachers Guide • 37 •

Unit 5–1

Student page 13

Outcomes

Indicators

N3.3, S3.3

The student is able to: • find repetitions of figures and objects within decorative patterns (e.g. tiling, fabrics), objects (e.g. beehives, blocks of flats, stacked Tetrapaks™) and formations (e.g. dance routines, marching groups). • informally explain why they think a figure won’t tile.

Skills • tracing • cutting • tessellating • problem-solving

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • 2-D shapes— square, equilateral triangle, rectangle • paper • scissors

Language • shapes • trace • tessellate • rectangle

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Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.3) Warm up

• Organise the class into small groups. • Distribute a range of 2-D shapes to the groups to play with. Allow several minutes for the groups to manipulate the shapes. • Ask the class what they discovered about their shapes. Accept all reasonable answers.

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• Read the directions in the workbook to the class. From their manipulation of the materials, ask the class which of the three shapes they think will fit together so that the rectangle is completely filled. • Encourage students to support their guesses with reasoned argument. • Provide, or have students cut shapes the same size as those on the page to fill the rectangle to check their guesses. It may assist if students trace around the shapes as they use them. Suggest to students that using a different colour for each shape may help them fit each shape with the previous one. • Completion of the tiling will provide the answers to the question.

Challenge

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What to do

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• The focus for this unit is multiplication tables and addition of two numbers each less than 10.

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• Ask students how they will work out which point is outside the lines. • Listen to students’ suggestions. Ask the class to check to see if they work. • Discuss correct suggestions.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 22 – 23. • 38 • New Wave Maths Book D – Teachers Guide

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Unit 5–2

Student page 14

Outcomes

Indicators

N3.3

The student is able to: • add and subtract whole numbers using their own written method or a conventional algorithm, explaining the method by reference to place value.

Skills • reordering numbers • adding

Resources

Language

• Base 10 MAB • calculator

• add • reorder • addition

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Memory Masters (N3.3)

Notes

Number (N3.3)

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• The focus for this unit is multiplication tables and addition of two numbers each less than 10.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3) Warm up

• Reordering numbers develops an understanding of the commutative property of addition (allowing addends to be placed in any order). Students are not required to know the term commutative. • Ask students why some pairs of numbers are easier to add than others. List the ideas. • Focus on pairs that make ten and pairs of the same number; e.g. 7 + 7. • Ask what would be the easiest way to add these three numbers—8 + 5 + 2. Have students explain their answers. Again focus on reordering so that 8 + 2 are paired, then add 5. • Ask students if it is always necessary to rewrite the order of the numbers before adding or is mental reordering sufficient.

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• Direct students to work out the first example in Exercise 3. • Discuss their answers. Those with no problems can proceed to complete the rest of the exercise, while the teacher can spend time helping any students with difficulties. • Repeat the explanation for the examples with five numbers in Exercise 4. Students can then complete the activity.

Challenge

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• Students will need to experiment with adding, subtracting, multiplying and dividing to find ways of making other numbers using a single number and the four operations. • As a hint, ask how can 1 be made from the number four using one or all of the operations; e.g. 4 ÷ 4. • Leave the students to experiment either by themselves or in small groups. • Share answers with the class and encourage the finding of different combinations. • Praise all efforts that reach the required results.

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New Wave Maths Book D – Teachers Guide • 39 •

Unit 5–3

Student page 15

Indicators

Outcomes N3.3, N3.4

The student is able to: • build sequences of simple shapes such as triangles, squares, ‘L’ or ‘T’ shapes, which increase in size systematically and write the equivalent number pattern.

Skills • making square numbers • drawing • reporting

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • square tiles

Language • square number • grid • overlapping

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Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.4) Warm up

• Introduce the lesson by making arrays of different dimensions using square tiles; e.g. 3 x 2, 4 x 4, 3 x 3, 5 x 1, 2 x 2, 4 x 3 etc. • Ask the students to describe the shapes that the arrays make. (Rectangular or square?) • Focus attention on the square arrays. – Does anyone know what these arrays represent? (Square numbers.)

What to do

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• The focus for this unit is multiplication tables and addition of two numbers each less than 10.

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Challenge

• Ask students how they might show a triangular number. • Ask them to show further triangular numbers, seeing if they are able to find a pattern. • Share findings with the class.

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• Complete the activity. • Students should be able to describe all square numbers as being represented by a square.

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• 40 • New Wave Maths Book D – Teachers Guide

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Unit 5—Answers

Student pages 13 – 15 Unit 5–1

Memory Masters 1. (a) 28 (b) 0 (c) 12 (d) 10 (e) 16 (f) 2 (g) 7 (h) 11 (i) 14 (j) 15 Number 2. (a) 89 (b) 78 (c) 88 (d) 99 (e) 99 (f) 67 Main Activity 3. (a) 12 (c) 17 (e) 18 (g) 19 (b) 16 (d) 14 (f) 7 (h) 15 4. (a) 22 (b) 22 (c) 23 (d) 25 Challenge Answers will vary. Possible solutions working left to right: 2+2+2=6 2x2+2=6 2x2x2–2=6

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Memory Masters 1. (a) 0 (b) 20 (c) 4 (d) 0 (e) 21 (f) 9 (g) 8 (h) 5 (i) 17 (j) 10 Number 2. (a) 141 (b) 127 (c) 125 (d) 143 (e) 141 (f) 121 Main Activity 3. The rectangle tessellates. Challenge Point 2 is outside the lines. Mark alternative lines across the maze.

Unit 5–2

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• Repeat the activity again using a 16 cm x 10 cm shape to fill instead of a 16 cm x 9 cm. Students should discover the square will now fit and not the rectangle.

Consolidation 5–2

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Memory Masters 1. (a) 0 (b) 18 (c) 36 (d) 8 (e) 30 (f) 18 (g) 13 (h) 6 (i) 7 (j) 8 Number 2. (a) 146 (b) 158 (c) 159 (d) 129 (e) 169 (f) 147 Main Activity 3. Answers may vary according to students’ diagrams. 1 2 1, 4, 9, 16, 3 4 25, 36, 49; The numbers 5 all form a 7 square on the grid. (Some students may 6 notice that 1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9 and so on.) Challenge Teacher check. Triangular numbers can be shown as triangles; 1, 3, 6, 10, 15 etc. etc.

• Complete similar activities using one two-digit number between 10 and 20 and two single-digit numbers.

Consolidation 5–3

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• Complete a table to show the square number pattern. Number

Number of squares

Multiplication

1x1=1

2x2=4

3x3=9

and so on.

New Wave Maths Book D – Teachers Guide • 41 •

Unit 6–1

Student page 16

Outcomes

Indicators

WM3.2, WM3.3, M3.3

The student is able to: • estimate the time taken to complete an event. • pose mathematical questions. • test conjectures. • classify events into those that take about, more than or less than one hour, half an hour or five minutes.

Skills • estimating • making conjectures

Resources

Language • estimate • time • guess • seconds • minutes • hour • half an hour • between • less than • more than

• stopwatch

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Notes

Main Activity (WM3.2, WM3.3, M3.3 )

• This activity is designed for students working collaboratively in groups. Allow enough time so students can discuss their options and for ideas to evolve. Investigative tasks such as these are a good opportunity for students to ‘take a risk’ with maths. • When completing investigative tasks, some students may be more successful in mixed-ability groups rather than same-ability groups. • Some groups will be able to work independently while others may need guidance. The stimulus questions below may prompt such groups. – What activities do you do that take … ~ less than five minutes (e.g. cleaning teeth, making a sandwich, making your bed, travelling to school). ~ between five minutes and half an hour (e.g. morning break at school, walking the dog, playing a game, eating dinner, watching a television program). ~ more than half an hour (e.g. watching a movie, a maths lesson, lunch break at school). • For Question 3, students will need access to stopwatches. Students may discuss the speed at which the tables are spoken. Remind them the tables need to be understood and that it is not a race. • At the end of the lesson, ask groups to report on their estimates for the five events in Question 1. Write the estimates on the board and discuss them. If the answers are varied for one or more of the events, plan to test the estimates by bringing in a toaster, asking a student to record how long it takes to read a novel (discuss the length of a ‘novel’) and asking the P.E. teacher or relevant students the length of a netball game. Students may also wish to test the estimate of walking to a friend’s house three blocks away (out of school hours). Discuss what a ‘block’ is. Also remind students to be safe and tell adults where they are going (or ask an adult to go with them). • Allow each group to discuss and evaluate its ability to problem-solve and its success as a group. A group or self-assessment form could be completed. This information will be helpful for creating groups for future open-ended, investigative tasks.

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What to do

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• 42 • New Wave Maths Book D – Teachers Guide

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Unit 6–2

Student page 17

Outcomes

Indicators

M3.1, N3.4, N3.3, C&D3.1, C&D3.2, C&D3.3, C&D3.4

Skills • making a tally • analysing data

The student is able to: • justify their choice of more or less likely by referring to past experience or known information. • record frequency data carefully using simple formats based on tallies or organised lists and take care with their measurements. • use diagrams such as Venn diagrams and two-way tables to represent a twoway classification. • report the frequency information provided in a tally produced by a classmate.

Resources

Language

• Base 10 MAB • calculator • street • stopwatch • coloured pencils

• tally • most likely • least likely

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Memory Masters (M3.1, N3.4)

Notes

Number (N3.3)

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• The focus for this unit is conversion of measurements and number patterns.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.1, C&D3.2, C&D3.3, C&D3.4) Warm up

• Ask students how they would make a tally. Ensure all students remember the four vertical lines before the fifth cross stroke. • Outline the activity—all the class will be participating in a traffic survey of vehicles passing by on a road beside the school. • The survey time will be for 15 minutes.

Challenge

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• Students can work individually or in small groups. • Remind students that all vehicles passing the school must be recorded. In schools with very busy roads, this is best done in groups. In some cases, one student may need to spot while another records. • Tell students that they will be doing a second survey at a later date, or later in the day. • Remind students that for safety reasons they must remain on school grounds at all times. • Once the recordings are completed, ask students to complete the questions in their workbook. Share the answers with the whole class.

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• Students are to colour the diagram using only four colours. No two adjacent sections are to have the same colour. • Share findings with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 106 – 107 and 112 – 113. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 43 •

Unit 6–3

Student page 18

Indicators

Outcomes

The student is able to: • add and subtract whole numbers using their own written method or a conventional algorithm, explaining the method by reference to place value. • use the decimal point in representing quantities and money.

N3.1a, N3.3

Skills • adding • reordering • recognising decimal numbers

Memory Masters (N3.1a)

Resources • Base 10 MAB • calculator

Language • patterns • adding • change • order • easier • tallest • shortest

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Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3, N3.1a) Warm up

• Ask students how the following numbers may be reorganised to make adding easier. (a) 6 + 5 + 3 + 5 + 4 = (b) 2 + 5 + 2 + 5 + 2 = • In (a) regroup so that 6 + 4 and 5 + 5 are together. Both make 10. • In (b) regroup 2s and 5s together so that there are 3 lots of 2 and 2 lots of 5, or 3 x 2 + 2 x 5.

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• The focus for this unit is conversion of money.

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• Direct students to reorganise the numbers in Exercise 3 so they can be added easier. Remind students that in each case there are only one or two different numbers and these may be grouped together before adding. • Check on progress to ensure understanding. • Discuss their approaches in the description. Whose were similar? Whose were different? • In order to arrange the four people shown in Exercise 4 from tallest to shortest, students need to understand the size of each number. • Which digit would you look at first to determine the largest? The ones digit (the one on the left). What can you tell from these digits? (They are all the same.) • Which digit would you look at next? The one in the tenths column, or first to the right of the decimal point. • Can you tell the largest? (1.77) Which is next? There are two with 5. What should you do now? (Use the final number.) • Which is next largest? 1.56, then 1.52 and finally 1.45. • Record these answers.

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Challenge • Students enter the largest number they can say into a calculator. Check for correctness and discuss answers as a class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 42 – 43. • 44 • New Wave Maths Book D – Teachers Guide

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Unit 6—Answers

Student pages 16 – 18 Unit 6–2

Unit 6–1 Main Activity 1. Most likely answers will be: (a) less than five minutes (b) more than half an hour (c) between five minutes and half an hour (d) less than five minutes (e) more than half an hour 2. Teacher check 3. Teacher check

Memory Masters 1. (a) 1000 (b) 60 (c) 7 (d) 12 (e) 1000 (f) 12 (g) 20 (h) 8 (i) 20 (j) 60 Number 2. (a) 220 (b) 220 (c) 330 (d) 230 (e) 310 (f) 210 Main Activity 3. Teacher check Challenge

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• Students estimate how many times they can do an activity in a set amount of time; e.g. ‘In three minutes, how many times can you • bounce a ball • skip with a skipping rope • hop • say ‘hippopotamus’ etc.

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Memory Masters 1. (a) $3 (b) $9 (c) $5 (d) $1 (e) $7 (f) 200c (g) 400c (h) 900c (i) 500c (j) 800c Number 2. (a) 300 (b) 500 (c) 400 (d) 600 (e) 300 (f) 100 Main Activity 3. (a) Descriptions will vary; 27 (b) Descriptions will vary; 24 (c) Descriptions will vary; 36 (d) Descriptions will vary; 33 (e) Descriptions will vary; 39 (f) Descriptions will vary; 36 (g) Descriptions will vary; 18 (h) Descriptions will vary; 30 (i) Descriptions will vary; 28 (j) Descriptions will vary; 32 4. Bob, Coleen, Rochelle, Brett Challenge Teacher check

Consolidation 6–2

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• Repeat the activity recording the vehicles that pass by his or her house during a 15-minute given time. Compare findings.

Consolidation 6–3

• Complete similar activities using six one-digit numbers, but where at least four are different numbers.

New Wave Maths Book D – Teachers Guide • 45 •

Unit 7–1

Student page 19

Outcomes S3.4, N3.3, S3.3

Skills • showing congruency • symmetry • matching

Indicators

Resources

The student is able to: • imagine and draw different crosssections of simple 3-D shapes and then check and improve the drawings by observing the crosssection. • informally describe the symmetry of a figure or arrangement. • visualise and describe cross sections of familiar objects.

• Base 10 MAB • calculator • 2-cm cubes • models and shapes cut in half (see warm up)

Language • solid objects • divided • exactly the same • congruent

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Notes

Memory Masters (S3.4, N3.3) Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.2, S3.3, S3.4) Warm up

• Provide a number of models and shapes that have been cut in half; e.g. cube cut in half, box cut in half, shapes cut in half or with a line drawn to show where they may be cut in half. • Organise class into small groups. • Distribute 2-cm cubes to the groups. • Demonstrate where the prepared models may be separated into two objects that are exactly the same. • Repeat this with the shapes. • Explain to the students that the identical parts are called ‘congruent’ parts. Congruent parts are exactly the same as each other. • Allow the students a few minutes to play with the blocks. • Ask students to make models with the blocks, so that when divided into two parts they will be exactly the same. • Students discuss their findings within their groups and report to the whole class.

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• The focus for this unit is identifying shapes and halving.

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What to do

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• Students complete Exercises 3 and 4 in their workbooks. • After checking these activities, show students a number of congruent shapes—shapes that are exactly the same as each other. Explain that both shapes must be exactly the same size and shape. • Ask students to find the congruent shapes in Exercise 5 of their workbook.

Challenge • Remind students that squares are special rectangles. • Students are to keep a record of how they find the total number of rectangles. • Ask a number of students to share their findings.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 20 – 21. • 46 • New Wave Maths Book D – Teachers Guide

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Unit 7–2

Student page 20

Outcomes N3.3, N3.1a

Skills • adding • recognising number values

Indicators

Resources

The student is able to: • remember basic addition facts and many multiplication facts and calculate mentally basic multiplication facts they don’t recall. • add and subtract whole numbers using their own written method or a conventional algorithm, explaining the method by a reference to place value. • read and write any whole number into the thousands.

• place value charts (see pages 205 – 206)

Language • patterns • order • sum • smaller

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Notes

Memory Masters (N3.3)

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• The focus for this unit is special sets of numbers—odd and even.

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3, N3.1a) Warm up

• Display the following set of numbers for students to see (blackboard, whiteboard or overhead) 8, 2, 9, 7, 6, 5, 3, 1, 4. Ask students to find the total of the numbers—45. • Ask students to explain how they reached the total. If a student found the total by rearranging the numbers—probe to find out why, and which numbers, were rearranged. • Students may then be asked to find the total of these numbers 7, 2, 6, 7, 6—28. • Ask students how they reached the total. Focus on answers that involve rearranging. • Reinforce to students that when rearranging numbers to add together, making a total of 10 or pairing numbers to allow doubling makes adding easier. These are skills that people use when adding mentally.

What to do

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• Students work through Exercise 3, finding the totals and explaining what patterns they used to assist in their addition. • Issue students with place value charts or strips. • Inform students the first thing they need to do to compare the size of two numbers is to look at the place value of the number on the left.The number in the higher place value will be the larger number of the two. If both numbers are the same, check the next number in the next lower place value to find the larger number. • This process can be repeated until the smaller of the two numbers is found and then crossed off in the workbook. • Students may be issued with cards with numbers on them, to make the numbers under the place value strip to assist in identifying the smaller number.

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Challenge • Find out from students the key to solving magic squares; i.e. knowing the total of the rows, columns and diagonals and that these totals are the same. • Solve the magic square.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 36 – 37. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 47 •

Unit 7–3

Student page 21

Indicators

Outcomes

The student is able to: • use a unit consistently and carefully to measure and compare volume. • identify and make devices which have a mathematical basis and which people have used in the past to perform practical tasks.

N3.1a, N3.3, M3.2, WM3.1

Skills • making models • observing • reporting • making pendulums • timing • recording

Memory Masters (N3.1a)

Resources • Base 10 MAB • calculator • 20-mm cubes • string and modelling clay • stopwatch

Language • volume • pendulum • bob

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Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2, WM3.1) Warm up

• Organise the class into small groups. Distribute a quantity of 20-mm cubes to each group. • Allow students to play with the cubes—encourage them to share verbally with each other about what they are doing. • Hold up one cube and ask students about the cube: — Describe this object to me. — How much space does it take up? — How many ways can I arrange the cube? (Rolling, sliding, rotating are repeat positions.) • Repeat with two cubes and three cubes. • Ask students if they know the correct word to describe the space taken by the cube(s). (Volume.)

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• The focus for this unit is comparison of numbers and place value.

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What to do

• Direct students to complete Exercise 3. • For Exercise 4 ask students to explain what they know about pendulums. • Explain they are conducting an experiment to see, if when using a bob of the same mass, changing the string length changes the speed of the swing of the pendulum. • Make the pendulums—suggest 20 cm, 30 cm, 40 cm and 50 cm lengths for the string—and attach to a ruler between two desks. • Set the stopwatch and begin swinging one pendulum; record results. • Repeat for the other three pendulums. • The activity may be repeated several times—always ensure the angle of the string is the same at the start.

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Challenge • Students will need to experiment with adding, subtracting, multiplying and dividing to find ways of making other numbers using a single number and the four operations. • As a hint, ask how can 1 be made from the number four using one or all of the operations; e.g. 4 ÷ 4. • Leave the students to experiment either by themselves or in small groups. • Share answers with the class and encourage the finding of different combinations. Praise all efforts that reach the required results.

• 48 • New Wave Maths Book D – Teachers Guide

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Unit 7—Answers

Student pages 19 – 21 Unit 7–1

1. (a) square (b) triangle (c) rectangle or oblong (d) circle (e) pentagon (f) 5 (g) 3 (h) 2 (i) 10 (j) 6 2. (a) 540 (b) 630 (c) 560 (d) 720 (e) 420 (f) 490 3.

Challenge 18 Size of rectangle 1x1 8

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Total 6

2x1

2x2

3x1

2x3

1. (a) O (b) E (c) O (d) O (e) E (f) E (g) E (h) O (i) O (j) E 2. (a) 100 (b) 300 (c) 200 (d) 400 (e) 100 (f) 400 3. Descriptions will vary; 4. (a) 436 (a) 15 (b) 938 (b) 23 (c) 927 (c) 19 (d) 819 (d) 17 (e) 399 (e) 19 (f) 6399 (f) 20 (g) 267 (g) 20 (h) 492 (h) 17 (i) 2461 (i) 16 (j) 14 Challenge

(also square) (also square)

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Unit 7–2

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1. (a) 56 (b) 82 (c) 41 (d) 53 (e) 91 (f) 40 (g) 50 (h) 50 (i) 80 (j) 40 2. (a) 63 (b) 82c (c) 88c (d) 93c (e) 66 (f) 62 3. (a) The volume is the same for each model. (8 cm3) (b) Yes. Teacher check 4. Teacher check table.The longer the string length, the less swings per minute. Challenge Answers will vary. Possible solutions, working left to right: 2+2+2÷2=3 2x2x2x2–2÷2=7 2x2=4

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• Trace around pattern or attribute blocks. Draw lines on the shapes to show congruent parts. (Some may have more than one line, as in Exercise 4.)

Consolidation 7–2

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• Complete similar activities using six one-digit numbers, but where at least four are different numbers.

Consolidation 7–3

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• Construct shapes of the same volume using ten 20-mm cubes. • Make pendulums using different weight bobs but strings of the same length. What do students notice?

New Wave Maths Book D – Teachers Guide • 49 •

Unit 8–1

Student page 22

Outcomes

Indicators

N3.1a, N3.3, C&D3.2, C&D3.3, C&D3.4

The student is able to: • remember basic division and multiplication facts. • record frequency data carefully using simple formats based on tallies or organised lists. • display frequency data in bar graphs. • explain their own data displays to their peers, talking about the features represented.

Skills • multiplying • dividing • timing • recording • graphing • comparing

Resources • Base 10 MAB • calculator • ruler • timer • graphs (see page 236)

Language • speed test • multiply • divide • timer • graph • record

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Notes

Memory Masters (N3.1a) Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3, C&D3.2, C&D3.3, C&D3.4) Warm up

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• Either provide students with a graph template or have them make a graph to record the number correct and a separate graph to show the time taken. • Remind students they should keep their graphs in a safe place as they will be required to use them again to check on their progress. • It is recommended that the students complete this speed test at least once a term and up to twice a term, or make it a once-weekly test for five or six weeks. • Instruct students to rule six columns of ten lines in their pad. • Once this is completed, suggest the pad is placed over the workbook page with the column to be worked on showing to the immediate left of the pad. As each column is completed, the pad is moved to the right to expose the next column. • When everyone is ready, tell the students you will give the order to start.They are to work as quickly as possible and to raise their hand when finished. You will call the time taken to complete the task, which the students will write on their page. Suggest they leave the ones they have trouble completing. • Mark the work when everyone is finished. • Students graph the number correct and the time taken. Assist as required. • Direct students to learn the facts they had wrong or left out.

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• Explain to the students they will be doing a speed test involving multiplication and division basic facts. This will include timing how long it takes and graphing the results.

What to do

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• The focus for this unit is rounding to the nearest ten.

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Challenge • Students may either draw arrangements or use five squares to make arrangements to check that the perimeter is 12 units. • All arrangements are to be drawn or made with a full side, on a full side not corner to corner or part side to part side. • Display findings on class pin-up board for others to view. For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 60 – 61. • 50 • New Wave Maths Book D – Teachers Guide

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Unit 8–2

Student page 23

Outcomes

Indicators

N3.3, N3.4, C&D3.2, C&D3.4

The student is able to: • build sequences of simple shapes which increase in size systematically and write the equivalent number pattern. • record frequency data carefully using simple formats based on tallies or organised lists and take care with their measurements. • explain their own data displays to their peers, talking about the features represented.

Skills • counting • recording • describing patterns

Resources

Language • counting • inside • regions • intersections • shapes

• Base 10 MAB • calculator

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Notes

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Memory Masters (N3.3)

• The focus for this unit is completion of number sentences.

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Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.4, C&D3.2, C&D3.4) Warm up

© R. I . C.Publ i cat i ons What to• do f orr evi ew pur posesonl y• • Discuss with students what a region is. • Ask students to give examples of regions. • Find out from the students what they understand an intersection to be.

Challenge

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• Direct students to look at the shapes drawn in their workbook. • Ask students to count the internal regions in each shape, recording their results in the table. • Ask students to count the intersections in each shape recording their results in the table. • Each student is to examine the findings in the table and then describe the pattern that is developing. • Ask students to share their description with the class.

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• Students should be encouraged to experiment with colours to see what is the least number that can be used to colour the shapes as directed. • Suggest that students start with two colours initially and increase the number they use as required. • Check working to make sure that no two adjacent regions are coloured the same.

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New Wave Maths Book D – Teachers Guide • 51 •

Unit 8–3

Student page 24

Indicators

Outcomes N3.4, N3.3, N3.1a

The student is able to: • distinguish and order whole numbers. • remember basic addition facts and many multiplication facts and calculate mentally basic multiplication facts they don’t recall.

Skills • completing number sentences

Memory Masters (N3.4)

Resources • Base 10 MAB • calculator • jelly beans

Language • symbols • equal • not equal • relationships • not true • less than • greater than • statements • accurately

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Number (N3.3)

Main Activity (N3.1a, N3.3) Warm up

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• The focus for this unit is number patterns.

What to do

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• Allow students to use calculators if required to complete Exercise 3. • Students may continue on with Exercise 4 using the terms not true, less than or greater than. • Exercise 5 requires students to use their knowledge of basic facts to complete the number sentences.

Challenge

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• Ask students what the symbol for ‘equals’ or ‘equal to’ is. • Ask students how they would show that two items are not equal. Use traffic symbols, no smoking symbols etc. to reinforce the slash across the sign to signify ‘not’.

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• Students may need to view a large jar filled with jelly beans (this may be used as a class fundraiser by having students pay a small amount for a guess) to complete this exercise. • Ask students to devise a mathematical strategy to help them reach their answer. • Students are to write their strategy. Have students share their strategies. • The winner of the jelly beans may be the student with the closest answer to the actual number or the student with the best mathematical strategy.

• 52 • New Wave Maths Book D – Teachers Guide

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Unit 8—Answers

Student pages 22 – 24 Unit 8–1

1. (a) 7 (b) 6 (c) 5 (d) 7 (e) 3 (f) 2 (g) 4 (h) 1 (i) 5 (j) 3 2. (a) $11 (b) 12c (c) 31c (d) 21c (e) $23 (f) 34c 3.

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Inside regions always have one more than the number of intersections. Challenge (a) 2 (b) 2 (c) 3

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1. (a) 30 (b) 30 (c) 60 (d) 80 (e) 40 (f) 50 (g) 70 (h) 60 (i) 90 (j) 40 2. (a) 12 (b) 11 (c) 24 (d) 32 (e) 13 (f) 22 3. 18 30 16 35 36 21 40 28 45 24 64 42 18 6 20 72 25 12 56 12 49 16 32 45 42 27 14 24 48 36 9 3 3 6 8 6 5 7 6 8 6 4 8 7 5 4 6 3 9 5 4 7 10 3 9 4 8 7 5 2 4. Teacher check Challenge Teacher check

Unit 8–2

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• When students have improved in speed and accuracy, devise another speed test of different multiplication and division basic facts.

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1. (a) 10 (b) 10 (c) 3 (d) 12 (e) 4 (f) 6 (g) 22 (h) 7 (i) 12 (j) 50 2. (a) 14 (b) 12 (c) 15 (d) 14 (e) 13 (f) 19 3. (a) = (c) ≠ (e) ≠ (b) = (d) = (f) ≠ 4. (a) not true (e) greater than (b) greater than (f) less than (c) not true (g) less than (d) greater than (h) less than 5. (a) 8 (e) 5 (i) 7 (b) 9 (f) 9 (j) 7 (c) 42 (g) 72 (k) 4 (d) 6 (h) 8 (l) 24 Challenge Answers will vary. One possible solution: Estimate how many jelly beans in one layer. Multiply by the number of layers.

Consolidation 8–2

• Extend the pattern to 10 regions drawn in the same configuration. Ask students to suggest an answer to the number of internal regions and the number of intersections, using their findings to date.

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Consolidation 8–3

• Provide opportunities for similar activities where students use =, ≠, not true, less than or greater than to complete statements.

New Wave Maths Book D – Teachers Guide • 53 •

Unit 9–1

Student page 25

Outcomes

Indicators

M3.1, N3.4, N3.3, S3.1, M3.4b

The student is able to: • attempt to provide a bird’s-eye view of familiar locations such as their classroom. • order and show a sense of the proximity of things in locating key features on maps. • interpret order and proximity from maps. • draw informal maps and plans which show a sense of scale, that is, look ‘roughly right’.

Skills • drawing • scaling • planning

Resources • Base 10 MAB • calculator

Language • plan • model • bird’s-eye view • scale

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Notes

Memory Masters (M3.1, N3.4) Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.1, M3.4b) Warm up

• Introduction to the drawing of the plan needs to focus on what a plan represents and understanding of a ‘bird’s-eye view’. Ask students: – When drawing a plan, what do you need to keep in mind about the equipment? (Size and position, safety—space between items.) – What items would you expect to find in an adventure playground? – What would you like to have in your adventure playground? – How would you arrange the items in the adventure playground? Circular fitness track? Equipment randomly placed or similar items placed together. – How can you be sure that anyone reading your plan will know what each item is? Does it really matter? Do you name the item and/or draw a sketch and/or description of the item?

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Teac he r

• The focus for this unit is conversion of units of measure and number patterns.

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What to do

• Direct students to think about their plan and draw a rough plan, before drawing the actual plan of their adventure playground in the workbook. • Finish with a brief presentation to small groups, who decide on the best for presentation to the class.

Challenge

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• Discuss materials required and how the model is to be displayed. • Set students to complete their model.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 2 – 3. • 54 • New Wave Maths Book D – Teachers Guide

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Unit 9–2

Student page 26

Outcomes

Indicators The student is able to: • use their own methods or a conventional algorithm to multiply whole numbers by single-digit numbers. • identify patterns in the multiplication tables and use to make predictions.

N3.1a, N3.3, N3.4

Skills • completing number sentences • using a calculator • multiplying

Resources

Language

• Base 10 MAB • calculator • 2-cm cubes

• statement • problem

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Notes

Teac he r

Memory Masters (N3.1a)

• The focus for this unit is changing cents to dollars and dollars to cents.

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Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3, N3.4) Warm up

• Allow students a few minutes to explore their calculator. • Direct students to enter in 100 followed by x followed by 3 followed by = and see what result they obtain. (300) • Continue with these sequences: —4 x 100 = —300 x 2 = —3 x 100 = —200 x 3 = —100 x 5 = —2 x 400 = • Ask students what they noticed about the answer in each case. (They each finished with 00.)

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What to do

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• Direct students to work through the two exercises in the workbook without using the calculator. • For students having difficulties, direct them to use their calculator for one or two examples, then try again without using the calculator.

Challenge

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• Distribute 2-cm cubes to students. • Ask students to make the model shown on the page in the workbook. • Now ask students to make a guess as to how many cubes would be required to make a double-scale model. • Ask students to check to see if their guess was correct. • Students choose their own method of verifying their guess.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 62 – 63. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 55 •

Unit 9–3

Student page 27

Indicators

Outcomes N3.4, N3.3, M3.4a, M3.2, C&D3.3

The student is able to: • find the perimeter of a polygon by measuring each side and adding the lengths. • use a uniform unit to compare the areas of two regions where the units are reasonably small relative to the shape. • display frequency data in bar graphs.

Skills • measuring perimeters • graphing • finding area

Memory Masters (N3.4)

Resources • Base 10 MAB • calculator

Language • measure • perimeters • shapes • grid • counting • squares • line graph • height

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Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.4a, M3.2, C&D3.3) Warm up

• Discuss the meaning of the words ‘perimeter’ and ‘area’. — The perimeter is the distance around the edge of a shape. — The area is the internal size of the shape. • Ask students to trace the perimeter of their Workbook and point to the area of a page.

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• When satisfied that students understand the difference, ask them to open their workbooks. Using the grid and the shapes on the page, find the perimeter in units, and the area in square units, of each shape. • Once the perimeter is found, students are to use the grid to make a single line graph to show the length of each perimeter. The line is to be drawn beside each shape. • Students then answer the questions in their workbook.

Challenge

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• This activity may be completed independently or in small groups. • Allow the students time to discuss how they might measure their height without using standard measuring equipment. • Once a method is decided, students measure their height and record in detail how they did it. Discuss findings as a class.

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What to do

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• The focus for this unit is doubling and halving.

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For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 80 – 81 and 100 – 101. • 56 • New Wave Maths Book D – Teachers Guide

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Unit 9—Answers

Student pages 25 – 27

Unit 9–1

1. (a) $2 (b) $8 (c) $4 (d) $6 (e) $12 (f) 700c (g) 100c (h) 600c (i) 300c (j) 1500c 2. (a) 138 (b) 144 (c) 194 (d) 167 (e) 147 (f) 207 3. (a) 6 (i) 3 (b) 100 (j) 100 (c) 7 (k) 100 (d) 2 (l) 8 (e) 5 (m) 100 (f) 100 (n) 4 (g) 100 (o) 100 (h) 100 (p) 7 4. (a) 300 (i) 700 Challenge (b) 900 (j) 400 40 (c) 200 (k) 600 (d) 500 (l) 100 (e) 100 (m) 600 (f) 700 (n) 900 (g) 800 (o) 500 (h) 300

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1. (a) 100 (b) 1000 (c) 24 (d) 14 (e) 31 (f) 18 (g) 16 (h) 16 (i) 35 (j) 80 2. (a) 75 (b) 78 (c) 77 (d) 95 (e) 93 (f) 97 3. Teacher check Challenge Teacher check

Unit 9–2

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• Draw a detailed plan of their ideal games room with objects such as couches, beanbags,TV, Playstation™ etc.

Consolidation 9–2

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1. (a) 10 (b) 16 (c) 8 (d) 6 (e) 14 (f) 2 (g) 9 (h) 4 (i) 50 (j) 20 2. (a) 969 (b) 939 (c) 578 (d) 737 (e) 696 (f) 889 3. (a) shape 2 (b) shape 2 (8 squares) shape 4 (9 squares) shape 1 (12 squares) shape 3 (16 squares) Challenge Teacher check

• Provide opportunities for similar activities. Students could multiply single-digit numbers by 10 or 1000, with or without a calculator.

Consolidation 9–3

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• Draw other squares and rectangles on grid paper, counting the distance around to measure perimeter and the number of squares to measure area.

New Wave Maths Book D – Teachers Guide • 57 •

Unit 10–1

Student page 28

Outcomes

Indicators

N3.4, N3.3, N3.1a, M3.2

The student is able to: • round numbers up or down or to the nearest 10 or 100 to serve a specific purpose such as estimation. • tell the time on digital and analog clocks.

Skills • rounding • reading and writing time

Memory Masters (N3.4)

Resources • Base 10 MAB • calculator

Language • subtract • number line • round

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Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a, M3.2) Warm up

• Ask how many students are in the class. • Ask what answer you would give if you were asked approximately how many students are in the class. (Usually rounded to the nearest five.) • Explain that rounding is a common means of giving an approximation. It may be used when estimating answers; for example – 6 x 99 is approximately 6 x 100 = 600. • Ask all the students to line up in one straight line. Ask those who are nearly nine or have just turned nine to move to the centre of the line. Those who will still be 8 until later in the year go to one end and those who will be 10 early the following year to go to the other end. The line now has an approximation or rounding of ages to 8, 9 or 10 years. • Remind students of the rules to follow when a number is halfway between the rounding points; round down: round up: 0 – 4 5 – 9 Therefore 5 rounded to the nearest ten is 10. 0 – 40 50 – 90 45 rounded to the nearest ten is 50. 0 – 400 etc. 500 – 900 etc. 50 rounded to the nearest hundred is 500.

