New Wave Maths - Teachers Guides: Level E - Ages 9-10 by Teacher Superstore

RIC–1088 12.3/590

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

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Robert Dayman 2003

RIC-1088 ISBN 978-1-86311-709-8 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 E – Teachers Guide • iii •

Contents Introduction ....................................................................................................................................................... 1 Appreciating Mathematics ........................................................................................................................ 2 Learning Environment ................................................................................................................................. 3 Language and Mathematics ...................................................................................................................... 4 Mixed Abilities .................................................................................................................................................. 4 General Content Outline ..................................................................................................................... 5–9 Technology ....................................................................................................................................................... 10 Assessment ...................................................................................................................................................... 11 Cross-curriculum Linkages ...................................................................................................................... 12

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Teachers Notes and Answers How to Use the Teachers Notes ........................................................................................................ 14 Materials List ................................................................................................................................................... 15 Overview of Activities Term One—Units 1–10 ..................................................................................................................... 16 Term Two—Units 11–20 .................................................................................................................... 17 Term Three—Units 21–30 ................................................................................................................ 18 Term Four—Units 31–0 ..................................................................................................................... 19 Lesson Notes, Consolidation and Answers Term One—Units 1–10 ............................................................................................................ 20–59 Term Two—Units 11–20 ........................................................................................................... 60–99 Term Three—Units 21–30 ................................................................................................. 100–139 Term Four—Units 31–40 ................................................................................................... 140–179

Additional Activities

Space Activities ............................................................................................................................... 182–183 Measurement Activities ......................................................................................................................... 184 Number Activities ..................................................................................................................................... 185

Assessment ©R . I . C.Publ i cat i ons •f orr evi ew pur posesonl y•

Reference to Student Outcomes .................................................................................................... 188 Record Sheets – Blank ............................................................................................................... 189–193 Proforma – Blank ...................................................................................................................................... 194

Photocopiable Resources

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Grid Paper ......................................................................................................................................... 196–199 Number Charts and Cards ..................................................................................................... 200–204 Spinners .............................................................................................................................................. 205–206 2-D Shapes ....................................................................................................................................... 207–209 Classification Chart .................................................................................................................................. 210 Sorting Circles ................................................................................................................................ 211–212 Sorting Activity – Zoo ............................................................................................................................ 213 Sorting Activity – Farm .......................................................................................................................... 214 Attribute Labels ............................................................................................................................. 215–217 Venn Diagram ...................................................................................................................................................................... 218 Carroll Diagram ................................................................................................................................................................. 219 Line or Bar Graph ..................................................................................................................................... 220 Picture Talk .................................................................................................................................................... 221 Coins ................................................................................................................................................................ 222 Clocks – Blank ............................................................................................................................................ 223 School Map ................................................................................................................................................... 224

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

Expectations of Knowledge of Basic Facts .................................................................................. Primary School Mathematics ............................................................................................................. Developing Mathematical Awareness ........................................................................................... Concrete to Mental ................................................................................................................................. Mathematical Learning Areas .............................................................................................................

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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. 3. Students should develop their capacity to use mathematics in solving problems individually and collaboratively. 4. Students should learn to communicate mathematically. 5. Students should learn techniques and tools which reflect modern mathematics. 6. Students should exercise the processes through which mathematics develops.

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:

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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; 4. acquire the mathematical knowledge, ways of thinking and confidence to use mathematics to: (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 (c) engage in the mathematical study needed for further education and employment. 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 (c) communicate mathematically to a range of audiences. 7. appreciate: (a) that mathematics is a dynamic field with its roots in many cultures; and (b) its relationship to social and technological changes.

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New Wave Maths Book E – 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.

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These are summarised as part of the Appreciating Mathematics substrand found in The Curriculum Framework 1998 (page 180): 1. Show a disposition to use mathematics to assist with understanding new situations, solving

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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 8. modelling positive attitudes towards mathematics.

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

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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 supportive, 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.

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• 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. • 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. • Inclusively and difference Learning experiences should respect and accommodate differences between learners.

© R. I . C.Publ i cat i ons • Independence and collaboration Learning experiences should students to o learn both from, and with, others as •f orr evi ew p uencourage r pos es n l y • well as independently.

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• Supportive environment 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 • fostering processes which enhance learning.

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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 E – 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

Teachers need to be aware of the individual differences of all students and provide learning •f orr e vi ewa levelpofu r pand oindependence seso n l y •To do this, experiences which develop success for each student.

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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.

• 4 • New Wave Maths Book E – Teachers Guide

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General Content Outline Goals and Guidelines After completing and understanding Year 4 well, students should then move onto Year 5. 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; 2. selection of appropriate strategies;

use of materials; ©purposeful R . I . C .Pu bl i c t i ons selection of appropriate operations to a solve problems; reflection in actions to formulate ideas; •f orr evi e puideas r p se so l y• expression ofw mathematical ino words, pictures andn symbols; 3. 4. 5. 6.

7. construction of lists, tables and graphs;

8. estimation of number and measurement activities;

9. identification of patterns and relationships;

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

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13. justification of conclusions and inferences.

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

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New Wave Maths Book E – Teachers Guide • 5 •

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.

Working Mathematically

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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. 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.

Problem-solving

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:

2. Prepare a plan to solve the problem – working from the known to the unknown, draw

diagrams, tables, charts to assist. 3. Carry out the plan – using different strategies as appropriate. 4. Review final solution to check and discuss its reliability and validity.

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1. Understand the problem – rewording, breaking into smaller parts may assist.

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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.

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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 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.

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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;

4. constructing models;

5. estimating and measuring;

6. recording and calculating;

7. inferring, predicting and hypothesising; and

r o e t s Bo r e p ok u S 8. discussing what they are doing.

Following this the students express, represent and interpret their workings by: 1. discussing findings and interpretations;

3. using symbols and words;

4. drawing pictures, diagrams and graphs; 5. constructing models; 6. translating between relationships; 7. making lists and tables; 8. drawing conclusions; 9. interpreting results; and 10. communicating findings.

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2. identifying patterns and relationships;

Then follows a. period ofP consolidation ofc understanding through © R. I C. ubl i at i on sfurther activities that embody the mathematical idea. Students should apply and extend their understanding through work in familiar, and then more novel, contexts. •f or r evi ew pur posesonl y•

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New Wave Maths Book E – 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. 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.

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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. 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

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3 + 2 =

Year 3

4–1=

3 lots of 4 =

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

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• 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:

17 +2

43 + 24

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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54 – 22

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• Addition and subtraction requiring regrouping should be done with the assistance of concrete materials, particularly Base 10 MAB; for example: 76 – 25 = 18 + 19 = 21 + 14 + 37 = 256 + 48 =

100 – 60 =

329 + 257 =

700 – 300 =

638 – 73 =

• Use Base 10 MAB and other concrete materials for multiplication; for example: 42 = x 21 40 x 2 = 34 x 2 =

6 x 100 =

200 =

30 x

x 40 = 90

r o e t s Bo r e p ok u S • Division is set out as shown in the examples below. 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

18 x 3

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

8r1 6 49

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6 4 24

5 76

Year 5

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1 + 1 = 4 4

2 + 3 = 8 8

3 – 1 = 4 4

7 – 3 = 10 10

1 + 1 = 1 3 3

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• 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.