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• The focus for this unit is odd and even numbers.

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What to do

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• Proceed to round the numbers given to the nearest 10 in Exercise 3 using the number line provided to assist. • Record the times shown in Exercise 4, ignoring whether it is morning or afternoon.

Challenge

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• Students will need to experiment with adding, subtracting, multiplying and dividing to find ways of making other numbers using a single number and the four operations. • As a hint, ask how can 1 be made from the number nine using one or all of the operations; e.g. 9 ÷ 9. • Leave the students to experiment either by themselves or in small groups. • Share answers with the class and encourage the finding of different combinations. • Praise all efforts that reach the required results.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 90 – 91. • 58 • New Wave Maths Book D – Teachers Guide

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Unit 10–2

Student page 29

Outcomes

Indicators The student is able to: • justify their choice of more or less likely by referring to past experience or known information.

N3.4, N3.3, C&D3.1

Skills • understanding chance events • recording possible outcomes

Resources

Language

• Base 10 MAB • calculator

• chance

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Notes

Teac he r

Memory Masters (N3.4)

• The focus for this unit is comparison of numbers and place value.

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Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.1) Warm up

• Ask the class what they understand by the word ‘chance’. • Discussion should be encouraged to be as wide as possible. Work toward the likelihood or possibility of an event taking place. • Develop the understanding that there is a wide range of examples of chance events: —What is the chance that 1 + 1 = 2? —What is the chance you will shave your hair off tonight? —What is the chance that you will be released to go home on time? —What is the chance that you will be caught talking in class?

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What to do

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• Ask students to think about the activity in their workbook. Either work through the activity as a whole class sharing a number of suggestions; work in small groups, choosing the best explanation for sharing with the class; or work independently and share responses from a variety of students. • Remember with chance, unless it is totally likely or unlikely, answers will vary greatly according to individual opinion or experience.

Challenge

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• Most students will, as a matter of course, add the numbers in sequence. • Ask students to stop and think of an alternative means of adding the numbers. • Share suggestions with the class. • Ask students to try the suggestions to see if they are quicker. • Accept all suggestions—final choice on speed may depend on the individual.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 106 – 107. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 59 •

Unit 10–3

Student page 30

Indicators

Outcomes N3.1a, N3.3

The student is able to: • read and write any whole number into the thousands.

Skills • reading whole numbers • writing whole numbers • writing whole numbers in words

Resources • Base 10 MAB • calculator • flashcards with numbers and number names

Language • tens • round • numeral • hundred

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Memory Masters (N3.1a) Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a) Warm up

• Distribute flashcards with numbers and number names (also include the word ‘and’) to

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Teac he r

• The focus for this unit is rounding.

groups of students. Direct them to read the numbers; e.g. • Use the number names to compile numbers; e.g.

What to do

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Challenge

• When completing a Magic Square, there is usually a clue included. Ask students to find the clue. • Ask how this clue may be used to complete the Magic Square. • Students complete the Magic Square. Allow the use of calculators to assist in addition as it requires regrouping.

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• Direct students to Exercises 3 and 4. Students should complete the exercises individually, though some may require assistance. • Students who are having difficulty could make the numbers using Base 10 MAB before writing them as numerals or words.

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For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 34 – 35. • 60 • New Wave Maths Book D – Teachers Guide

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Unit 10—Answers

Student pages 28 – 30 Unit 10–2

1. (a) O (b) O (c) O (d) E (e) E (f) O (g) E (h) E (i) E (j) O 2. (a) 348 (b) 394 (c) 362 (d) 274 (e) 167 (f) 647 3. (a) 190 (f) 210 (b) 200 (g) 190 (c) 190 (h) 200 (d) 200 (i) 210 (e) 210 (j) 210 4. (a) 3.00 (c) 8.30 (e) 1.30 (g) 10.30 (b) 11.15 (d) 4.45 (f) 12.10 (h) 4.40 Challenge Answers will vary. Possible solutions, working left to right: 9÷9=1 9+9+9+9+9÷9=5 9x9–9÷9=8

1. (a) 23 (b) 39 (c) 79 (d) 59 (e) 29 (f) 7 (g) 6 (h) 7 (i) 9 (j) 0 2. (a) 125 (b) 181 (c) 427 (d) 426 (e) 237 (f) 262 3. Answers will vary Challenge 45.The following diagram shows a quick means of solving the problem. It involves pairing addends to make 10.

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Unit 10–1

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26 44 36 50

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• Record the attendance for students at their school for two weeks and round the figures to the nearest ten. • Write the time shown on clock faces which do not have all numbers showing, or those with strokes in place of numbers.

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1. (a) 20 (b) 30 (c) 30 (d) 70 (e) 50 (f) 20 (g) 40 (h) 60 (i) 30 (j) 80 2. (a) 616 (b) 411 (c) 543 (d) 281 (e) 521 (f) 443 3. (a) 255 (f) 2003 (b) 410 (g) 909 (c) 689 (h) 4732 (d) 505 (i) 8544 (e) 1360 (j) 380 4. (a) four hundred and seventy-two (b) three hundred and one (c) five hundred and fifty-five (d) one thousand, seven hundred and thirty-three (e) four thousand and forty-five Challenge

Consolidation 10–2

• Students can brainstorm further questions to answer with ‘unlikely’, ‘likely’ or ‘certain’. These could be written on the blackboard/whiteboard for students to copy, answer and discuss in small groups.

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• Use flashcards as described in the warm up to read and write numbers in numerals and words.

48 38 46 24

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New Wave Maths Book D – Teachers Guide • 61 •

Unit 11–1

Student page 31

Outcomes

Indicators The student is able to: • inspect a prism or pyramid, put it aside and then select 2-D shapes to match the faces of polyhedrons. • integrate conventional names of shapes and component parts of shapes into his/her descriptions of things. • describe and compare the spatial features of various mathematical objects which he/she can see and handle.

N3.3, S3.2, S3.4

Skills • describing 3-D shapes • identifying 3-D shapes • working with a partner

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • 3-D shapes

Language • solid • 3-dimensional shapes • pyramid • prism • cylinder • square • rectangular

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Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.2, S3.4) Warm up

• Distribute sets of a variety of 3-D shapes to groups within the class. • Direct students to handle the shapes and talk about each shape. • Ask one person from each group to report back to the class on the attributes they discovered about one of the shapes they had been examining.

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• Direct students to Exercise 3. Focus using the descriptions just given of 3-D shapes. Compare and contrast some of the descriptions that have been given. • Students write a detailed description of each of the three shapes—using a model if necessary. • Students share one of their descriptions to see if a partner can guess which shape is being described.

Challenge

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What to do

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Teac he r

• The focus for this unit is basic facts of division and subtraction.

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For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 32 – 33. • 62 • New Wave Maths Book D – Teachers Guide

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Unit 11–2

Student page 32

Outcomes

Indicators

N3.3, S3.2

The student is able to: • match standard geometric models with realistic drawings and conventional diagrams.

Skills • making models • observing • matching models

Resources

Language

• Base 10 MAB • calculator • 2-cm cubes

• cubes • build(ing) • model • plan • construction

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Notes

Memory Masters (N3.3) Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.2) Warm up

• Organise the class into small groups. Distribute a quantity of 2-cm cubes to each group. • Allow students to play with the cubes—encourage them to share verbally with each other about what they are doing.

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Teac he r

• The focus for this unit is basic facts of division and subtraction.

Challenge

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• Direct students to the diagram shown in Exercise 3. Ask students to build a copy of the model shown. • Discuss the number of cubes used in each section and how this matches the plan given for building the cube. • Students can now complete the exercise individually, with a partner or in a small group, depending on how confident they feel. • Discuss and compare the plans drawn. Point out errors (if any) and explain how to correct them. • Students can build, draw and make a plan of their own model.

o c . che e r o t r s super

• Students build a variety of constructions using eight 2-cm cubes. The surface area can be noted next to each construction. • Students compare answers and constructions. Different constructions will be found to have the same surface area.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 18 – 19. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 63 •

Unit 11–3

Student page 33

Indicators

Outcomes

The student is able to: • chooses suitable units of measure and measuring devices.

N3.3, M3.1

Skills • measuring • recording

Memory Masters (N3.3)

Resources

Language

• Base 10 MAB • calculator • scale, ruler, measuring cups and jugs, stopwatch, tape measure, trundle wheel • materials for students to measure

• mass • length • volume • capacity • time • height • distance • duration • measuring units; e.g. grams, kilograms, minutes etc. • materials used to measure; e.g. scales, stopwatch etc.

r o e t s Bo Notes r e p ok u S

Number (N3.3)

Main Activity (M3.1)

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• Discuss that different items weigh different amounts, some activities take different lengths of time and that different objects have different capacities. Different things use different measuring tools. For example, we wouldn’t measure how long it takes us to brush our teeth with a ruler, or, if we used time, we wouldn’t use hours—we would use a timer, and refer to the measure in minutes and seconds. – Use different situations to ask students what tool and unit of measure they would use. – Can some measuring situations use more than one tool and/or unit? • Revise units of measurement with the class. For example, 1000 g = 1 kg, 100 cm = 1 m. • Revise terms used for different measurements. For example, capacity – quantity or amount which can be held or contained by a container.

What to do

• Read through the question and model with the students how to complete the first item. Draw the answers from the students. • Ask the students to complete the second example on their own, then discuss their answers as a group. Once you are satisfied the students understand the concept, they can complete the activity independently. • Students move into pairs or small groups to review their table. Any differences between answers should be justified and supported by the students.

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Challenge

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Warm up

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Teac he r

• The focus for this unit is basic facts of division and subtraction.

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• Students should be encouraged to try various numbers between 1 and 10 to complete the puzzle. In order to prove or disprove a theory, it must be performed several times. • Once students have had adequate opportunity to complete the challenge, allow them to move into pairs or small groups to discuss their findings. If solutions differ, students should compare their method to others and state which method they felt was more accurate and why. • Discussion about the task allows students time for reflection about how well they performed how they can improve in future problem-solving activities.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 74 – 75. • 64 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 11—Answers

Student pages 31 – 33

Unit 11–1

1. (a) 6 (b) 4 (c) 7 (d) 1 (e) 5 (f) 6 (g) 1 (h) 6 (i) 2 (j) 9 2. (a) 28 (b) $39 (c) 88c (d) 69 (e) 96 (f) 36 3. Teacher check Challenge Teacher check

r o e t s Bo r e p ok u S

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Teac he r

1. (a) 4 (b) 7 (c) 2 (d) 7 (e) 9 (f) 4 (g) 1 (h) 3 (i) 0 (j) 7 2. (a) 96 (b) 88 (c) 48 (d) 84 (e) 64 (f) 69 3. Teacher check descriptions. Shape A – square-based pyramid Shape B – rectangular prism Shape C – cylinder Challenge In a square pyramid, the base is square and it has four triangular sides. In a triangular pyramid, the base is triangular and it has three triangular sides.

Unit 11–2

Item

(a) the mass of feathers

(b) the length of a blackboard

(d) the capacity of a bucket

Consolidation 11–2

• Design a plan for a model using 2-cm cubes and give it to a partner to construct.

o c . che e r o t r s super Consolidation 11–3

metre rule or tape measure

graduated cylinder

mL/L container

(e) the time taken to get to school

minutes

stopwatch

(f) your height

cm or m

metre rule

(g) the length of your thumb

mm or cm ruler/tape measure

(h) the length of this page

mm or cm ruler/tape measure

(i) the length of a netball court

(j) the distance of 20 times around the oval

trundle wheel

(k) the duration of your school’s lunch

minutes

watch/clock

scale

(l) the mass of a softball

Measurement

scale

Answers may vary

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Unit of Measure Measured with …

• Repeat the activity using a 3-D shape of each student’s choice. Each student reads his/her description to see if a partner can guess.

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1. (a) 5 (b) 4 (c) 3 (d) 3 (e) 4 (f) 9 (g) 8 (h) 1 (i) 3 (j) 3 2. (a) 85 (b) 92 (c) 52 (d) 96 (e) 96 (f) 78 3. Answers may vary.

• Students brainstorm other items to list the unit of measure, and what to measure it with before taking the measurement.

trundle wheel

Challenge 3. Yes. Teacher check explanation.

R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 65 •

Unit 12–1

Student page 34

Outcomes

Indicators

N3.3, N3.1a, M3.3

The student is able to: • read and write any whole number into the thousands. • distinguish and order whole numbers. • make sensible numerical estimates based on provided units.

Skills • writing numbers • estimating • rearranging

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • place value chart (see page 205)

Language • digits • largest • smallest • estimate • count • minutes • actual • rearrange

r o e t s Bo r e p ok u S

Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a, M3.3) Warm up

• Discuss place value with students. • Issue place value charts for students to use. • Ask students to use the chart to help them decide on the largest and smallest number that can be made from 3, 4 and 5. Discuss answers.

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• Direct students to Exercise 3 in the workbook. • Ask students which four different digits they would use to make the largest four-digit number they could. • Students write the number in their workbook. (9876) • Repeat for the smallest four-digit number. (1234) • For Exercise 4 students need to rearrange three digits to make different numbers. There are six possible combinations for each. • Encourage students to be systematic. Work through the first example—641, 614. Ask what number would be the next best to go first. (4, the next largest) • Instruct students to write 461 and 416. Finally, write 164 and 146. • Students can complete the rest of the exercise. • Follow the same procedure to complete Exercise 5. There are 24 possible combinations for rearranging the four digits.

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Challenge

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What to do

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Teac he r

• The focus for this unit is basic facts of division and subtraction.

o c . che e r o t r s super

• Ask students to think about the number 100. What does it mean, what do they know about it? • Students write all the things they know about the number 100. • Share the knowledge with the class. • Record the categorised knowledge of 100 for the whole class to see.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 34 – 35 and 88 – 89. • 66 • New Wave Maths Book D – Teachers Guide

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Unit 12–2

Student page 35

Outcomes

Indicators The student is able to: • use diagrams such as Venn diagrams and two-way tables to represent a two-way classification. • explain their own data displays to their peers, talking about the features represented.

N3.3, C&D3.3, C&D3.4

Skills • recording • completing arrow diagrams

Resources

Language • tree diagram • parent • grandparent • great-grandparent • arrow diagram

• Base 10 MAB • calculator • counters (10 per student)

r o e t s Bo r e p ok u S

Memory Masters (N3.3)

Notes

Number (N3.3)

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Teac he r

• The focus for this unit is basic facts of division and subtraction.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.3, C&D3.4) Warm up

• Discuss family members with the students. Brainstorm responses to the following questions and record on the blackboard/whiteboard. – Who are the members of your family? (e.g. mum, dad, brother, sister) – What names are given to these people? (e.g. parents, siblings) – What are the names given to people who are in your extended family but don’t necessarily live with you? (e.g. grandparents, aunties, uncles, cousins, great-grandparents) • Talk about the concept that our grandparents are our parents’ parents and that our greatgrandparents are our grandparents’ parents. • Draw the diagram on the board and model its completion with your own family.

What to do

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Challenge

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• Allow students the opportunity to complete the diagram in Exercise 3. Some students may not be able to complete this activity during class time because they do not know the required information. Forewarn the class of the activity so they can gather the information from home. • Discuss the purpose of an arrow diagram as shown in Exercise 4. An arrow diagram is one which uses arrows to show a relationship. • Use an example to illustrate the use of an arrow diagram; e.g. the door is the same shape as the desk and the mat. • Ask the students to think of their own examples of people they know to complete the arrow diagram.

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• Give each student ten counters and ask them to lay them out as shown in the diagram. • Read through the question to ensure students have the understanding of what is required. • Allow students the opportunity to complete the activity. Students are likely to have a number of different responses. • Ask a few students to share how they solved the problem. If they used four counters, ask them if they can work out a way to use only three counters to turn the triangle upside down. • Some students may have difficulty with such a concept. Position them next to a student who will be able to model their approach to solving problems. Students need to be encouraged to tackle the problem a number of times if they are having difficulty.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 110 – 111. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 67 •

Unit 12–3

Student page 36

Indicators

Outcomes

The student is able to: • separate objects and collections into equal parts to show unit fractions. • ‘count’ orally in (common) fractional amounts.

N3.3, N3.1b

Skills • representing fractions

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • coloured pencils • ruler • fraction cakes • coloured rods

Language • fractions • one-fifth • one-tenth • one-half • one-eighth • one-fourth • one-third

r o e t s Bo Notes r e p ok u S

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1b) Warm up

• Place the class in small groups. • Distribute fraction cakes, coloured rods and other materials that can be used to show fractions. • Allow the students time to play with the materials. • Ask the students to use their materials to show halves, fourths, thirds, tenths and eighths. • Ask students which material representation did they find the easiest to use. • Ask students to give the reasons for their choice.

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Teac he r

• The focus for this unit is basic facts of multiplication and addition.

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• Direct the students to the activity in their workbook. • Ask students what is shown in the first row of diagrams (thirds, fifths, and sixths). • Ask students which is the easiest diagram to use to show fifths. Shade one-fifth of the diagram showing the five equal parts (fifths). • Complete the activity on the page, reminding students that some diagrams may be divided into the required number of parts but are not equal parts.To accurately represent fractions, the parts must be equal.

Challenge

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What to do

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• Ask the students to show how many parts the circle can be divided into if three lines do not intersect. • How many parts can the circle be divided into if just two lines intersect? • Students use a ruler to draw three straight lines to see the greatest number of parts the circle can be divided into. • Discuss findings.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 48 – 49. • 68 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 12—Answers

Student pages 34 – 36

Unit 12–1

1. (a) 3 (b) 1 (c) 8 (d) 5 (e) 2 (f) 8 (g) 8 (h) 6 (i) 9 (j) 3 2. (a) 26 (b) 24 (c) 13 (d) 13 (e) 23 (f) 28 3. Teacher check 4. Teacher check Challenge

r o e t s Bo r e p ok u S

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Teac he r

1. (a) 4 (b) 1 (c) 6 (d) 3 (e) 1 (f) 1 (g) 7 (h) 9 (i) 6 (j) 4 2. (a) 18c (b) 12c (c) $12 (d) $19 (e) $16 (f) $17 3. (a) 9876 (b) 1234 4. (a) 461, 416, 164, 146 (b) 952, 925, 592, 529, 295, 259 (c) 831, 813, 381, 318, 183, 138 (d) 764, 746, 674, 647, 476, 467 (e) 875, 857, 785, 758, 587, 578 5. (a) 8741, 8714, 8471, 8417, 8174, 8147, 7841, 7814, 7481, 7418, 7184, 7148, 4871, 4817, 4781, 4718, 4187, 4178, 1874, 1847, 1784, 1748, 1478, 1487 (b) 9652, 9625, 9562, 9526, 9265, 9256, 6952, 6925, 6592, 6529, 6295, 6259, 5962, 5926, 5692, 5629, 5296, 5269, 2965, 2956, 2695, 2659, 2596, 2569 Challenge Teacher check

Unit 12–2

(b) One-tenth

(e) One-fourth

(f) One-fifth

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• Provide further opportunities to rearrange three and four digits into as many combinations as they can.

Consolidation 12–2

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1. (a) 15 (b) 32 (c) 0 (d) 8 (e) 24 (f) 9 (g) 15 (h) 10 (i) 8 (j) 3 2. (a) 21 (b) 21 (c) 32 (d) 11 (e) 23 (f) 33 3. (a) One-fifth

• Brainstorm ideas with the students for other things to display on tree and arrow diagrams.

Consolidation 12–3

o c . che e r o t r s super (g) One-third

• Provide opportunities for further activities to choose and shade the correct diagrams to show given fractions.

(h) One-fourth (i) One-fifth

(j) One-third

Challenge 7

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New Wave Maths Book D – Teachers Guide • 69 •

Unit 13–1

Student page 37

Outcomes

Indicators

N3.3, S3.2

The student is able to: • match standard geometric models with realistic drawings and conventional diagrams.

Skills

Resources • Base 10 MAB • calculator • 3-D shapes

• drawing • observing

Memory Masters (N3.3)

Language • sketch • 3-D models • cube • cylinder • square • pyramid • tetrahedron • rectangular prism • cone

r o e t s Bo r e p ok u S

Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.2) Warm up

• Provide students with a variety of 3-D shapes to explore. • Organise students into small groups and encourage them to discuss the shapes as they manipulate them. • To ensure students know what each shape named in the workbook looks like, display the shapes listed in the workbook and tell the class the name of each shape.

• Direct the students to sketch the shapes in their books. • If students ask about the rear of the shape that is obscured, suggest they can ignore it or show the edges using dotted lines.

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Challenge

• Suggest that students may wish to draw the shape onto a larger sheet of paper that is square in shape. • The internal shaded figure’s corners (vertices) are at the sides of the square. • Ask students to think carefully about how they can determine the fractional part of the whole that is shaded. • Allow students to experiment without giving them a direct means to the solution. • For teachers’ information only—cut the unshaded pieces and arrange over the shaded section—shows both cover the same area—hence shaded part is one-half of the whole.

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What to do

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Teac he r

• The focus for this unit is basic facts of multiplication and addition.

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For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 12 – 13. • 70 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 13–2

Student page 38

Outcomes

Indicators

N3.3, N3.1a

The student is able to: • use their own methods or a conventional algorithm to multiply whole numbers by 10. • mentally estimate the results of a calculation in order to check the reasonableness of calculator results. • produce and use standard partitions of two- and three-digit numbers.

Skills • estimating • multiplying • trading

Resources

Language • pattern • hundred • tens • ones • totals • estimate • actual

• Base 10 MAB • calculator

r o e t s Bo r e p ok u S

Notes

Memory Masters (N3.3) Number (N3.3)

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Teac he r

• The focus for this unit is basic facts of multiplication and addition.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3, N3.1a) Warm up

• Ask students to estimate … 4 lots of 24. (Nearly 100.) 3 lots of 31. (Nearly 90.) 4 lots of 19. (Nearly 80.) • If estimating 10 lots of a number, how accurate would you expect your answer to be? (Probably exact.)

What to do

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Challenge

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• Try these – estimate … 10 lots of 15. 150. Check using your calculator. 10 lots of 36. 360. Check using your calculator. 10 lots of 71. 710. Check using your calculator. • Discuss why the estimates are likely to be exact. (Multiplying by 10 requires a 0 in the ones place.) • Set class to work to complete the Exercise 3. • Check progress and ensure estimates are being checked. • In Exercise 4, students take out Base 10 MAB materials as directed—23 tens and 47 ones. Trade the pieces of wood to find the least required to make the same total (2 hundreds, 7 tens and 7 ones or 16 pieces.) • Complete the rest of the exercise as a class, or individually for able students.

o c . che e r o t r s super

• Ask students how they would find the answer to this question: ‘What is half of a half of a half?’ • Students may discuss this with a partner or small group before reporting ideas back to the class. • Many students will offer the idea of a diagram to shade or cut out.

R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 71 •

Unit 13–3

Student page 39

Indicators

Outcomes N3.3, M3.2, M3.4a, WM3.3

The student is able to: • construct things to a specified length in metres and centimetres. • measure the perimeter of an irregular shape. • make conjectures which reflect their understanding of measurement. • discard conjectures which fail their tests.

Skills • finding circumference • investigating • checking/verifying

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • hoops • ruler • chalk

Language • circumference • accurately • explanation • method • diagrams • drawings

r o e t s Bo Notes r e p ok u S

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2, M3.4a, WM3.3) Warm up

• Discuss with class the meaning of ‘circumference’. Focus discussion on the measuring of the distance around circular shapes; i.e. the perimeter of a circle.

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• Provide small groups of students with a hoop, a ruler and some chalk. • Explain to the groups they are to find the circumference of the hoop as accurately as they can. • Each group is to keep written notes on the process they used and the steps taken to find the circumference of the hoop. • If more than one method was used, records for each method are to be kept. It may be that different group members use their own book to record one method used on behalf of the group. • Use the space provided to draw diagrams to assist in the explanation.

Challenge

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What to do

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Teac he r

• The focus for this unit is basic facts of multiplication and addition.

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• Inform students that a new chocolate made of 24 cubes needs to be packaged. • Ask them to draw possible packaging solutions and recommend one. • Discuss the recommendations with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 78 – 79. • 72 • New Wave Maths Book D – Teachers Guide

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Unit 13—Answers

Student pages 37 – 39

Unit 13–1 1. (a) 36 (b) 0 (c) 25 (d) 14 (e) 40 (f) 12 (g) 9 (h) 11 (i) 4 (j) 10 2. (a) 1256 (b) 1488 (c) 1479 (d) 1668 (e) 1686 (f) 1676 3. (a) cube (d) triangular pyramid

(b) cylinder

/2. Teacher check explanation

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Teac he r

(f) cone

Challenge

1. (a) 35 (b) 3 (c) 30 (d) 16 (e) 27 (f) 17 (g) 10 (h) 4 (i) 13 (j) 7 2. (a) 645 (b) 669 (c) 718 (d) 548 (e) 958 (f) 846 3. Teacher check estimations (a) 240 (f) 470 (k) 290 (p) 740 (b) 250 (g) 320 (l) 350 (q) 930 (c) 210 (h) 640 (m) 330 (r) 680 (d) 380 (i) 430 (n) 450 (s) 560 (e) 530 (j) 590 (o) 520 (t) 700 4. (a) 230 + 47 = 277 16 pieces (b) 460 + 35 = 495 18 pieces (c) 260 + 54 = 314 8 pieces (d) 720 + 83 = 803 11 pieces (e) 350 + 29 = 379 19 pieces Challenge

r o e t s Bo r e p ok u S (e) rectangular prism

Unit 13–2

/8 ; A diagram could assist students in answering the question. 1

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• Make models of 3-D shapes using toothpicks and plasticine.

Consolidation 13–2

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1. (a) 24 (b) 10 (c) 0 (d) 45 (e) 6 (f) 16 (g) 7 (h) 12 (i) 8 (j) 13 2. (a) 585 (b) 943 (c) 794 (d) 862 (e) 995 (f) 767 3. Answers may vary. Possible solution: Mark one point on the hoop with chalk. Roll the hoop once and measure this length with the ruler. Challenge Answers may vary

• Provide opportunities to complete similar exercises multiplying whole numbers by 10. • Practise trading Base 10 MAB for fewer pieces.

o c . che e r o t r s super

R.I.C. Publications® www.ricpublications.com.au

Consolidation 13–3

• Discuss other measuring devices that could be used to measure the circumference of circular objects; e.g. a length of string, tape measure.

New Wave Maths Book D – Teachers Guide • 73 •

Unit 14–1

Student page 40

Outcomes

Indicators The student is able to: • link the action of sharing into a number of equal portions with the language of unit fractions.

N3.3, N3.1b

Skills

Resources • Base 10 MAB • calculator • ruler

• identifying equivalent fractions • understanding fractions

Memory Masters (N3.3)

Language • midpoint • • thirds • • fifths • • measure • • equal parts • fractional parts

r o e t s Bo r e p ok u S

fourths sixths tenths whole

Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1b) Warm up

• Ask students to place their ruler on their desk. • Ask the following questions: —Where is the midpoint of your ruler? —How can you be sure? —What is another name for the midpoint? (Half or halfway.) —Can you find the point(s) that show a fourth(s) of the ruler? —How can you be sure? —What can you tell me about the four parts of the ruler?

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Teac he r

• The focus for this unit is basic facts of multiplication and addition.

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• Use the same process to find the midpoint and fourths for Exercise 3. (a). • Complete the activities on the page following the directions given. • Ask students to explain to the class what they discovered through this activity—seeking an explanation of equivalence of fractions.

Challenge

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What to do

o c . che e r o t r s super

• Explain to the students that when comparing fractions we normally look at the denominator (bottom number). The bigger it is the smaller the fraction; e.g. 1/2 is bigger than 1/4 is bigger than 1/8 etc. • Illustrate this concept with concrete materials or drawings. • Where we have the same numerator (top number) in each fraction, this method works. • If the fractions have different numerators, we need to find another way to check. • Use the example given in the Challenge to show how to accurately tell which fraction is larger. Students need to explain how they reached a conclusion.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 48 – 49. • 74 • New Wave Maths Book D – Teachers Guide

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Unit 14–2

Student page 41

Outcomes

Indicators

N3.3, M3.2, C&D3.2, C&D3.3

The student is able to: • use a uniform unit of length carefully to make their own graduated scale to measure things. • use a unit consistently and carefully to measure and compare mass. • record frequency data carefully using simple formats based on tallies or organised lists. • summarise data based on tallying.

Skills • measuring • tallying • balancing

Resources

Language

• Base 10 MAB • calculator • paper streamers • string • nails • balance scales • 20-mm cubes • bottle tops • gumnuts • coloured rods

r o e t s Bo r e p ok u S

• measure • unit • tally • total • perimeter • length • balance • scale

Notes

Memory Masters (N3.3) Number (N3.3)

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Teac he r

• The focus for this unit is basic facts of division and subtraction.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2, C&D3.2, C&D3.3) Warm up

• Provide groups of students with paper streamers and string and allow them to measure various body parts. • Provide each group with a balance scale and 20-mm cubes, bottle tops, gumnuts, nails and coloured rods. Allow exploratory balancing.

Challenge

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• Direct students to choose a unit of measure to use to measure the length of the body parts listed in Exercise 3; e.g. centimetres, hand span, thumb length, ruler length, Base 10 MAB long. • Measure each body part using paper streamers or string. • Check how many units of their choosing were required to measure the length of the streamer. • Use a tally to record the number used and to find the total. • In Exercise 4, use a tally to find the total of each object’s mass when balanced against the items shown in the table.

o c . che e r o t r s super

• Students are to collect data to prove or disprove whether the length of a person’s foot is the same as the distance from his/her elbow to the wrist. • Discuss with students how they would collect data to prove or disprove this. How many students’/teachers’ measurements would they need? • This activity could be completed in small groups. Findings can be discussed when data has been collected.

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New Wave Maths Book D – Teachers Guide • 75 •

Unit 14–3

Student page 42

Indicators

Outcomes N3.3, N3.1a, N3.2

The student is able to: • round numbers up or down or to the nearest 10 or 100 to serve a specific purpose such as estimation. • use their own methods or a conventional algorithm to multiply whole numbers by single-digit numbers. • select an appropriate division to deal with sharing and grouping situations.

Skills • estimating • calculating • multiplying • dividing • sharing

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • toothpicks or similar

Language • calculator • estimate • between • problems

r o e t s Bo Notes r e p ok u S

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a, N3.3, N3.2) Warm up

• Ask students how they would estimate the answers to the following multiplication sums: 49 x 6; 83 x 4; 65 x 9; 51 x 3. • Using the concept of rounding provides a rough estimate of the answer. • Another way of providing a guide to the accuracy of an answer is to find out the range the correct answer may fall between. • Ask students how this might be done. • If students can not give a satisfactory answer, explain that by rounding to the ten less than the number and multiplying by the single-digit number, then rounding to the ten greater than the number and multiplying by the single-digit number provides a lower and upper range within which the actual answer will fall; e.g. 47 x 6—40 x 6 = 240; 50 x 6 = 300—the answer lies between 240 and 300.

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• The focus for this unit is basic facts of division and subtraction.

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What to do

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• Ask students to round Exercise 3(a) as the example shows in the workbook. Check with the calculator. • Continue for the rest of the exercise. Students who are able may continue unguided. • Allow students to experiment to find answers to Exercise 4 (a) and (b). Base 10 MAB or calculators can be used.

Challenge

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• Provide students with toothpicks. • Ask them to make the arrangement in their workbook. • Direct students to try different options for removing six toothpicks to leave two squares. • Encourage students to record their attempts to follow their progress and reasoning.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 64 – 65. • 76 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 14—Answers

Student pages 40 – 42

Unit 14–1 1. (a) 35 (b) 0 (c) 21 (d) 9 (e) 40 (f) 6 (g) 15 (h) 8 (i) 13 (j) 12 2. (a) 341 (b) 242 (c) 231 (d) 731 (e) 243 (f) 151 3. (a) Half of a half is a fourth; two-fourths equal one-half (b) Half of a third is a sixth; two-sixths equal one-third (c) Three-ninths equal one-third (d) Half of a fifth is a tenth; one-fifth equals two-tenths (e) whole half eighth

r o e t s Bo r e p ok u S

Challenge

1. (a) 4 (b) 6 (c) 6 (d) 3 (e) 9 (f) 2 (g) 3 (h) 8 (i) 4 (j) 1 2. (a) 434 (b) 381 (c) 262 (d) 124 (e) 433 (f) 545 3. Teacher check 4. Teacher check Challenge Teacher check

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fourth

Unit 14–2

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• Use a fraction cake or a fraction chart for students to identify equivalent fractions.

Consolidation 14–2

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1. (a) 9 (b) 5 (c) 4 (d) 1 (e) 5 (f) 7 (g) 8 (h) 3 (i) 2 (j) 4 2. (a) 312 (b) 541 (c) 332 (d) 242 (e) 282 (f) 154 3. (a) between 400 and 450 420 (b) between 400 and 480 448 (c) between 630 and 700 644 (d) between 360 and 450 378 (e) between 180 and 240 222 (f) between 280 and 320 292 (g) between 200 and 250 240 4. (a) 19 (b) 26 Challenge

• Measure other body parts such as a hand, foot or arm length with a streamer or string and a unit of measure. • Repeat Exercise 4 using different objects to balance against.

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Consolidation 14–3

• Provide further opportunities for students to practise rounding techniques to estimate answers for multiplication problems.

New Wave Maths Book D – Teachers Guide • 77 •

Unit 15–1

Student page 43

Outcomes

Indicators

N3.3, M3.2, M3.4a

The student is able to: • use a uniform unit to compare the areas of two regions where the units are reasonably small relative to the shape. • find the perimeter of a polygon by measuring each side and adding the lengths.

Skills • enlarging • doubling • drawing • investigating

Teac he r

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • pencil

Language • guide • draw • coy • shape • dimensions • doubled • height • width • area

r o e t s Bo r e p ok u S

Notes

• The focus for this unit is basic facts of division and subtraction.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2, M3.4a) Warm up

• Demonstrate to the students how to double the dimension of a shape using a simple rectangle; e.g. 2 x 4 units.

What to do

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Number (N3.3)

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Challenge

• Students use grid paper to make rectangles with a perimeter of 16 units. • Remind them that a square is also a rectangle. • Students decide which rectangle has the greatest area and which has the least area. • This activity could be done in pairs or individually. • Report findings to the class.

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• Direct the students to draw the new shape starting at the bottom left-hand side to ensure the new drawing fits on the grid. • Remind students that when doubling, the dimensions of all lines are double; i.e. twice as long. • Check final drawings. Discuss errors with individuals and assist in making corrections. • Students answer the questions in Exercise 3(a) and (b). Discuss findings as a class.

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• 78 • New Wave Maths Book D – Teachers Guide

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Unit 15–2

Student page 44

Outcomes N3.3, N3.1a

Skills • adding • writing numbers • ordering

Indicators

Resources

Language

The student is able to: • add and subtract whole numbers using their own written method or a conventional algorithm, explaining the method by reference to place value. • read and write any whole number into the thousands.