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

34 x 23

• 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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40 720

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3 60.24

50 267

New Wave Maths Book E – Teachers Guide • 9 •

Year 6 • Addition and subtraction operations extended to cater for student ability and individual and practical needs in whole number and decimals. Fractions added and subtracted with unlike denominators. • Multiplication in whole numbers and decimals limited by ability and needs. • Division extends to the introduction of division greater than ten; for example, 659 ÷ 43.

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.

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Chance and Data

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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.

Students are able to use and understand the language of chance and from this make a statement about the likelihood an event will occur. 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. collection activities should lead to classification, organisation, summarising and displaying in a variety of ways.

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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. 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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• 10 • New Wave Maths Book E – Teachers Guide

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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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New Wave Maths Book E – 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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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.

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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:

• 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.

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• 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. 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.

• 12 • New Wave Maths Book E – Teachers Guide

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

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.

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• Comprehensive Judgments on student progress should be based on multiple kinds and sources of evidence. Assessment is a crucial aspect of the mathematics learning process. Assessment provides the 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 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; workor project-based assessment; collected samples of students’ independent work; individual homework assignments; group reports; anecdotal records; self-assessment; and peer assessment. 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)

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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 E – 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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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.

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Science provides for a variety of measurement activities with particular emphasis on the measurement component. 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 E – Teachers Guide

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

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Teachers Notes and Answers

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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

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New Wave Maths Book E – Teachers Guide • 15 •

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

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

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. A space for you to record notes relevant to the lesson has been provided. This space could be used for any purpose. Some suggestions:

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

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• record any problems you or your students experienced during the lesson;

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• record any improvements you made to the lesson;

• record individual student's progress or development;

Skills relevant to the main activity have been listed.

• 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.

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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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.

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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.

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.

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The answers for the Challenge activities are generally an example of one possible solution, as many solutions are often possible.

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 E – Teachers Guide

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

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. • Denotes items produced in New Wave Maths Teachers Guide as a blackline master which are available on pages 198 to 234. Teachers may photocopy and use them with their class(es). adhesive tape

fraction grid • page 208

plastic polygons

analog clock/watch

fraction squares

playing cards

atlas/street directory

geoboards

polyominoes

attribute blocks balance balloons balls Base 10 MAB boxes

popsticks

geostrips

reading or library books

glue

recycled materials

heavy card – coloured or plain

restaurant menus

– A4 and A3

canteen price list

ruler

height measuring stick

scissors

historical reference books or the

shapes – 2-D

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beads

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Internet

shapes – 3-D

bundles of 10s and 1s

house plan

small squares

calculator

interlocking cubes

spinners • pages 209–210

calendar • page 211

joiners

square tiles

candles

kitchen scales

square shapes

canteen price list

large circles • page 231

stopwatch

cardboard strips

lead pencil

straws

cardboard tubes

LEGO

streamers

catalogues

light card – coloured or plain

string

class members

– A4 and A3

sundials

symmetrical patterns/drawings

coloured counters

markers (cones)

tangrams • pages 220–223

coloured pencils

measuring containers (mL/L)

tape measure

compass – for drawing and for

measuring stick or tape

tessellating sets

metre rule

timetables

containers – various shapes and sizes

mirror/mira

toothpicks or equivalent

cotton

modelling clay

tracing paper

money (coins/notes)

trundle wheel

nets • pages 224–230

TV program guide

curve-stitch sheets • pages 232–233

newspapers

unit cubes

diaries

number chart 1–100 • page 202

wire

number squares

wool

objects for weighing activities

1-cm dot grid paper • page 198

dominoes

overhead projector

1-cm grid paper • page 199

eggtimer

paper – coloured or plain

2-cm cubes

direction

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dice – 6- and 10-sided digital clock/watch

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magazines

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clock stamp

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elastic bands

– A4 and A3

felt-tip pens

pattern blocks

fraction cake

pegboards

fraction/decimal number line • page 208

pipe-cleaners place value charts • pages 205–206

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New Wave Maths Book E – Teachers Guide • 17 •

Term One Week Unit Outcomes 1

Page

S3.1—Use directional language to find a path.

N3.1a—Produce and use partitioning for a given number.

M3.4a—Draw and compare shapes of a set area.

N3.1a—Trade, count and write numbers into the thousands.

C&D3.4—Read frequencies from a bar and line graph and answer relevant questions.

N3.1a, N3.3—Show, read and write decimal numbers. Add and subtract whole numbers with

reference to place value.

S3.2—Look closely at a 3-D shape and decide which 2-D shapes could be used to make it.

N3.3, N3.4—Use calculation, manipulation and recording techniques to investigate patterns.

M3.2—Use a ruler to measure length, a measuring jug to measure capacity and a balance

to measure mass.

N3.4—Find and complete patterns using addition and multiplication strategies.

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C&D3.2, C&D3.3, M3.2—Measure and record data in a bar graph. Represent data in a line graph.

N3.1a, M3.1—Write measures using decimal points. 5

WM3.2, WM3.3, WM3.4, N3.4—Make conjectures about patterns and relationships in the times tables.

N3.3—Add whole numbers using the simplest method.

N3.3—Estimate answers to written problems. 7

S3.2—Use 2-cm cubes to make and draw as many arrangements as possible. N3.1b, N3.3—Add and write fractions correctly.

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S3.2—Use 2-cm cubes to make and draw as many arrangements as possible. 8

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16 17 18

N3.3—Use mental strategies to multiply and divide whole numbers.

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M3.1, S3.3—Use regular shapes to tile an area. 6

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20 21 22

C&D3.2, C&D3.3, C&D3.4—Collect, collate and record data in a two-way table.

N3.2—Use the relationship between multiplication and division to solve problems.

S3.1, S3.2—Work collaboratively to design and make models to represent a fete.

N3.3—Solve multiplication problems using the commutative and associative processes.

M3.4a—Use grid paper to generate polygons of a set perimeter in order to compare area.

10 10

N3.1a, C&D3.3—Summarise and analyse a set of data.

C&D3.2, M3.2, M3.4a—Read and record 24-hour times in a timetable.

N3.3—Use mental strategies to solve addition problems.

• 18 • New Wave Maths Book E – Teachers Guide

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

Term Two Week Unit Outcomes

Page

1 11

S3.2—Analyse and describe 3-D shapes.

N3.3—Use chosen strategies to solve subtraction problems.

M3.2, N3.1a—Measure and record height in centimetres. Use data to answer questions.

2 12

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

C&D3.3—Record selections using an arrow diagram.

N3.1b—Read and write fractions.

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S3.2—Draw and highlight nets that could be used to construct a box.

N3.3—Use a constant to add to or subtract from a number sentence to keep it balanced.

M3.2, S3.2—Compare and record surface area and volume using a specified number of cubes to

make 3-D models.

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N3.1b—Use concrete materials to separate objects into equal parts.

WM3.2, WM3.4, C&D3.2, C&D3.3—Pose, ask and contribute mathematical questions prompted by

a specific stimulus.

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N3.3—Use rounding to estimate sums and products.

39 40 41 42

5 15

M3.2—Compare angles to find right angles, acute angles and obtuse angles.

N3.3—Estimate answers through the use of rounding techniques.

M3.2—Read and write 12- and 24-hour times.

© R. I . C.Publ i cat i ons S3.3, C&D3.3—Observe the symmetry of the letters of the alphabet and record in the Venn diagram. • f orr evi ew pur posesonl y• N3.4—Complete sequences to solve arithmetic problems.