• Base 10 MAB • calculator • number cards 0 – 9

• adding • digits • order • smallest • largest

r o e t s Bo r e p ok u S

Memory Masters (N3.3)

Notes

Number (N3.3)

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Teac he r

• The focus for this unit is basic facts of division and subtraction.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3, N3.1a) Warm up

• Write two pairs of numbers between 10 and 100 on the blackboard/whiteboard; e.g. 46 + 53. • Ask students for different ways these numbers can be added—record the methods given on the board. For example; add ones then tens; add tens then ones; add 6 to 53 then add 40; or add 3 to 46 then add 50. • Discuss with the students whether the order of addition affects the final result.

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Challenge

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• Have students work through the first example—share the methods used. • Complete the rest of the exercise. • Have the numbers 0 – 9 on separate cards, one set between two. Ask the students to select three numbers to make a new number. For example; 2, 0 and 1. Students could either present these cards as 201, 210, 120, 102, 12 or 21. – Does the way we order the numbers make a difference to the number? – How does the order of each number affect the final number we make? – How many different ways could you arrange the numbers? • Look at and discuss the order of numbers; for example, four is larger than three but smaller than five. Ask students to order their cards from smallest to largest then largest to smallest. • Ask students to use the numbers 6, 2, 9 and 4 to make as many numbers as they can.There are 24 possible combinations. If students work methodically, writing the numbers from smallest to largest, they should find all possible combinations in the correct order.

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• Students will need to experiment with adding, subtracting, multiplying and dividing to find ways of making other numbers using a single number and the four operations. • As a hint, ask how can 1 be made from the number four using one or all of the operations; e.g. 4 ÷ 4. • Leave the students to experiment either by themselves or in small groups. • Share answers with the class and encourage the finding of different combinations. Praise efforts that reach the required results. For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 62 – 63 and 34 – 35. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 79 •

Unit 15–3

Student page 45

Indicators

Outcomes N3.3, M3.2

The student is able to: • tell the time on digital and analog clocks.

Skills

Resources • Base 10 MAB • calculator • clock

• reading and writing time

Memory Masters (N3.3)

Language • clock • nearest • minutes • past the hour

r o e t s Bo Notes r e p ok u S

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2) Warm up

• Use a cardboard clock with moveable hands to show different times for students to read. • Discuss with the class how to round time to the nearest five minutes for ease of reading. • Show times between sets of five minutes and ask the class to read the time to the nearest five minutes. • Ask the class how they decide which five minutes to read the clock to. (Look at the minute hand on the clock.)

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Teac he r

• The focus for this unit is basic facts of division and subtraction.

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• Students can now attempt the activity with guidance from the teacher. • Many students may be unfamiliar with an analog clock and will need assistance in drawing hands in the correct places. • Work with those having difficulties and allow others to proceed at their own rate.

Challenge

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What to do

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• Explain to the students that if there is an answer, there must have been a question. • What might the question have been to obtain an answer of 24? Answers should be varied and include such things as age and date, as well as more common mathematical statements. • Accept all questions, displaying them for students to see. • Ask if there is anything common about any of the questions—students may suggest that certain questions are addition sums, others are subtraction sums etc. • Ask whether there might be a means of recording all the questions so they are easy to read—group all addition, subtraction etc. in like groups.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 90 – 91. • 80 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 15—Answers

Student pages 43 – 45

Unit 15–1 1. (a) 5 (b) 8 (c) 8 (d) 1 (e) 1 (f) 7 (g) 5 (h) 9 (i) 6 (j) 7 2. (a) 87c (b) $95 (c) 98c (d) 75c (e) $90 (f) 75c 3. Dimensions Shape

Double size

12 48

Teac he r Least area = 7 units

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(a) The area is four times larger. (b) The shape doubled in height and doubled in width; i.e. 2 x 2 = 4 so multiplying the original area by 4 —12 x 4 = 48 Challenge Greatest area = 16 units

1. (a) 2 (b) 9 (c) 2 (d) 6 (e) 1 (f) 9 (g) 5 (h) 1 (i) 4 (j) 9 2. (a) 96 (b) 96 (c) 70 (d) 78 (e) 72 (f) 91 3. (a) 76 Teacher check ways to add (b) 57 (c) 78 (d) 79 (e) 66 (f) 38 (g) 84 (h) 97 4. 2469, 2496, 2649, 2694, 2946, 2964, 4269, 4296, 4629, 4692, 4926, 4962, 6249, 6294, 6429, 6492, 6924, 6942, 9246, 9264, 9426, 9462, 9624, 9642 Challenge Answers will vary. Possible solutions working left to right: 6+6÷6+6=8 6x6+6+6÷6=8 6+6+6+6+6+6+6+6÷6=8

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Height Width Area

Original

Unit 15–2

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• Students draw a shape on grid paper to draw a copy with double the dimensions. Compare the height, width and area.

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1. (a) 3 (b) 9 (c) 5 (d) 8 (e) 2 (f) 7 (g) 5 (h) 7 (i) 4 (j) 5 2. (a) 126 (b) 712 (c) 455 (d) 152 (e) 312 (f) 592 3. (a) 8.25 (d) 3.15 (g)11.45 (b) 2.05 (e) 7.25 (h) 9.15 (c) 11.55 (f) 9.05 (i) 4.50 Challenge Answers will vary. Possible answers are: 3 x 8, 6 x 4, 12 + 12, 30 – 6, 48 ÷ 2

Consolidation 15–2

• Work in pairs to show each other different ways of adding two-digit numbers. • Choose four numbers between one and ten and repeat Exercise 4.

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Consolidation 15–3

• Direct students to check the time during the day on an analog clock displayed in the classroom.

New Wave Maths Book D – Teachers Guide • 81 •

Unit 16–1

Student page 46

Outcomes

Indicators

N3.3, N3.1a, N3.2

Skills • estimating • calculating • multiplying • dividing • sharing

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator

Language • answer • between

r o e t s Bo r e p ok u S

Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a, N3.3, N3.2) Warm up

• Explain that rounding to the ten less than the number and multiplying by the single-digit number, then rounding to the ten greater than the number and multiplying by the singledigit number provides a lower and upper range within which the answer will fall; e.g. 47 x 6—40 x 6 = 240; 50 x 6 = 300— the answer lies between 240 and 300.

• Ask students to round Exercise 3(a) as the example shows in the workbook. • Continue for the rest of the exercise. Students who are able may continue unguided. • Allow students to experiment to find answers to Exercise 4(a) and (b). Base 10 MAB or calculators can be used.

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Challenge

• The activity is given without the starting number. Ask students how they might work out what the starting number was. Accept all reasonable answers. • Direct students to work out the original number using one of the methods given or one of their own.

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What to do

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Teac he r

• The focus for this unit is addition of three digits.

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For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 64 – 65. • 82 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 16–2

Student page 47

Outcomes

Indicators

N3.3, S3.2, WM3.2

The student is able to: • construct and draw 3-D models given specific criteria. • represent a problem with concrete materials and manipulate the materials to find a solution.

Skills • constructing 3-D models • recording data • drawing models

Resources

Language

• Base 10 MAB • calculator • 20-mm cubes

• similar • surface area • identical • table • model • face • corner

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Notes

Memory Masters (N3.3)

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Teac he r

• The focus for this unit is addition of three digits.

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.2, WM3.2) Warm up

• Have the students make the model shown in the workbook. Ask the following: — How many cubes in each layer? — How many layers? — What is the total number of cubes? — What is the surface area of the top? • Some discussion may be required to ensure an understanding of surface area.

What to do

Challenge

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• Show how the details from the questions above have been recorded in table form in the workbook. • Students make models of their own choosing, ensuring that the layers of their models are identical. • Students record the details of each model. • Share results in small groups or from various students with the whole class. Focus discussion on the questions outlined above with support from the actual model.

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• Students can work individually or in small groups. • Encourage students to experiment with different arrangements to those used in the previous activity. • Share findings with the class or in small groups. • Find out who has the highest amount of different shapes that meet the required arrangements. Share these with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 18 – 19. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 83 •

Unit 16–3

Student page 48

Indicators

Outcomes

The student is able to: • round numbers up or down to the nearest 10 or 100 to serve a specific purpose such as estimation. • add and subtract whole numbers using their own written method or a conventional algorithm, explaining the method by reference to place value.

N3.3, N3.1a

Skills • estimating • subtracting • analysing

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator

Language • subtract • estimate • differences • sums • closest • less than

r o e t s Bo Notes r e p ok u S

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a, N3.3) Warm up

• Ask students to write a subtraction number sentence that will give an answer close to 20. The answer is not to be exactly 20. • Share these, asking students what number the answer is close to. (20) • Ask students how they made a decision to choose 20 as the answer that the sum was near to (other than because they were told to!). • Remind students that the easiest way to obtain a reasonable accurate answer is by rounding to the nearest ten or, in Exercises 4 and 5, to the nearest 100.

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Teac he r

• The focus for this unit is addition of three digits.

• Students work through Exercise 3 with teacher assistance as required. • If satisfied with the level of understanding, have students continue with Exercises 4 and 5.

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Challenge

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• When completing a Magic Square, there is usually a clue included. Ask students to find the clue. • Ask how this clue may be used to complete the Magic Square. • Students complete the Magic Square. Encourage them to complete their working mentally, but calculators should be used if students are having difficulties.

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• 84 • New Wave Maths Book D – Teachers Guide

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Unit 16—Answers

Student pages 46 – 48

Unit 16–1

1. (a) 14 (b) 16 (c) 10 (d) 18 (e) 8 (f) 12 (g) 16 (h) 14 (i) 18 (j) 12 2. (a) 13 (b) 12 (c) 19 (d) 13 (e) 13 (f) 24 3. Teacher check Challenge Teacher check

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Teac he r

1. (a) 16 (b) 15 (c) 10 (d) 18 (e) 16 (f) 14 (g) 19 (h) 10 (i) 17 (j) 19 2. (a) 42 (b) 11 (c) 11 (d) 12 (e) 12 (f) 14 3. (a) 160 and 200 (f) 180 and 210 (b) 560 and 640 (g) 300 and 360 (c) 270 and 360 (h) 400 and 450 (d) 100 and 120 (i) 360 and 400 (e) 420 and 490 (j) 350 and 400 4. (a) 20 (b) 26 Challenge 14. (33 – 5 = 28, 28 ÷ 2 = 14)

Unit 16–2

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• Provide students with further opportunities to practise rounding techniques to estimate answers to multiplication problems.

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1. (a) 11 (b) 17 (c) 14 (d) 18 (e) 12 (f) 15 (g) 13 (h) 15 (i) 18 (j) 10 2. (a) 15 (b) 16 (c) 16 (d) 12 (e) 17 (f) 13 3. (a) 80 4. (a) 200 (b) 20 (b) 300 (c) 20 (c) 700 (d) 30 (d) 300 (e) 30 (e) 400 (d) and (e) are (a) is <300 closest to 30 5. (a) 600 Challenge (b) 700 24 3 18 (c) 400 21 9 15 (d) 200 12 27 6 (e) 300 (c), (d) and (e) are <500

Consolidation 16–2

• Working in pairs, students take turns building and copying a model using 20-mm cubes.

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Consolidation 16–3

• Provide further opportunities for students to practise rounding to estimate answers for subtraction problems.

New Wave Maths Book D – Teachers Guide • 85 •

Unit 17–1

Student page 49

Outcomes

Indicators

WM3.2, S3.3

The student is able to: • pose, ask and contribute mathematical questions prompted by a specific stimulus. • informally describe the symmetry of a figure.

Skills • observing • designing • posing questions • making conjectures

Resources • coloured pencils • mirror/mira (optional)

Language • symmetry • symmetrical • patterns • right, left • same • design

r o e t s Bo r e p ok u S

Notes

Main Activity (WM3.2, S3.3)

• This activity is designed for students working collaboratively in groups. Allow enough time for students to discuss their options and for ideas to evolve. Investigative tasks such as these are a good opportunity for students to ‘take a risk’ with maths. • When completing investigative tasks, some students may be more successful in mixed-ability groups rather than same-ability groups. • Some groups will be able to work independently while others may need guidance. The stimulus questions below may prompt such groups. – What do you notice about the patterns and colours on the butterflies and moths? – How many wings do they have? – Are the wings the same size, the same colours and do they have the same patterns? – What does the word ‘definition’ mean? • If students are having difficulty grasping the concept of ‘symmetry’ distribute small mirrors and ask students to place them along the centre of a butterfly. What do they notice? • To further explain the concept of symmetry, ask students to fold a sheet of paper in half and cut a shape from it. When they open out the paper, they will notice the shape is symmetrical, with the line of symmetry being the fold. • With Question 2, students may wish to walk around the classroom or inside and outside the school grounds to identify other objects that are symmetrical. Have shapes available for students to view. They may notice that some shapes have a number of lines of symmetry, such as some letters of the alphabet—O and X. • Objects that are symmetrical may include some windows, chairs or sunglasses. • Students may wish to copy a butterfly or moth design onto white card and colour it for display.

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What to do

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Challenge

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• Are people symmetrical?

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• 86 • New Wave Maths Book D – Teachers Guide

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Unit 17–2

Student page 50

Outcomes

Indicators The student is able to: • understand the terms ‘multiple’, ‘factor’ and ‘prime’ and use them appropriately.

N3.3, N3.2

Skills • recognising prime and composite numbers • dividing

Resources

Language • multiply • prime • composite • factors • divisible

• Base 10 MAB • calculator • grid paper (see page 199)

r o e t s Bo r e p ok u S

Notes

Memory Masters (N3.3)

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Teac he r

• The focus for this unit is addition of three digits.

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.2) Warm up

• Ask students if they know what prime and composite numbers are. If not, explain to them that prime numbers have two factors, only 1 and the number itself. Composite numbers have more than two factors. • Give some examples; e.g., 2, 7 and 11 are prime numbers while 4, 10 and 15 are composite numbers. • Show students how to use diagrams of arrays to help them determine whether numbers are prime or composite; e.g. 7 1 x 7 = factors 1 and 7;

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1 x 10; 2 x 5 = factors 1, 2, 5, and 10; i.e. composite

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i.e. prime

• Ask students whether they noticed anything about prime numbers, excluding 2. They are all odd; but odd numbers are not always prime numbers.

What to do

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• Direct students to examine the arrays in the workbook and use these to help them determine whether the numbers are prime or composite. • Exercise 4 can be used as an extension of the above activity by using grid paper to draw arrays of the numbers showing the factor requested; e.g. 2—all arrays will be in rows of 2. • Some students may recognise that odd numbers are not divisible by two and go straight to the answer. • The same procedure may be applied for divisibility by 5 in Exercise 5. Alternatively, students may be asked to use their calculator to assist. • Some students may recognise immediately that numbers ending in 0 or 5 are divisible by 5.

Challenge • Students investigate multiples of 3 to see if they can find and prove their answers. • If students think they have a solution, provide them with larger numbers to test their theory; e.g. 111, 156, 94 and 79. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 87 •

Unit 17–3

Student page 51

Indicators

Outcomes

The student is able to: • measure time by interpreting a calendar. • record frequency data carefully using simple formats based on tallies or organised lists and take care with their measurements.

N3.3, M3.2, C&D3.2

Skills • reading a calendar • writing a list • ordering

• Base 10 MAB • calculator

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2, C&D3.2)

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• Work through the activity with the whole class, checking to see that the dates are accurately recorded. • Once the dates have been marked on the calendar, students should then list the dates in order and write next to the date why it is important. The list will vary and students may not be able to list all the dates they have marked. • Check work as students proceed.

Challenge

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• Students may work out how many days they have been alive, using their calculator to help. • This activity can be extended to hours, minutes and seconds is desired.

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• Ask the class for a list of dates they think are important during the year. Write the list on the blackboard/whiteboard.

What to do

• calendar • list • date

r o e t s Bo Notes r e p ok u S

• The focus for this unit is completion of a number sentence with basic facts of multiplication and a single-digit addend.

Warm up

Language

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Teac he r

Memory Masters (N3.3)

Resources

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• 88 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 17—Answers

Student pages 49 – 51

Unit 17–1

1. (a) 12 (b) 11 (c) 16 (d) 13 (e) 12 (f) 20 (g) 14 (h) 17 (i) 15 (j) 16 2. (a) 761 (b) 531 (c) 962 (d) 711 (e) 764 (f) 761 3. (a) composite (b) prime (c) composite (d) composite 4. (a) 5 (c) 8 (e) 10 (g) 3 (i) 6 (b) 26 (d) 17 (f) 20 (h) 29 (j) 14 They are even numbers and end in 0, 2, 4, 6 or 8. 5. (a) 13 (c) 18 (e) 25 (g) 40 (i) 55 (b) 27 (d) 60 (f) 30 (h) 32 (j) 45 They can be odd or even and end in 0 or 5. Challenge Add all the digits; for example, 324 — 3 + 2 + 4 = 9. Keep adding until a single-digit number is formed. If the number is 3, 6 or 9 then the original number is divisible by 3. Another example, 325 — 3 + 2 + 5 = 10 — 1 + 0 = 1. Answer is 1 and not a multiple of three, so original number is not divisible by 3.

r o e t s Bo r e p ok u S

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Teac he r

1. A shape is symmetrical if it is identical on either side of a line dividing it into two parts. 2. Answers will vary 3. Teacher check Challenge People appear to be symmetrical but a closer look reveals they are not. For example; earlobes can be higher or lower, a mole or scar can be on one side of the face and not the other and so on.

Unit 17–2

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• Students identify and draw the lines of symmetry in the numbers one to 10.

Consolidation 17–2

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1. (a) 7 (b) 18 (c) 11 (d) 4 (e) 13 (f) 15 (g) 12 (h) 9 (i) 2 (j) 12 2. (a) 1615 (b) 1426 (c) 1666 (d) 1439 (e) 1917 (f) 1236 3. Teacher check 4. Teacher check Challenge Teacher check

• Colour prime and composite numbers on a 1 – 100 chart (see page 202), using different colours for each.

Consolidation 17–3

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• Mark important dates on a personal calendar and list in order from the beginning of the year.

New Wave Maths Book D – Teachers Guide • 89 •

Unit 18–1

Student page 52

Outcomes

Indicators

N3.3, N3.1a, M3.2

The student is able to: • use the decimal point in representing quantities and money. • enter and read amounts of money and measurements on a calculator, truncating calculator displays to the nearest cent or unit. • use a uniform unit to measure length.

Skills • recognising monetary equivalents • measuring • using decimal notation

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • ruler

Number (N3.3)

Main Activity (N3.1a, M3.2)

Notes

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Teac he r

• cents, dollars • measure • lengths • centimetres • metres • value • diagrams

r o e t s Bo r e p ok u S

• The focus for this unit is completion of a number sentence with basic facts of multiplication and a single-digit addend.

Warm up

Language

What to do

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• Ask students to complete Exercise 3, changing cents to dollars. Encourage use of a calculator if required. • Show students how to use their ruler correctly to measure a line. Start with the zero point at the end of the line, then read the measurement value at the other end of the line. • Check, as students measure the lines in Exercise 4, to ensure they are measuring correctly. • Record lengths in centimetres. • Use the diagrams in Exercise 5 to add the value of the equivalent lengths of MAB longs to the metre rule. Remind students that the Base 10 MAB long is worth ten centimetres in this exercise. (Diagram is not drawn to scale.) • Check orally with the class on each set of diagrams before having students record answers.

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Challenge

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• Students use their calculator to show cents or centimetres as dollars or metres; e.g. 20c; 33 cm; 56c; 82 cm; 74c. Enter the number into the calculator then divide (÷) by 100.

o c . che e r o t r s super

• A calculator is required for this activity. • Students multiply a variety of two-digit numbers by 101. Multiply the same numbers by 100. • Compare the results. What do they notice? • Students describe a quick way to multiply a number by 101. (24 x 101 is the same as 24 x 100 + 24.) • Students could discuss this with a partner or in a small group and report findings to the class. • Some students may not be able to grasp the concept until explained by another. For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 40 – 43. • 90 • New Wave Maths Book D – Teachers Guide

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Unit 18–2

Student page 53

Outcomes

Indicators

N3.3, C&D3.1, C&D3.2, C&D3.3,C&D3.4

The student is able to: • describe outcomes as having an equal chance or being equally likely. • record frequency data carefully using simple formats based on tallies or organised lists. • summarise data based on tallying. • comment upon their predictions in light of the results of their own data collection.

Skills • tossing coins • recording data • comparing results • graphing

Resources

Language • record • tally table • total • most • graph

• Base 10 MAB • calculator • 10c coins • 20c coins • 50c coins • $1 coins

r o e t s Bo r e p ok u S

Notes

Teac he r

Memory Masters (N3.3)

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• The focus for this unit is completion of a number sentence with basic factors of multiplication and a single-digit addend.

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.1, C&D3.2, C&D3.3, C&D3.4) Warm up

© R. I . C.Publ i cat i ons orr evi ew pur posesonl y• What to• do f • Ask students what result they would expect if they tossed a coin; head or tails? • Ask them why they think the particular result they chose would occur. • Explain that unless the coin is biased, the result is purely chance and there is an even likelihood of a tail or head being thrown.

Challenge

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• Issue coins to groups of students and ask them to toss each coin 10 times, recording the tally of heads or tails that each coin shows. • Find the total for each coin and answer the questions. • Repeat the activity and write a summary giving a comparison of the two sets of results. Explain similarities or differences. • Graph the results of the two sets of throws. Add more squares to the end of the graphs if required.

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• Ask students: What is the chance of tossing three heads in a row? • They may need to draw a tree diagram and list all the possibilities to help solve this.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 112 – 113 and 118 – 119. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 91 •

Unit 18–3

Student page 54

Indicators

Outcomes

The student is able to: • build sequences of simple shapes such as triangles, squares, ‘L’ or ‘T’ shapes, which increase in size systematically and write the equivalent number pattern. • understand that multiplication can be used for repeated addition. • use their own methods or a conventional algorithm, multiply by whole numbers by 100.

N3.3, N3.4

Skills • making models • following directions • problem-solving • calculating • multiplying

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • 20-mm cubes • balance scales • 20c coins

Teac he r

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.2, N3.3)

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• Direct students to build models with the array base as directed in their workbook. Then place layers on top of the base array as directed. • Find how many cubes were used in each model. Record the number in the space provided. • Ask students if they can find a way of determining the total number of cubes used without counting them individually. If no way is suggested, explain they can multiply the base array (length x width) by the number of layers (height). • Exercise 4 requires students to solve the problems posed.This may be done by discussion or by allowing groups to experiment to find answers. • Check answers among groups by sharing findings. • Exercise 5 may be completed mentally or by using a calculator.

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• Provide small groups of students with 20-mm cubes. Direct students to build models and discuss within the group what they have built. • Explain that an array is a uniform grouping with length and width.

Challenge

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Number (N3.3)

What to do

• base • array • layers • mass • volume • equivalent • number pattern • cubes

r o e t s Bo Notes r e p ok u S

• The focus for this unit is completion of a number sentence with basic facts of multiplication and a two-digit addend.

Warm up

Language

o c . che e r o t r s super

• Encourage students to think of the means they would use to solve this problem. • Discuss the options given. • Direct students to solve the frog problem keeping records of how they reached their findings.

• 92 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 18—Answers

Student pages 52 – 54

Unit 18–1

1. (a) 28 (b) 11 (c) 7 (d) 19 (e) 29 (f) 20 (g) 30 (h) 10 (i) 26 (j) 8 2. (a) 19 (b) 18 (c) 18 (d) 37 (e) 16 (f) 37 3. Answers will vary 4. Teacher check 5. Teacher check Challenge 1 chance in 8.

r o e t s Bo r e p ok u S

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Teac he r

1. (a) 8 (b) 18 (c) 9 (d) 23 (e) 11 (f) 22 (g) 15 (h) 5 (i) 14 (j) 20 2. (a) 16 (b) 36 (c) 29 (d) 37 (e) 25 (f) 15 3. (a) $0.60 (c) $0.20 (e) $0.30 (g) $0.70 (b) $0.90 (d) $0.45 (f) $0.80 (h) $0.65 4. (a) 10 cm (b) 15 cm (c) 11 cm (d) 14 cm (e) 12 cm 5. (a) 1.3 m (b) 1.5 m (c) 1.1 m (d) 1.4 m (e) 1.2 m Challenge For example, 24 x 101 is the same as (24 x 100) + 24.

Unit 18–2

• Repeat the activity, throwing the coins 20 times instead of 10 and compare results.

Consolidation 18–3

o c . che e r o t r s super

• In pairs, students can make up a measuring problem for themselves and other students to solve, following the format in Exercise 4.

Metres

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• Provide further opportunities to write cents in dollar terms and convert centimetres to metres.

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1. (a) 22 (b) 35 (c) 11 (d) 28 (e) 7 (f) 8 (g) 29 (h) 11 (i) 7 (j) 27 2. (a) 224 (b) 627 (c) 326 (d) 327 (e) 326 (f) 428 3. (a) 24 cubes (b) 30 cubes (c) 54 cubes (d) 40 cubes (e) 18 cubes 4. (a) 16 kg (b) Teacher check Challenge (c) 5 x mass 8 days (d) 9 x mass 10 9 5. (a) 400 (f) 900 8 (b) 1000 (g) 800 7 6 (c) 1600 (h) 800 5 (d) 200 (i) 1600 4 3 (e) 4900 2 1 0

1 2 3 4 5 6 7 8 Days

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New Wave Maths Book D – Teachers Guide • 93 •

Unit 19–1

Student page 55

Outcomes

Indicators

N3.3, S3.3

The student is able to: • informally describe the symmetry of a figure or arrangement.

Skills • drawing lines of symmetry

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • mira • ruler

Teac he r

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.3)

Notes

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Number (N3.3)

What to do

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• Ask students if they can find a line of symmetry in the isosceles triangle in Exercise 3(a). • If students are having difficulties, distribute miras (if they are available) for students to use. When the mira is placed on a line of symmetry both sides of the shape are superimposed. Students may then draw the line of symmetry along the front of the mira. • Continue with the other shapes. Note: There may be more than one line of symmetry.

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• Ask students what they see when they look in a mirror. (They see a mirror image or reflection.) • Explain that the mirror is a plane of symmetry. For things to be symmetrical, objects, shapes or images on either side of the line or plane of symmetry must be exactly the same; i.e. a mirror image. • Ask students to find examples in the classroom that either show symmetry or are symmetrical; e.g. a desk top with a central line of symmetry, two desks equal distance from a line of symmetry.

Challenge

• lines of symmetry • isosceles triangle • scalene triangle • equilateral triangle • square • rectangle • hexagon • quadrilateral • pentagon

r o e t s Bo r e p ok u S

• The focus for this unit is completion of a number sentence with basic facts of multiplication and a single-digit addend.

Warm up

Language

o c . che e r o t r s super

• Students can search the newspaper, magazines or the Yellow Pages to find business logos that have symmetry. • These can be copied or pasted into a book or on a chart for display.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 20 – 21. • 94 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 19–2

Student page 56

Outcomes

Indicators The student is able to: • understand that multiplication can be used for repeated addition situations. • mentally estimate the results of a calculation in order to check the reasonableness of calculator results. • use a constant function on a calculator to solve repeated subtraction problems.

N3.3, N3.2, N3.4

Skills • estimating • calculating

Resources

Language • patterns • estimate • sequences

• calculator

r o e t s Bo r e p ok u S

Memory Masters (N3.3)

Notes

Teac he r Number (N3.3)

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• The focus for this unit is completion of a number sentence with multiplication of three digits.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.2, N3.3, N3.4) Warm up

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• What to do • Revise estimation of addition of a series of numbers—pairs of numbers making ten, rounding, adding doubles etc. • Ask students; ‘Where a set of numbers are all the same, is there another way of estimating?’ • By counting the number of occurrences of the digit and multiplying the digit by the number of times it occurs, the actual answer is obtained. This is the simplest method of estimating.

Challenge

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• Use the calculator to work through Exercise 3, after estimating the answer first. • Exercise 4 is similar, but has a number greater than 10 as the starting point. This number may be rounded to the nearest ten if not already a multiple of 10. • Once estimates have been worked out, complete the answer by using the calculator. • Exercise 5 follows the same process, except the students are required to take the constant from the starting number.

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• The object of this exercise is self-explanatory. Remind students what a factor is—a number that will divide evenly into a number. • Students will need to experiment to find the ‘perfect number’. • Students should keep records of their workings to show how they reached their answer.

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New Wave Maths Book D – Teachers Guide • 95 •

Unit 19–3

Student page 57

Indicators

Outcomes N3.3, M2.2

The student is able to: • use a ruler or tape measure to measure and make things to the nearest centimetre. • use a uniform unit consistently to measure the area of an irregular shape.

Skills • measuring • using a ruler • ordering • finding area

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • counters • ruler

Language • length • area • shortest • longest • order

r o e t s Bo Notes r e p ok u S

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M2.2) Warm up

• Discuss with students how they might measure the length of an object. – What types of things might you need to know the length of? – What equipment could you use to measure the length of an object? – Does the equipment change according to the length you wish to measure?

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• Ask students what would be the most appropriate equipment to use to measure the lengths of the lines in Exercise 3. Some students may choose to use a ruler, while others may use Base 10 MABs. • Revise how to measure, starting from zero and recording the number on the ruler where the object finishes. • Allow students time to record the measures of each line, to the nearest centimetre. • Once students have recorded the length for each line, they can order them from shortest to longest. Some students will use the visual aid of the lines to write them in order, while other students will use the numerical measure to order the lines. Either method is acceptable, providing students order the lines correctly. • For Exercise 4, discuss the concept of area with the students. Area is the space within a boundary. This is an irregular shape, which is difficult to work out the area of and why the use of concrete materials has been recommended. Answers will therefore vary according to the size of the counters. Smaller counters will generally allow students to be more accurate with coverage. • Allow students to use counters or other objects to solve the problem. Discuss with the group how they managed. Was it easy or difficult to work out the area of this shape in this way? • Students can discuss the problem of using counters, as they leave gaps and do not tessellate. This is why area is measured in squares.

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What to do

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Teac he r

• The focus for this unit is completion of a number sentence with the multiplication of three digits.

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Challenge • This activity will require students to think laterally. Some students will solve the problem easily, while others may have difficulty in working out the best way to approach the problem. • Some students may feel more comfortable solving the problem by making a model with paper or card and using folding techniques. All approaches should be encouraged and developed. Problem-solving strategies vary with the individual. • At the end of the session, allow students the opportunity to share their thought processes. For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 80 – 81. • 96 • New Wave Maths Book D – Teachers Guide

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Unit 19—Answers

Student pages 55 – 57

Unit 19–1 1. (a) 31 (b) 7 (c) 15 (d) 18 (e) 20 (f) 7 (g) 24 (h) 38 (i) 17 (j) 37 2. (a) 370 (b) 520 (c) 430 (d) 190 (e) 180 (f) 510 3. (a) (b)

isosceles triangle

rectangle

(f)

(g)

regular pentagon

regular hexagon

quadrilateral (h)

equilateral triangle

Challenge Teacher check

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square

scalene triangle

(e)

Teac he r

(d)

1. (a) 8 (b) 20 (c) 6 (d) 0 (e) 16 (f) 12 (g) 12 (h) 0 (i) 24 (j) 27 2. (a) $435 (b) $336 (c) $603 (d) 504c (e) 354c (f) 486c 3. (a) 20 Teacher check estimates (b) 35 (c) 15 (d) 40 (e) 30 4. (a) 60 Teacher check 5. (a) 14 estimates (b) 104 (b) 45 (c) 74 (c) 24 (d) 62 (d) 45 (e) 106 (e) 48 (f) 67 (f) 40 (g) 98 (g) 43 (h) 68 (h) 11 Challenge 6=1+3+2

r o e t s Bo r e p ok u S none

(c)

Unit 19–2

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• Draw a picture or diagram on one side of a mid-line on a sheet of grid paper. Another student can complete the drawing using a mira to assist if necessary.

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1. (a) 0 (b) 18 (c) 32 (d) 35 (e) 0 (f) 50 (g) 18 (h) 40 (i) 30 (j) 24 2. (a) 266 (b) 675 (c) 768 (d) 464 (e) 276 (f) 195 3. (a) 9 cm (b) 11 cm (c) 8 cm (d) 14 cm (e) 7 cm Order: e, c, a, b, d 4. Teacher check Challenge

Consolidation 19–2

• Provide further opportunities to use the constant function on a calculator to add and subtract.

o c . che e r o t r s super

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Consolidation 19–3

• Use 1-cm grid paper (see page 199) to measure the area of the shape in the workbook. Count half-squares to assist. Compare answers.

New Wave Maths Book D – Teachers Guide • 97 •

Unit 20–1

Student page 58

Outcomes

Indicators The student is able to: • mentally estimate the results of a calculation in order to check the reasonableness of calculator results. • check their answers against their estimates and reconsider both their methods and their calculations if results seem unreasonable.

N3.3, WM3.4

Skills • estimating • ordering • calculating

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator

• estimate • add • actual • order • objects • height • lowest • highest

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3, WM3.4) Warm up

Notes

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r o e t s Bo r e p ok u S

• The focus for this unit is completion of a number sentence with the multiplication of three digits.

Teac he r

Language

• Discuss with students what they would do if asked to estimate the total of two numbers such as 327 and 486. • Focus the discussion on rounding to the nearest hundred or ten to provide a rough estimate. • Try with 412 and 21; 742 and 196; and 47, 31 and 19.

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• Direct students to estimate the answers to the sums provided. • After writing the estimation, find the actual total of each sum. A calculator may be used to assist. • Discuss the closeness of estimation and actual answers. Why are some very close and others not so close? • Discuss the need for reasonable closeness of estimates but the acceptance of variance due to the size of original numbers—29 + 28 will be closer to the actual answer than 26 + 26. • Discuss with students ways they feel that they might be able to estimate more accurately. The suggestions should be documented and compared for suitability of use. • Use another method and check to see if this is more accurate. • Direct students to read each question in Exercise 5. After whole-class discussion, students write an explanation for each answer.

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What to do

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Challenge • Draw this shape on the blackboard/whiteboard without students seeing how you drew it. • Ask students if they can draw the shape without lifting their pencil from the paper. • This challenge is related to topology—specifically, the traversability of networks. • Students may like to look up the Konigsberg Bridge Problem on the Internet and see how Euler solved a similar problem. For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 64 – 65. • 98 • New Wave Maths Book D – Teachers Guide

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Unit 20–2

Student page 59

Outcomes

Indicators

N3.3, C&D3.3, N3.2, S3.4

The student is able to: • use diagrams such as Venn diagrams and two-way tables to represent a two-way classification. • understand the terms ‘multiple’, ‘factor’ and ‘prime’ and use them appropriately. • integrate conventional names of shapes and component parts of shapes into their descriptions of things.

Skills • counting numbers • recording in Venn diagrams • recognising multiples

Resources

Language

• calculator

r o e t s Bo r e p ok u S

• Venn diagram • counting numbers • multiples • shapes • straight • curved • edges

Memory Masters (N3.3)

Notes

Teac he r Number (N3.3)

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• The focus for this unit is completion of a number sentence with multiplication of three digits.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.3, N3.2, S3.4) Warm up

What to do

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• Explain that a Venn diagram is a means of recording information on a number of items that is exclusive to each item and also common to more than one of the items. The diagram therefore shows portions of overlap to allow for accurate recording of the information. • Ask students: — ‘What are counting numbers?’ (1, 2, 3…) — ‘What are multiples?’ (a multiple of a number is that number multiplied by other whole numbers.) — ‘Give examples of multiples of 3 and 2.’

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• Complete Exercise 3, reminding students that any numbers that are not multiples of two, three or both are written outside the diagram. • Look at the drawings of the shapes in Exercise 4. Ask students what they can tell about them. Accept all answers, then focus on the fact that some have only straight edges, some have only curved edges and some have both. • Use this information to draw a copy of the diagrams in the correct place in the Venn diagram. • Discuss placements of shapes when students have placed shapes contrary to expected.

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Challenge

o c . che e r o t r s super

• Ask students to design their own Venn diagram to help sort out class members according to what they are wearing; e.g. red top, black shoes.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 110 – 113. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 99 •

Unit 20–3

Student page 60

Indicators

Outcomes

The student is able to: • link the action of sharing into a number of equal portions with the language of unit fractions. • separate objects and collections into equal parts to show unit fractions.

N3.3, N3.1b

Skills • representing fractions • dividing

Memory Masters (N3.3)

Resources

Language • equal sets • thirds • eighths • fourths

• calculator

r o e t s Bo Notes r e p ok u S

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1b) Warm up

• Ask the class to stand, or take them outside to start this activity. • Ask the class to form groups of five. Any leftover students are to remain to one side. • Ask the groups to then have three-fifths of each group move a step clear of the rest of the group. • Check to see or ask to confirm that each group is now two groups, one with three members and one with two members. • Repeat this activity for: —groups of eight with six-eighths staying together. —groups of four with one-fourth staying together. —groups of six with five-sixths staying together. • Form a group of nine. Form three equal groups from the nine. • Form a group of 10. Form two equal groups, then five equal groups from the 10. • Form a group of six. Form three equal groups, then two equal groups from the six.

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Teac he r

• The focus for this unit is completion of a number sentence with multiplication of three digits.

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What to do

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• Complete Exercise 3 by grouping the lollies as indicated.

Challenge

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• Students should be provided with a variety of materials such as a fraction cake, fraction charts and paper for diagrams. • Ask students to find 1/2 of 1/3 and 1/3 of 1/2 using the materials to assist in making their final determination. • Write an explanation to show how the answer was reached.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 46 – 47. • 100 • New Wave Maths Book D – Teachers Guide

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Unit 20—Answers

Student pages 58 – 60

Unit 20–1

1. (a) 1 (b) 27 (c) 18 (d) 16 (e) 28 (f) 0 (g) 100 (h) 24 (i) 0 (j) 18 2. (a) 13 (b) 24 (c) 14 (d) 12 (e) 27 (f) 19 3. Counting numbers to thirty 1, 5, 7, Multiples 11, 13, Multiples of of 17,19, two three 23, 25, 2, 4, 8, 6, 12, 18, 3, 9, 15, 29 10, 14, 16, 24, 30 21, 27 20, 22, 26, 28

r o e t s Bo r e p ok u S 4.

Straight edges

Challenge Teacher check

Curved edges

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Teac he r

1. (a) 0 (b) 35 (c) 36 (d) 30 (e) 0 (f) 28 (g) 45 (h) 40 (i) 60 (j) 21 2. (a) 12 (b) 14 (c) 19 (d) 17 (e) 23 (f) 14 3. Teacher check estimates and findings. (a) 221 (b) 182 (c) 101 (d) 207 (e) 118 4. Teacher check 5. (a) Teacher check (b) Teacher check Challenge

Unit 20–2

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(d)

(e)

• Discuss other real-life situations as in Exercise 5, where it would be better to overestimate or underestimate.

Consolidation 20–2

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1. (a) 80 (b) 80 (c) 60 (d) 30 (e) 40 (f) 0 (g) 20 (h) 70 (i) 50 (j) 60 2. (a) 28 (b) 15 (c) 14 (d) 24 (e) 13 (f) 18 3. (a) (b)

• Complete a Venn diagram using the counting numbers to 30 to sort into multiples of three and five.

Consolidation 20–3

o c . che e r o t r s super

• Show different unit fractions by shading diagrams.

Challenge They are the same.

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New Wave Maths Book D – Teachers Guide • 101 •

Unit 21–1

Student page 61

Outcomes

Indicators

N3.3, S3.3

The student is able to: • identify repetitions of component parts in symmetrical objects/ arrangements and demonstrate by moving one component over another. • informally describe the symmetry of a figure or arrangement.

Skills • identifying symmetry • completing symmetrical diagrams

Memory Masters (N3.3)

Resources • calculator • paper • scissors • mira

Language • line of symmetry • symmetrical pattern

r o e t s Bo r e p ok u S

Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.3) Warm up

• Hold up a sheet of paper that is either square, rectangular or an equilateral triangle. • Ask the class if anyone can show the rest of the class a line of symmetry on the paper. • Have a student show the line of symmetry that he/she has discovered. • Discuss line symmetry, eliciting from the class that both halves of the object or drawing on each side of the line of symmetry are exact mirror images.

• Direct students to the diagrams in their workbook and ask them to complete the drawings so that the dotted line is a line of symmetry. • Students who have difficulty should be encouraged to use a mira to assist in drawing the missing half of the diagram.

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Challenge

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What to do

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Teac he r

• The focus for this unit is multiplication of basic facts and subtraction of a single-digit number.

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 20 – 21. • 102 • New Wave Maths Book D – Teachers Guide

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Unit 21–2

Student page 62

Outcomes

Indicators

N3.3, N3.1a

The student is able to: • distinguish and order whole numbers using money values. • regroup money to the fewest number of notes or coins.

Skills • counting money • converting money

Resources

Language

• calculator • plastic money (notes and coins)

• total value

r o e t s Bo r e p ok u S

Memory Masters (N3.3)

Notes

Number (N3.3)

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Teac he r

• The focus for this unit is multiplication of basic facts and subtraction of a single-digit number.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a) Warm up

• Distribute money among groups of students. • Allow the students a brief period of free play with the money. • Ask students to make combinations of different values with the money to share with their group. The group is to agree on the total value shown.

Challenge

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• Direct students to their workbook to find the combined total of the coins shown. • Ask students to provide oral explanations of how they reached their answers—addition, multiplication, counting by the coin total or any other means. • Exercise 4 requires the students to use a combination of notes to make the note value shown. • Share the combinations within a group or choose students to share with the class. Each time ask if other students have different combinations they feel may be correct.

• The challenge is an extension of the activity above. • The same value note or coin may be used more than once. • Students are to record their workings and provide an explanation of how they reached their answer. • Share a range of results from different students, or work through the answer as a wholeclass exercise.

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R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 103 •

Unit 21–3

Student page 63

Indicators

Outcomes

The student is able to: • use a uniform unit of length carefully to make their own graduated scale to measure things in their environment. • make conjectures which reflect their understanding of measurement.

N3.3, M3.2, WM3.3

Skills • problem-solving • measuring • reasoning

Memory Masters (N3.3)

Resources • calculator • measuring equipment as required

Language • accurately • measure • height • directly

r o e t s Bo Notes r e p ok u S

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2, WM3.3) Warm up

• Form groups of five. Ask students to order themselves from shortest to tallest without standing back to back. • Discuss how each group completed the task.

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• Students are posed a challenge to accurately measure their height without using conventional measuring devices directly against their body. Measuring devices may be used to find the measure but not directly against the body. • Challenge students to find more than one means of measuring their height in this manner. • Students record an explanation of how they determined their height and why they consider it to be accurate. • Diagrams may be useful in helping with the explanation.

Challenge

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• Explain to students that this activity is centred on finding special number patterns that total 12. • Remind students that they are only able to use the numbers 1 – 7 inclusive for this exercise. • Students should keep notes and copies of attempts for showing their thoughts in reaching a solution.

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What to do

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Teac he r

• The focus for this unit is multiplication of basic facts and subtraction of a single-digit number.

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• 104 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 21—Answers

Student pages 61 – 63

Unit 21–1

1. (a) 32 (b) 1 (c) 28 (d) 11 (e) 24 (f) 22 (g) 5 (h) 44 (i) 1 (j) 2 2. (a) 788 (b) 798 (c) 787 (d) 699 (e) 889 (f) 797 3. (a) 40c (b) 70c (c) $2.50 (d) $1.40 4. (a) $20, $20, $10 (b) $10, $10 (c) $5, $5 Challenge 6. ($50, $20, $20, $5, $2, $2)

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1. (a) 11 (b) 25 (c) 1 (d) 5 (e) 23 (f) 32 (g) 0 (h) 17 (i) 8 (j) 37 2. (a) 1824 (b) 1123 (c) 1247 (d) 1533 (e) 1227 (f) 1631 3. Teacher check Challenge Yes. Symmetrical patterns will vary.

Unit 21–2

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• Display and discuss the symmetrical patterns created in the ‘Challenge’ section.

Consolidation 21–2

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1. (a) 18 (b) 6 (c) 3 (d) 40 (e) 2 (f) 29 (g) 0 (h) 16 (i) 2 (j) 33 2. (a) 1367 (b) 1985 (c) 1268 (d) 1578 (e) 1689 (f) 1679 3. Answers will vary. Possible solutions: Comparing against a known height or making a mark of height on a wall and measuring with a tape measure. Challenge One possible answer:

• Work out the least number of notes and coins needed to make a variety of totals.

Consolidation 21–3

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R.I.C. Publications® www.ricpublications.com.au

• As a class, discuss students’ explanations as to how they determined their height and how accurate the different methods were.

New Wave Maths Book D – Teachers Guide • 105 •

Unit 22–1

Student page 64

Outcomes

Indicators The student is able to: • follow a rule to generate a number sequence based on an addition or subtraction strategy.

N3.3, N3.4

Skills • adding • identifying patterns ‘• ordering

Memory Masters (N3.3)

Resources • calculator • coloured pencils • 1 – 100 grid (see page 202)

Language • grid • add • numbers • circling • patterns • area • lowest • highest

r o e t s Bo r e p ok u S

Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.4) Warm up

• Ask students collectively, or individually, to count on by a given number from a set starting number; e.g. by 2 starting at 9; by 5 starting at 3; by 10 starting at 7; and by 3 starting at 4. Students may refer to a 1 – 100 grid. • Stop counting after six or so numbers are in the sequence.

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• Direct students to the grid in their workbook. • Explain they are to circle or colour the number 11. • Count on four and circle or colour this number. What is the number? (15) • Count on four and circle or colour this number. (19) • Continue to count on by four, circling or colouring each number until the end of the grid is reached. • Write all the numbers circled or coloured. • Discuss and describe any patterns that may be found in the grid. • For Exercise 4, students use a blank 1 – 100 grid and use different starting numbers to add on to. • Describe the patterns they make.

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Challenge

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What to do

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Teac he r

• The focus for this unit is multiplication of basic facts and subtraction of a single-digit number.

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• Pose the following question to students: ‘If you start at 99 and subtract 7 continually, will you land on 50? If so, after how long?’ • Students can work individually or in a group to solve this problem. • Discuss the methods used as a class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 72 – 73. • 106 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 22–2

Student page 65

Outcomes N3.3, C&D3.1, C&D3.2, C&D3.3, C&D3.4

Skills • estimation • graphing

Indicators

Resources

The student is able to: • describe outcomes as having an equal chance or being equally likely. • record frequency data carefully using simple formats based on tallies or organised lists. • display frequency data in bar graphs. • comment upon their predictions in light of the rest of their own data collection.

• Base 10 MAB • calculator • spinner (see pages 210 – 211) • coloured pencils

Language • spinner • chance • likely to occur • record • estimate • graph • explain

r o e t s Bo r e p ok u S

Memory Masters (N3.3)

Notes

Number (N3.3)

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Teac he r

• The focus for this unit is multiplication of basic facts and subtraction of a single-digit number.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.1, C&D3.2, C&D3.3, C&D3.4) Warm up

• Make a spinner, or provide a spinner to pairs of students. Allow students to practise spinning before settling to the completion of the task. • Ask how many numbers or different colours there are on the spinner. What is the chance of one of these occurring with a spin? (One chance in six or one chance in four, depending on the spinner used.)

Challenge

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• Direct students to take turns with a partner to spin the spinner 60 times. Each time the spinner stops, record the number or colour shown on the graph in the workbook. • Suggest to students they use a different colour to show each number or use the same colour for the graph as the one shown on the spinner. • Before starting, ask students to make an estimate of how many times they think each number or colour will occur. Record this on the workbook. • When 60 spins have been made, explain the results shown on the graph. • How do students’ findings compare with their estimates? Why is this so?

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• Pose the following problem to students: ‘In an experiment, a spinner was spun 55 times and the following data was collected: R G B Y Draw what you think the spinner looked like.’

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 118 – 119. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 107 •

Unit 22–3

Student page 66

Indicators

Outcomes N3.3, N3.1a

Resources

The student is able to: • count up and down in tens from any starting number. • add and subtract whole numbers using their own written method or a conventional algorithm, explaining the method by reference to place value.

Skills • calculating • multiplying • dividing

Memory Masters (N3.3)

• Base 10 MAB • calculator

Language • number chain • thousands • hundreds • tens • ones

r o e t s Bo Notes r e p ok u S

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a, N3.3) Warm up

• Provide this example on the blackboard/whiteboard to model how students can solve the problem: – 1000 + 100 – 10 4662 3662 4562 3652

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Teac he r

• The focus for this unit is basic facts of multiplication with addition of a single-digit number.

3572

– 1000

4572

+ 10

4562

– 100

4652

– When we add 100, which number is affected in the original number? – When we subtract 1000, which number would you expect to change? Why? How? • Follow this questioning technique through the process of solving the problem. By the end of the modelling session, students should have developed a clear understanding of what is happening throughout the process of the number chains.

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What to do

• Students can use a calculator to complete Exercise 3 if necessary. • Exercise 4 is different in that students are given the solutions and are required to work out what has been added or subtracted. This requires different skills and should be discussed as a group. Ask students to share how they think they could go about solving the problem. The less able students will appreciate the insight, as they may not know where to begin. With a little guidance, all students should be able to complete the task to a satisfactory level. • Students can discuss how they went about solving the problem. Compare their approach with others in the class. This provides students with a bank of strategies they can use the next time they tackle a similar problem. • Students can then go on to complete Exercise 5. They should be encouraged to work mentally, then check answers using a calculator.

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+ 1000

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Challenge • Questions like these allow students the opportunity to think laterally and you will probably be surprised with the broad range of answers students will devise. • It is important to note that some students will complete the activity with basic answers for the solution, while other students will work at a more complex level. All answers put forward should be praised, provided their answer is seven. • Some students may wish to write more than five different ways and should be allowed to continue as long as they offer different solutions. • 108 • New Wave Maths Book D – Teachers Guide

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Unit 22—Answers

Student pages 64 – 66 Unit 22–2

1. (a) 1 (b) 7 (c) 0 (d) 2 (e) 3 (f) 4 (g) 2 (h) 1 (i) 4 (j) 5 2. (a) 319 (b) 146 (c) 445 (d) 424 (e) 319 (f) 438 3. (a) 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99 (b) The numbers line up along a diagonal. Also, the numbers line up in the first, third, fifth, seventh and ninth columns. 4. (a) Answers will vary (b) Answers will vary Challenge Yes, you will land on 50 after subtracting the number 7, seven times.

1. (a) 1 (b) 8 (c) 1 (d) 10 (e) 3 (f) 8 (g) 6 (h) 0 (i) 2 (j) 11 2. (a) 316 (b) 513 (c) 525 (d) 212 (e) 457 (f) 436 3. (a) 1 in 4 or 1 in 6 times, depending on the number of sides on the spinner. 4. Teacher check 5. Teacher check 6. Teacher check 7. Teacher check Challenge

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Unit 22–1

+ 1000

2652

2500

√ 10 25 000

200

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– 1000

x 10 √ 10 x 10

3652

25 000

250 000

2000

– 10

4552

√ 100

Consolidation 22–2

• Repeat the activity at a later date and compare results.

Consolidation 22–3

• Provide further opportunities to complete similar number chains, working out answers mentally before checking with a calculator.

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√ 1000 √ 10

4562

• Repeat Exercise 4, but give it to a partner to complete after giving the starting number and number to count by.

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1. (a) 16 (b) 13 (c) 9 (d) 2 (e) 11 (f) 4 (g) 12 (h) 15 (i) 39 (j) 17 2. (a) 611 (b) 242 (c) 329 (d) 538 (e) 425 (f) 623 3. – 100 + 1000 + 10 – 1000

+ 100

3662 25

x 10

x 100

2 500 000

x 10

√ 10

3562 250

x 100

25 000

200 x 100

2000

x 100

√ 1000

20 000

Challenge Answers will vary. Some examples are: 7, 14 ÷ 2, 3 + 4, 10 – 3, 21 ÷3, 7 x 1 etc.

R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 109 •

Unit 23–1

Student page 67

Outcomes

Indicators

N3.3, S3.3

Skills • drawing enlargements and reductions • halving • doubling

Memory Masters (N3.3)

The student is able to: • use multiple copies of figures to create patterns based on systematic movements of the shape and informally describe the movement used.

Resources • Base 10 MAB • calculator

Language • shape • dimensions • halved • doubled

r o e t s Bo r e p ok u S

Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.3) Warm up

• Demonstrate to the class how to halve dimensions of a shape using a simple rectangle; e.g. 2 x 4. • Ask students what they need to do if the original shape only covers one unit in length. • Does halving change if the line is diagonal? Demonstrate to show that half diagonals are made the same way as whole vertical or horizontal lines.

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• Direct the students to draw the shape with its dimensions halved in the top right-hand corner of the grid. • Once completed, remind students that when they are doubling the dimensions, all lines must be twice as long. • Direct students to draw the shape with its dimensions doubled, starting at the bottom left-hand side of the grid.

Challenge

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What to do

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Teac he r

• The focus for this unit is basic facts of multiplication with addition of a single-digit number.

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• The students will need to follow the instructions given. In each case they should record the numbers they use through each stage so their results may be readily checked. • Students will need to work through the instructions several times to be confident they can answer the problem correctly.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 18 – 19. • 110 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 23–2

Student page 68

Outcomes

Indicators The student is able to: • use the decimal point in representing quantities and money. • regroup money to the fewest number of notes or coins.

N3.3, N3.1a

Skills • subtracting money as decimals • writing money

Resources

Language

• Base 10 MAB • calculator • money

• change • notes • coins • smallest

r o e t s Bo r e p ok u S

Notes

Memory Masters (N3.3)

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Teac he r

• The focus for this unit is basic facts of multiplication with addition of a single-digit number.

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a) Warm up

• Students should be familiar with handling money. If not, or as an introduction, allow students to play with artificial money for a few minutes. • Ask the students how they can mentally work out: – How much change they would get from $1 if they spent 20c? – How much change from $2 if 20c is spent? – How much change from $5 if 20c is spent? • Encourage students to explain how they were able to work out the answer for change from $1 through to change from $2 and $5.

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• Use this knowledge to complete Exercise 3. • For Exercise 4, if students are unable to readily transfer their knowledge from the previous activity, let them build on from the cost of the item. • Exercise 5 may require students to use money to reach the answers.

Challenge

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• Ask students to use a store catalogue to determine what the most common amount a price ends with; e.g. 69c, 99c, 95c. Explain why they think this is the case.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 40 – 41. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 111 •

Unit 23–3

Student page 69

Indicators

Outcomes N3.3, S3.2

The student is able to: • talk about what they can and can not see of an object from different positions and attempt to draw what they see rather than what is known to be there. • match standard geometric models with realistic drawings and conventional diagrams.

Skills • following instructions • constructing models

Teac he r

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • 2-cm cubes

Language • model • cube • side view • front view • top view

r o e t s Bo Notes r e p ok u S

• The focus for this unit is basic facts of multiplication with addition of a single-digit number.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.2) Warm up

• Distribute 2-cm cubes to groups of students and allow time for them to build models. • Ask students what they think is meant by a side, top and front view of a model they have made. • Direct students to identify and discuss the side, top and front views of models they make.

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• Direct students to look carefully at the model in Exercise 3 and to make one as shown. • Students identify and match the side, top and front views to check accuracy. Groups of students or partners could check each others’ models. • In groups or with a partner, construct the model in Exercise 4 and draw the side, top and front views. Ask a student(s) to draw these on the blackboard/whiteboard for the class to check against theirs. • Students make the model from the views shown in Exercise 5 and draw in the space provided. Assist as required as this activity can be quite difficult for some students to complete.

Challenge

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What to do

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Number (N3.3)

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• Ask students to count the number of cubes used to build the model in Exercise 4. • Direct students to build the model with double the length, width and height. • Discuss what they notice about the volume (number of cubes) used to build the new model.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 16 – 17. • 112 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 23—Answers

Student pages 67 – 69 Unit 23–2

Unit 23–1

1. (a) 10 (b) 17 (c) 9 (d) 20 (e) 8 (f) 18 (g) 17 (h) 7 (i) 29 (j) 14 2. (a) 335 (b) 288 (c) 189 (d) 490 (e) 588 (f) 768 3. (a) $4.50 (c) $1.50 (b) 50c (d) $9.50 4. (a) $7.80 (c) $17.80 (b) $2.80 (d) $1.80 5. (a) 6 (b) 3 (c) 6 (d) 7 Challenge Answers will vary

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Teac he r

1. (a) 10 (b) 13 (c) 12 (d) 7 (e) 12 (f) 26 (g) 38 (h) 10 (i) 16 (j) 13 2. (a) 216 (b) 295 (c) 558 (d) 525 (e) 441 (f) 711 3. Teacher check 4. Teacher check Challenge Yes

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• Draw a simple shape of their own on grid paper (see page 199), then halve and double its dimensions.

Consolidation 23–2

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1. (a) 18 (b) 13 (c) 8 (d) 13 (e) 20 (f) 13 (g) 2 (h) 15 (i) 28 (j) 15 2. (a) 234 (b) 126 (c) 468 (d) 472 (e) 873 (f) 259 3. Teacher check 4.

• Provide further opportunities to work out amounts of change from given values.

Consolidation 23–3

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• Make further models with 2-cm cubes and draw the side, top and front views.

Challenge The number of cubes needed to build the new model will be eight times the original number.

R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 113 •

Unit 24–1

Student page 70

Outcomes

Indicators

N3.3, C&D3.2, C&D3.3, C&D3.4

The student is able to: • remember basic addition and subtraction facts and calculate mentally. • record frequency data carefully using simple formats based on tallies or organised lists. • display frequency data in bar graphs. • explain their own data displays to peers.

Skills • adding • subtracting • timing • recording • graphing

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • stopwatch/timer • blank graph (see page 236)

Language • speed test • addition • subtraction • graph • timer • record

r o e t s Bo r e p ok u S

Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3, C&D3.2, C&D3.3, C&D3.4) Warm up

• Explain to the students they will be doing a speed test involving addition and subtraction basic facts. This will include timing how long it takes and graphing the results.

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• Either provide students with a graph template or have them make a graph to record the number correct and a separate graph to show the time taken. • Remind students they should keep their graphs in a safe place as they will be required to use them again to check on their progress. • It is recommended that students complete this speed test at least once a term and up to twice a term, or make it a once-weekly test for five or six weeks. • Instruct students to rule six columns of ten lines in their pad. • Once this is completed, suggest the pad is placed over the workbook page with the column to be worked on showing to the immediate left of the pad. As each column is completed, the pad is moved to the right to expose the next column. • When everyone is ready, tell the students you will give the order to start.They are to work as quickly as possible and to raise his/her hand when finished. You will call the time taken to complete the task, which students write on their page. Suggest they leave the ones they have trouble completing. • Mark the work when everyone is finished. • Students graph the number correct and the time taken. Assist as required. • Direct students to learn the facts they got wrong or left out.

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What to do

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Teac he r

• The focus for this unit is basic facts of multiplication with addition of a single-digit number.

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Challenge • When completing a Magic Square, there is usually a clue included. Ask students to find the clue. • Ask how this clue may be used to complete the Magic Square. • Students complete the Magic Square, encourage them to complete their working mentally, but calculators should be used if students are having difficulties.

• 114 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 24–2

Student page 71

Outcomes

Indicators

N3.3, C&D3.3, C&D3.4

The student is able to: • use diagrams such a Venn diagrams and two-way tables to represent a two-way classification. • interpret straightforward one- and two-way tables.

Skills • completing a Carroll diagram • collecting information • collating data • recording • analysing data

Resources

Language • Carroll diagram

• Carroll diagram (see page 221)

r o e t s Bo r e p ok u S

Notes

Teac he r

Memory Masters (N3.3)

• The focus for this unit is basic facts of multiplication with subtraction of a single-digit number.

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Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.3, C&D3.4) Warm up

• Give a little background information to the class on Lewis Carroll, author and mathematician. Perhaps read ‘The Walrus and the Carpenter’ and acknowledge ‘Alice in Wonderland’ or ‘Through the Looking Glass’. Carroll was also a highly respected mathematician. One of the methods we use for classifying information bears his name—Carroll diagram. • Draw an outline of the Carroll diagram from the workbook on the blackboard/whiteboard or use an overhead projector and explain to the students how the information is collected for each cell on the diagram. • Use the students in the class to complete the Carroll diagram to show blonde- and brownhaired boys and girls.

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What to do

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• Ask the students how they will record the information in the cells. Tally or total are the most likely options. If using a tally then a total will be required for easy reading. • Ask students how they would record the students who do not have blonde or brown hair. • Students who suggest extending the Carroll diagram should be encouraged to either demonstrate to the class or draw their own diagram to show their extended results. • Direct students to complete the second Carroll diagram.

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Challenge

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• Students may need reminding of the properties of a Magic Square—all rows, columns and diagonals add to the same total—in this case, 15. • Students are left to find their own solutions but must record the steps used so they are able to explain how they reached their solution, or to assist in understanding difficulties they may have had.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 110 – 111. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 115 •

Unit 24–3

Student page 72

Indicators

Outcomes N3.3, N3.1, N3.2

The student is able to: • understand that multiplication can be used for repeated addition situations. • partition two-digit numbers to assist in adding and subtracting them mentally.

Skills • multiplying • adding • doubling

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • counters/cubes

Language • double • add • multiply

r o e t s Bo Notes r e p ok u S

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.2, N3.3) Warm up

• Distribute counters or cubes to groups of students. Ask students to nominate numbers between 1 and 20. Students then make that amount with the counters or cubes, placing them in a row. • Ask students to place another row next to the first. Ask what they have made. (Multiplying by two, adding the same number again or doubling.)

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• Direct students to Exercise 3. Complete the activity mentally or by using counters to assist. • Discuss how they could double the numbers mentally in Exercise 4. All numbers in this section can be doubled by doubling each digit; e.g. 23—2 + 2 = 4 and 3 + 3 = 6; i.e. 46. • Exercise 5 requires mental or written regrouping for most of the numbers. Students may wish to use counters, cubes or a calculator to assist if necessary. • Ask students what they noticed about all the answers in each exercise. After discussion, students write an answer for Exercise 6. (When numbers are doubled, the answers are always even, regardless of whether the original number was odd or even.)

Challenge

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What to do

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Teac he r

• The focus for this unit is basic facts of multiplication with subtraction of a single-digit number.

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• Ask students what they think consecutive numbers are. • Give examples of two consecutive numbers such as 41 and 42. • Present students with the problem: ‘The answer to a problem is 107. Find the two consecutive numbers, when added together, make 107.’ • Students give a brief explanation of how they found the two numbers.

• 116 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 24—Answers

Student pages 70 – 72

Unit 24–1

1. (a) 14 (b) 6 (c) 20 (d) 6 (e) 2 (f) 0 (g) 4 (h) 1 (i) 9 (j) 35 2. (a) 647 (b) 876 (c) 798 (d) 799 (e) 998 (f) 766 3. Teacher check 4. Teacher check Challenge Answers will vary. One solution is:

r o e t s Bo r e p ok u S 2 9 4 7 5 3 6 1 8

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Teac he r

1. (a) 22 (b) 11 (c) 20 (d) 30 (e) 7 (f) 6 (g) 34 (h) 36 (i) 14 (j) 6 2. (a) 13 (b) 38 (c) 16 (d) 18 (e) 19 (f) 14 3. 9 18 7 15 15 15 13 14 11 10 9 12 6 6 8 13 10 12 8 9 9 7 9 11 12 8 17 14 13 4 16 12 12 14 10 11 12 13 16 17 10 11 8 10 14 14 7 13 11 2 9 1 8 7 2 6 7 1 8 9 3 5 8 5 5 4 6 6 8 9 6 7 9 7 9 5 4 7 6 8 1 6 9 6 7 3 3 8 5 1 9 9 3 9 2 4 3 4 4 2 4. Teacher check Challenge

Unit 24–2

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• When students have improved speed and accuracy, devise another speed test of different addition and subtraction basic facts.

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1. (a) 20 (b) 15 (c) 10 (d) 30 (e) 0 (f) 2 (g) 30 (h) 16 (i) 13 (j) 0 2. (a) 857 (b) 899 (c) 777 (d) 997 (e) 798 (f) 737 3. (a) 2 (d) 4 (g) 6 (b) 8 (e) 10 (h) 12 (c) 14 (f) 16 (i) 18 4. (a) 20 (f) 40 (k) 60 (p) 80 (b) 22 (g) 44 (l) 66 (q) 88 (c) 24 (h) 42 (m) 62 (r) 82 (d) 26 (i) 46 (n) 64 (s) 84 (e) 28 (j) 48 (o) 68 (t) 86 5. (a) 34 (e) 98 (i) 58 (m) 90 (b) 32 (f) 72 (j) 78 (n) 92 (c) 96 (g) 30 (k) 100 (o) 38 (d) 56 (h) 76 (l) 70 (p) 36 6. All the answers are even, regardless of whether the original number is odd or even. Challenge 53 + 54 = 107

Consolidation 24–2

• Brainstorm other ideas to record data in a Carroll diagram.

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R.I.C. Publications® www.ricpublications.com.au

Consolidation 24–3

• Provide further opportunities to double numbers mentally or in written form where regrouping is required.

New Wave Maths Book D – Teachers Guide • 117 •

Unit 25–1

Student page 73

Outcomes

Indicators The student is able to: • use multiple copies of figures to create patterns based on systematic movements of the shape and informally describe the movement used. • respond to questions such as ‘Is there another way you could check your answer?’ by doing it in a different way.

N3.3, S3.3, WM3.4

Skills • verifying • rotating • explaining • supporting

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • 2-D shapes as required • 1-cm grid paper (see page 199)

Language • verify • turning • one-quarter • direction • original • shape

r o e t s Bo r e p ok u S

Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.3, WM3.4) Warm up

• Ask the students to stand behind their chair facing their desk. • Ask students to follow the directions you give them exactly. • ‘Turn one-quarter of a turn to your left. Where are you now?’ (Right side of their bodies should be facing the desk.) • ‘Turn one-quarter of a turn to your left. Where are you now?’ (Back of their bodies should be facing the desk.) • ‘Turn one-quarter of a turn to your left. Where are you now?’ (Left side of their bodies should be facing the desk.) • ‘Turn one-quarter of a turn to your left. Where are you now?’ (Students should be facing the desk.) • Repeat the activity with four one-quarter turns to the right. ‘Where do you finish now?’ (Same place.)

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• The focus for this unit is multiplication of basic facts with subtraction of a single-digit number.

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What to do

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• In Exercise 3, mark where the shaded part will be when turned through a quarter turn as shown. Continue until four quarter turns have been made. Where is the shaded part now? (Where it started.) • Students repeat the activity using their own shape. • Students finish by writing their explanation as to why they think the shape or object will always return to its original position.

Challenge

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• Students draw a simple maze on 1-cm grid paper and write some instructions to guide a ‘robot’ through the maze. • Inform students that 1 square = 1 step.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 24 – 25. • 118 • New Wave Maths Book D – Teachers Guide

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Unit 25–2

Student page 74

Outcomes

Indicators The student is able to: • use their own methods or a conventional algorithm to multiply whole numbers by single- and double-digit numbers. • identify patterns in the multiplication tables and use to make predictions.

N3.3, N3.4

Skills • calculating • completing patterns

Resources

Language • pattern • calculator • grid

• Base 10 MAB • calculator

r o e t s Bo r e p ok u S

Memory Masters (N3.3)

Notes

Number (N3.3)

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Teac he r

• The focus for this unit is basic facts of multiplication with subtraction of a single-digit number.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3, N3.4) Warm up

• Students take out calculators and have a few minutes of free play exploring their calculator. • Ask students to make up a pattern using their calculators. Write the pattern and ask a partner to work through the activity to see if he/she can also find the pattern.

What to do

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Challenge

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• Direct students to their workbook and ask them to use their calculators to write the answers to the pattern in Exercise 3. • When the calculator activity is completed, ask the students to describe any patterns they can see. • Share answers with small groups and select one from each group to share with the class or select a few students to share their answers with the class.

• The grid activity involves making rows to total 20. • The numbers given are to be used once only in the grid. • When completed, ask students to see if they can find a pattern in the grid. • The total of each pair added to the 8 must equal 12. Therefore the highest and lowest of the numbers given are paired. • Ask students why they think six was not used. (Would need to be used twice to make 12.)

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New Wave Maths Book D – Teachers Guide • 119 •

Unit 25–3

Student page 75

Outcomes

Indicators

N3.3, M3.2, C&D3.3, C&D3.4

The student is able to: • use a uniform unit of length carefully to make their own graduated scale to measure things in their environment. • display frequency data in bar graphs. • read frequencies from a bar graph and hence describe the data.

Skills • measuring perimeters • recording • graphing • analysing data

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • string • ruler

Language • measure • length • perimeters • graph

r o e t s Bo Notes r e p ok u S

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2, C&D3.3, C&D3.4) Warm up

• Divide the class into groups of five. • Provide each group with string. • Ask students to decide within their groups where they consider their wrist measure is to be taken.

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• Students measure each other’s wrist and record the measure. • The recorded measures are then graphed using the graph in the workbook. • When the graphs are complete, ask each group to decide which student(s) has (have) the longest wrist measurement. • Ask which student(s) has (have) the smallest wrist measurement. • Check with each group to see whether these students are also the tallest and shortest.

Challenge

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What to do

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Teac he r

• The focus for this unit is basic facts of multiplication with subtraction of a single-digit number.

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• Students will need to find a way to measure their height and arm span. • Once the two measures have been made, the student decides if there is a relationship between the two measures. • Students will need to explain why they think there is a relationship or there is no relationship.

• 120 • New Wave Maths Book D – Teachers Guide

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Unit 25—Answers

Student pages 73 – 75

Unit 25–1

1. (a) 24 (b) 1 (c) 7 (d) 3 (e) 1 (f) 40 (g) 12 (h) 0 (i) 2 (j) 28 2. (a) 1379 (b) 1795 (c) 1846 (d) 1799 (e) 1478 (f) 1897 3. (a) 111 (j) 1110 (s) 2109 (b) 222 (k) 1221 (t) 2220 (c) 333 (l) 1332 (u) 2331 (d) 444 (m) 1443 (v) 2442 (e) 555 (n) 1554 (w) 2553 (f) 666 (o) 1665 (x) 2664 (g) 777 (p) 1776 (y) 2775 (h) 888 (q) 1887 (z) 2886 (i) 999 (r) 1998 Answers may vary but should include the fact that the numbers increased by 111 each time; between 100 – 1000 the pattern is 111, 222 etc.; between 1000 – 2000 the final three numbers make the pattern 1110, 1221, 1332 etc.; between 2000 – 3000 the final three numbers make the pattern 2109, 2220, 2331 etc. 11 8 1 Challenge 10 8 2 Starting with the largest and smallest, 9 8 3 the numbers decrease by one and 8 8 4 7 8 5 increase by one.

r o e t s Bo r e p ok u S

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1. (a) 7 (b) 2 (c) 7 (d) 20 (e) 1 (f) 13 (g) 0 (h) 25 (i) 8 (j) 28 2. (a) 664 (b) 414 (c) 352 (d) 342 (e) 256 (f) 456 3. Teacher check Challenge Teacher check

Unit 25–2

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• In pairs, direct each other in turn to move one-, two-, three- or four-quarter turns to the left or right.