6 16

N3.3—Estimate answers through the use of rounding techniques.

45 46 47 48

S3.4, S3.2—Visualise and name the 2-D shapes created by cross-section cuts as shown.

N3.2, N3.4—Use the grid to locate and list prime numbers.

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N3.3—Use own methods to solve subtraction problems.

C&D3.1, C&D3.2, C&D3.3—Collect, record and summarise data. Rank events from most to

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M3.2—Use a standard calendar to locate and calculate particular days.

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least likely.

N3.2, N3.3—Complete number sentences as required.

53 54

9 19

S3.3—Complete a symmetrical drawing.

N3.1b, N3.1a—Read and write whole numbers and fractions.

M3.2, N3.3—Use a uniform unit of length to measure and order height. Add and subtract

whole numbers.

10 20

N3.2, N3.3—Understand that multiplication and addition are interrelated.

C&D3.3, N3.3—Use Venn diagrams to organise and classify data.

N3.3—Add fractions with like denominators.

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New Wave Maths Book E – Teachers Guide • 19 •

Term Three Week Unit Outcomes

Page

1 21

S3.3—Complete symmetrical patterns.

M3.3—Estimate and compare regions, containers and objects for area, capacity and mass.

M3.4a—Estimate and measure the perimeter of irregular shapes.

2 22

N3.4—Use a number-letter code to write a message.

C&D3.4, C&D3.2—Collect, tally and summarise data.

N3.1a—Read, write and partition numbers into hundreds and thousands.

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3 23

S3.3—Complete and draw symmetrical pictures.

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

M3.1—Use a pentomino to complete a tessellating design. S3.1—Draw a plan of the classroom to scale.

C&D3.2, C&D3.3—Record frequency data in a table to help devise a menu.

70 71

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4 24

N3.1a, N3.3—Read and write numbers into the thousands. Estimate sums and products by rounding. 72 5 25

WM3.1, WM3.2, M3.2—Pose, ask and contribute mathematical questions prompted by a

specific stimulus.

N3.3—Use partitioning to multiply whole numbers.

M3.2—Use measuring tools to measure body parts and compare lengths.

6 26

S3.3—Look at a shape and decide whether or not it will tile. Support answers with a clear explanation. 76

C&D3.1—Describe and order events according to the likelihood of it happening.

and original shapes.

N3.1b—Order, read and write fractions. Match fraction equivalents.

M3.2—Build models according to directions and compare volume.

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8 28

N3.3—Partition and round numbers to solve addition problems.

C&D3.1, C&D3.2, C&D3.3, C&D3.4—Use chance events to record and analyse data.

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N3.1b—Read, write and order fractions. Show fraction equivalences.

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81 82 83 84

9 29

S3.2, M3.2—Use 2-D regular shapes to show those that tessellate and those that do not.

N3.3—Subtract money.

M3.2—Use an arbitrary unit to compare the capacity of various containers. 10 30

N3.3—Add money.

C&D3.1, C&D3.2, C&D3.3, C&D3.4—Record and interpret frequency data in two-way tables

87 88

and graphs.

S3.4—Name an object based on spatial features used in a description.

• 20 • New Wave Maths Book E – Teachers Guide

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

Term Four Week Unit Outcomes

Page

1 31

S3.1—Draw a plan, keeping in mind the relative size of items on the drawing.

N3.1a—Calculate and read decimal numbers.

M3.2, N3.1a, N3.3—Measure using consistent units, record and order data.

Add numbers each less than 10.

2 32

N3.1a, N3.3—Read, write and calculate amounts of money.

C&D3.1, C&D3.2—Record data, interpret and justify results of chance events.

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N3.4—Recognise, record and complete number patterns involving operations.

3 33

S3.3—Prove or disprove the statement by finding repetition in pattern.

N3.2, N3.3—Read and calculate whole numbers in word problems.

4 34

N3.2, N3.3—Partition two-digit numbers to assist with multiplying.

C&D3.1—Describe chance outcomes referring to probability.

M3.3—Use clues to estimate time of day or year.

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M3.2, M3.4a—Use uniform units to measure and order perimeter and area of polygons.

99 100 101 102

5 35

S3.4—Draw variations of 2-D shapes, maintaining common features such as number of sides.

103

N3.1b—Find unit fractions of a whole number.

104

M3.4a—Read and make a daily schedule. 6 36

S3.1—Place features on a map in sensible locations.

C&D3.2, C&D3.3—Collect, record and summarise data using a tally accurately.

© R. I . C.Publ i cat i ons C&D3.4, N3.3—Interpret and summarise data shown in a pictograph. f o rr evi eaw p uand r p o eso nl 7 37 • S3.3, y• M3.4b—Roughly draw plan to scale note itss symmetrical features.

N3.3—Use conventional algorithms to solve the problems.

106 107 108 109 110

M3.2—Read and write the time from analog and digital clocks.

111

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

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to solve a problem.

C&D3.2, C&D3.3—Collect and classify data using two-way tables.

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105

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112 113 114

9 39

S3.1, M3.4b—Demonstrate an understanding of proximity when drawing pathways on a map.

115

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

116

M3.3, M3.2—Estimate, measure and record time taken to complete tasks.

117

10 40

N3.1a—Subtract decimals, measures and amounts of money.

118

C&D3.4—Read and summarise information displayed in a graph.

119

N3.4—Use the constant function on a calculator to solve multiplication number sentences.

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120

New Wave Maths Book E – 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 to describe a route.

Skills • mapping • following directions • using a compass • speaking and listening

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • compass

Language • add • directions • south • north • east • west • 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

• Ask the students which direction is east. How do they know? (The sun rises in the east.) What are the three other main compass directions? (North, south and west) Where is north; south; and west? • What instrument is used to show compass directions? (A compass) In which direction does it point? (North) • The activity they will be doing requires them to follow compass directions. • Have a compass for students to try.

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

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• On the grid you will find a cross, this is your starting point. • The directions are given as S = south, N = north, E = east and W = west. • Your direction to move is given as 4S, meaning four squares south. You commence counting with the first square you move into in the direction given as square one. Do not count the X square. • Follow the directions given to find at which restaurant Brett ate. • When you have found the restaurant, make your own set of directions to find a restaurant to dine at, starting at the square X. • Ask a friend to follow your directions to discover the restaurant.

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• A code is used to deter others from access to a computer. Many of us have difficulties in remembering codes. Find the simple four-letter code that is also a man’s name that was used in the early days of computing. • Provide the hint of looking at the keyboard as a source of assistance.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 6–7. • 22 • New Wave Maths Book E – Teachers Guide

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

Unit 1–2

Student page 2

Outcomes

Indicators

N3.3, N3.1a

The student is able to: • produce and use standard partitions of two- and three-digit numbers.

Skills • recognising place value • recording place value

Resources

Language

• calculator • Base 10 MAB • place value chart (see pages 205–206) • compass for drawing • compass to show direction

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• place value • tens • numeral • ones • hundreds • tenths • circle • compass • hundredths • add • thousandths • decimal places • expanded form

Notes

Memory Masters (N3.3)

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

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 to the class the large place value chart found on pages 205–206. • Ask the class what the place value is of the various places on display. Choose places at random and repeat several times. • Write on the blackboard/whiteboard the numerical value of each place as shown on page 2 of the workbook. • Using the example in the workbook, explain to the class that any number may be written in an expanded form; e.g.. 362 as 300 + 60 + 2. Note: It may also be written as 36 tens and 2 ones. 362.584 as 300 + 60 + 2 + 0.5 + 0.08 + 0.004. • The decimal representation may be written as fraction equivalents; e.g. 0.5 as 5/10, 0.08 as 8/100 and 0.004 as 4/1000. • The fractional form and the whole number component may then be written as factors; e.g. 300 as (3 x 100); 60 as (6 x 10); 2 as (2 x 1); 5/10 as (5 x 1/10); 8/100 as (8 x 1/100); and 4/1000 as (4 x 1000).