Consolidation 25–2

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1. (a) 6 (b) 1 (c) 10 (d) 0 (e) 4 (f) 11 (g) 21 (h) 0 (i) 16 (j) 38 2. (a) 786 (b) 794 (c) 984 (d) 583 (e) 854 (f) 804 3. Teacher check Challenge The taller the person, usually the longer the arm span.

• Repeat the activity, replacing the number 37 with another number. Ask students to describe patterns found, if any.

Consolidation 25–3

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• Repeat the activity, measuring foot or hand length.

New Wave Maths Book D – Teachers Guide • 121 •

Unit 26–1

Student page 76

Outcomes

Indicators The student is able to: • describe and continue number sequences based on addition or subtraction but involving more than adding or subtracting a constant amount. • identify the starting number and the constant multiplier needed to generate a number sequence. • follow a rule to generate a number sequence based on an addition, subtraction or multiplication strategy.

N3.3, N3.4

Skills • completing patterns • making patterns • working cooperatively

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator

Language • patterns • code • rule

r o e t s Bo r e p ok u S

Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.4) Warm up

• Discovering hidden patterns can be quite a challenge for some students. It requires a clear understanding of operations and how they work to change numbers and the effect numbers have on other numbers. It takes time to train the brain to recognise a pattern and then to be able to explain how it works requires the skill of communication. Discuss with students: – What is a pattern? (A repeated design. A number pattern is a sequence of numbers formed by following a rule.) – Where would we see or find a pattern? – Some patterns involve numbers—have you seen any number patterns before? Where? • Provide students with several examples of number patterns. Multiplication tables, repeated addition, repeated subtraction are all good examples of simple patterns.These will help to switch the students’ thinking onto patterns. Seating in sporting stadiums and theatres and movie cinemas also follow a pattern. Students will probably be quite familiar with these patterns. Demonstrating some of these patterns on the board will help students become familiar with the concept of patterns and encourage them to seek out patterns in their environment.

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• The focus for this unit is basic facts of multiplication with addition of a single-digit number.

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What to do

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• Direct students to the first example on their page. Can anyone see a pattern? What is it? What numbers do we need to write in the blank spaces to complete the pattern? • Ask students to discuss what they think the rule could be for this pattern. All patterns need to follow a rule or else they wouldn’t be a pattern. Students can record the rule in the space provided on the page. • Direct students to complete the next example.This pattern is a little more complicated and students may need some guidance. Once students become more confident, they can move on to complete each example. Discussion with partners should be encouraged. Sometimes ‘bouncing’ ideas off someone else can help to clarify a thought. • In Exercise 4, students make patterns of their own and swap with classmates. Some patterns may become quite involved, while other students will opt to keep the patterns quite simple.

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Challenge • Secret codes are fun and often follow a rule.This activity allows students to follow on from the previous activity, applying their skills to a new situation. • Students can develop one or more codes to share with classmates. An invitation to share lunch or play together at recess could be written in code. For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 70 – 71. • 122 • New Wave Maths Book D – Teachers Guide

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Unit 26–2

Student page 77

Outcomes

Indicators The student is able to: • record frequency data carefully using simple formats based on tallies or organised lists. • summarise data based on tallying.

N3.3, C&D3.2, C&D3.3

Skills • tallying • recording • surveying • analysing • interpreting

Resources

Language

• Base 10 MAB • calculator

• tally • total • survey

r o e t s Bo r e p ok u S

Memory Masters (N3.3)

Notes

Number (N3.3)

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Teac he r

• The focus for this unit is basic facts of multiplication with addition of a single-digit number.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.2, C&D3.3) Warm up

• Explain to the class they will be completing a survey. • Ask if they understand what a survey is. If not explain what it is and what it might be used for. (A survey asks the views of people in order to write a report about what people think or do.) • Ask students how they would record their findings while conducting the survey. If the five-group tally (IIII) is suggested, continue, if not, explain that for ease of counting, we use a five-group tally record with the fifth in the group drawn across the first four recordings.

What to do

Challenge

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• Direct the class to Exercise 3. • Ask what will be the best way to collect the information. Select the best and set the class to find and record the information. • Exercise 4 can be completed by collecting the information in the same manner as suggested above and recording the results on the page. • Using the results, work through the questions asked about the tally with the students. Have the students explain how they reached their answers.

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• Ask the students, how they think they would find the correct placement of the numbers in the circles. • Encourage them to propose ideas they can then test rather than just trial and error. • Check answers as students complete the challenge, but don’t give away the result to the whole class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 112 – 113. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 123 •

Unit 26–3

Student page 78

Indicators

Outcomes N3.3, N3.1b

The student is able to: • describe and record simple fractional equivalents in words and diagrams.

Skills

Resources • Base 10 MAB • calculator • coloured rods

Language • diagrams • equivalent fraction

• representing fractions • identifying equivalent fractions

Memory Masters (N3.3)

r o e t s Bo Notes r e p ok u S

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1b) Warm up

• Distribute coloured rods or similar concrete materials to small groups of students. • Allow time to play with the materials. • Direct students to match different sized rods to see how many of the smaller rods equal one of the larger rods. • Find out which smaller rods can be matched equally to a specific larger rod. • Ask the students for their explanation of the matching of smaller rods to a larger rod. If an explanation comes forward explaining the matching as fractions, use this as the cue to move into the activity. • If a fractional explanation is not forthcoming, explain to the students that where two rods equals one larger rod, the two smaller rods are half the larger rod. Continue with thirds and fourths. • Ask the students to take out two rods the same and match them with four smaller rods. Repeat this with three rods matched with six smaller rods. • Explain this process as finding equivalent fractions—half equals two-fourths, one-third equals two-sixths, two-thirds equals four-sixths.

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• The focus for this unit is multiplication of basic facts with addition of a single-digit number.

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What to do

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• Students may now be directed to their workbook to complete the activity. Students having difficulties should continue with the rods to assist.

Challenge

• Ask the students to suggest what operation they are going to use to solve the problem. • Ask how they intend to solve the problem using this operation. • Ask if there are other ways this problem may be solved. • Set the class to finding their answer.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 48 – 49. • 124 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 26—Answers

Student pages 76 – 78

Unit 26–1

1. (a) 5 (b) 20 (c) 33 (d) 11 (e) 30 (f) 22 (g) 28 (h) 11 (i) 7 (j) 14 2. (a) 1432 (b) 1111 (c) 1502 (d) 1686 (e) 1452 (f) 1383 3. Teacher check 4. Teacher check Challenge One possible solution:

r o e t s Bo r e p ok u S 1

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1. (a) 20 (b) 30 (c) 26 (d) 6 (e) 10 (f) 22 (g) 14 (h) 5 (i) 9 (j) 15 2. (a) 1758 (b) 1616 (c) 1759 (d) 1826 (e) 1227 (f) 1858 3. (a) 10, 12, 14, 16 add 2 (b) 16, 20, 21 add 1, add 4 (c) 27, 32, 37 add 5 (d) 22, 26, 30 add 4 (e) 15, 17, 20 add 3, add 2 (f) 18, 15, 12 subtract 3 (g) 27, 23, 19 subtract 4 (h) 64, 128, 256 multiplying by 2 (doubling) (i) 12, 10, 15 2 x 1, 3 x 1, 2 x 2, 3 x 2, 2 x 3, 3 x 3, 2 x 4, 3 x 4, 2 x 5, 3 x 5 (j) 20, 8, 25 5 x 1, 2 x 1, 5 x 2, 2 x 2, 5 x 3, 2 x 3, 5 x 4, 2 x 4, 5 x 5 4. Teacher check Challenge Answers will vary

Unit 26–2

(a) 1/2 = 2/4

(b) 3/5 = 6/10

(d) 1/4 = 3/12

(e) 1/2 = 6/12

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(f) 2/3 = 4/6

(g) 2/3 = 8/12

(h) 1/5 = 2/10

Consolidation 26–2

• Brainstorm other data they can tally about the students in their class.

Consolidation 26–3

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(i) 1/2 = 5/10

(j) 1/4 = 2/8

• Use a 1 – 100 or 0 – 99 grid (see page 202) to explore number patterns.

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1. (a) 7 (b) 21 (c) 40 (d) 16 (e) 36 (f) 28 (g) 49 (h) 8 (i) 19 (j) 19 2. (a) 1333 (b) 2017 (c) 1327 (d) 1683 (e) 1915 (f) 1537 3. (k) 3/4 = 6/8

• Display diagrams of equivalent fractions on a chart for students to refer to.

(l) 3/4 = 9/12

(m) 1/2 = 3/6

(n) 1/3 = 2/6

(o) 1/3 = 4/12

Challenge 38 and 45

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New Wave Maths Book D – Teachers Guide • 125 •

Unit 27–1

Student page 79

Outcomes

Indicators

N3.3, S3.3, M3.2

The student is able to: • use multiple copies of figures to create patterns based on systematic movements of the shape and informally describe the movement used. • use square-grid paper to find the area of shapes.

Skills • drawing • scaling • recording

Teac he r

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • tangram set (see page 224)

Language • grid • dimensions • halved • doubled

r o e t s Bo r e p ok u S

Notes

• The focus for this unit is multiplication of basic facts with addition of a single-digit number.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.3, M3.2) Warm up

• Discuss halving and doubling dimensions with the class. Emphasise what happens to a line if it is halved or if it is doubled. • Demonstrate on the blackboard/whiteboard or by using audiovisual equipment to show students. • Select students to demonstrate halving and doubling to the class.

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• Direct students to Exercise 3. • Explain to the class that all lines in bold print are to be redrawn half the length shown on the page. • Students draw the half-dimension shape on the same grid as the original shape. • Once completed, ask the students to complete the table in Exercise 4. – How high was the original shape? Write the answer in the table. – How wide was the original shape? Write the answer in the table. – What was the area of the original shape? Students may need to be guided to determine the area by counting the shaded squares. Write the answer in the space. • Repeat this for the half-dimension shape. • Draw the double-sized shape in Exercise 5.

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Challenge

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What to do

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Number (N3.3)

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• Explain to the students that a regular seven-piece tangram was originally a square. • By rearranging the seven pieces, students should be able to reconstruct the square. • Encourage students to experiment with spatial arrangements to remake the square.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 18 – 19. • 126 • New Wave Maths Book D – Teachers Guide

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Unit 27–2

Student page 80

Outcomes

Indicators The student is able to: • place unit fractions in order and explain the order either in objects, diagrams or words. • read and write fractional notation to represent unit fractions.

N3.3, N3.1a, N3.1b

Skills • ordering fractions • writing numbers

Resources

Language • fractions • smallest • largest • numbers • digits

• Base 10 MAB • calculator • coloured rods

r o e t s Bo r e p ok u S

Notes

Memory Masters (N3.3)

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• The focus for this unit is basic facts of multiplication with a small addend.

Number (N3.1a, N3.3)

Main Activity (N3.1b)

Warm up

What to do

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• Organise the class into small groups for this activity. • Distribute coloured rods or similar to the groups. (Strips of paper showing fraction parts or a fraction grid may also assist.) • Use the red 10-cm rod to represent a whole. • Show 1/8, 1/2 and 1/4 using the other rods. Compare the size of each of these to arrange from smallest to largest. How else could they have decided which was smallest? [Using the denominator (bottom number).] • Continue using the concrete materials to make the fractions, and compare the size of the fractions directly before placing them in the correct order. • When arranging digits in a different order to make new numbers, it may be useful for the students to have cards with the numbers displayed so they can physically alter the position of the digits to represent new numbers. • Students should record the new numbers as they are made. The physical representation allows students to readily check to see if they have already made the number. • Suggest to the students, or ask if they can suggest a way to organise their data. (Start with one digit at the beginning, then arrange the other digits in differing order. Swap the first digit and repeat.) or (Arrange numbers from largest to smallest, or smallest to largest using the same procedure.)

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Challenge • Encourage students to undertake this activity by themselves. • Check that the numbers are the largest and the smallest possible before working out the difference. • Check by using a calculator. • Ask students if they can find another way to check if their working is correct. (Subtracting with regrouping.) For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 48 – 49. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 127 •

Unit 27–3

Student page 81

Indicators

Outcomes WM3.1, WM3.2, C&D3.1, C&D3.2

The student is able to: • work mathematically to describe how chance is built into some familiar games. • pose mathematical questions prompted by specific stimulus. • describe outcomes as having equal chance or being equally likely. • discuss what data to collect to answer a question.

Skills • posing questions • collecting data • analysing data • making conjectures

Resources • calculator • packs of playing cards

Language • chance • probability • playing cards • odd • clubs • hearts • diamonds • picture cards

• • • • •

random fraction even suits spades

r o e t s Bo Notes r e p ok u S

What to do

• This activity is designed for students working collaboratively in groups. Allow enough time so they can discuss their options and for ideas to evolve. Investigative tasks such as these are a good opportunity for students to ‘take a risk’ with maths. Note: Students remove any jokers or wild cards before starting the activity. • When completing investigative tasks, some students may be more successful in mixed-ability groups rather than same-ability groups. • Some groups will be able to work independently while others may need guidance. The stimulus questions below may prompt such groups. – How many cards in a pack of cards? (52) – How many different colours are the cards? (2 – red and black) – Do you know the name of each different suit? (clubs, hearts, spades, diamonds) – How many cards are there of each suit? (13) – What are the picture cards? (Jacks, Queens, Kings) – How many picture cards are there in a pack? (12) • Once students have recorded the facts, they will need to know how to write their answers. For example, how many even cards? Answer is written as:

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Main Activity (WM3.1, WM3.2, C&D3.1, C&D3.2)

w ww number of even cards

26 52

total number of cards in the pack

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This fraction can also be broken down to ‘one-half ’. • Discuss the answers of each part of Question 1. • Students complete Questions 2 and 3. Allow each group to share their favourite game with the class. Students explain how ‘chance’ is involved in the game. • Allow each group to discuss and evaluate its ability to problem-solve and its success as a group. A group or self-assessment form could be completed. This information will be helpful for creating groups for future open-ended, investigative tasks.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 20 – 21. • 128 • New Wave Maths Book D – Teachers Guide

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Unit 27—Answers

Student pages 79 – 81

Unit 27–1 1. (a) 30 (b) 15 (c) 20 (d) 12 (e) 6 (f) 18 (g) 25 (h) 29 (i) 31 (j) 21 2. (a) 370 (b) 828 (c) 903 (d) 1312 (e) 1588 (f) 1285 3. Teacher check 4. Width

Area

6 3

20 5

Dimensions halved

3. (a) 1/8

(b) 1/10

/4 1 /3 1 /4 2 /3 1 /2 1

/2 1 /2 3 /8 7 /8 7 /10 1

(d) 5/10

(f) 2/10 (g) 1/2 (h) 2/10 (i) 1/2

/8 5 /8 5 /10 2 /3 1 /3 2

5 (e) 3/8 /6 (j) 1/4 4. (a) 321, 312, 231, 213, 132, 123 (b) 4321, 4312, 4231, 4213, 4132, 4123, 3421, 3412, 3241, 3214, 3142, 3124, 2431, 2413, 2341, 2314, 2143, 2134, 1432, 1423, 1342, 1324, 1243, 1234 Challenge 97543210 – 1234579 96308631

/3 3 /4 9 /10 3 /4 1 /2 2

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5. Teacher check Challenge

1. (a) 8 (b) 14 (c) 7 (d) 31 (e) 29 (f) 24 (g) 40 (h) 17 (i) 19 (j) 24 2. (a) 7.7 cm (b) 8.7 cm (c) 9.8 cm (d) 5.6 cm (e) 8.9 cm (f) 6.8 cm

r o e t s Bo r e p ok u S

Height

Original Shape

Unit 27–2

Note: Students will first need to remove the joker or wild cards from the pack.

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(b) 26/52 = 1/2

• Provide opportunities for further activities to practise ordering fractions.

(d) 13/52 = 1/4 2. Teacher check 3. Teacher check

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Consolidation 27–2

1. (a) 26/52 = 1/2

• Draw simple diagrams or shapes on grid paper to enlarge or reduce.

Consolidation 27–3

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R.I.C. Publications® www.ricpublications.com.au

• Students calculate the probability of rolling a number on a die. This can be extended to rolling two numbers on two dice (such as rolling two sixes).

New Wave Maths Book D – Teachers Guide • 129 •

Unit 28–1

Student page 82

Outcomes

Indicators The student is able to: • round numbers up or down to the nearest 10 or 100 to serve a specific purpose such as estimation. • estimate sums and products by rounding.

N3.1a, N3.3

Skills • estimating • adding • rounding

Teac he r

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator

Language • estimate • answers • sums • closest • more than

r o e t s Bo r e p ok u S

Notes

• The focus for this unit is basic facts of multiplication with subtraction of a single-digit number.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a, N3.3) Warm up

• Ask students to write an addition number sentence that will give an answer close to 20. The answer is not to be exactly 20. • Share these, asking students what number the answer is close to. (20) • Ask students how they made a decision to choose 20 as the answer that the sum was near to (other than because they were told to!). • Ask students, ‘If I have 49 and 49 what number is the total close to?’ (100). ‘How did you reach the answer?’ (50 + 50). • ‘What about 23 and 29?’ (40 and 50 may well be given. Accept both.) • ‘What is the sum of 36 and 38 close to?’ (80) • Explain that the process involves rounding and gives us an estimate.

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Number (N3.1a, N3.3)

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What to do

• Work through Exercise 3, using the same method of rounding to the nearest ten to estimate the answers. Find which estimate is closest to 110. • Check working for understanding—give more practise when and if required. • Students continue with Exercise 4. • Exercise 5 requires rounding to the nearest 100. Students may need practise prior to attempting the exercise. If so, repeat the process for rounding to the nearest 10 but use numbers between 100 and 1000.

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Challenge

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• Students need to find combinations of four numbers from 1 to 9 that add to 17. • Each number can only be used once. • Students should keep records of their workings to show how they reached their answer.

• 130 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 28–2

Student page 83

Outcomes

Indicators

N3.1a, N3.3, C&D3.1, C&D3.2, C&D3.3, C&D3.4

The student is able to: • describe outcomes as having an equal chance or being equally likely. • record frequency data carefully using simple formats based on tallies or organised lists. • display frequency data in bar graphs. • comment upon their predictions.

Skills • throwing a die • recording • decision-making • graphing

Resources

Language • chance • die • record • bar graph

• Base 10 MAB • calculator • six-sided die for each student

r o e t s Bo r e p ok u S

Memory Masters (N3.3)

Notes

Number (N3.1a, N3.3)

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Teac he r

• The focus for this unit is basic facts of multiplication with subtraction of a single-digit number.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.1, C&D3.2, C&D3.3, C&D3.4) Warm up

• Use a coin with heads and tails to introduce the idea of chance. Ask students: – How many chances do you have of tossing a coin and it coming up heads? (1 out of 2) – How many chances do you have of tossing a coin and it coming up tails? (1 out of 2) • Use the game ‘Heads and Tails’ to reinforce the idea of chance. – All students stand. – Teacher has a coin ready to toss. – Students put their hands on their heads if they think the coin will come up heads, or on their bottoms if they think the coin will come up tails. – Toss the coin and call out which side the coin landed on. – Students who selected correctly can remain standing for another turn.Those who picked the incorrectly must sit down. – Keep playing until there is only one student left standing.

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What to do

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• Hand out a die to each student. Ask: – How many chances do you have of rolling a one? (1 in 6) How do you know this? – How many chances do you have of rolling a six? (1 in 6) • Students can record their answer to this in the space provided. • Have students consider how many times they think they would roll a six out of sixty throws. Discuss with a partner and support their ideas clearly. • Direct students to complete the graph by throwing the die sixty times. Students should ensure they mark each throw as it occurs to keep an accurate record of the event. • Once students have completed their sixty throws, allow opportunity for students to discuss their findings in small groups. How close were they to their original idea of what would happen?

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Challenge • Ask students what they think would happen to the results if they had 120 throws of the die. • Allow students time to complete and record 120 throws.This will need to be recorded on a separate sheet of paper and students may like to work in pairs for this activity. • Once students have completed their 120 throws, allow them the opportunity to discuss the results and compare them to their results from sixty throws. Was there any difference? For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 106 – 107 and 118 – 119. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 131 •

Unit 28–3

Student page 84

Indicators

Outcomes

The student is able to: • describe and record simple fractional equivalences in words or diagrams. • place unit fractions in order and explain the order either in objects, diagrams or words.

N3.3, N3.1a, N3.1b

Skills • ordering fractions • identifying equivalent fractions

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • coloured rods • toothpicks

Language • diagram • equal • not equal • fraction • greater • triangle

r o e t s Bo Notes r e p ok u S

Number (N3.1a, N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1b) Warm up

• Arrange the class into small groups. • Distribute coloured rods to each group. • Allow students time to have free play with the rods. • Show the students the 10-cm rod and explain to them that this will represent one whole and the rest of the rods will represent fractions of one whole. • Ask: —Which rod represents 1/2? (Yellow) —How do you know? Direct the students to place two of this rod below the one rod to show two halves equal to one whole. • Repeat this process for 1/3, 1/4, 1/5, 1/6, 1/8 and 1/10 to make a fraction grid.

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Teac he r

• The focus for this unit is multiplication of basic facts with subtraction of a single-digit number.

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What to do

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• Direct students to the fraction grid in their workbook and ask them to look at the activities in Exercise 3. • Using the fraction grid, show which are equal and which are not equal. • Complete Exercise 4, using the fraction grid to determine which of the fraction pairs is the greater.

Challenge

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• Distribute toothpicks to students. • Ask them to make the arrangement shown in the workbook. • Experiment by removing two toothpicks to find an arrangement that leaves two triangles. • Either draw each arrangement tried or write a description so that it is understood by others what arrangements were attempted.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 50 – 51. • 132 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 28—Answers

Student pages 82 – 84

Unit 28–1

1. (a) 0 (b) 2 (c) 24 (d) 0 (e) 20 (f) 0 (g) 0 (h) 15 (i) 15 (j) 15 2. (a) 2.2 kg (b) 5.2 m (c) 1.6 km (d) 4.3 cm (e) 6.2 kg (f) 4.1 L 3. One chance in six of throwing a six. Teacher check 4. Answers will vary 5. In the short term, anything can happen. In the long term, expect 1 in 6.

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Teac he r

1. (a) 18 (b) 23 (c) 1 (d) 15 (e) 12 (f) 6 (g) 15 (h) 10 (i) 7 (j) 0 2. (a) 3.3 m (b) 2.3 m (c) 5.3 m (d) 2.2 m (e) 4.2 m (f) 5.3 m 3. (a) 90 (b) 160 (c) 120 (d) 170 (e) 150 closest to 110 — (c) 4. (a) 210 (b) 190 (c) 100 (d) 210 (e) 130 more than 150 — (a), (b), (d) 5. (a) 1600 (b) 1500 (c) 2100 (d) 1300 (e) 2100 closest to 2000 — (c) and (e) Challenge 1 9 6 17 5 7 2 4 8 3

Unit 28–2

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Consolidation 28–2

• Complete a similar activity using a 10-sided die.

Consolidation 28–3

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4. (a) 1/2

(d) 1/6

(g) 1/8

(b) 3/6

(e) 5/8

(h) 4/5

2 (c) /4 Challenge

(f) 1/3

(i) 5/8

• Provide further opportunities for students to practise rounding techniques to estimate answers for addition problems.

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1. (a) 10 (b) 3 (c) 10 (d) 6 (e) 3 (f) 8 (g) 21 (h) 0 (i) 36 (j) 8 2. (a) $2.00 (b) $1.00 (c) $6.00 (d) $2.00 (e) $2.00 (f) $5.00 3. (a) ≠ (e) ≠ (b) = (f) = (c) = (g) ≠ (d) = (h) =

R.I.C. Publications® www.ricpublications.com.au

• Display diagrams of equivalent fractions on a chart for students to refer to.

New Wave Maths Book D – Teachers Guide • 133 •

Unit 29–1

Student page 85

Outcomes

Indicators

N3.3, N3.1a, S3.2, S3.4

Skills • drawing four-sided shapes • analysing shapes • observing

Teac he r

Memory Masters (N3.3)

The student is able to: • match standard geometric models with realistic drawings and conventional diagrams. • integrate conventional names of shapes and component parts of shapes into their descriptions of things.

Resources • Base 10 MAB • calculator • ruler • pencil

Language • different • four-sided shapes • same

r o e t s Bo r e p ok u S

Notes

• The focus for this unit is multiplication of basic facts with subtraction of a single-digit number.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.2, S3.4) Warm up

• This activity may be completed individually or in small groups. • Hold a general discussion about four-sided shapes. Ask questions such as: —Are all sides equal? —Are some sides equal? —What do we call shapes with all sides equal and parallel? —What do we call shapes with opposite sides equal and parallel? —Do you know the names of other four-sided shapes? —What is the one name we use to include all four-sided shapes? (quadrilateral)

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Number (N3.1a, N3.3)

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• Students draw as many different four-sided shapes as they can, using the space in the workbook. • Suggest they use a ruler to get straight lines. • After allowing a reasonable drawing time, direct the students to examine all their shapes carefully. • Ask the two questions in the workbook: —What is the same about the shapes? —What is different about the shapes? • Encourage a variety of responses. • Ask students to complete the answers to the questions in their workbook.

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Challenge

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What to do

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• Ask students how many different types of triangles they can make. • After allowing a reasonable drawing time, direct the students to examine all their triangles carefully. • Ask the same two questions in the workbook: —What is the same about the shapes? —What is different about the shapes? • Encourage a variety of responses.

• 134 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 29–2

Student page 86

Outcomes

Indicators

N3.3, N3.1a

The student is able to: • use the decimal point in representing quantities and money. • mentally estimate the results of a calculation in order to check the reasonableness of calculator results. • estimate sums and products by rounding to single-digit numbers.

Skills • estimating • subtracting • comparing

Resources

Language • difference between • decimal • measurements • estimate • subtraction

• Base 10 MAB • calculator

r o e t s Bo r e p ok u S

Memory Masters (N3.3)

Notes

Number (N3.1a, N3.3)

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Teac he r

• The focus for this unit is basic facts of multiplication with addition of a single-digit number.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a, N3.3) Warm up

• Write a number with two decimal places on the blackboard or whiteboard. • Ask students to round it to the nearest whole number. • Repeat with a smaller number with two decimal places. • Ask students what the difference between the two rounded numbers is. • Ask students to find the difference between the two original numbers either mentally, in written form or using a calculator. • Explain to students that rounding assists in making an estimate.

What to do

Challenge

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• Direct students to work through Exercise 3(a) only and wait for a whole-class check of working out and the answer. • Direct the class to continue, focus assistance on those having difficulties. Suggest they use the number in the ones column only to make their estimate and to use the calculator to find the difference.

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• Explain to the class that all subtraction answers can be checked by adding, but don’t tell them how. • Ask students to look at the work they have completed in Exercise 3(a) – (j) and see if they can work out how to check the accuracy of their answers by adding. • Calculator use should be encouraged as should working in small groups. • Share student findings.

R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 135 •

Unit 29–3

Student page 87

Indicators

Outcomes

The student is able to: • use a uniform unit of length carefully to make their own graduated scale to measure things in their environment. • use the decimal point in representing quantities and money. • find the perimeter of a polygon by measuring each side and adding the lengths.

N3.3, N3.1a, M3.2, M3.4a

Skills • measuring • using a ruler • converting measures

Teac he r

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • ruler

Language • length • centimetres • metres • measure • polygons • perimeters • standard • accurate

r o e t s Bo Notes r e p ok u S

• The focus for this unit is basic facts of multiplication with addition of a single-digit addend.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2, N3.1a, M3.4a) Warm up

• Remind the students that when they are using their ruler for measuring, the 0 mark, not the end of the ruler is the point to be measured from.

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• Ask students to place their ruler on the workbook ready for measuring in Exercise 3. Check or ask a partner to check that the ruler is placed correctly. • What is the length of the page? Record this in the workbook to the nearest centimetre, then in metres. • When measuring their desk, what will they need to do? (Place their ruler in the correct position and carefully mark the 30 cm point until the length of the desk has been measured.) • Measure the length of the desk and record it as centimetres and metres. • In Exercise 4, students use their ruler to measure the perimeters of the four shapes in the workbook. • Record perimeters as centimetres and as metres. • Can anyone think of another way to measure the perimeters? Accept all reasonable answers and suggest that students attempt to use them to see if they are simpler.

Challenge

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What to do

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Number (N3.1a, N3.3)

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• In pairs or small groups, students could discuss and then try out a method(s) to measure around their desk without a standard measuring device. • Check the method(s) used, using a ruler or tape measure to see how accurate they were.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 101 – 102. • 136 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 29—Answers

Student pages 85 – 87

Unit 29–1

1. (a) 21 (b) 34 (c) 4 (d) 15 (e) 30 (f) 23 (g) 35 (h) 25 (i) 20 (j) 3 2. (a) 7.1 km (b) 5.6 km (c) 7.3 km (d) 8.1 km (e) 4.3 km (f) 8.5 km 3. Teacher check estimates (a) 4.23 m (f) 4.31 m (b) 4.33 m (g) 4.41 m (c) 3.22 m (h) 5.31 m (d) 4.9 m (i) 2.8 m (e) 1.7 m (j) 1.8 m Challenge Teacher check

r o e t s Bo r e p ok u S

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Teac he r

1. (a) 4 (b) 24 (c) 0 (d) 41 (e) 30 (f) 2 (g) 12 (h) 1 (i) 1 (j) 7 2. (a) 8.5 m (b) 7.4 m (c) 8.1 m (d) 8.3 m (e) 6.1 m (f) 7.2 m 3. Teacher check (a) 4 sides 4 angles (b) length of sides size of angles Challenge 6—equilateral, isosceles, scalene, right-angled, acute and obtuse

Unit 29–2

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• Students could re-create the quadrilaterals they made on coloured card and make a display on a chart or table.

Consolidation 29–2

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1. (a) 37 (b) 8 (c) 26 (d) 20 (e) 44 (f) 34 (g) 13 (h) 39 (i) 8 (j) 27 2. (a) 55.9 m (b) 68.9 m (c) 44.6 m (d) 54.6 m (e) 76.9 m (f) 74.8 m 3. (a) 29 cm and 0.29 m (b) Teacher check 4. (a) 16 cm (c) 12 cm 0.16 m 0.12 m (b) 9 cm (d) 13 cm 0.09 m 0.13 m Challenge Teacher check

• Provide further opportunities for estimating decimal subtraction problems, prior to using a calculator or a written calculation.

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R.I.C. Publications® www.ricpublications.com.au

Consolidation 29–3

• Draw 2-D shapes on 1-cm grid paper. Count the distance around the squares or measure with a ruler and write the perimeter in centimetres and metres.

New Wave Maths Book D – Teachers Guide • 137 •

Unit 30–1

Student page 88

Outcomes

Indicators

N3.3, N3.1a

The student is able to: • remember basic addition facts and many multiplication facts and calculate mentally basic multiplication facts they don’t recall. • partition two-digit numbers to assist in adding and subtracting them mentally. • use their own methods or a conventional algorithm to multiply whole numbers by single-digit numbers.

Skills • halving • analysing • calculating • multiplying

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • toothpicks

Language • halve • difference • calculator • multiply

r o e t s Bo r e p ok u S

Notes

Number (N3.1a, N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3) Warm up

• Take a sheet of paper, hold it up in front of the class, then cut or tear it in half. • Ask the class to explain to you what you did. Focus on answers that involving halving. • Ask students to explain how you find half of a number, when you can’t physically cut it in half. • Lead to dividing by 2. Give examples to confirm understanding; e.g. half of 6, 10, 18, 30. • Ask, ‘Can odd numbers be divided in half?’ (Yes, but the answer is not a whole number.)

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Teac he r

• The focus for this unit is basic facts of multiplication with addition of a single-digit number.

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• Direct students to complete Exercise 3. • Ask if anyone has been able to find the difference between the answers in the first column and those in the second column. (The difference is odd numbers in the first column and even numbers in the second.) • Ask students to take out their calculator and enter 6 followed by x followed by 100 followed by = into their calculator. ‘What is the answer?’ (600) Record the answer in the workbook. • Repeat this for each example. ‘What did you notice in each case?’ (All answers finished with two zeros or the original number was changed to hundreds.)

Challenge

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What to do

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• Ask students to make up the diagram shown with toothpicks. • Experiment by removing toothpicks, so when four are removed, there are four squares left. • Students leave the final display on the desk to be checked.

• 138 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 30–2

Student page 89

Outcomes

Indicators

The student is able to: • read frequency data carefully using simple formats based on tallies or organised lists. • summarise data based on tallying. • use diagrams such as Venn diagrams and two-way tables to represent a two-way classification. • interpret straightforward one- and two-way tables. • understand the terms ‘multiple’, ‘factor’ and ‘prime’.

N3.3, N3.1a, C&D3.2, C&D3.3, C&D3.4, N3.2

Skills • tallying • recording • organising numbers • discovering patterns

Resources

Language

• Base 10 MAB • calculator • 25 counters – one colour for each group • 25 counters – another colour for each group • ice-cream container

r o e t s Bo r e p ok u S

• equal number • tally • total • Carroll diagram • patterns • multiples • add • equal • even • common multiples

Memory Masters (N3.3)

Notes

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Teac he r

• The focus for this unit is basic facts of multiplication with addition of a single-digit number.

Number (N3.1a, N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.2, C&D3.3, C&D3.4, N3.2) Warm up

• Ask the class to work out the minimum number of socks they would need to take from a drawer if they were in total darkness, to be sure they had a pair. There are five socks in the drawer, three of one colour and two of another colour. • The answer is 3. • Arrange the class into groups for this activity.

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Challenge

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• Distribute 50 counters, 25 of each of two different colours to each group. Counters are to be placed in an ice-cream container. • Ask students to take out 30 counters, without looking into the container. How many of each colour did they expect to have? How many of each colour did they pick out? • Record these results in the workbook. • Were the results close to what students expected? • After each person has had a turn, check if anyone has results close to expected. • Repeat this activity several times, either on this day or subsequent days. Compare results. • Revise the term ‘multiples’ with the class. • Complete Exercise 4 as directed. • Answer the questions in each group. • Share results of each group with the whole class.

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• Students are to experiment to see if they can find a way of joining the dots as directed.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 110 – 111. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 139 •

Unit 30–3

Student page 90

Indicators

Outcomes N3.3, N3.1a, N3.2, N3.4

The student is able to: • understand the terms ‘multiple’, ‘factor’ and ‘prime’ and use them appropriately. • identify patterns in the multiplication tables and use to make predictions. • distinguish and order whole numbers.

Skills • following directions • ordering • counting

Teac he r

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • coloured pencils

Language • eighth • twenty-first • thirty-fifth • fiftieth • multiples • sets • smallest • largest • digit

r o e t s Bo Notes r e p ok u S

• The focus for this unit is multiplication of basic facts with addition of a single-digit number.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.2, N3.4, N3.1a) Warm up

• Ask the class to line up in one line, or as they are seated, ask them to count in order—first, second, third etc. • Ask who was fifth, ninth, second, sixteenth.

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• Direct students to colour the squares on the page in Exercise 3. • To direct attention to multiples of 3, 4 and 5 have the class count by threes, fours and fives. • Students mark the multiples of 3, 4 and 5 on the square trail as directed in Exercise 4. • Ask students: – Which, if any squares, are marked by all three multiples? Are there any possibilities? Try adding two more squares and continue the pattern. – Which squares are common multiples of 3 and 4, 3 and 5, 5 and 4? • Before commencing Exercise 5 revise arranging numbers in order. – When arranging numbers in order we look at the first digit (the one on the left)—the digit in the highest place value. (In these examples, the digit in the hundreds place.) – When ordering from smallest to largest, the digit that has the lowest value indicates the smallest number. – If all first digits are the same, then the digit in the second place (in these examples, the digit in the tens place) may indicate the smallest number. • Complete Exercise 5, ordering from smallest to largest.

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What to do

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Number (N3.1a, N3.3)

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Challenge • Students should be able to use the strategy of marking the multiples of each of the three numbers given, following on from Exercise 4. • Once started it will become apparent that every second multiple of two is a multiple of four, and every second multiple of three is a multiple of two.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 72 – 73 and 34 – 35. • 140 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 30—Answers

Student pages 88 – 90

Unit 30–1

1. (a) 27 (b) 18 (c) 1 (d) 51 (e) 13 (f) 22 (g) 21 (h) 19 (i) 23 (j) 17 2. (a) 5.26 m (b) $4.15 (c) 3.62 L (d) 5.21 km (e) 3.32 kg (f) 3.43 m 3. Answers will vary 4. Multiples of four Multiples of three

r o e t s Bo r e p ok u S odd

none

3, 9, 15, 21, 27, 33, 39

even

4, 8, 12, 16, 20, 24, 28, 32, 36, 40

6, 12, 18, 24 30, 36

5. (a) All multiples of four are even; even multiples of three are also multiples of six. (b) 4 is even (c) Yes (d) 3 (e) 12, 24, 36 Challenge Not possible

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Teac he r

1. (a) 41 (b) 12 (c) 37 (d) 21 (e) 36 (f) 27 (g) 27 (h) 11 (i) 46 (j) 4 2. (a) $3.20 (b) $6.10 (c) $1.60 (d) $2.20 (e) $3.30 (f) $3.20 3. (a) 1 2 Odd numbers in the first column. (b) 3 4 Even numbers in the second column. (c) 5 6 (d) 7 8 (e) 9 10 (f) 21 12 (g) 33 20 (h) 41 22 4. (a) 600 (d) 400 (b) 700 (d) 900 (c) 200 (d) 800 (d) 300 (d) 500 Each answer is 100 times larger. Challenge

Unit 30–2

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• Practise halving even numbers from 1 – 100 mentally.

Consolidation 30–2

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1. (a) 42 (b) 6 (c) 30 (d) 10 (e) 43 (f) 15 (g) 25 (h) 17 (i) 39 (j) 43 2. (a) 2.7 (b) 1.8 (c) 6.3 (d) 5.7 (e) 3.9 (f) 3.4 3. & 4.

• Complete a Carroll diagram using multiples of two and three instead of four and three.

Consolidation 30–3

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• Mark in multiples of two, three and four on a 1 – 100 grid. • Practise ordering further sets of three-digit numbers in order.

5. (a) 365 593 824 (b) 299 510 752 (c) 128 413 463 (d) 296 832 884 (e) 941 969 983 Challenge 12, 24, 36, 48, 60, 72, 84, 96 2, 3 and 4 have a lowest common multiple of 12. The others are multiples of 12. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 141 •

Unit 31–1

Student page 91

Outcomes

Indicators The student is able to: • attempt to provide a bird’s-eye view of familiar locations such as their classroom. • draw informal maps and plans which show a sense of scale, that is, look ‘roughly right’.

N3.1a, N3.3, S3.1, M3.4b

Skills • mapping • planning • drawing

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • ruler • pencil

Language • bird’s-eye view • plan

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Notes

Number (N3.1a, N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.1, M3.4b) Warm up

• Work together as a class to draw a plan of the classroom. Explain to the students they need to imagine they are a fly on the ceiling of the room. Draw the plan of the classroom on the blackboard, involving students through discussion about position and size of furniture. Point out to the students that the teacher’s desk is larger than theirs, so it should be drawn to show this on the plan. • Ask students to close their eyes and visualise their bedroom as if they were a fly on the ceiling. — What shape is your bedroom? — Where are the windows and doors in your bedroom? — Where is the bed in your room? — What other furniture do you have in your bedroom?

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Teac he r

• The focus for this unit is multiplication of basic facts with subtraction of a single-digit number.

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What to do

• Once students have a clear image in their mind about the layout of their bedroom, they can begin to draw the plan in the space provided on the page. • It is best if students draw the plan lightly in pencil to begin with and complete the detail once they are happy with the accuracy of the diagram.

Challenge

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• Some students may feel more comfortable using real materials to solve this problem, while others will able to visualise and work out the problem using written or mental skills. Students should be given the opportunity to solve the problem using their own techniques. • Provided students end up with only three litres of water, all methods of solving the problem should be praised. • It is important at the end of the session to encourage discussion, focused on how students solved the problem. Those students who have difficulty with these types of problems will appreciate the insight into how other students found easier ways to solve the problem.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 2 – 3. • 142 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 31–2

Student page 92

Outcomes

Indicators

N3.1a, N3.3

The student is able to: • remember basic addition facts and many multiplication facts and calculate mentally basic multiplication facts they don’t recall.

Skills • completing number sentences • multiplying

Resources

Language • number facts

• Base 10 MAB • calculator

r o e t s Bo r e p ok u S

Notes

Memory Masters (N3.3)

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Teac he r

• The focus for this unit is basic facts of multiplication with subtraction of a single-digit number.

Number (N3.1a, N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3) Warm up

• The activities on this page are aimed to establish the commutative property of multiplication in the student’s minds. When working the basic facts, the order does not affect the final answer. • Point out to students that knowing this means they have, in effect, less basic facts to learn. If they know 4 x 2 then they also know 2 x 4. • Provide students with a series of examples to reinforce this point: —3 x 4 = 4 x = —2 x 5 = x2= —2 x 6 = 6 x = —4 x 5 = x4=

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What to do

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• Students complete Exercises 3 and 4 in their workbook. Those who have difficulty with their basic facts may be encouraged to use their calculator or a reference chart to find the final answer.

Challenge

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• Questions like these allow students the opportunity to think laterally and you will probably be surprised with the broad range of answers students will devise. • It is important to note that some students will complete the activity with basic answers for the solution, while other students will work at a more complex level. All answers put forward should be praised, provided their answer is two. • Some students may wish to write more than five different ways and should be allowed to continue as long as they offer different solutions.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 60 – 61. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 143 •

Unit 31–3

Student page 93

Indicators

Resources

The student is able to: • choose to make numerical measurements of objects to order the objects. • make sensible numerical estimates based on provided units. • use the result of measuring with a physically-present unit to try to improve their estimates with successive objects.

• Base 10 MAB • calculator • balance or kitchen scales • drawing pins, stones, 20-mm cubes, 10mm cubes, leaves, chalk, erasers, pegs, paper clips, counters, pencils

Outcomes N3.3, N3.1a, M3.2, M3.3

Skills • measuring • estimating • recording • using a balance scale

Memory Masters (N3.3)

Language • balance • scales • measure • record • mass • estimate • weigh • heaviest • lightest

r o e t s Bo Notes r e p ok u S

Number (N3.1a, N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2, M3.3) Warm up

• Organise students into small groups, with each group provided with a kitchen scale and an assortment of the materials to be weighed. Some of the materials may need to be collected or provided by the students. • Remind students to treat the scales with care. • Direct the students to estimate the mass of a cupful of each object and to record their estimate before they weigh the cup of objects. • Suggest to students they determine within their group what a cupful means and to follow this guide when filling their cup for measuring.

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Teac he r

• The focus for this unit is basic facts of multiplication with subtraction of a single-digit number.

What to do

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• Work through one example; e.g. 20-mm cubes with the whole class, showing how to estimate and measure and how to record their results. • Students work through the weighing activity recording their results. • Students, in their group, are to answer the three questions from their records.

Challenge

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• This activity can be completed individually. • Students choose one of the objects used in Exercise 3 to estimate and measure how many will be needed to balance with his/her lunchbox.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 96 – 97. • 144 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 31—Answers

Student pages 91 – 93

Unit 31–1

1. (a) 16 (b) 10 (c) 12 (d) 13 (e) 16 (f) 2 (g) 1 (h) 5 (i) 22 (j) 30 2. (a) 78.7 L (b) 75.4 L (c) 78.2 L (d) 58.6 L (e) 59.6 L (f) 79.7 L 3. (a) 3 = 21 (g) 5 = 45 (m) 8 = 48 (b) 4 = 32 (h) 7 = 63 (n) 9 = 54 (c) 6 = 36 (i) 0 = 0 (o) 9 = 72 (d) 7 = 14 (j) 7 = 28 (p) 4 = 8 (e) 3 = 30 (k) 5 = 10 (q) 10 = 50 (f) 9 = 72 (l) 8 = 56 (r) 0 = 0 4. (a) 6 = 24 (e) 9 = 36 (b) 2 = 6 (f) 8 = 16 (c) 3 = 27 (g) 0 = 0 (d) 7 = 42 (h) 5 = 15 Challenge Teacher check; possible solutions: 2, 1 + 1, 4 – 2, 4 ÷ 2, 2 x 1

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1. (a) 4 (b) 1 (c) 15 (d) 4 (e) 2 (f) 2 (g) 17 (h) 33 (i) 40 (j) 11 2. (a) 64.2 m (b) 65.2 m (c) 67.2 m (d) 74.7 m (e) 74.2 m (f) 56.1 m 3. Teacher check Challenge Pour from the 5 L container into the 2 L container. The remainder will be 3 L.

Unit 31–2

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• Students plan ‘a bedroom of their dreams’—one they would love to have if they had unlimited space and money and no parental control. Display completed plans.

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1. (a) 5 (b) 8 (c) 12 (d) 3 (e) 0 (f) 23 (g) 31 (h) 1 (i) 21 (j) 3 2. (a) 8.65 m (b) 5.76 m (c) 7.38 m (d) 5.57 m (e) 8.98 m (f) 6.88 m 3. Teacher check Challenge Teacher check

Consolidation 31–2

• Provide further opportunities to complete problems involving the commutative property of multiplication.

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Consolidation 31–3

• Use pairs of the objects listed in Exercise 3 to balance with each other. Estimate the amount before balancing.

New Wave Maths Book D – Teachers Guide • 145 •

Unit 32–1

Student page 94

Outcomes

Indicators

N3.3, N3.1a, S3.3, C&D3.3

Resources

Skills • identifying symmetry • classifying • recording

Memory Masters (N3.3)

• axes • symmetry • horizontal, vertical • top, bottom • halves • Venn diagram

• Base 10 MAB • calculator • mirror/mira • pencil • ruler

The student is able to: • informally describe the symmetry of a figure or arrangement. • use diagrams such as Venn diagrams and two-way tables to represent a two-way classification.

Language

r o e t s Bo r e p ok u S

Notes

Number (N3.1a, N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.3, C&D3.3) Warm up

• Draw the letter A as shown, on the blackboard/whiteboard. Ask a student to show any or all axes of symmetry. • Explain to students that there is only one axis of symmetry and it divides the letter A into left and right halves. This axis of symmetry is called a vertical axis of symmetry. • Draw the letter E on the blackboard/whiteboard. Ask a student to show any or all axes of symmetry. • Explain to students that there is only one axis of symmetry and it divides the letter E into top and bottom halves. This axis of symmetry is called a horizontal axis of symmetry. • Draw the letter H on the blackboard/whiteboard. Ask a student to show any or all axes of symmetry. • Explain to students that there are two axes of symmetry—one horizontal axis and one vertical.

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Teac he r

• The focus for this unit is basic facts of multiplication with subtraction of a single-digit number.

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What to do

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• All the letters of the alphabet are shown. Students are to find those that have only a vertical or a horizontal axis of symmetry and those that have both. Some letters will not have any axis of symmetry. • When students have found all the axes of symmetry, the letters are to be recorded in the correct place on the Venn diagram. Those with only one axis of symmetry are placed in the circle segments marked ‘vertical’ or ‘horizontal’. Those with both vertical and horizontal axes of symmetry are placed in the overlapping segment of the circles and those with no axis of symmetry are placed in the surrounding rectangle.

Challenge

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• Ask students what they think rotational symmetry means. (A point or a line about which a shape may be rotated. For example; 1. 2. 3. 4.

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• Ask, ‘Which letters have rotational symmetry?’. For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 110 – 111. • 146 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 32–2

Student page 95

Outcomes

Indicators

N3.3, N3.1a, C&D3.1, C&D3.2, C&D3.3, C&D3.4

The student is able to: • justify their choice of more or less likely by referring to past experience or known information. • record frequency data carefully using simple formats based on tallies or organised lists and take care with their measurements. • display frequency data in bar graphs. • comment upon their predictions in light of the results of their own data collection.

Skills • throwing dice • tallying • recording data • analysing information

Resources

Language • record • table • tally sheet • graph • most often • least often

• Base 10 MAB • calculator • 2 dice (for each student or small group)

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Notes

Memory Masters (N3.3) Number (N3.1a, N3.3)

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Teac he r

• The focus for this unit is basic facts of multiplication with subtraction of a single-digit number.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.1, C&D3.2, C&D3.3, C&D3.4) Warm up

• Distribute two dice to individual students or to small groups of students. • Ask the students what answers they obtain if they multiply the numbers shown on the dice when they are rolled.

Challenge

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• Tell the students they will be rolling the dice 100 times each, multiplying the numbers shown on the dice and recording, using a tally on the tally sheet in their workbook. • If working in groups, each person takes turns in throwing the dice. • Once 100 throws have been made and recorded, write the total each number was rolled under the total column. • Use this information to complete the graph provided. Note that the graph is in increments of five—students will need to show the number thrown by making the graph represent the numbers between those shown when the total is not a multiple of five. • When students answer the questions, encourage them to share their reasoning with the rest of the class.

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• Direct students to roll two dice 72 times and add the numbers. • Keep a record of the results. • Ask students to find which number(s) came up the most often. • Discuss why this was so and share reasoning with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 118 – 119. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 147 •

Unit 32–3

Student page 96

Indicators

Outcomes N3.1a, N3.3, N3.2

The student is able to: • use their own methods or a conventional algorithm to divide a whole number by a one-digit number. • select an appropriate division to deal with sharing and grouping situations. • produce and use standard partitions of two- and three-digit numbers.

Skills • sharing • dividing • converting Base 10 MAB

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator

Number (N3.1a, N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3, N3.2, N3.1a)

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Teac he r

• share • lots of • least • totals • smallest

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• The focus for this unit is the commutative property of addition (order does not affect the result) by completing number sentences.

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• Organise the class into small groups. • Distribute Base 10 MAB to the groups and allow students a few minutes free play. • Ask students to take out Base 10 MAB to show 20.Then ask students to share this between two people. How much did each person receive? • Take out Base 10 MAB to show 15. Ask students to share this between three people. What do you need to do with the Base 10 MAB? (Exchange.) Now can you share the Base 10 MAB? How much did each person receive? • Repeat this for 25 shared among five people; 18 shared among three people; and 24 shared among 6 people.

What to do

• Ask students to work through Exercise 3. Assist as required. • Remind students to talk through their working as this helps commit the process to memory and also allows a quick check by you. • For Exercise 4, ask students to take out Base 10 MAB to represent 26 tens and 45 ones.; i.e. 260 + 45 = 305. If the students have not already exchanged the wood for its smallest representation, ask if it is possible to show the two lots of Base 10 MAB with as few pieces. • Students are to combine the Base 10 MAB to find a total and show this total with the least number of Base 10 MAB possible. (3 flats and 5 ones) • Repeat for the rest of Exercise 4.

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Warm up

Language

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Challenge • Provide students with play money and ask them to make up $87. • Ask students to check to see if this is the least number of notes and coins they need. If not, ask them to exchange notes and coins as required to make $87 with the least number of notes and coins.

• 148 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 32—Answers

Student pages 94 – 96

Unit 32–1 1. (a) 7 (b) 0 (c) 3 (d) 0 (e) 0 (f) 1 (g) 9 (h) 29 (i) 12 (j) 11 2. (a) 34.5 L (b) 71.8 L (c) 42.6 L (d) 63.6 L (e) 31.7 L (f) 43.3 L 3.Not symmetrical B F G J L N P Q

Vertical A T

M U V

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1. (a) 11 (b) 3 (c) 0 (d) 4 (e) 26 (f) 3 (g) 34 (h) 19 (i) 1 (j) 9 2. (a) 3.59 cm (b) 5.14 m (c) 4.24 kg (d) 5.18 km (e) 3.35 L (f) 4.18 kg 3. Answers will vary (a) Most likely a 6 or a 12. More chances ( 4/36 ) of rolling these.

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Horizontal

Challenge Rotational symmetry: H, I, N, O, S, X, Z

(b) 1, 9, 16, 25 or 36. Less chance ( 1/36 ) of rolling these. Challenge Answers will vary

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Unit 32–2

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• Choose a different font for the upper case letters of the alphabet. Students sort according to vertical or horizontal symmetry.

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1. (a) 3 (b) 6 (c) 9 (d) 8 (e) 5 (f) 4 (g) 8 (h) 6 (i) 5 (j) 1 2. (a) $2.55 (b) $2.38 (c) $2.44 (d) $2.37 (e) $2.38 (f) $7.18 3. (a) 8c each (b) $7 each (c) $6 each (d) $4 each (e) $9 each (f) 6 (g) 8 (h) 8 (i) 7 (j) 5 4. (a) 260 + 45 = 305 3 flats + 5 ones = 8 pieces (b) 360 + 25 = 385 3 flats + 8 tens + 5 ones =16 pieces (c) 280 + 44 = 324 3 flats + 2 tens + 4 ones = 9 pieces (d) 740 + 63 = 803 8 flats + 3 ones = 11 pieces Challenge 5 ($50, $20, $10, $5, $2)

Consolidation 32–2

• Record the results of the ‘Challenge’ in a bar graph.

Consolidation 32–3

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R.I.C. Publications® www.ricpublications.com.au

• Continue to practise trading Base 10 MAB for fewer pieces and using it to assist in sharing amounts of money.

New Wave Maths Book D – Teachers Guide • 149 •

Unit 33–1

Student page 97

Outcomes

Indicators

N3.3, N3.1a, S3.3

The student is able to: • use multiple copies of figures to create patterns based on systematic movements of the shape and informally describe the movement used.

Skills • describing • identifying positional change

• Base 10 MAB • calculator • ruler • pencil • large square card

Number (N3.1a, N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.3)

Notes

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• Use a large square of card, plastic or other rigid material with a mark or pattern in one corner; e.g. . • Place the square on a flat vertical surface where all students are able to see it. • Slide the square to the left—ask what positional change has the square made. • Slide up, down and to the left each time asking for a description of the positional change. • Start again; this time rotate the square clockwise through 90° (1/4 turn).The term clockwise may need explaining as many students are unfamiliar with analog clocks today. • Ask students to explain what positional change the square has made. • Repeat this through three more 90° clockwise turns; each time asking for a description of the movement of the square. • Students should note that the fourth rotation brings the square back to the starting position. Its change from its original position now is none. • Do the same with a series of random rotations of 90°, 180°, 270° and 360° both clockwise and anticlockwise, eliciting an explanation from the students each time.

What to do

• positional change • shape • flip • pattern • reflect • square • slide • directions • turn • rotate • vertical • forward • backward • clockwise • anticlockwise

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• The focus for this unit is the completion of a number sentences showing the commutative property of addition.

Warm up

Language

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Teac he r

Memory Masters (N3.3)

Resources

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• Students complete Exercises 3 and 4.

Challenge

• Students are to think of 20 ways that will give an answer of 30. • Allow time for the problem to be solved. • Share the answers from the class—classify them into addition, subtraction, multiplication or division operations.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 24 – 25. • 150 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 33–2

Student page 98

Outcomes

Indicators The student is able to: • round numbers up or down or to the nearest 10 or 100 to serve a specific purpose such as estimation.

N3.3, N3.1a

Skills • rounding • using a number line

Resources

Language • patterns • rounding • tens • hundreds • number line

• Base 10 MAB • calculator • number line (see page 208)

r o e t s Bo r e p ok u S

Memory Masters (N3.3)

Notes

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Teac he r

• The focus for this unit is equality or inequality of number combinations.

Number (N3.1a, N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a) Warm up

• Draw or display a number line for the class to see (or use the one in the workbook). • Ask students individually or collectively to identify numbers shown on the number line. • Ask students to identify placement of numbers not shown on the number line. • Ask which number on the number line the number they are placing is closest to. • Repeat this several times. Explain that if we have a number and we use the nearest whole number, ten or hundred to represent the number this is called rounding. • Remind (or ask) students of the rules to follow when a number is halfway between the rounding points; e.g. 5, 50. – round down 0 – 4, 0 – 40, 0 – 400 – round up 5 – 9, 50 – 90, 500 – 900

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What to do

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• Direct attention to Exercise 3 in the workbook and ask students to round the numbers in the first activity to the nearest hundred. • Complete the second activity by rounding firstly to the nearest ten, then to the nearest hundred (use the original number in both cases—do not round from the answer to tens to hundreds).

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Challenge

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• Using the work just completed, students apply this knowledge to verify their explanation about rounding. • Attention may need to focus on money issues such as: If we always round up or always down, is this fair?

R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 151 •

Unit 33–3

Student page 99

Indicators

Outcomes N3.3, N3.1a, M3.2, M3.1, M3.3

The student is able to: • choose to make numerical measurements of objects to order the objects. • choose ‘same size’ objects to use as a repeated unit for measuring. • make sensible numerical estimates based on provided units.

Skills • balancing • weighing • estimating mass

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • balance scales • scales • 1-cm cubes • 2-cm cubes • basketball • tennis ball • squash ball • soccer ball

Teac he r

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2, M3.1, M3.3) Warm up

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Number (N3.1a, N3.3)

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• For Exercise 3, ask students to list objects they found weighed close to one kilogram. • Record in the space provided • For Exercise 4, use a set of scales to find the mass of the balls listed in the workbook. • Record results in the space provided. • For Exercise 5, use the balance scales to find how many 1-cm cubes are required to balance a 2-cm cube. • Record results. • Repeat for 2, 3, 4, 5 to 10 2-cm cubes, recording findings each time. • Describe any patterns that emerge.

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• Provide a small group of students with a balance scale, a one kilogram weight and allow them to balance objects against the one kilogram weight.

Challenge

• scales • weighing • approximately • mass • pattern • balance

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• The focus for this unit is equality or inequality of number statements, including the use of less than < and greater than >.

What to do

Language

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• Ask students to consider the question carefully. • Some students will resolve the problem immediately, while others will need to work through the problem. • Encourage students to record their approach to solving the problem.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 84 – 85 and 96 – 97. • 152 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 33—Answers

Student pages 97 – 99

Unit 33–1

1. (a) = (b) ≠ (c) ≠ (d) = (e) = (f) ≠ (g) = (h) ≠ (i) = (j) ≠ 2. (a) $6.45 (b) $5.67 (c) $6.92 (d) $5.82 (e) $8.67 (f) $6.23 3. (a) 400 (f) 700 (k) 400 (p) 800 (b) 600 (g) 300 (l) 300 (q) 300 (c) 900 (h) 500 (m) 900 (r) 400 (d) 400 (i) 400 (n) 100 (s) 500 (e) 500 (j) 900 (o) 700 (t) 800 3. (a) 460, 500 (e) 840, 800 (i) 530, 500 (b) 330, 300 (f) 740, 700 (j) 610, 600 (c) 940, 900 (g) 290, 300 (k) 100, 100 (d) 650, 700 (h) 470, 500 (l) 170, 200 Challenge Teacher check

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Teac he r

1. (a) 6 (b) 7 (c) 9 (d) 4 (e) 3 (f) 6 (g) 8 (h) 4 (i) 7 (j) 4 2. (a) 6.95 kg (b) 5.65 kg (c) 6.93 kg (d) 6.86 kg (e) 7.72 kg (f) 7.86 kg 3. Note: Vocabulary used in describing the positional changes may var y. 1/ 4 turn clockwise, flip/reflect vertically, flip/reflect vertically, 1/4 turn clockwise, 1/4 turn anticlockwise and flip/reflect vertically. 4. Teacher check Challenge Teacher check

Unit 33–2

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• Create a pattern on dotted grid paper (see page 198) to flip, slide and turn.

Consolidation 33–2

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1. (a) NT (b) > (c) > (d) < (e) NT (f) < (g) > (h) < (i) NT (j) < 2. (a) 9.53 kg (b) $6.87 (c) 8.69 L (d) 9.38 km (e) 9.51 m (f) $8.28 3. Teacher check 4. Teacher check 5. Teacher check Challenge They both weigh the same—a tonne.

• Provide further opportunities for students to round numbers to the nearest ten and hundred.

Consolidation 33–3

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• Complete a similar activity to Exercise 4 but use a variety of books, shoes or full cereal packets etc.

New Wave Maths Book D – Teachers Guide • 153 •

Unit 34–1

Student page 100

Outcomes

Indicators

N3.3, N3.1a, M3.2

The student is able to: • use their own methods or a conventional algorithm to multiply whole numbers by single-digit numbers. • measure the area of squares.

Skills • problem-solving • multiplying • finding area

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • 1-cm grid paper (see page 199)

Teac he r

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3, M3.2)

Notes

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Number (N3.1a, N3.3)

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What to do

• Work through Exercise 3, using Base 10 MAB if necessary. • Supply each student with 1-cm grid paper. • Ask students to draw a square 2-cm by 2-cm on the grid paper. Ask students how many squares there are inside—4. Ask if anyone can see a relationship between the length of the two sides and the area. • Repeat by drawing squares 3-cm by 3-cm; 5-cm by 5-cm; 7-cm by 7-cm and 10-cm by 10-cm. • Ask students to work the answers to Exercise 4 either using grid paper, or from the examples given.

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• Ask students what the answer is to the following problem: 4 people paid $5 each to rent several videos. How much did they pay altogether? ($20) What operation (process) did you use to find the answer? (multiplication or repeated addition). • Explain they will be multiplying larger numbers. To help, Base 10 MAB can be used. • Organise the class into groups with sufficient Base 10 MAB for each group. • Ask the groups to show 6 lots of 18 with Base 10 MAB. Then trade to find the answer. • Show 16 lots of 21 with Base 10 MAB. Trade to find the answer. Check with a calculator. • Show 12 lots of 27 with Base 10 MAB. Trade to find the answer. Check with a calculator.

Challenge

• concrete materials • how much • total • how many • area • polygons • estimate

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• The focus for this unit is completion of open number sentences using basic facts of multiplication.

Warm up

Language

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• Students should be given ‘a free rein’ to attempt this problem. It is anticipated they will draw on their knowledge of rounding to find the estimate. However, encourage divergence of thought and give credit for the effort given.

• 154 • New Wave Maths Book D – Teachers Guide

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Unit 34–2

Student page 101

Outcomes

Indicators

N3.3, N3.1a, C&D3.1, C&D3.3

The student is able to: • describe outcomes as having an equal chance or being equally likely. • order a few easily-understood situations from least likely to most likely. • display frequency data in bar graphs where one axis shows the whole numbers.

Skills • recording • collating information • collecting data

Resources

Language • likelihood • chance • order • bar graph

• Base 10 MAB • calculator • calendar • bar graph (see page 236)

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Notes

Memory Masters (N3.3) Number (N3.1a, N3.3)

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• The focus for this unit is addition of multiples of ten less than 100.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.1, C&D3.3) Warm up

• The questions in Exercise 3 focus student attention on probability of events occurring. At this stage no record or request for birth date information has been requested. • Start the lesson with general discussion on probability by asking questions similar to: – Is it likely to rain today? Students explain their answers. – Is it likely that you will walk home this afternoon? Students explain answers—some may be unsure, delve to find out their uncertainty and try to get a probable answer.

What to do

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Challenge

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• Answer the questions in Exercise 3 before completing Exercise 4. • For Exercise 4, use an overhead projector, blackboard or computer to generate a simple bar graph. Explain the vertical axis heading and the horizontal axis heading. • Outline the procedure for recording blocks on the bar to show the number of items generated in each column. • The following questions could be asked: – How would you show a zero result? – What would you do if you have more items in a column than the vertical axis shows? (Extend the axis.) – How will we collect the information on class members’ birth dates for recording on a bar graph? (Choose the most efficient method.) • Collect and display the data. Check the results.

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• Ask the students to make a reasonable guess and give justification for their answer. • Ask students how they can check their guess. • Provide students with a pack of cards to test their guess and come to a final conclusion, explaining how they reached their conclusion.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 106 – 107. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 155 •

Unit 34–3

Student page 102

Indicators

Outcomes

The student is able to: • inspect a prism or pyramid, put it aside and then select 2-D shapes to match the faces or polyhedron. • integrate conventional names of shapes into their descriptions of things.

N3.3, N3.1a, S3.2, S3.4

Skills • recording • observing • looking for patterns

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • 3-D shapes • modelling clay • fishing line

Language • equivalent • shape • corners • edges • faces • patterns • cube • square pyramid • triangular prism • rectangular prism • hexagonal prism

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Number (N3.1a, N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.2, S3.4) Warm up

• Distribute a number of 3-D shapes to small groups in the class. • Ask students to examine the shapes, then in their groups, describe the features of the shapes. • Ask a member of each group to share some of the features they found with the whole class. • Direct student attention to the models again and ask them to count the faces. You may need to explain what a face is. • Ask them to count the edges. • Ask them to count the corners (vertices).

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• The focus for this unit is subtraction of multiples of ten less than 100.

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• Direct students to their workbook and have them count the corners, edges and faces of the shapes shown in the diagrams. Students may either use the diagrams or models of the diagrams to complete the table. • Encourage students to examine their results to see if they are able to find any patterns. They may not be able to. • If any patterns are found, or are thought to be found, have the students share their findings with the class.

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Challenge

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• Ask students to make a cube out of modelling clay. • Ask students to cut the cube in half (20-cm of fishing line tied to two pieces of dowel and pulled tight makes an effective cutting tool) to see what shapes they can make. • Ask them if there is another or way(s) they could have cut the cube in half. If alternative suggestions are given, have students find what shapes are obtained. • Encourage students to share their findings.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 28 – 29. • 156 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 34—Answers

Student pages 100 – 102

Unit 34–1

1. (a) 20 (b) 70 (c) 90 (d) 60 (e) 90 (f) 90 (g) 60 (h) 90 (i) 70 (j) 60 2. (a) $3.80 (b) $4.60 (c) $3.50 (d) $3.40 (e) $3.70 (f) $2.30 3. (a) Teacher check (b) Teacher check (c) Teacher check 4. Teacher check Challenge

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/52

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1. (a) 3 x 3 (b) 7 (c) 9 (d) 42 (e) 4 x 4, 2 x 8 (f) 7 x 7 (g) 6 x 6, 9 x 4 (h) 5 (i) 9 (j) 72 2. (a) $4.43 (b) $3.41 (c) $4.16 (d) $4.38 (e) $4.14 (f) $4.12 3. (a) $216 (b) $184 (c) $252 (d) 310 sheep (e) 208 marbles 4. 4 x 4 = 16 cm2 8 x 8 = 64 cm2 6 x 6 = 36 cm2 Challenge 1000 x 250 = 250 000 (Round 986 to 1000 and 254 to 250)

Unit 34–2

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Cube

Square Pyramid

Rectangular Prism Triangular Prism Hexagonal Prism

8 5 8 6 12

12 8 12 9 18

6 5 6 5 8

• Students could make up their own word problems involving multiplication of whole numbers less than 100 by a digit 10 or less. Swap with a classmate.

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1. (a) 10 (b) 10 (c) 60 (d) 30 (e) 0 (f) 0 (g) 20 (h) 40 (i) 0 (j) 10 2. (a) 5.80 m (b) 2.90 km (c) 5.30 g (d) 2.60 L (e) 4.90 kg (f) 3.40 m 3. Shape Corners Edges Faces

Consolidation 34–2

• Brainstorm a list of random events to discuss, record answers and show on a bar graph.

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• Repeat the activity using the same and/or other 3-D shapes at a later date, for reinforcement.

Vertices + Faces = Edges – 2 Challenge Square, rectangle, triangular prism

R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 157 •

Unit 35–1

Student page 103

Outcomes N3.3, N3.1a, S3.2, S3.3

Skills • problem-solving • making models • concluding

Memory Masters (N3.3)

Indicators

Resources

• Base 10 MAB • calculator • modelling clay • clay cutter (fishing line)

Teac he r

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.2, S3.3)

Notes

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Number (N3.1a, N3.3)

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What to do

• Distribute modelling clay and clay cutters (fishing line) and ask students to make their own 3-D models and then cut the model to make two halves that look the same. • Ask students to draw their original model and the two halves in their workbooks. • Ask students to repeat this with at least two other 3-D models they make. • From their observations, ask students to draw a conclusion about the original model and the halves they cut—looking at whether they were correct and if any of the models are more likely to make two halves the same than others.

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• Organise the class into small groups. • Provide each group with a range of 3-D shapes to play with. • Ask students to discuss features of the models—corners (vertices), edges, faces, whether it is a pyramid or a prism, whether it slides or rolls etc. • Ask students whether they think they could cut a model in half so that both halves are exactly the same. • Ask them to sort the shapes into those they think they can and those they think they can’t cut in half, so that both halves are exactly the same.

Challenge

• three-dimensional • half • exactly • same • original • conclusion

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• The focus for this unit is multiplication of a multiple of ten less than 100 by a single-digit number.

Warm up

Language

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• Students are to select either a library book or their class reader for this activity. • Explain to the class they are to find out which letters of the alphabet occur most frequently in the book. • Direct students not to count every letter in the book, but to find a way they think is representative of the whole book. • Class discussion on how this may be done may be encouraged or set students to work with their own ideas. • Students’ explanations may be shared with the whole class. For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 30 – 31. • 158 • New Wave Maths Book D – Teachers Guide

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Unit 35–2

Student page 104

Outcomes

Indicators

Resources

The student is able to: • read and write any whole number into the thousands. • count up and down in tens from any starting number.

N3.3, N3.1a

Skills • writing whole numbers • counting • adding

Language

• Base 10 MAB • calculator • flashcards with numbers and number names • place value chart (see pages 205 – 207)

• first • third • fourth • last • tallest

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Memory Masters (N3.3)

Notes

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• The focus for this unit is subtraction of multiples of ten less than 100.

Number (N3.1a, N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a) Warm up

• Distribute flashcards with numbers and number names (also include the word ‘and’) to

• Use the number names to make up numbers; e.g.

four

hundred

781

and

1362 . twelve

Challenge

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• Direct students to Exercise 3. Students should complete the exercise individually, but assist as required. • Students who are having difficulty could make the numbers using Base 10 MAB before writing them as words. • Students can complete Exercise 4 individually, using Base 10 MAB or a place value chart to assist if necessary.