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• Work with the class as a whole and show the expanded form for each of the numbers in turn shown on page 2. • Ask the students for the expanded form, write it on the blackboard/whiteboard and have students copy each step into their workbook.

Challenge

• Students are to draw a circle but may not use a compass. (Note: This compass is different from one that shows direction.) Show students the different types of compass. • Ask students to think about how they might do this. (Use a template.) Draw the circle and write about how you drew the circle. • Share different ideas with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 44–45. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 23 •

Unit 1–3

Student page 3

Outcomes

Indicators The student is able to: • given an area, use grid paper to generate possible shapes where the shapes fit along the grid lines.

N3.3, M3.4a

Skills • measuring • scale drawing • finding area • finding perimeter

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB

Language • add • shapes • area • square • units • grid • table • edges • perimeter

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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 (M3.4a) Warm up

• Discuss with students the prefix ‘hexa’ meaning six. • Talk about hexagon (six sides), hexagonal (six angles). • Introduce hexominoes (six squares).

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• Today’s activity is to draw shapes of 6 cm2; that is shapes made up of six squares.The squares must be joined by a full side to at least one other square. Avoid drawing diagonals as it is difficult to work out the perimeter. • Draw as many shapes as possible, numbering them as you go, that are different from others you have drawn. If you have no more room on the page, stop. • When you have drawn all the shapes and numbered them you will need to complete the table below the grid. • Count the number of edges for each shape in turn and write the number on the table next to the shape number. Next to this write the perimeter in centimetres in the space provided. • How can you find the perimeter in centimetres? (Measure with a ruler or, knowing that one square’s length is equal to one centimetre, count the number of unit squares along the outside edge [perimeter] of the shape.) • When you have completed this, write what you noticed about the perimeters in the different shapes. • Share these findings across the class.

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

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• Which hexominoes fold up to form a box; i.e. which ones are nets for cubes?

• 24 • New Wave Maths Book E – Teachers Guide

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

Unit 1—Answers

Student pages 1–3 Unit 1–1

(b) 21 (g) 4 (b) 89

(d) 0 (i) 5 (d) 98

(e) 60 (j) 6 (e) 77

1. (a) 6 (b) 14 (c) 72 (d) 45 (e) 54 (f) 6 (g) 6 (h) 5 (i) 9 (j) 6 2. (a) 86 (b) 84 (c) 94 (d) 87 (e) 87 (f) 75 3. (a) = 50 + 4 + 0.8 + 0.03 + 0.009 = 50 + 4 + 8/10 + 3/100 + 9/1000 = (5 x 10) + (4 x 1) + (8 x 1/10) + (3 x 1/100)+ (9 x 1/1000 ) (b) = 70 + 1 + 0.6 + 0.02 + 0.007 = 70 + 1 + 6/10 + 2/100 + 7/1000 = (7 x 10) + (1 x 1) + (6 x 1/10) + (2 x 1/100) + (7 x 1/1000 ) (c) = 200 + 8 + 0.4 + 0.03 + 0.006 = 200 + 8 + 4/10 + 3/100 + 6/1000 = (2 x 100) + (8 x 1) + (4 x 1/10) + (3 x 1/100) + (6 x 1/1000 ) Challenge Teacher check

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1. (a) 0 (f) 5 2. (a) 78 (f) 87 3. R3

Unit 1–2

4. Teacher check Challenge FRED

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• Provide students with grid paper (see page 198). Work with a partner to write directions and swap to find the path.

Consolidation 1–2

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1. (a) 12 (b) 49 (c) 32 (d) 81 (e) 48 (f) 3 (g) 8 (h) 8 (i) 7 (j) 6 2. (a) 149 (b) 149 (c) 169 (d) 208 (e) 196 (f) 157 3. (a) Teacher check. (b) The perimeters are not all the same. Challenge Teacher check

• Write further examples for students to complete in their pad or on the blackboard/whiteboard; e.g. 109.570, 482.762, 371.892, 549.776 etc.

Consolidation 1–3

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R.I.C. Publications®

• Students use 1-cm cubes to create their hexominoes in 3-D.

New Wave Maths Book E – 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.

Skills • counting • trading • subtracting • following instructions

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB

Language • subtract • trade • blocks • flats • least • same order

• • • • • •

weights larger smaller units longs change

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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

• Organise the class into small groups. • Distribute the Base 10 MAB and allow a brief period of free play. • Direct the students to place separately on their desk one block, one flat, one long and one unit. • Ask students how many units are required to make one long. Show the teacher and check. • Ask students how many longs make one flat. Show the teacher and check. • Ask students how many flats make one block. Show the teacher and check. • Ask: What did you find out about the number of smaller pieces of wood required to make the next larger piece? (They all required ten of the smaller pieces.) • The relationship may change depending on how you define one piece. For example, if the cube is defined as one then a flat is 0.1 or 1/10. • Explain to students that our number system has a relationship of 10 between each place. Because of this the Base 10 MAB are very useful to help develop understanding of our place value system and working out the four algorithms – addition, subtraction, multiplication and division.

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

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What to do

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• Work through the Exercises 3, 4 and 5 with the whole class, following the instructions in the workbook. • Use further examples of your own if required.

Challenge

• Remind students they are to record all moves either with diagrams or in words so their solution and attempted solutions may be followed. • Share some of the solutions with the class.

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

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

Unit 2–2

Student page 5

Outcomes

Indicators

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

The student is able to: • read frequencies from a bar graph and hence describe the data.

Skills

Resources

Language

• calculator • Base 10 MAB

• patterns • graph • weigh • between • mass • line graph

• reading graphs • recording information gathered from graphs

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Memory Masters (N3.4)

• • • • •

subtract heaviest difference highest justify

Notes

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

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&D3.4) Warm up

• Explain to the class that we use graphs for a variety of reasons and graphs are represented in many forms. If available, display copies of graphs as charts or on an overhead or computer. • Perform a newspaper search for graphs. List the different types found. • Today’s activity uses two types of graph: one is a bar graph which shows data information in the form of a bar or solid block; the other is a line graph which shows data using a line that joins points of plotted information.

What to do

Challenge

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• Ask students to look at the bar graph. What information does the graph show? (A display of the mass of a group of children.) The mass is shown on which axis? (Vertical) The children are shown on which axis? (Horizontal) • Read the questions to Exercise 3 to the class. Ask for the answer and the reason for giving that answer. Students record their answers. • When finished with the bar graph, direct student attention to the line graph. • Students complete the questions related to the line graph.