• Explain that consecutive numbers are numbers that immediately follow each other; e.g. 4 and 5; 10 and 11. • Ask students to complete the activity, finding as many whole numbers less than 10 that are the sum of two consecutive whole numbers.

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For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 34 – 35 and 38 – 39. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 159 •

Unit 35–3

Student page 105

Indicators

Outcomes

The student is able to: • use a uniform unit of length carefully to make their own graduated scale to measure things in the environment. • choose to make numerical measurements of objects to order the objects. • make sensible numerical estimates based on provided units.

N3.3, N3.1a, M3.2, M3.3

Skills • measuring • recording • estimating

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator • tape measure (30- or 50-m) • trundle wheel

Language • measure • trundle wheel • length • furthest • measurements • metres • centimetres • estimate • actual

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Number (N3.1a, N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2, M3.3) Warm up

• This activity may be completed as part of a physical education lesson. • Explain to the class that the activity will be completed outside. It is a measuring activity but should allow for some fun as well. • Organise the class into groups of six if possible. • Provide the equipment required and move to the oval or an area large enough to allow the throwing to take place. A length of 50 to 60 m may be required.

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• The focus for this unit is addition of multiples of ten less than 100.

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• Explain the activity to the students. Each person in the group will throw the softball in turn. After each throw, the length of the throw from the point of delivery to the point where the ball landed will be measured and recorded on the table in the workbook. • When all members of the group have thrown and their throws measured and recorded, students will need to make sure they have the required six throws. Groups of less than six can add a students’ throws from another group. Compare results among groups. • Write the name of the student in the table who had the longest throw. • Exercise 4 may also be completed outside, but can be just as readily finished inside. • Ensure that all students understand the meanings of the terms they are measuring, by asking for an explanation of each or telling the students. • As each turn is described, ask students to make an estimate of their own body measure. • Complete the actual measures in pairs or small groups. • Students then decide and explain the accuracy of each of the measures.

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Challenge • Students will need to experiment and try different combinations to find a way to draw three squares all of different sizes, using only eight straight lines. • Students should record and explain their attempts, to share at the completion of the exercise.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 78 – 79. • 160 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 35—Answers

Student pages 103 – 105

Unit 35–1

1. (a) 50 (b) 30 (c) 20 (d) 70 (e) 40 (f) 10 (g) 10 (h) 20 (i) 40 (j) 20 2. (a) $83.50 (b) $87.50 (c) $78.80 (d) $85.60 (e) $68.70 (f) $76.50 3. (a) Three thousand, four hundred and thirty-seven. (b) Nine thousand, nine hundred and twenty-two. (c) Five thousand and eighty-nine. (d) One thousand and six. 4. 10 more 10 less 100 more 100 less (a) 235 245 225 135 335 (b) 867 857 967 767 877 (c) 122 222 132 112 22 (d) 406 416 396 506 306 510 600 400 (e) 500 490

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1. (a) 40 (b) 100 (c) 60 (d) 80 (e) 60 (f) 90 (g) 80 (h) 80 (i) 60 (j) 100 2. (a) $56.40 (b) $98.40 (c) $35.40 (d) $66.60 (e) $59.70 (f) $74.60 3. Yes. Teacher check Challenge E will occur most often. Count a set number of letters. Tally the number of each letter. It is likely that vowels will occur most frequently and ‘e’ is the most commonly used vowel.

Unit 35–2

Challenge Answers may vary. Possible solutions are: 1 + 2 = 3 2 + 3 = 5 3 + 4 = 7 1 + 2 + 3 = 6 4 + 5 = 9 2 + 3 + 4 = 9

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• Identify and draw cross-sections of 3-D shapes on diagrams.

Consolidation 35–2

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1. (a) 40 (b) 70 (c) 70 (d) 80 (e) 80 (f) 80 (g) 80 (h) 80 (i) 90 (j) 90 2. (a) 65.59 m (b) 67.70 m (c) 55.90 m (d) 58.11 m (e) 56.60 m (f) 46.41 m 3. Teacher check 4. Teacher check Challenge

• Use the flashcards as described in the Warm up to read and write numbers in numerals and words.

Consolidation 35–3

o c . che e r o t r s super

R.I.C. Publications® www.ricpublications.com.au

• Complete a similar activity to Exercise 3 using a tennis ball or basketball.

New Wave Maths Book D – Teachers Guide • 161 •

Unit 36–1

Student page 106

Outcomes

Indicators

WM3.2, WM3.4, N3.1a, N3.2, N3.3

The student is able to: • generate mathematical questions. • decide what information in a problem needs to be represented. • make lists or tables of data. • check their answers with estimates. • read and write any whole number into the thousands. • choose appropriate operations to solve a problem.

Skills • estimating • calculating • posing questions • surveying • collecting data • analysing data • making conjectures

Resources • calculator

Language • problem • open-ended • ‘snack-pack’ • average • estimate • population • survey • standard • money

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• cost • multiply • fruit • data • tallying

Notes

Main Activity (WM3.2, WM3.4, N3.1a, N3.2, N3.3)

• This activity is designed for students working collaboratively in groups. Allow enough time so they can discuss their options and for ideas to evolve. Investigative tasks such as these are a good opportunity for students to ‘take a risk’ with maths. • When completing investigative tasks, some students may be more successful in mixed-ability groups rather than same-ability groups. • Some groups will be able to work independently while others may need guidance. The stimulus questions below may prompt such groups. – What do we need to find out to solve the problem? – How many days in a week? – What is the number of people in the school? (students, teachers, assistants, administration etc.) – What does it mean by ‘a piece of fruit’? For example, is one strawberry or one grape a piece of fruit? Is a snack-pack® considered to be a piece of fruit? – Do people eat more fruit on weekdays or weekends? • Groups may decide they need to survey a lower, middle and upper class to estimate the amount of fruit eaten by students of different ages. • Students may also wish to create survey forms for the adults to complete regarding the amount of fruit eaten in one week. • Students will be working with quite large numbers to complete the activity. Calculators are essential. For some groups, estimating and calculating with large numbers may be challenging. This task can also be made easier or extended by changing it to the amount of fruit eaten ‘by people in your class’ or ‘people in your local community’. • Groups may wish to collate and summarise their findings and present them as a poster with a series of graphs, diagrams and information. • When each group has finished the activity, compare the results. By how much do they vary? Why do students think this is? • Allow each group to discuss and evaluate its ability to problem-solve and its success as a group. A group or self-assessment form could be completed. This information will be helpful for creating groups for future open-ended, investigative tasks.

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What to do

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• 162 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 36–2

Student page 107

Outcomes

Indicators

N3.1, N3.4, N3.3, C&D3.3, C&D3.4, WM3.2

Skills • completing tables • calculating

The student is able to: • use diagrams such as Venn diagrams and two-way tables to represent a two-way classification. • explain their own data displays to their peers, talking about the features represented. • use arrays and tree diagrams to make organised counts.

Resources

Language

• Base 10 MAB • calculator

• add • diagram • tree diagram

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Memory Masters (N3.1, N3.4)

Notes

Number (N3.3)

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• The focus for this unit is completion of number patterns.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.3, C&D3.4, WM3.2) Warm up

• Discuss with students what a tree diagram is and what it looks like. Use a family tree as an example. • When tennis, squash, badminton or other knockout tournaments are organised, a tree diagram is used to show who plays who as the tournament continues. The winners of the first round go through to the second round, to pair and play each other and so on. • Draw part of a tree diagram to show students while explaining it to them. • To use a tree diagram, it works best if there are 2, 4, 8, 16, 32, 64, 128 etc. participants.

What to do

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• Explain to the students that there is a 16-member knockout competition in the workbook. Using the information given, students are to complete the diagram by filling in the missing names. • Ask students to complete Exercise 4. It is presumed that each round of matches will be played at the same time. — If each round of matches takes 35 minutes to complete, how many minutes (hours and minutes) are required for match play? — If there is a 10-minute break between each round of matches, how many minutes of break time is there? — What is the total time required to complete the competition? • Students can attempt to work this out individually or with teacher guidance.

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• Students should be directed to the tree diagram on this page of their workbook to look for possible solutions. • Allow students to explore options for themselves.

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New Wave Maths Book D – Teachers Guide • 163 •

Unit 36–3

Student page 108

Indicators

Outcomes N3.1a, N3.4, N3.3

The student is able to: • read and write any whole number into the thousands. • regroup money to the fewest number of notes or coins. • mentally estimate the results of a calculation in order to check the reasonableness of calculator results. • use their own methods or a conventional algorithm to multiply whole numbers by single-digit numbers.

Skills • reading numbers • converting money • estimating • calculating

Resources • Base 10 MAB • calculator • plastic coins and paper notes • calculator • number cards

Language • money • notes • coins • values • estimate • calculate

r o e t s Bo Notes r e p ok u S

Memory Masters (N3.1a, N3.4) Number (N3.1a, N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a, N3.3) Warm up

• Prepare various numbers, similar to those on the student page, written on flashcards. Display the numbers at random and ask students as a whole group to read the numbers.

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• Ask students to read the numbers in Exercise 3. Students can then move into pairs to complete the activity. • Any numbers students are having difficulty saying should be checked by the teacher and modelled if necessary. • Introduce the topic of money. If possible, it will help to have plastic coins and paper notes for the students to refer to. Hold up the notes and coins at random and ask the students to name them. – Ask the students if they can tell you the best way to make various monetary amounts. For example, ‘What would be the best way I could make $1.60?’ – Does it matter what notes and coins we use to make up amounts? – Why do you think it is best to use the least number of notes and coins when paying for items at the shop? • Discuss that 100 cents equals one dollar. • Read through Exercise 4 and work through the first example on the board, extracting the information from the students. • Allow the students time to complete the second example by themselves. Once they have finished, stop the class and ask several students how they used the smallest number of coins and notes. Discuss which answer is the most efficient. Students can use the plastic coins and paper notes as an aid if they choose. • Allow the students to complete the remainder of the activity. • For Exercise 5, revise how many days there are in a year and the fact that there are also leap years. They will also need to consider how long since their last birthday in order to work this problem out. Allow students to choose their own method to solve the problem—some students may choose to use a calculator.

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What to do

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• The focus for this unit is conversion of metres to centimetres and completion of patterns.

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Challenge • Students may find it easier to work with number cards, as they can be rearranged accordingly until the students are satisfied with their solution, which can then be recorded on the page. • It is important to allow students to discuss, perhaps in small groups, how they worked out the puzzle. The approach that students take will vary greatly, and while some students may struggle with a problem such as this, using those students who found the most efficient approach as a model will provide the weaker students with insight for future purposes. • 164 • New Wave Maths Book D – Teachers Guide

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Unit 36—Answers

Student pages 106 – 108

Unit 36–1 1. & 2. Answers will vary

Unit 36–2 1. (a) 4 (b) 60 (c) 17 (d) 9 (e) 17 (f) 30 (g) 22 (h) 17 (i) 500 (j) 16 2. (a) 88 (b) 133 (c) 75 (d) 146 (e) 77 (f) 197 3. Bill Sam

Bill

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Jan Sue

Chris

Tony Alan

Sarah

Julia Ian

Jan

Bill

Sue

Tony

Julia

Ian

Paul

Simon

Anne Shelley Bruce

Julia

Anne

Shelley

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Bill

4. It is presumed that each round will be played at the same time; i.e. 4 rounds of matches x 35 = 140 mins; plus 3 x 10 between matches = 170 mins or 2 hours 50 minutes. Challenge Teacher check

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Consolidation 36–2

• Discuss and compare the diagrams prepared in the challenge section.

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6 9

• Students can work in a group to estimate other openended problems such as ‘How many nappies does a baby use in one week?’ or ‘How many sheets of paper are thrown away in your classroom in one week?’.

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1. (a) 200 (b) 120 (c) 50 (d) 240 (e) 30 (f) * (g) @ (h) B (i) F (j) * 2. (a) 22 (b) 11 (c) 530 (d) 110 (e) 206 (f) 400 3. Teacher check 4. (a) $36.25 — $20, $10, $5, $1, 20c, 5c (b) $19.80 — $10, $5, $2, $2, 50c, 20c, 10c (c) $52.45 — $50, $2, 20c, 20c, 5c (d) $64.90 — $50, $10, $2, $2, 50c, 20c, 20c (e) $25.50 — $20, $5, 50c 5. Teacher check Challenge Add the numbers = 81. Divide by 9 as there are 9 numbers = 27; i.e. totals should make 27.

Consolidation 36–3

• Complete further activities, reading numbers aloud to a partner and showing the least number of coins needed for different values.

11 10

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New Wave Maths Book D – Teachers Guide • 165 •

Unit 37–1

Student page 109

Outcomes N3.1a, N3.3, S3.1, M3.4b

Skills • mapping • planning • drawing

Indicators

Resources

The student is able to: • order and show a sense of the proximity of things in locating key features on maps. • draw informal maps and plans which show a sense of scale, that is, look ‘roughly right’.

• Base 10 MAB • calculator • 1-cm grid transparency (see page 199)

Language • grid • map • route • features • number • signs

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Notes

Teac he r

Memory Masters (N3.1a)

• The focus for this unit is conversion of cents to dollars and dollars to cents.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.1, M3.4b) Warm up

• On the blackboard/whiteboard, or using an overhead projector, display a grid similar to the grid in the workbook. • Tell the class that you are going to show the route you use to come to school, or go to the shops or something similar.

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• Mark destination and departure points on the map. • Following the grid lines, explain to the students the direction, distance and changes in direction that you need to make when leaving your departure point to arrive at your destination. • Suggest to students to use a light pencil for the first draft so that adjustments can be made if required. • Ask students to mark their departure point, (home) and their destination (school) on their map, leaving sufficient space to mark the route they take between these two points. • Students then mark in their route. Direct students to note key features along the route. • Ask students to give an oral presentation to the whole class or a small group of their route.

Challenge

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What to do

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Number (N3.3)

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For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 2 – 3. • 166 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 37–2

Student page 110

Outcomes

Indicators

N3.3, N3.4

The student is able to: • use constant addition on a calculator to generate multiples of a number.

Skills

Resources

Language

• Base 10 MAB • calculator

• subtraction • zero • pattern

• calculating • subtracting • describing patterns

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Notes

Teac he r

Memory Masters (N3.3)

• The focus for this unit is doubling and halving of numbers.

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Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.4) Warm up

• Ask students to take out their calculators and experiment with them to make number patterns. • Ask for some of the students to describe what they did and what they discovered. • Ask students to press the 5 button, then the + button, then the = button six times. Ask, ‘What is the number on your screen? (30) What pattern did you see?’ (Multiples of 5) • Let students experiment again. • Share some of their results. • Ask students to press 6 and then 7 (67), then the – button, then the 5 button, then the = button and repeat from 5 on. Ask students to describe what they found.

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What to do

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• Read Exercise 3 and have students work through the examples. • Ask students why they reached zero this time but not with the example you gave above of 67 – 5. (Need to start with a multiple of the number that is being subtracted.) • Students complete Exercise 4. Note: Different models of calculator may use different keystrokes.

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Challenge

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• This activity is similar to the process above except that students are now using the constant function on the calculator. • Students may be able to explain this in such terms, but most likely as multiples of 5 and multiplying each answer by 5. • Suggest students experiment with numbers of their own choosing.

R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 167 •

Unit 37–3

Student page 111

Indicators

Outcomes N3.1a, N3.3, M3.2

The student is able to: • use a uniform unit of length carefully to make their own graduated scale to measure things in their environment. • measure time intervals using natural units of time, artificial non-standard units, timers or standard units. • display frequency data in bar graphs. • read frequencies from a bar graph.

Skills • making a model • recording results • graphing • timing

Resources • Base 10 MAB • calculator • paper • stopwatch

Language • record • results • graph • seconds • surface area • longer

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Memory Masters (N3.1a) Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2) Warm up

• Provide each student with a sheet of paper and ask them to make a paper aeroplane. • Explain that the aeroplane will be used in an exercise to time its flight.

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• Organise the class into groups with each group having a stopwatch. • Once the planes have been made, students can trial their planes outside. For best results, a large enclosed area is preferred. • Time and record each flight. Each aeroplane is to be given five trials. • Distances of flights may also be recorded if another dimension is to be added to the activity. • Analyse the flight times and try and find a way to modify the aeroplanes to keep them in the air for a longer period of time. • Remake the aeroplanes using another sheet of paper. Make a note of the changes made. • Run five more trials, timing each. • Record the results and compare them with the first trial. • Give an explanation for the results.

Challenge

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What to do

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Teac he r

• The focus for this unit is odd and even numbers.

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Pose the following challenge to the students: • Extending the information you have gained from your trials, do you think an aeroplane with more wing surface area will fly longer than one with smaller wing surface area? • Make an aeroplane using the same type of paper either with more or less wing surface area than the one you have previously used. • Test your prediction by running further trials. Remember that flight conditions must be as close to the same each time if you are to accurately compare your results—you need to control the variables, changing only one at a time.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 118 – 119. • 168 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 37—Answers

Student pages 109 – 111

Unit 37–1

1. (a) 8 (b) 14 (c) 20 (d) 12 (e) 6 (f) 30 (g) 1 (h) 7 (i) 40 (j) 12 2. (a) $8 (b) $9 (c) $9 (d) $12 (e) $13 (f) $19 3. Teacher check 4. Teacher check Challenge Addition of five each time; multiples of five. Multiplication by five each time.

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1. (a) $3.20 (b) $5.10 (c) $2.90 (d) $1.70 (e) $8.40 (f) 460c (g) 280c (h) 540c (i) 950c (j) 370c 2. (a) 42c (b) $80 (c) $150 (d) $480 (e) 300c (f) 88c 3. Answers will vary Challenge Possible answers working left to right: 3 x 3 x 3 + 3 ÷ 3 = 10 3 x 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 ÷ 3 = 10

Unit 37–2

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• Repeat the activity, choosing a different destination and route.

Consolidation 37–2

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1. (a) E (b) O (c) O (d) O (e) O (f) O (g) O (h) E (i) O (j) E 2. (a) 78 (b) 148 (c) 211 (d) 987 (e) 1456 (f) 787 3. Answers will vary 4. Answers will vary Challenge Generally, yes, but aerodynamics will vary depending on design.

• Provide further opportunities to use the constant function on a calculator to explore number patterns.

Consolidation 37–3

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• Discuss and compare findings for the ‘Challenge’ as a class.

New Wave Maths Book D – Teachers Guide • 169 •

Unit 38–1

Student page 112

Outcomes

Indicators

N3.1, N3.3, N3.4

The student is able to: • identify patterns in the multiplication tables and use to make predictions.

Skills

Resources • Base 10 MAB • calculator

Language • number • pattern

• following directions • recognising number patterns

Memory Masters (N3.1)

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Notes

Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.4) Warm up

• Ask students what ‘divisible by’ means. (A number must be a multiple of the number they are dividing by.) • Ask students to give examples of numbers that are divisible by 2, 3 and 4.

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• Ask students to look at the table in Exercise 3 and to circle the numbers that are divisible by 2. • Check work, and if satisfied, ask students to cross the numbers on the same table that are divisible by 5. • Ask the students if they can describe the numbers that are circled and crossed. (They are multiples of 10.) • This leads directly to Exercise 4 which should be able to be readily completed on the second table. • Direct students to complete the patterns on the 1 – 100 grid in Exercise 5, following the directions. • Ask students to share their findings with the class.

Challenge

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What to do

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Teac he r

• The focus for this unit is recognising the smaller numbers in a pair and place value.

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• Ask students to make a rule that will tell everyone which numbers are divisible by 3. • Ask students to check their rule on the three numbers given. • Ask students to share their rule with the class. • Ask other students whether they agree that the rule is accurate. Students should be able to justify their answers.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 72 – 73. • 170 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 38–2

Student page 113

Outcomes

Indicators

N3.1a, N3.3, C&D3.2, C&D3.3

The student is able to: • record frequency data carefully using simple formats based on tallies or organised lists and take care with their measurements. • use diagrams such as Venn diagrams and two-way tables to represent a two-way classification.

Skills • sorting • recording • making a tally

Resources

Language • sort • table • tally • total

• Base 10 MAB • calculator

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Notes

Teac he r

Memory Masters (N3.1a)

• The focus for this unit is ordinal numbers and rounding to the nearest 100.

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Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (C&D3.2, C&D3.3) Warm up

• Discuss with the class how each of us are individuals and have different characteristics. Some may be similar to others, but everything won’t be the same unless we are identical twins. • Ask students for some of the characteristics of each person that distinguishes one from another. • Ask students how we could record this information so we could show the differences, but also recognise that there are similarities between each person as well.

What to do

Challenge

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• Explain to the students that a Carroll diagram is one means of recording such information. • Collect and record the required information in the two diagrams for Exercises 3 and 4. • Ask students how to represent a tally, then have them complete Exercise 5 either by collecting the information themselves, or centrally controlling the collection by a show of hands.

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• Remind students that when solving problems, they need to record the information they are working with. • In this case, unless the data is recorded, there is little likelihood of a correct solution being reached. • Students set about their work, recording, then sharing with either small groups or the whole class. • Discuss the methods used and gain an understanding of which may be the best.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 110 – 113. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 171 •

Unit 38–3

Student page 114

Indicators

Outcomes N3.3, N3.1a

The student is able to: • use their own methods or a conventional algorithm to divide a whole number by a one-digit number. • produce and use standard partitions of two- and three-digit numbers.

Skills • problem-solving • dividing • counting

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator

Language • divide • receive • between • among • share

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Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.3, N3.1a) Warm up

• Distribute Base 10 MAB among groups of students. Allow the students time (approximately five minutes) to play with the Base 10 MAB. • Tell the class that you have $182 to be shared among seven class members. Ask them how they would solve this problem. • Using the Base 10 MAB, test their suggestion. • Check results. Remind students that sharing is division. • Using Base 10 MAB, the first step is to exchange the single hundred for 10 tens.The eighteen tens are shared among the seven students. Each student receives 4 tens and the remaining 4 tens are exchanged for ones. The 42 ones are shared among the seven students with each receiving 6 ones. Each of the 7 students now has $46 shown as 4 tens and 6 ones.

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Teac he r

• The focus for this unit is completion of open number sentences.

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What to do

• Students who understand the process should proceed Exercises 3 and 4, using Base 10 MAB, pen and paper and/or a calculator. • Students who are having difficulties should be guided through the process using Base 10 MAB to develop understanding. They may be directed to use their calculator to find the answers to the remaining sums once understanding is established.

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Challenge

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• Ask students to make a reasonable guess and give justification for their answer. • Ask students how they can check their guess. For some students, this may involve making selections from a bag containing the specific counters. Other students will use their experience with previous chance and probability activities and solve the problem in mental and written form. • Share findings as a whole class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 106 – 107. • 172 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 38—Answers

Student pages 112 – 114

Unit 38–1 1. (a) 34 (b) 7 (c) 42 (d) 88 (e) 78 (f) 500 (g) 400 (h) 700 (i) 0 (j) 800 2. (a) 198 (b) 281 (c) 611 (d) 221 (e) 545 (f) 154 3.1 2 3 5 6 7 8 9

x10 x x x 21 22 23 24 25 x 26 27 28 29 30x 4

1. (a) 300 (b) 600 (c) 800 (d) 200 (e) 700 (f) 400 (g) 200 (h) 600 (i) 500 (j) 800 2. (a) 26 (b) 86 (c) 80 (d) 75 (e) 91 (f) 592 3. Teacher check 4. Teacher check 5. Teacher check Challenge 78 rides

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11 12 13 14 15 16 17 18 19 20

4.1

Unit 38–2

9 10

11 12 13 14 15 16 17 18 19 20

Ride combinations

21 22 23 24 25 26 27 28 29 30

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A B C D E F G H I J K L M

BK BL

CD CE

CG CH

DE DF

DH DI

CK CL CM

DK DL DM

GH GI

GK GL GM

HL HM

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5. (a) They are multiples of 5; they appear in the 5th and 10th columns. (b) They are multiples of 2; they appear in the 2nd, 4th, 6th, 8th and 10th columns; multiples of 10 are also multiples of 2. Challenge 87, 762 Add the digits in the number; e.g. 8 + 7 = 15 — 1 + 5 = 6 If the digits add to 3, 6 or 9, the original number is divisible by three without leaving a remainder.

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Colour 1 = 1/2

Colour 2 = 1/4

Colour 3 = 1/4

• Use a 1 – 100 grid to find other number patterns involving multiples.

Consolidation 38–2

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1. (a) 1 (b) 9 (c) 13 (d) 11 (e) 6 (f) 80 (g) 10 (h) 90 (i) 30 (j) 10 2. (a) 17 (b) 17 (c) 15 (d) 23 r1 (e) 13 r5 (f) 11 3. (a) 18 (b) 27 (c) 7 (d) 3 4. (a) $4.50 (b) 336 (c) 288 (d) 208 (e) 338 Challenge

• Brainstorm other data to collect and record in a Carroll diagram.

Consolidation 38–3

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• Provide opportunities to practise solving similar word problems.

New Wave Maths Book D – Teachers Guide • 173 •

Unit 39–1

Student page 115

Outcomes

Indicators

N3.4, N3.3, S3.1

The student is able to: • order and show a sense of the proximity of things in locating key features on maps.

Skills • drawing • locating • logical thinking • working collaboratively

Teac he r

Memory Masters (N3.4)

Resources • Base 10 MAB • calculator • coloured pencils

Language • plan • layout • design • location • placement

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Notes

• The focus for this unit is number patterns.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (S3.1) Warm up

• Ask students to imagine the layout of their house. • Think about the location of furniture in each room. • Discuss why certain items are situated where they are; e.g. the table in the centre of a room with space to fit chairs; bed near a window or in a corner etc.

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• Direct students to look at the house plan in the workbook. • In small groups, discuss the best location for the items listed. Working in groups encourages students to justify their reasoning. • Students draw the items on the plan. • Share placement of items with the class. Discuss and justify ideas.

Challenge

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• When completing this puzzle, the three fields need to be exactly the same size and shape, and similar to the original shape. • Set students to work to find the solution to the puzzle. • Share results.

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What to do

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Number (N3.3)

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For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 2 – 3. • 174 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 39–2

Student page 116

Outcomes N3.3, N3.1a

Skills • rearranging • ordering • subtracting • problem-solving

Indicators

Resources

The student is able to: • read and write any whole number into the thousands.

• Base 10 MAB • calculator • number cards (see page 204)

Language • thousands • tens • digit • rearrange • smallest

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• • • • •

hundreds ones order largest subtract

Notes

Memory Masters (N3.3) Number (N3.3)

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Teac he r

• The focus for this unit is conversion of measuring units.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.1a) Warm up

• Have the numbers 0 – 9 on separate cards, one set between two. Ask the students to select two numbers to make a new number; e.g. 2 and 1. Students could either present these cards as 12 or 21. – Does the way we order the numbers make a difference to the number? – How many different ways could you arrange the numbers? • Repeat the activity with each student having three numbers. Follow the same procedure and questioning process as above. Students will have more options as they have more numbers to rearrange. • Look at and discuss the order of numbers; e.g. four is larger than three but smaller than five. Ask students to order their cards from smallest to largest then largest to smallest.

What to do

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• Read through Exercise 3 and model with the students how to arrange the numbers to make the largest possible number. Explain that as 4 is the largest number and by putting it in the thousands place, means it will make the largest number possible. Continue with the process until all four numbers have been suitably placed. • Ask the students to complete the second example on their own, then discuss their answers as a group. Once you are satisfied the students understand the concept, they can then complete the activity. • Repeat the process with making the smallest number in Exercise 4, until the activity is complete. • In Exercise 5, the use of correct position terminology will help the students understand what they need to subtract to make a number appear as zero.

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Challenge

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• Students may find it easier to work with their number cards, as they can be rearranged accordingly until the students are satisfied with their solution, which can then be recorded on the page. • It is important to allow students to discuss, perhaps in small groups, how they worked out the puzzle.The approach that students take will vary greatly, and while some students may struggle with a problem such as this, using those students who found the most efficient approach as a model will provide the weaker students with insight for future purposes.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 36 – 37. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 175 •

Unit 39–3

Student page 117

Indicators

Outcomes N3.1a, N3.3, M3.2, M3.3, C&D3.3

The student is able to: • measure time intervals using natural units of time, artificial nonstandard units, timers or standard units. • make sensible numerical estimates based on provided units. • display data in a bar graph.

Skills • timing • recording • graphing

Resources • Base 10 MAB • calculator • stopwatch • skipping rope • tape measure or trundle wheel • markers

Language • add • stopwatch • record • graph • metres • seconds • measure • accurately • minute

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Memory Masters (N3.1a) Number (N3.3)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.2, M3.3, C&D3.3) Warm up

• Inform the students they will be working outside in groups using a stopwatch to time a number of different activities. • Move to a grassed area at least 30 m long. Markers are set up 20 m apart for this activity. • Ask students to make an estimate of how long they think each activity will take before attempting it.

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• Students, in turn, attempt each activity and record their times in their workbook. • Ask students to check the accuracy of their estimates. • When all students have completed all the activities (once only, there are no retries to improve times) the class moves inside. • Once inside, students complete the bar graph to show the time taken for each activity.This exercise may need to be completed on another day.

Challenge

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What to do

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Teac he r

• The focus for this unit is conversion of money.

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• Hold a general discussion about timing devices—watches, clocks, sundial, metronome etc. • Ask students to invent a timing device that will accurately measure one minute. • If practical, make a working model to demonstrate the invention. • Students are required to keep notes and/or diagrams of their invention.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 88 – 89. • 176 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 39—Answers

Student pages 115 – 117

Unit 39–1

1. (a) 2000 g (b) 7000 g (c) 3200 g (d) 5100 g (e) 4400 g (f) 2.3 kg (g) 1.05 kg (h) 0.47 kg (i) 3.4 kg (j) 2.6 kg 2. (a) 24 (b) 17 (c) 324 (d) 244 (e) 225 (f) 322 3. (a) 4312 (d) 873 (g) 952 (b) 8743 (e) 9532 (h) 976 (c) 865 (f) 8421 (i) 8552 4. (a) 25 (d) 129 (g) 245 (b) 2788 (e) 1379 (h) 1567 (c) 3458 (f) 1247 (i) 2378 5. (a) – 50 (b) – 500 (c) – 5000 Teacher check other suggestions. Challenge One possible answer:

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Teac he r

1. (a) 5 (b) 29 (c) 11 (d) 13 (e) 10 (f) 25 (g) 9 (h) 15 (i) 20 (j) 50 2. (a) 692 (b) 1183 (c) 656 (d) 1626 (e) 1391 (f) 897 3. Teacher check Challenge

Unit 39–2

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• Add further items to draw in a suitable location on the house plan.

Consolidation 39–2

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1. (a) $4.32 (b) $8.14 (c) $2.99 (d) $3.33 (e) $5.92 (f) 833c (g) 187c (h) 734c (i) 666c (j) 937c 2. (a) 665 (b) 285 (c) 581 (d) 376 (e) 462 (f) 512 3. Answers will vary Challenge Devices will vary. Possible devices are: • a small candle with marks along its length; • a tin of water with a small hole in the bottom.

• Students can use number cards to pick three or four numbers at random for rearranging into largest and smallest numbers.

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R.I.C. Publications® www.ricpublications.com.au

Consolidation 39–3

• Repeat the activity using shorter or longer distances. Compare results.

New Wave Maths Book D – Teachers Guide • 177 •

Unit 40–1

Student page 118

Outcomes

Indicators

N3.3, N3.2

The student is able to: • match word problems with particular calculations. • in writing story questions that relate to a symbolic number sentence involving multiplication or division of whole numbers, show that they understand various meanings for these operations.

Skills • multiplying • dividing • adding • subtracting

Teac he r

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator

Language • double • halve • algorithm • word problem

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Notes

• The focus for this unit is doubling and halving numbers less than 100.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.2) Warm up

• Distribute Base 10 MAB among groups of students. • On the blackboard/whiteboard, write four different algorithms which use the same numbers; e.g. 9 + 3, 9 – 3, 9 ÷ 3, 9 x 3. • Ask students to represent each algorithm with Base 10 MAB. • Relate a word problem to the students for each algorithm; e.g. ‘I had 9 marbles and won 3 more in a game.’ Students work out which word problem matches the MAB representation.

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Number (N3.3)

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• Direct students to Exercise 3.The activity can be done individually or with teacher direction for those students who are still unsure. • Base 10 MAB can be used to solve each problem if necessary. • In Exercise 4, students make up their own word problems, individually or with a partner. • Share answers as a whole class.

Challenge

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What to do

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• Students may work this exercise in a practical manner if measuring containers are available. • Ask students to record all the steps they use in trying to solve the problem. • The ‘paint’ is able to be poured from any container to any other container.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 58 – 59. • 178 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 40–2

Student page 119

Outcomes

Indicators

N3.3, N3.1a, M3.3, WM3.4

The student is able to: • make informal statements about how confident they are about their estimates. • use the result of measuring with a physically-present unit to try to improve their estimates with successive objects. • check their answers against their estimates and reconsider both their methods and calculations if results seem unreasonable.

Skills • weighing • recording • estimating

Resources

Language • estimate • actual • mass • grams

• Base 10 MAB • calculator • objects small enough for a kitchen scale • kitchen scale

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Notes

Teac he r

Memory Masters (N3.3)

• The focus for this unit is identification of odd and even numbers.

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Number (N3.3, N3.1a)

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (M3.3, WM3.4) Warm up

• Introduce this activity by providing each pair of students with a selection of items light enough to be weighed on a kitchen scale. Ask them to order them from lightest to heaviest. – How did you work out the order of the objects? – How could you estimate the weight of an object? • Allow students to select six objects from around the classroom for the activity. • Demonstrate to the students how hefting can help them estimate the weight of the object before actual weighing takes place. Make a physical demonstration of hefting an object of which the weight is known with the object to be weighed. Use the kitchen scales to get an actual weight measure of the object. Discuss with the class the accuracy of each estimate. Is an estimate meant to be ‘spot on’?

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What to do

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• Ask students to follow the same procedure for their first three objects in Exercise 3, estimating first, then measuring. • Discuss how successful or unsuccessful their estimate was. Discuss how they could improve on their technique and write this in Exercise 4(a). • Direct students to complete the remainder of Exercise 3. Ask students to evaluate their new estimates and record their response in Exercise 4(b).

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Challenge

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• Students refer to Exercise 2(f) and write a word problem that would result in the algorithm. • This activity can be done individually or with a partner. • Share answers as a class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 96 – 97. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book D – Teachers Guide • 179 •

Unit 40–3

Student page 120

Indicators

Outcomes N3.1a, N3.3, N3.2

The student is able to: • understand that multiplication can be used for repeated addition situations.

Skills

Resources • Base 10 MAB • calculator

Language • repeatedly • number sentences

• adding repeatedly • multiplying

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Teac he r

Memory Masters (N3.1a)

• The focus for this unit is place value.

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results.

Main Activity (N3.2) Warm up

• Students take out their calculators and explore, making number patterns on the calculator.

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• Ask students if they know of any relationship between addition and multiplication. • To show the relationship, ask students to enter the following sequence into their calculator and describe what happened: 2+===== • Repeat this with: 3+==== 5+==== • Using the same process, find the missing numbers to complete the activities in Exercises 3 and 4. Note: Keystrokes may vary according to calculator model.

Challenge

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What to do

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Number (N3.3, N3.1a)

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• Using information from the activity in their workbook, ask students to explain, in their own words, the relationship between addition and multiplication. • Encourage students to use mathematical symbols to support their explanation.

• 180 • New Wave Maths Book D – Teachers Guide

R.I.C. Publications® www.ricpublications.com.au

Unit 40—Answers

Student pages 118 – 120

Unit 40–1

1. (a) O (b) O (c) O (d) O (e) E (f) E (g) E (h) O (i) E (j) O 2. (a) $8.43 (b) $6.63 (c) $8.69 (d) $55.50 (e) $79.80 (f) $89.86 3. Answers will vary 4. Teacher check Challenge Teacher check

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1. (a) 40 (b) 100 (c) 80 (d) 60 (e) 20 (f) 8 (g) 16 (h) 22 (i) 40 (j) 9 2. (a) 29 (b) 24 (c) 15 (d) 16 (e) 19 (f) 14 3. (a) 20 x 5 = 100 (b) 20 ÷ 5 = 4 (c) 20 – 5 = 15 (d) 20 + 5 = 25 4. Teacher check Challenge From the 10-litre can, fill the 3-litre can. Store this in the 5-litre can. Fill the 3-litre can again.There are now 4 litres left in the 10-litre can.

Unit 40–2

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Consolidation 40–2

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1. (a) 31 (b) 76 (c) 50 (d) 49 (e) 23 (f) 2 (g) 4 (h) 9 (i) 0 (j) 0 2. (a) 4.15 L (b) $4.14 (c) $2.24 (d) $3.80 (e) 2.70 g (f) 3.64 L 3. (a) 5 (f) 8 (k) 5 (b) 4 (g) 6 (l) 6 (c) 8 (h) 7 (m) 8 (d) 5 (i) 8 (n) 8 (e) 6 (j) 5 (o) 3 4. (a) 3 (f) 6 (k) 2 (b) 7 (g) 4 (l) 6 (c) 6 (h) 4 (m) 4 (d) 4 (i) 3 (n) 3 (e) 7 (j) 5 (o) 4 Challenge Teacher check. An example is: 5x4=4+4+4+4+4

• Repeat the activity at a later date, using different objects.

Consolidation 40–3

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• Students record their own number sentences by adding a number repeatedly on a calculator.

New Wave Maths Book D – Teachers Guide • 181 •

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New Wave Maths Book D – Teachers Guide • 183 •

Space Activities S3.1

1. Draw shapes or pictures in squares on 2-cm grid paper. Write coordinates A, B, C and so on along the horizontal axis and 1, 2, 3 and so on along the vertical axis. Write the coordinates for each shape or picture. Give to another student who lists the shape or picture found at each coordinate. 2. Use directional language of north, south, east and west to write directions to proceed through squares on a grid. 3. Locate and/or draw features on maps; e.g. adventure playground, shopping centre.

S3.2

S3.3

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1. Make 3-D shapes from modelling clay. Use fishing line, or similar, to make cross-sectional cuts of the shapes. Draw the new cross-section to show what shape is formed. 2. Make a shape using 12 2-cm cubes. Show how the shape can be bisected to form two shapes that are exactly the same (congruent). Repeat to make different shapes from the 12 cubes or use a different number of cubes. 3. Use modelling clay to make a 3-D shape; for example, a rectangular prism. The following two activities will require a large quantity of shapes as well as the materials to make further shapes. 4. Students make as many shapes as they can with four sides. In what way are they similar or different? 5. Students make shapes with 3 to 8 sides. Investigate the flexibility of the shapes. Change the size of the angles to see if that has any effect. Record findings. To complete the following two activities, a wide variety of drawing and construction materials, and assorted blocks is required. 6. Use selected, or directed selection of, shapes to make drawings of monsters, means of transport or other objects. Describe the shapes used to make different parts. 7. Choose equipment to draw circles in as many different ways as possible.

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1. Investigate lines of symmetry in 3-D shapes. Classify shapes according to whether they are symmetrical or not. 2. Fold and cut sheets of paper to investigate the congruence of the parts formed by the cuts or folds.

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3. Use an assortment of 2-D shapes to see which ones tessellate. Using a shape that tessellates, make a pattern to cover a page. Estimate how many shapes it will take to cover the page. Investigate which shapes can be used to make another shape by tessellating the first shape. For example, three equilateral triangles make a trapezium. 4. Use an overhead projector and 2-D shapes for this activity.

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Describe what is needed to be done to this shape

to make this shape

Using the overhead projector, continue with these shapes.

to a to a to a Students choose their own shapes to start with and experiment to find different final shapes.

• 184 • New Wave Maths Book D – Teachers Guide

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Space Activities cont. S3.4 1.

Using a variety of 3-D shapes, classify them according to their attributes. It may help to have headings on sheets of paper to assist in initial classification. Alternatively, use a Venn Diagram. See pages 219 – 220 of New Wave Maths Teachers Guide for a blackline master to use as a recording sheet. Some suggested headings—flat sides, rolls, shapes of sides, etc. 2. Classify shapes according to the number of faces, edges or vertices. Ask students to see if they can find any relationships between faces, edges and vertices. A range of 3-D objects is required for these activities. See pages 219 – 220 of the New Wave Maths for WA Teachers Guide for a blackline master to use as a recording sheet. 3. Classify the objects according to shape by placing them into the same collection as the solid 3-D shapes on which students are focusing. They may wish to have containers to place objects in or use sheets of paper; e.g. cube, cylinder, triangular prism (diagrams). Ask students to choose their own attributes to classify objects.

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A wide variety of 2-D shapes is required for this activity. 4. Sort polygons according to attributes; for example, Venn diagram – straight edges, curved edges, both straight and curved edges. See page 220 of the New Wave Maths Teachers Guide for a blackline master of a Venn diagram.

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New Wave Maths Book D – Teachers Guide • 185 •

Measurement Activities M3.1

1. Use a hand span or foot length to measure the length of the streamer for each body part. Which unit of measure required the greatest number to measure each body part? Choose another arbitrary unit of measure that will require more to measure the same body parts, and one that will require less. Record results as a graph to aid comparisons. 2. Select other items to measure; for example, desk length, width, width of classroom, own height, arm span, length or width of netball court and so on. Each time, have students select an arbitrary unit of measure to measure with. Record results and discuss findings. Results may be recorded in graph form or as a tally. 3. Using string or wool, cut two equal lengths. Use one length to make a long, thin shape and the other to make a shape which is almost circular. Use tiles or counters to see which needs the most to cover the surface. M3.2

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1. Using paper streamers, measure two or three different body parts (each student to measure the same body parts as other class members). Order the streamers from shortest to longest. Check to see if each student is in the same position each time. 2. Growing plants, measuring growth rates and predicting future growth rates allows a comparison of length with time over a daily or weekly period. 3. Collect a number of cardboard packages—cereal boxes, cake mix, etc. —and record the volume of each box by filling the box with sand, rice or beans. Measure the volume by cupfuls. Cut the boxes open and compare their surface areas. Check to see whether the box with the largest surface area also has the largest volume. 4. To measure volume by displacement there are three methods that may be used. Each is detailed as an example activity. (a) Select a large, clear container with a wide opening. Partly fill with water. Mark the water level in the wide container. Place an object (scissors, golf or tennis ball, stone etc.) into the water and mark the new water level. Measure the difference. (b) Select a large, clear container with a wide opening. Place the object in the bottom of the container. Partly fill with water and mark the water level. Measure the differences. (c) Select a suitably large, clear container and place it in a larger container. Fill the smaller container carefully to the top. Slowly place the object. Measure the overflow. 5. Choose a number of large, different objects; for example, chair, bin. Pick up one object at a time then order them from lightest to heaviest. Choose a number of smaller, different objects. Pick up one object at a time then order them from lightest to heaviest. Always swap the objects to the other hand when lifting to help in testing their mass. Repeat this activity many times using different objects. Use balance scales to check findings.

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Measurement Activities cont. M3.3 1. Estimate before measuring in all activities. Roll a lump of modelling clay into a ball. Weigh the modelling clay and record its mass. Roll the modelling clay out into a cylinder. Weigh the modelling clay again. What has happened to its mass? Repeat by changing the modelling clay into a number of different shapes. Explain the findings. 2. Use the newspaper to find the time that the moon rises, sets and the phases of the moon. 3. Set a marker—flag pole, tree or similar—in the playground and record the relative position of the moon over a period of a term. Take recordings each day. 4. Make a shadow stick and record the length of its shadow at set times each day. Make temperature readings each month. Use a similar time, such as every second Monday, to measure shadow length on the hour. M3.4a

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Use a geoboard, or square dot paper, to make shapes using: 4 squares; 5 squares; and 6 squares. Record the shapes made either on square dot paper or on square grid paper. See pages 198 – 199 of the New Wave Maths for WA Teachers Guide for a blackline master of dot and grid paper. Check the different positions that can be found for each shape made. Check the perimeter and the area for each set of shapes made from the given number of squares. Write the perimeter and area on each shape; for example, a diagram of shape on square dot paper, with area and perimeter marked. 2. Relate area and perimeter through previous activities.

3. Water conservation is very important to us all. Have students see how much water is wasted when a tap drips by collecting the drips for one to five minutes. Calculate what volume of water would be wasted in a day, a week or longer.

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1. Draw informal maps of the school grounds which show a sense of scale. 2. Make 3-D models of items that would be suitable to fit a certain scale; i.e. make a car out of modelling clay that would suit a particular figurine or toy.

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New Wave Maths Book D – Teachers Guide • 187 •

Number Activities N3.1a

1. Show two- and three-digit numbers with Base 10 MAB. 2. Use Base 10 MAB to combine (add) pairs of numbers. 3. Use Base 10 MAB to show the difference between two sets of numbers. Students may use direct comparison of the two sets or take the smaller amount away from the larger number.

N3.1b

1. Set up a class shop with priced items. Students can purchase items using coins and notes as provided on page 214 of New Wave Maths for WA Teachers Guide. Students take on the role of the shopkeeper and provide the correct change. 2. Students exchange coins for notes and notes for coins. Distribute appropriate amounts of money for activity to be done. Initially, students sort their money and exchange from central point (bank). This activity may be used as a class reward with students receiving money tokens for good work, behaviour and the like. Students can exchange the money at the end of the week, month or term.

3. Use working activities to read and write fractional notation. For example 1/2 cup, 3/4 tablespoon.

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Use day-to day situations to encourage student to define the operation required to solve the problem. For example, there are 30 students in our class, and I have 120 counters. How many counters will each student get? (40 counters) N3.3

Students choose and use various methods to solve algorithms. Compare and discuss levels of accuracy, rate efficiency of techniques chosen and consider the reliability of the technique. N3.4

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Search for patterns in all areas of mathematics and daily routines, classroom/school environment, at home and at large in the community.

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New Wave Maths Book D – Teachers Guide • 189 •

Student Outcomes Working Mathematically

Chance and Data

WM3.1 The student identifies familiar mathematical features C&D3.1 The student distinguishes certain from uncertain inherent in the activities and products of own and things and describes familiar, easily-understood other communities. events as having equal chances of happening or being more or less likely. WM3.2 The student poses mathematical questions prompted by a specific stimulus or familiar context C&D3.2 The student contributes to discussions to clarify and uses problem-solving strategies which include what data would help answer particular questions those based on representing key information in and take care in collecting, classifying, sequencing models, diagrams and lists. and tabulating data in order to answer those questions. WM3.3 The student understands mathematical conjectures as more than simply a guess, makes straightforward C&D3.3 The student displays and summarises data using tests of conjectures and discards those that fail the frequencies, measurements and many-to-one test. correspondences between data and representation.

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Space

S3.1

S3.2

The student understands a map or plan a 'bird'seye view' and uses order, proximity and directional language associated with quarter and half turns on maps and in descriptions of locations and paths.

Number

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WM3.4 The student uses alternative ways, when prompted, C&D3.4 The student reads and makes sensible statements to check working and choice of method. about the information provided in tallies and in simple tables, diagrams, pictographs and bar graph.

N3.1a The student reads, writes, says, counts with and compares whole numbers into the thousands, money and familiar measurements.

The student attends to the shape and placement N3.1b The student reads, writes, says and understands of parts when matching, making and drawing things, the meaning of fractions, flexibly partitioning and including matching 3-D models which can be seen rearranging quantities to show equal parts. and handled with conventional drawings of them N3.2 The student understands the meaning, use and and with their nets. connections between the four operations on whole S3.3 The student recognises repetitions of the same numbers, and uses this understanding to choose shape within arrangements and patterns and uses appropriate operations and construct and complete repetitions of figures and objects systematically to equivalent statements. produce arrangements and patterns. N3.3 The student adds and subtracts whole numbers S3.4 The student interprets common spatial language and amounts of money and multiplies and divides and uses it to describe and compare features of by one-digit whole numbers, drawing mostly on things. mental strategies for doubling, halving, adding to 100, and additions and subtractions readily derived from Measurement basic facts.

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M3.1

The student realises that using a uniform unit repeatedly to match an object gives a measure of the size of the object, and chooses suitable and uniform things to use as units and a common unit to compare two things.

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The student recognises, describes and uses patterns involving operations on whole numbers, and follows and describes rules for how terms in a sequence can be linked by multiplication or an addition- or subtraction-based strategy.

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M3.2

The student directly and indirectly compares and orders things by length, area, capacity, mass, time and angle, measures them by counting uniform units an uses standard scales to measure length and time.

M3.3

The student make sensible numerical estimates using units which they can see or handle and uses language such as 'between' to describe estimates.

M3.4a The student understands and measures perimeter directly and uses straightforward arithmetic to determine perimeters, elapsed time and other measurements which cannot be obtained directly. M3.4b The student attends informally to scale when making and using plans, maps and models.

• 190 • New Wave Maths Book D – Teachers Guide

Extracted from Mathematics Outcomes and Standards Framework – Student Outcome Statements, Education Department of Western Australia 1998.

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Working Mathematically—Record Sheet

Apply and Verify

Reason Mathematically

Mathematical Strategies

Comment

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Contextualise Mathematics

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New Wave Maths Book D – Teachers Guide • 191 •

Space—Record Sheet

Reason Geometrically

Represent Transformations

Comment

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Represent Shape

Represent Location

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Measurement—Record Sheet

Indirect Measure

Estimate

Direct Measure

Comment

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Understand Units

Outcome Category

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New Wave Maths Book D – Teachers Guide • 193 •

Chance and Data—Record Sheet

Interpret Data

Summarise and Represent Data

Comment

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Collect and Organise Data

Understand Chance

Outcome Category

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Number—Record Sheet

Reason about Number Patterns

Calculate

Understand Operations

Comment

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Understand Numbers

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New Wave Maths Book D – Teachers Guide • 195 •

Proforma This proforma has been provided for you to copy and use with your class. You can either: • select an activity and evaluate the whole class; or • select a small group of students and evaluate their work. The indicators are found on the relevant page in the New Wave Maths for WA Teachers Guide. 1. Photocopy this page. 2. Write the appropriate date, strand, outcome(s) and indicators. 3. Photocopy enough for one per student being assessed. 4. Inform the students they are being assessed on the activity they are about to complete. 5. Students complete the activity in the workbook. 6. Mark the work completed by the student. 7. Attach the proforma to the appropriate workbook page. 8. Record evaluation as required.

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Grid Paper.........................................................................................................................................198 – 201 Number Charts and Cards......................................................................................................202 – 204 Place Value Charts.........................................................................................................................205 – 207 Number Lines and Fraction Chart ....................................................................................208 – 209 Spinners..............................................................................................................................................210 – 211 Calendar – Any year..................................................................................................................................... 212 Clocks..............................................................................................................................................................................................213 Money................................................................................................................................................................ 214 Bingo Cards.......................................................................................................................................215 – 218 3-D Model Attribute Table..................................................................................................................... 219 Venn diagrams – Blank............................................................................................................................... 220 Carroll diagram............................................................................................................................................. 221 3-D Shapes...................................................................................................................................................... 222 Tangrams.............................................................................................................................................223 – 226 Nets.......................................................................................................................................................227 – 233 Paper Circles.................................................................................................................................................. 234 Graphs and Table – Blank...........................................................................................................235 – 236 New Wave Maths Book D – Teachers Guide • 197 •

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1-cm dot grid paper.

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New Wave Maths Book D – Teachers Guide • 199 •

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2-cm grid paper.

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1-cm triangle grid paper.

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New Wave Maths Book D – Teachers Guide • 201 •

100 Chart

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11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

r o e t s Bo r e 52 p 53 54 55 56 57 58 59 60 ok u S 62 63 64 65 66 67 68 69 70

41 42 43 44 45 46 47 48 49 50 51

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60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 • 202 • New Wave Maths Book D – Teachers Guide

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0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 0 2 4 6 8 10 12 14 0 3 6 9 12 15 18 21 0 4 8 12 16 20 24 28 0 5 10 15 20 25 30 35 0 6 12 18 24 30 36 42 0 7 14 21 28 35 42 49 0 8 16 24 32 40 48 56 0 9 18 27 36 45 54 63 0 10 20 30 40 50 60 70

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9 10 0 0 9 10 18 20 27 30 36 40 45 50 54 60 63 70 72 80 81 90 90 100

X 0 1 2 3 4 5 6 7 8 9 10

8 0 8 16 24 32 40 48 56 64 72 80

9 10 0 0 9 10 18 20 27 30 36 40 45 50 54 60 63 70 72 80 81 90 90 100

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New Wave Maths Book D – Teachers Guide • 203 •

Number Cards

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Hundreds

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•

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Tenths Hundredths

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New Wave Maths Book D – Teachers Guide • 207 •

Number Lines

25 21 22 23 24

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20 G l u 16 17 18 19 e

G l u e

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G l u e

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51 52 53 54

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100

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

12/3

11/3 11/4

11/2

Note: These number lines can be cut out and attached to make a single number line from 1 – 100.

13/4

Note: These number lines are designed to be used independently for any purpose.

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Fraction Chart 1/ 10 1/ 9 1/ 8 1/ 6 1/ 5

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Spinners

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New Wave Maths Book D – Teachers Guide • 211 •

January

February

March

15 22 29

12 19 26

16 23 30

13 20 27

10 17 24 31

14 21 28

11 18 25

15 22

15 22 29

12 19 26

16 23

16 23 30

13 20 27

10 17 24

10 17 24 31

14 21 28

11 18 25

30 31

16 23

14 21 28

10 17 24

15 22 29

11 18 25

16 23 30

12 19 26

10 17 24 31

13 20 27

11 18 25

14 21 28

12 19 26

15 22 29

11 18 25

12 19 26

13 20 27

14 21 28

15 22 29

16 23 30

16 23

13 20 27

10 17 24

14 21 28

11 18 25

15 22 29

12 19 26

16 23 30

13 20 27

14 21 28

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12 19 26

13 20 27

10 17 24 31

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14 21 28

11 18 25

15 22 29

October

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April

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15 22 29

12 19 26

16 23 30

10 17 24

15 22 29

12 19 26

16 23 30

13 20 27

11 18 25

10 17 24 31

14 21 28

12 19 26

11 18 25

15 22 29

13 20 27

12 19 26

16 23 30

14 21 28

13 20 27

10 17 24

15 22 29

14 21 28

11 18 25

16 23 30

• 212 • New Wave Maths Book D – Teachers Guide

Note: By writing the correct day for the current year next to the correct date, this calendar may be used for any year except a leap year.

Calendar

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Clocks 12

11 2

10 9

4 7

2 3

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3 4 5

2 3 4

3 4

8 7

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3 4

8 7

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1 2

3 4

8 7

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Money

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• 214 • New Wave Maths Book D – Teachers Guide

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27 100 14

16 100 28

r o e t s Bo r e p ok u 9 S 36 2 80 60 21

5 12

100 90

100

. 14 t 18

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Note: These cards can be cut along the dark, solid lines to make bingo cards.

o c . che e r o t r s 30 4s 100 15 3 up6er

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New Wave Maths Book D – Teachers Guide • 215 •

r o e t s Bo r e p o u 18 S 5 12 9 8 k9

16 . t21

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28 100 16

• 216 • New Wave Maths Book D – Teachers Guide

Note: These cards can be cut along the dark, solid lines to make bingo cards.

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100 12

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Teac he r

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r o e t s Bo r e p o u k 21 S 5 4 35 40 25

100

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Note: These cards can be cut along the dark, solid lines to make bingo cards.

o c . che e r o t 100 r 40 10 4 s 2 50 super

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New Wave Maths Book D – Teachers Guide • 217 •

r o e t s Bo r e p o u 4 25 100 21 25 k8 S

28 . t32

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• 218 • New Wave Maths Book D – Teachers Guide

Note: These cards can be cut along the dark, solid lines to make bingo cards.

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Teac he r

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Teac he r Attributes

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3-D Model

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Note: This table can be used to explore any attributes of 3-D shapes.

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New Wave Maths Book D – Teachers Guide • 219 •

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• 220 • New Wave Maths Book D – Teachers Guide

Note: These Venn diagrams can be used as required by the students to sort items into categories.

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Carroll diagram

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Note: This Carroll diagram can be used as required by the students to sort items into categories.

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New Wave Maths Book D – Teachers Guide • 221 •

triangular prism r o e t s Bo r e p ok u S cylinder

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cube

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• 222 • New Wave Maths Book D – Teachers Guide

sphere

square pyramid

Note: Students list or draw examples of these shapes found in their environment.

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Circular Tangram

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Enlarge to A3. These circles are used together as the circular tangram.

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New Wave Maths Book D – Teachers Guide • 223 •

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• 224 • New Wave Maths Book D – Teachers Guide

Enlarge to A3. Use this tangram puzzle to make various shapes.

Pythagorean Puzzle

Enlarge to A3. Use this Pythagorean puzzle to make various shapes.

r o e t s Bo r e p ok u S

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Tangram

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Teac he r

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Enlarge to A3. Use this circle puzzle to make various shapes.

Circle Puzzle

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Enlarge to A3. Use this broken heart puzzle to make various shapes.

•f orr evi ew pur posesonl y•

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New Wave Maths Book D – Teachers Guide • 225 •

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The Magic Egg Puzzle

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• 226 • New Wave Maths Book D – Teachers Guide

Enlarge to A3. Use this magic egg puzzle to make various shapes.

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Teac he r

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Cut out these nets to make an enclosed cone. Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

Cone Net

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New Wave Maths Book D – Teachers Guide • 227 •

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• 228 • New Wave Maths Book D – Teachers Guide

Cut out these nets to make an enclosed cylinder. Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

r o e t s Bo r e p ok u S

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Teac he r

Cylinder Net

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r o e t s Bo r e p ok u S

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Cube Net

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Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

o c . che e r o t r s super

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New Wave Maths Book D – Teachers Guide • 229 •

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Square Prism Net

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• 230 • New Wave Maths Book D – Teachers Guide

Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

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r o e t s Bo r e p ok u S

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Rectangular Prism Net

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Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

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New Wave Maths Book D – Teachers Guide • 231 •

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Triangular Prism Net

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• 232 • New Wave Maths Book D – Teachers Guide

Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

w ww

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m . u

Teac he r

Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

r o e t s Bo r e p ok u S

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Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

Triangular Pyramid Net

New Wave Maths Book D – Teachers Guide • 233 •

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Paper Circles

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• 234 • New Wave Maths Book D – Teachers Guide

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20 19 18 17 16 15 14 13

Units of Measure:

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10 9 8 7 6 5

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3 2 1

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Item

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Tally

Total

New Wave Maths Book D – Teachers Guide • 235 •

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Line or Bar Graph

(Title)

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(Title) • 236 • New Wave Maths Book D – Teachers Guide

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Parent Information

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Contents

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New Wave Maths Book D – Teachers Guide • 237 •

Parent Information Expectations of Knowledge of Basic Facts Year 1 An informal, general introduction to number and combinations.

Year 2 Discovery approach (manipulating concrete material) to finding and learning addition and subtraction facts.

Year 3

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Discovery and some recall of addition and subtraction facts. Use the terms 'add' or 'subtract' rather than 'plus' or 'minus'. Learn basic multiplication facts of 2, 3, 4 and 5 and multiples of 10, to 10 times 10.

Year 4

Year 5

Recall basic addition, subtraction, multiplication and division facts.

Years 6 and 7

Automatic response is desirable.

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Consolidate basic sums to 18 and differences taking from 18. Extend recall of basic multiplication and division facts to facts of 6, 7, 8 and 9 times tables.

The following suggestions can be used at home to assist your child in becoming more proficient at gaining automatic recall of the basic number facts. The ideas are not exclusive; many alternatives may be used.

4 5

7 8

'Snap' – played with flashcards. Play as for ordinary snap. A variation – write pairs of numbers on cards, or blank playing cards, without operation signs. Child may add, subtract, multiply or divide the pair of numbers to find a matching pair.

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'Flashcards' – with all combinations of basic facts, hold up, child responds with the answer. Flashcards can be easily made from light card (cereal packet) or by purchasing blank playing cards and writing basic facts on these.

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Throw two dice then either add, subtract or multiply the two numbers shown.

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Race the calculator. Call out a basic fact, while you work out the answer using the calculator your child attempts to race you to the correct answer, working mentally. 'Sums, Differences, Products' (add, take, multiply) – The game is played using a hundred chart. Call a pair of numbers from the basic facts. Your child covers the sum, the difference and the product of the two numbers called on the hundreds chart. Play for a given time; for example, five minutes, or until all of a set of basic facts have been used. 'Bingo' – The game is played as for ordinary bingo. You call a basic fact, use basic number fact sheet, your child covers the correct answer if it is on the card. First to cover the card or a line wins the game. When using board games encourage your child to add onto the total when throwing the die, or add the total of the dice, rather than counting on. 'Numero' is one of the best mathematical games available which can be used at home to develop mathematics skills.

• 238 • New Wave Maths Book D – Teachers Guide

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Parent Information Primary School Mathematics The Algorithm The following examples show the recommended method of recording the written algorithm for each of the four processes. These formats are not prescriptive, but are recommendations. In all cases, the ultimate aim is to arrive at the simplest (usually the shortest) form of recording the algorithm. A simplistic progression is shown for the development of each algorithm from Year 1 to Year 7.

Combining and Separating – Addition and Subtraction

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Year 1 – Concrete activities are recorded in number sentence form – first written recordings may possibly be made late in Year 1. The same form of recording is used in Years 2 and 3. In Year 3, adding and subtracting without regrouping are also recorded in vertical form. Vertical recording continues through to Year 7, with regrouping and increased difficulty of examples. Year Three Allow free exploration of addition initially. Focus on commenting with ones for ease of recording. 8

3 2 3 2 5 5c + 5 1 2 + 4 7 2 2c 1 3 + 1 5 7 9 7 + 8c 5 9 1 5c

When subtracting start with the top line number; for example, 3 take 6, can't be done. Regroup, 13 take 6 equals 7. 1 3

8 7 $7 0 6 – 2 5 – $3 0 7 6 2 $4 0

–

Year Four Students may still require the use of concrete materials at this stage.

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$2 . 3 7 + 5 6 + $2 . 4 8 9 3 $4 . 8 5 Students may still require the use of concrete materials at this stage. 21

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6 .8 0 m – 2 2 7 – 3 . 2 7 m 6 0 8 3 . 5 3 m Year Five

111

o c . che e r o t r s super 6 + 7 + 6 = 19 Write the ones (9) under the ones column and add the tens (1) to the tens column.

1756 2837 + 4276 8869

7 .5 1 + 2 1 .0 8 2 8 .5 9

2+1=3 5 5 5

Add the numerators 2 + 1 = 3 and write the denominator as it appears.

3 – 7 is not possible. So we exchange one 10 for ten ones to make 13 – 7, which we can do. 71

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5 171

6 .8 0 – 2 .9 3 3 .8 7

4– 2=2 5 5 5

Subtract the numerators 4 – 2 = 2 and write the denominator as it appears.

New Wave Maths Book D – Teachers Guide • 239 •

Parent Information Primary School Mathematics Grouping and Sharing – Multiplication and Division Commencing in Year 2, concrete activities are recorded, using the multiplication symbol in a number sentence from late Year 2 or as ready. Number sentence recording of concrete activities is carried on into Year 3. The written algorithm is introduced in its extended form in Year 4, working to the abbreviated form when the student understands the process. Year Three Allow free exploration of multiplication and division initially. Some students may require the assistance of concrete materials.

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5 4

x 7c = 28c x 3 3 1 5

1 0 30

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3 5 3 5 ➤ x 3 x 2 x 3 8 0 1 5 1 0 5 + 9 0 1 0 5 Year Five

6 x 6 = 36. Write the 6 in the ones column and carry the 3 tens to the tens column.This will be added after we multiply 6 by 7.

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76 x 58 608 + 3800 4408

0 .5 8 x 69 5 .2 2 + 3 4 .8 0 4 0 .0 2

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468 √ 3

Starting with the leftmost digit (4, really 400) share the four hundreds among the 3 people. How many hundreds does each person receive? (1) How many hundreds were shared altogether? (4) How many hundreds are left? (1)

Then repeat for the tens. Firstly exchange the 100 for 10 tens which gives 16 tens.The 16 tens are then shared among the 3 people. How many tens does each person receive? (5) How many tens were shared altogether? (16) How many tens are left? (1)

This is recorded as:

1 468 –300 168

This is recorded as:

15 3 468 –300 168 – 150 18

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576 x 6 3456

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Year Four Some students may require the assistance of concrete materials.

32 ÷ 4 = 8

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Follow the same process as before, ensuring the decimals are all in line. Estimate the result to determine the possible placement of the decimal point.

Then repeat for the ones. Firstly exchange the 10 for 10 ones which gives 18 ones. The 18 ones are then shared among the 3 people. How many ones does each person receive? (6) How many ones were shared altogether? (18) How many ones are left? (0)

It is assumed this process would be modelled with Multibase Arithmetic Blocks (MAB).

This is recorded as:

156 3 468 –300 168 –150 18 – 18 0

Note This is a guide only and students are encouraged to develop progressively through these stages as they are ready. If you have any concerns, please make an appointment to discuss them with me. • 240 • New Wave Maths Book D – Teachers Guide

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Parent Information Problem-solving Strategies To assist your child in solving mathematical problems, the following strategies may help.

1 Understand the problem. (a) Ask relevant questions to determine the operation, pattern, sequence, form of measurement or other mathematical means to begin to work out the problem. (b) Choose a plan or strategy to work out all or part of the problem. (c) Simplify the problem by breaking it into smaller parts and working out each small part. (d) Guess. (e) Work backwards.

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Use appropriate computation: addition, subtraction, multiplication, or division, to work out the problem.

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To help in working out the problem: (a) Make organised lists or tallies of data. (b) Make tables to show data. (c) Use physical models: objects; pictures; diagrams; graphs; or symbols. (d) Look for patterns and relationships.

Explain, generalise, prove relationships and patterns.

Guess and test facts, hypotheses or rules.

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Write or present conclusions clearly for others to be able to check your findings.

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Check results.

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Some Problem-solving Activities Around the Home

The following suggestions can be used at home to assist your child to become more proficient in problem-solving. The ideas are not exclusive; many alternatives may be used.

1 2 3 4 5

When planning the next family holiday, include your child in the planning, budgeting, activities, travelling time, itinerary and allow him/her to help solve any problems which may arise. When renovating your home—painting, replacing flooring, fencing, grassing or reticulating the garden—encourage your child to participate in the planning, costing, measuring and evaluation of the budget. When planning a party, include your child in the planning, catering, shopping and cooking. When fertilising your lawn, invite your child to help you work out how much fertiliser will be required for the area. Also work out the cost and the best way to approach the task to ensure even coverage. When planning your family's next big purchase, encourage your child to help work out a savings plan.

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New Wave Maths Book D – Teachers Guide • 241 •

Parent Information Concrete to Mental – Including the Calculator

Dear Parent(s) There are a number of different means of completing the four algorithms. Children start by using concrete materials to work through the algorithms to develop understandings. As their knowledge and understanding develop, students move to more abstract means of achieving the solutions to the algorithms. These solutions may be achieved by pencil and paper calculations or by working the solutions mentally (the ultimate aim).

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Children who have great difficulties in completing algorithms are to be encouraged to use the calculator to find the solutions after first estimating the answer. Estimation skills are essential in showing the development of mathematical knowledge. Should you encounter any problems, please contact me. Kind regards

Classroom Teacher

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Dear Parent(s)

There are a number of different means of completing the four algorithms.

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Concrete to Mental – Including the Calculator

Children start by using concrete materials to work through the algorithms to develop understandings. As their knowledge and understanding develop, students move to more abstract means of achieving the solutions to the algorithms. These solutions may be achieved by pencil and paper calculations or by working the solutions mentally (the ultimate aim).

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During these developmental phases, children will encounter algorithms that are complex, difficult or are a means to another step or the final solution. In such cases, the child should be encouraged to use a calculator to find the solution to the algorithm. The calculator is an invaluable aid in mathematics and its use is to be encouraged from the very beginning of a child's days at school. Children who have great difficulties in completing algorithms are to be encouraged to use the calculator to find the solutions after first estimating the answer. Estimation skills are essential in showing the development of mathematical knowledge. Should you encounter any problems, please contact me. Kind regards

Classroom Teacher

• 242 • New Wave Maths Book D – Teachers Guide

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Parent Information Mathematical Learning Areas Mathematics is comprised of a series of learning areas. These learning areas are outlined for teachers in the Student Outcome Statements document produced by the Education Department. There are seven learning areas, each of which is outlined briefly below.

Appreciating Mathematics Appreciate the role of mathematics in their own and other communities.

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Working Mathematically

Space

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Thinking about ideas, investigating, applying, verifying and reasoning mathematically. In brief, problem solving.

Knowledge of location (place), shape, transformations (changes), and reasoning geometrically (angles, constructions and other geometrical relationships).

Measurement

Understand units of measure, measure objects using measuring units, estimate measures and calculate measurements.

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Understand chance events. Collect and organise data and information. Summarise and represent data. Interpret data.

Number

Understand number and the relationships, order, count, place value. Understand addition, subtraction, multiplication and division and be able to calculate using these operations. Work out number patterns.

Pre-Algebra

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Understand symbols and graphs. Represent variation. Solve equations and inequalities.

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New Wave Maths Book D – Teachers Guide • 243 •

Parent Information Homework Policy

Dear Parent(s) As part of my Homework Policy I encourage students to regularly undertake given exercises in the reinforcement of mathematical concepts learnt at school. These activities will be within the expected competency level of the children; however, there may be times when, due to unforeseen circumstances, your child does encounter difficulties with the homework. Please take the time to assist with the processes involved, but please encourage your child to 'have a go'.

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Occasionally your child will receive a problem to solve. Encourage your child to explore the problem using the problemsolving strategies sheet. Again, encourage your child to 'have a go'. It is the process of investigation and working mathematically that is the focus of these activities. For this reason it is essential that all steps are written out as the problem is solved. Your encouragement and positive support are crucial to the continued development of your child's mathematical skills.

Kind regards

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Should you encounter any problems, please contact me.

Classroom Teacher

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Dear Parent(s)

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Homework Policy

As part of my Homework Policy I encourage students to regularly undertake given exercises in the reinforcement of mathematical concepts learnt at school. These activities will be within the expected competency level of the children; however, there may be times when, due to unforeseen circumstances, your child does encounter difficulties with the homework. Please take the time to assist with the processes involved, but please encourage your child to 'have a go'.

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Classroom Teacher

• 244 • New Wave Maths Book D – Teachers Guide