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• Using the temperature graph, students are to use mathematical reasoning to come up with what they think the mean temperature for September is likely to be. • There is no correct answer. Student’s justification or rationalisation of their answer is what you will be looking for. • Remind students to record their thinking in full.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 116–119. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 27 •

Unit 2–3

Student page 6

Outcomes

Indicators

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

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

Skills • reading place value • recording place value

Resources • calculator • Base 10 MAB • coloured pencils • place value chart (see pages 205– 206) • counters

Language • change • • kilograms • • place value • • hundredths • • hundreds • • tenths • • operation • • predictions • • least number

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grams subtract ones tens thousands table addition pattern

Notes

Memory Masters (M3.1, 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, N3.3) Warm up

• Organise the class into small groups. • Distribute Base 10 MAB and individual place value charts (or display large place value chart). • Ask the students how they might show a decimal point if they were using Base 10 MAB to make a decimal number. Accept all answers—students may make their own choice.

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• Ask the students, in their groups, to use the Base 10 MAB to make all the numbers given in Exercise 3. Check all groups as they are working, questioning why they have used certain wood, what they are using to show the decimal, and to read the numbers shown to you. • Ask members of groups to read the numbers shown to the class and to describe the wood they have used. Ask if any group has a different representation; if so, explain it. Remind students that the wood has no specified value until it is given one for each different activity. • Use the place value chart to find the three places to the left of the hundredths place and write these in the workbook. • Ask students to enter 59 into their calculator. If you add two tens, or 20, which place value column do you think will be affected? Write this in the column provided. Use your calculator to add 20 and see which columns are affected. Write this in the column provided. • Continue with each example, allowing confident students to move on by themselves.

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Challenge

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What to do

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• The focus for this unit is conversion of grams to kilograms and kilograms to grams.

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• Students are required to keep a record of their trials and a description of their final answer. • Remind students that two areas with a common line boundary can not have the same colour.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 36–39. • 28 • New Wave Maths Book E – Teachers Guide

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

Unit 2—Answers

Student pages 4–6 Unit 2–1

1. (a) 20 (b) 35 (c) 9 (d) 16 (e) 36 (f) 40 (g) 0 (h) 15 (i) 37 (j) 14 2. (a) 25 (b) 32 (c) 29 (d) 18 (e) 67 (f) 14 3. (a) Holly (b) 50 kg (c) Michelle and Kelly (d) 40 kg (e) 30 kg 4. (a) 19th and 20th (b) 26 ºC, 27 ºC, 27.5 ºC, 28 ºC, 28.5 ºC, 29 ºC (c) 3 ºC Challenge Teacher check (Answer should indicate an understanding of ‘mean’)

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1. (a) 18 (b) 35 (c) 64 (d) 27 (e) 42 (f) 6 (g) 8 (h) 1 (i) 4 (j) 7 2. (a) 52 (b) 21 (c) 23 (d) 22 (e) 62 (f) 62 3. Teacher check; 3 x flats, 3 x longs, 6 x units 4. 300 longs 30 flats 4. 6 longs 5 longs, 3 units Challenge Answers will vary; one possible solution – 7 moves.

Unit 2–2

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• Play trading games using Base 10 MAB.

Consolidation 2–2

Consolidation 2–3

• Collect data and display on a suitable graph.

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1. (a) 4783 g (b) 2941 g (d) 3860 g (e) 1900 g (g) 8.99 kg (h) 1.743 kg (j) 2.548 kg (b) 247 2. (a) 221 (d) 441 (e) 252 3. Teacher check. 4. tens, ones, tenths 5. (a) Teacher check., tens (b) Teacher check., hundreds (c) Teacher check., hundreds (d) Teacher check., tens (e) Teacher check., ones (f) Teacher check., hundreds Challenge 4

• Students collect numbers from their everyday environment to record on their place value chart (see pages 206–206). Numbers could be: year, house number, favourite number, cost of something etc.

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New Wave Maths Book E – Teachers Guide • 29 •

Unit 3–1

Student page 7

Outcomes

Indicators

N3.3, S3.2

The student is able to: • inspect a prism or pyramid, put it aside and then select 2-D shapes to match the faces of polyhedron.

Skills • observing • manipulating • analysing • recording

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • 3-D shapes • straws and modelling clay or similar

Language • multiply

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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 (S3.2) Warm up

• Distribute a range of 3-D shapes for students to observe. • Discuss with students the shapes (2-D) used to make up the 3-D shapes. Discuss the ‘faces’ of each shape. For example, a triangular pyramid is made up of four triangles.

• Look at the photographs displayed in the student workbook. • Ask students to name the 3-D shapes (rectangular prism, cube, cone and hexagonal prism). • Explain to students they are to colour the 2-D shapes below each photograph which could be used to make the shape in the photograph. • Students complete the activity.

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Challenge

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What to do

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

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For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 10–13. • 30 • New Wave Maths Book E – Teachers Guide

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

Unit 3–2

Student page 8

Outcomes

Indicators

N3.3, N3.4

The student is able to: • use their own methods or a conventional algorithm to multiply whole numbers by single-digit numbers. • build sequences of simple shapes such as triangles, squares, ‘L’ or ‘T’ shapes, which increase in size systematically, and write the equivalent number pattern. • identify particular terms in a sequence.

Skills • recognising patterns • calculating • manipulating

Resources

Language • multiply • calculator • pattern • square numbers • square arrays • grid • hexagonal numbers

• calculator • Base 10 MAB • square tiles

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Notes

Memory Masters (N3.3)

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

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

• Explain to the students they will be exploring a number of patterns using their calculator and then following the pattern mentally, if possible. • Arrange the class into small groups. Each student should have his/her calculator.

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• Working with the whole class, ask them to enter 37 x 3 = into the calculator. What answer did you obtain? (111) Write this in your book. • Next enter 37 x 6 = into the calculator. What answer have you found? (222) • What might you expect to find when you enter in 37 x 9? Try it. (333) • Try with 37 x 12. You will make? (444) • Where possible, without using the calculator, write the continuation of the pattern for Exercise 3 (a) to (o). • Ask them to predict the result for 60 x 37 then check their results. • Did you find the results you expected? Explain your findings. • Exercise 4 requires the distribution of square tiles to each group. • Using the tiles, find out which of the numbers shown can be made into a square array. Draw each array that you make on the grid. (An array is made by joining the tiles together by full side contact. There are many different arrays that may be made. In this exercise the aim is to find square arrays.) • Square arrays show square numbers.

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Challenge • Having made square numbers in the main activity, students are to follow the concrete practice to make an educated guess on how they may show pentagonal numbers. • Armed with this knowledge, students are to find the first three pentagonal numbers. • Record all thinking and findings to share with the teacher and the class.

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

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 31 •

Unit 3–3

Student page 9

Outcomes

Indicators The student is able to: • use uniform units to measure, length, capacity and mass to needed levels of accuracy.

N3.3, M3.2

Skills • measuring

Memory Masters (N3.3)

Resources • calculator

• ruler • bucket of water • calibrated measuring jug • plastic cup, aerosol cap, small tin and other containers • balance scales and weights • duster, dictionary, pad, exercise book and others

Language • multiply • measure • nearest centimetre • calibrated • measuring jug • balance scales • capacity • mass • length

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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 (M3.2) Warm up

• Organise class into small groups and provide each group with the materials as described. • Remind students that when measuring using a ruler, the 0 mark is the starting point for measuring, not the end of the ruler, and is placed level with the end of the line to be measured.

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• Ask students to measure the lines to the nearest centimetre. • Half fill a bucket with water for each group or pair of groups. Students fill the containers carefully and transfer the water to the measuring jug so they are able to measure the capacity of the container to the nearest 10 mL. • Use the balance scales to balance the given objects against the weights to find the mass of the objects to the nearest 100 grams.

Challenge

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• Students are to record their chosen method for measuring the length of wood giving justifiable reasons for their choice. • Show how their method is used to demonstrate its accuracy. Draw diagrams to show the method.

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What to do

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

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• 32 • New Wave Maths Book E – Teachers Guide

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

Unit 3—Answers

Student pages 7–9 Unit 3–1

1. (a) 13 (f) 9 2. (a) 69 (f) 88 3. (a)

(b) 12 (g) 6 (b) 48

(d) 7 (i) 3 (d) 88

(e) 14 (j) 9 (e) 99

(b)

(c)

(b) 12 (g) 3 (b) 92

(d) 10 (i) 7 (d) 58

(f) 1110 (g) 1221 (h) 1332 (i) 1443 (j) 1554

(k) 1665 (p) 2220 (l) 1776 (m) 1887 (n) 1998 (o) 2109

(e) 15 (j) 1 (e) 96

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1. (a) 10 (f) 4 2. (a) 81 (f) 98 3. (a) 555 (b) 666 (c) 777 (d) 888 (e) 999 4. 9, 25, 16

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Challenge Teacher check

Unit 3–2

Challenge Pentagonal numbers may be shown as a pentagon. The first three are

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• Students can cut up boxes etc. to find the 2-D shapes used to make them.

Consolidation 3–2

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1. (a) 7 (b) 13 (c) 9 (d) 14 (e) 11 (f) 2 (g) 2 (h) 6 (i) 8 (j) 3 2. (a) 248 (b) 568 (c) 159 (d) 328 (e) 637 (f) 246 3. (a) 14 cm (b) 6 cm (c) 9 cm (d) 11 cm 4. Teacher check 5. Teacher check Challenge Teacher check. Answers will vary: One possible solution is: Use something that I know the length of; e.g. a sheet of A4 paper, or a part of the body such as a finger or foot.

• Look for patterns of triangular, square, pentagonal etc. numbers in the times tables chart (see page 203).

Consolidation 3–3

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R.I.C. Publications®

• Students develop their own calibrated tools for measuring; i.e. make own ruler, measuring jug etc.

New Wave Maths Book E – Teachers Guide • 33 •

Unit 4–1

Student page 10

Outcomes

Indicators The student is able to: • identify the starting number and the constant multiplier needed to generate a number sequence.

N3.3, N3.4

Skills

Resources • calculator • Base 10 MAB • pencil

• finding patterns • recording

Memory Masters (N3.3)

Language • divide • patterns • rule • multiplication grid • diagonal

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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

• As an introduction to this exercise, remind the class that mathematics is based on patterns and logic. • What is a pattern? (A pattern is a set of numbers or objects that are generated by following a specific rule.)

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• Look at Exercise 3 (a): What pattern can you see? (Counting by sevens, seven times table.) What will the next four numbers in the pattern be? Write these on the page to complete the pattern. • Follow the same process for each of the other patterns in Exercise 3. • Exercise 4 requires students to complete the multiplication grid for the tables shown. Remind students that the number to be placed in the square is the answer to the multiplication of the number below the square to the number beside the square. For example, in the top right-hand square the number below is 5 and the number beside is 10. The answer to 5 x 10 is 50. Write 50 in the square. Check student response. • Complete the grid. • Ask students to describe the patterns generated on each diagonal.

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

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• Students have been given a pattern, they are to give an answer to the question of how many dots will there be in the fifth row without drawing a continuation of the pattern. • When arriving at the answer, students are required to write an explanation of their reasoning. • Students then check their answer by completing the rows.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 70–73. • 34 • New Wave Maths Book E – Teachers Guide

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

Unit 4–2

Student page 11

Outcomes

Indicators The student is able to: • specify how frequencies or measurements are to be made. • display frequency data in bar graphs where one axis shows the whole numbers. • accurately measure height using calibrated tools.

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

Skills • measuring • recording

Resources

Language

• calculator • Base 10 MAB • measuring stick or tape • pencil • ruler • coloured pencils • newspapers

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• divide • measure • height • record • bar graph • title • temperature • construct • line graph

Memory Masters (N3.3)

Notes

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

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, M3.2) Warm up

• Organise the class into groups of five. • Explain the use of the measuring stick and ensure students know how to use it. • This activity can be turned into an open-ended task by asking each group to devise its own method of measuring members’ height using any of the equipment provided – tapes, measuring stick, chalk, metre rulers, string, streamer etc.

What to do

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• Students measure each other’s height and record measurements on the table provided. • Using an overhead or blackboard/whiteboard, draw up (or have prepared) a graph outline like the one in the workbook. • Show students how to mark the height axis in centimetres or multiples of 2 or 5 centimetres as required, within the range of measured heights. If the range is greater than 16 cm then multiples of 2 or other will be required to construct the graph. • Show students, or ask a student to demonstrate, how to construct a bar for the bar graph. • Ask students to show their own height as the first bar on the graph. Check work. • Allow students to plot the rest of the graph helping or demonstrating to those who have trouble. Alternatively, encourage peer tutoring. Bars may be coloured. • When the bar graphs are completed, use a pre-prepared graph to show how a line graph is made. In this case it is the meeting point of the axis, not the column or bar that shows the point on the graph. • Students find all points, with teacher and their peers checking, then join each adjacent point, using a ruler, to make a line graph.

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Challenge • Look through a newspaper and cut out any examples of graphs that you can find.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 116–119. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 35 •

Unit 4–3

Student page 12

Outcomes

Indicators

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

The student is able to: • use the decimal point in representing quantities and money. • understand that units are needed when direct comparison is not possible or we wish to know ‘how big …’ or ‘how much bigger …’.

Skills • converting • calculating

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • place value chart (see pages 205– 206)

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.

Notes

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Number (N3.3)

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• Demonstrate to the whole class using the blackboard/whiteboard to show how kilograms and grams can be written in kilograms using a decimal point. • Explain to the class that when converting kilograms and grams to kilograms, kilometres and metres to kilometres, and litres and millilitres to litres that there will be a maximum of three decimal places. One of each of the smaller measures is one one-thousandth of the larger measure; hence one metre, or one gram or one litre is 0.001 km, kg or L. • Explain the origin of the prefix ‘kilo’ meaning 1000. • Provide opportunities for students to develop a feel for the relative size of measures to help them make the appropriate conversion. • When converting it is easiest to separate the larger measure from the smaller measure to confirm the placement of the decimal point between the two measures. • Say to students, when faced with 200 m, do you need to write this as 0.200 km? If not, how may it be written? (Yes as 0.2 km) This is because the zero on the end of a decimal number has no hundredths, no thousandths. Remember that if the number is 0.202 or 0.002 then the zeros do have value and must be shown.

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Main Activity (N3.1a, M3.1)

What to do

• convert • divide • measures • pattern • sum • whole numbers • total

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• The ‘Today’s number is …’ activity ask students to list all they know about a particular number; e.g. Today’s number is … 12 2 + 2 + 2 + 2 + 2 + 2 = 12, 3 x 4 = 12, 24 ÷ 2 = 12, 120 ÷ 10 = 12, 20 – 8 = 12, 2 x 6 = 12, 2 x 2 x 3 = 12, 100 – 88 = 12 etc.

Warm up

Language

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• Direct students to complete both exercises. Check for accuracy and understanding.

Challenge • In the exercise students are to find the total of the numbers 1 to 10 inclusive. Having found the total they are then to explore alternative simpler methods of finding the total. • All attempts are to be recorded and have explanations attached. • Share findings. • Apply findings to find the total of 1 to 99 inclusive.

• 36 • New Wave Maths Book E – Teachers Guide

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

Unit 4—Answers

Student pages 10–12 Unit 4–1

1. (a) 8 (b) 10 (c) 13 (d) 6 (f) 3 (g) 9 (h) 0 (i) 1 2. (a) 41 (b) 53 (c) 51 (d) 51 (f) 41 3. (a) 28, 35, 42, 49 (d) 26, 32, 38, 44 (b) 44, 55, 66, 77 (e) 29, 37, 45, 53 (c) 21, 25, 29, 33 4. Patterns in the diagonals are:

(e) 6 (j) 4 (e) 92

1. (a) 5 (b)10 (c) 10 (d) 8 (e) 16 (f) 0 (g) 6 (h) 9 (i) 2 (j) 3 2. (a) 201 (b) 102 (c) 101 (d) 403 (e) 301 (f) 302 3. Teacher check 4. Teacher check Temperature Readings for the Week 5.

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40 35 30 25 20 15 10

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0, 10 = +10 0, 9, 20 = +9, +11 0, 8, 18, 30 = +8, +10, +12 0, 7, 16, 27, 40 = +7, +9, +11, +13 0, 6, 14, 24, 36, 50 = +6, +8, +10, +12, +14 5, 12, 21, 32, 45 = +7, +9, +11, +13 10, 18, 28, 40 = +8, +10, +12 15, 24, 35 = +9, +11 20, 30 = +10 Challenge 5 dots in the fifth row. Teacher check reasoning.

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Unit 4–2

5 0

Mon Tue Wed Thu

Fri

Sat

Days of the Week

Challenge Teacher check

Sun

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• Provide opportunities for students to develop their own patterns.

Consolidation 4–2

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1. Answers will vary; some possible solutions 2x7 10 + 4 20 – 6 28 ÷ 2 2. (a) 330 (b) 220 (c) 120 (d) 130 (e) 320 (f) 410 3. km m km (a) 5 256 5.256 5.256 km (b) 4 827 4.827 4.827 km (c) 2 300 2.300 2.3 km (d) 8 904 8.904 8.904 km (e) 9 400 9.400 9.4 km 4. L mL L (a) 5 804 5.804 5.804 L (b) 3 750 3.750 3.75 L (c) 8 600 8.600 8.6 L (d) 7 300 7.300 7.3 L (e) 1 580 1.580 1.58 L Challenge The sum is 45. Teacher check: One possible method is: 1 + 9 = 10 x 4 = 40 + 5 = 45 applied to adding the numbers from 1 to 99 (100 x 49) + 50 = 4950

• Record and graph other data as selected by the students.

Consolidation 4–3

• Provide further practice for students to write measures using decimal places.

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New Wave Maths Book E – Teachers Guide • 37 •

Unit 5–1

Student page 13

Outcomes

Indicators

WM3.2, WM3.3, WM3.4,

The student is able to: • generate questions from data in a table. • use their understanding of tables to make conjectures about patterns and relationships.

N3.4

Skills • poses questions • analyses data • makes conjectures

Resources • calculator (optional)

Language • tables • patterns • eight • multiply • addition • subtraction • relationship

• • • •

increase decrease nine multiples

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What to do

• This activity is designed for students working collaboratively in groups. Allow enough times so that 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 to investigate the first part of the activity. – How do the units increase/decrease? – How do the tens increase/decrease? – What happens if you add the two-digit numbers in the answers? – What happens if you subtract the two-digit numbers in the answers? – Which numbers are odd and which are even?

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Main Activity (WM3.2, WM3.3, WM3.4, N3.4)

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• The eight and nine times tables are related. This means there is a pattern between them. Can you find it? (HINT: If you can’t see the pattern by looking at the chart, write both sets of tables side by side.) • From the last exercise, students may deduce the relationship between the eight and nine times tables. The relationship is: 1 x 9 = 1 x 8 + 1 2x9=2x8+2 3 x 9 = 3 x 8 + 3 and so on. • Groups may wish to collate and summarise their findings and present them as a poster with a series of graphs, diagrams and information. • 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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• 38 • New Wave Maths Book E – Teachers Guide

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Unit 5–2

Student page 14

Outcomes

Indicators

N3.1a, N3.3

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

Skills • grouping

Resources

Language • round • nearest 10 • add • rearrange • table

• calculator • Base 10 MAB

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Notes

Memory Masters (N3.1a)

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• The focus for this unit is rounding to the nearest 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 (N3.3) Warm up

• Ask students to add the following pairs of numbers; 6 + 4; 8 + 2; 1 + 9; 3 + 7; 5 + 5; 7 + 7; 4 + 4; 8 + 8. What did you find when you added these numbers? (Added to 10; easy to add; simple adding pairs.) • Explain to students that when adding lists of numbers it is easier to add combinations that total 10 or add common pairs of numbers.

Challenge

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• Exercise 3 requires the list of numbers to be rearranged so that the numbers are paired for easier addition. • Share the pairings with the class. • Exercise 4 requires students to add the number shown at the beginning of each row to the numbers in the boxes of the row in turn. Write the total below the number in the box.

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• Students are to record their answer so that it is self-explanatory to another person. Use the method of their choice to show the answer.

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New Wave Maths Book E – Teachers Guide • 39 •

Unit 5–3

Student page 15

Outcomes

Indicators

N3.3, M3.1, S3.3

The student is able to: • refer to gaps and overlaps in explaining differences in the number of units taken to cover a region. • choose shapes that can cover a region with no gaps. • refer to the size of the unit used to explain differences in the number of units taken to cover an object. • provided with multiple copies of a simple figure, decide whether or not it will tile.

Skills • tessellating • summarising

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • 2-D regular shapes • paper • ruler

Language • add • regular shapes • tile • space

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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 (M3.1, S3.3) Warm up

• Ask students if they can explain the difference between a regular and an irregular shape. (Regular shape has all sides and internal angles equal – square, equilateral triangle etc.) • Distribute a set of regular shapes, including a square, an equilateral triangle, a regular pentagon, a regular hexagon and any others available. Allow students to examine the shapes to see that sides and internal angles are equal.

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• Select one of the regular shapes you have examined and use it to tile the space provided in your pad. Lightly trace around the edges of the shape and repeat until the space has been covered. • Where parts of the shape overhang the edges of the space provided, truncate, or cut off, the overhanging parts. • Once you have completed the task with one regular shape, choose another regular shape, trace over the space in your workbook onto a separate piece of paper and repeat the activity with your second shape. • This activity may be repeated a number of times. • Write about your tiling, describing the best regular shape(s) to use to cover the space effectively.

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What to do

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

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• Do all quadrilaterals (four-sided shapes) tessellate?

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 22–23. • 40 • New Wave Maths Book E – Teachers Guide

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Unit 5—Answers

Student pages 13–15 Unit 5–1 1. (a) 80 (b) 40 (c) 50 (d) 90 (f) 20 (g) 70 (h) 10 (i) 30 2. (a) 687 (b) 568 (c) 869 (d) 956 (f) 959 3. Teacher check rearrangement of numbers. (a) 17 (c) 19 (e) 18 (g) 17 (b) 15 (d) 27 (f) 24 (h) 31

(e) 30 (j) 90 (e) 848

(i) 16 (j) 23

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(b)

(c)

Challenge You are the only person in the race, thus first and last.

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1. Patterns in the 9 times table include: • The units digits increase by one. • The tens digits increase by one. • When the two digits that form the answer are added together they make a nine or a multiple of nine. • The difference between the digits that form the answer produce a field of odd numbers: 9, 7, 5, 3, 1, 1, 3, 5, 7 2. A pattern in the eight times table is that the unit digits forming the answers decrease by two and then begin again: 8, 6, 4, 2, 0, 8, 6, 4, 2, 0 Challenge The relationship is: 1x9=1x8+1 2x9=2x8+2 3 x 9 = 3 x 8 + 3 and so on.

Unit 5–2

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• Use the flip array or tables charts to find other patterns. Focus on sets of tables to discover any relationships between them; for example, 8 and 4 times tables.

Consolidation 5–2

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1. (a) 24 (b) 63 (c) 40 (d) 36 (e) 63 (f) 2 (g) 9 (h) 9 (i) 9 (j) 6 2. (a) 865 (b) 385 (c) 884 (d) 971 (e) 671 (f) 785 3. Teacher check. Students should clearly verify their findings using the diagrams drawn. Challenge No

• Provide students with further opportunities to add a constant number such as those in Exercise 4.

Consolidation 5–3

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R.I.C. Publications®

• Search through magazines to find examples of tiling patterns.

New Wave Maths Book E – Teachers Guide • 41 •

Unit 6–1

Student page 16

Outcomes

Indicators

N3.3, N3.2

The student is able to: • understand the terms ‘multiple’, ‘factor’, and ‘prime’ and use them appropriately.

Skills • counting • drawing • comparing

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • 1-cm grid paper (see page 199) • square tiles or 2-cm cubes

Language • subtract • odd and even • numbers • rectangular array • prime and composite numbers • divisible • consecutive

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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 (N3.2) Warm up

• Ask students to explain what they know about odd and even numbers. • Explain to students that we can show odd and even numbers by drawing a 2 column, or a 2 row array. An odd number will have one column/row with one extra square. An even number will have columns/rows with even number of squares. • Demonstrate showing 2, 3, 4, and 5, on the blackboard/whiteboard or overhead projector.

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

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What to do

• Ask students to draw the arrays for the numbers shown. Supply 1-cm grid paper if students find this easier to use. From the array determine whether the numbers are odd or even. • Prime and composite numbers may also be shown by using arrays. In these cases arrays may be made up of multiple columns to show the different factors; e.g. 9 may be shown as 3 x 3 array; 12 by a 2 x 6, 3 x 4 array. • Prime numbers are those that can only be shown by a 1 x the number itself array; e.g. 7 can only be shown by a 1 x 7 array. • A prime number has exactly two factors, 1 and itself. (Hence 1 is not a prime number—only one factor.) • Draw the arrays for the numbers shown and decide which are composite and which are prime.

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Challenge • Students will need to explore the range of prime numbers to determine whether there are any other consecutive prime numbers. • Some students may be able to use their knowledge of numbers and logic to find the answer quickly. • In all cases a recorded description of findings must be given.

• 42 • New Wave Maths Book E – Teachers Guide

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

Unit 6–2

Student page 17

Outcomes

Indicators

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

The student is able to: • suggest a suitable way to classify data in order to answer straightforward questions. • interpret straightforward one- and two-way tables.

Skills • working collaboratively • recording • counting • analysing data

Resources

Language

• calculator • Base 10 MAB • watch

• subtract • tally • total • most common • least common

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Memory Masters (N3.3)

Notes

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

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&D3.4) Warm up

• Revise keeping a tally. Students to explain and demonstrate on the blackboard/whiteboard how a tally is kept. Remind students that the diagonal tally mark represents the fifth item in the count. Tally sets are in lots of five for ease of counting. (1111)

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• The activity today will be to keep a tally of all vehicles that pass the school grounds in a 15-minute period. • Read the question with the students and ask them what information needs to be collected. Set up the table to reflect this information. Suggested colours could be red, green, yellow, black, white, blue etc. • You will be working in small groups. (If the traffic is very heavy suggest that the group splits into two with one group tallying traffic travelling one way and the other group tallying the traffic travelling in the opposite direction. It may also help to have one member spotting and another tallying; a third member may act as backup in case a vehicle is missed.) • When the 15-minute time limit is up everyone moves back inside. Within each group ensure every member has recorded a copy of the tally and the total for each car colour. • Use the information recorded to answer the questions. • When convenient, repeat the exercise on different days and at different times through the day to compare results.

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• Students will need to devise their own method to find the answer to the question. • Students will need to decide on a means of representing their information for the rest of the class to see. • A report discussing the results is then to be written to accompany the display of the results.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 112–113. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 43 •

Unit 6–3

Student page 18

Outcomes

Indicators The student is able to: • estimate sums by rounding.

N3.3

Resources • calculator • Base 10 MAB

Skills • estimating • calculating • recording

Memory Masters (N3.3)

Language • subtract • approximately • double • halve • add • take away

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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 (N3.3) Warm up

• Ask students to explain what they understand by the term ‘approximately’. • Settle on the understanding that it is a close estimate or an educated guess of an answer. • Ask students if they know of a means of determining a close estimate or an approximation of an answer. • From possible suggestions, settle on the use of rounding. • Ask students what place they would round to if the numbers they were working with were individually: 657 (hundreds), 2756 (thousands), 706 (hundreds), 94 (tens) or 7989 (thousands). • Revise the rules for rounding – greater than 5, 50, 500 or greater round up to the nearest ten, hundred or thousand respectively. If 4, 40, 400 or less round down. • Note: If you required a piece of wood to be 2.3 m long you would not round down to 2 m but rather round up.

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

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What to do

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• The activities on the page ask for an approximation of the answer to addition or subtraction sums. Using the information from this lesson to date, work out, using rounding, the approximate answer to 3 (a). Check working. • Students continue to complete the exercises, provide assistance as required.

Challenge

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• Follow the directions given. Record your findings for each part of the exercise and for each subsequent check, using a different starting point. • Share your results with the class and your teacher.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 64–65. • 44 • New Wave Maths Book E – Teachers Guide

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Unit 6—Answers

Student pages 15–18 Unit 6–1

1. (a) 30 (f) 8 2. (a) 239 (f) 516 3.

(b) 7 (g) 2 (b) 628

(d) 18 (e) 72 (i) 4 (j) 2 (d) 147 (e) 718

1. (a) 56 (b) 48 (f) 1 (g) 8 2. (a) 263 (b) 272 (f) 192 3. Teacher check Challenge Teacher check

(d) 9 (e) 28 (i) 6 (j) 3 (d) 161 (e) 291

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Unit 6–2

Prime numbers: 2, 3, 5, 7, 11, 13 Composite numbers: 4, 8, 9, 10, 12 Challenge No. All even numbers are divisible by two.

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• Provide further opportunities for students to investigate prime and composite numbers.

Consolidation 6–2

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1. (a) 18 (b) 42 (c) 56 (d) 54 (e) 36 (f) 4 (g) 8 (h) 7 (i) 1 (j) 7 2. (a) 238 (b) 166 (c) 159 (d) 357 (e) 287 (f) 536 3. Teacher check estimates. Actual answers: (a) $56.20 (b) 16 432 vehicles (c) 4570 sheep (d) $833 (e) 1566 supporters Challenge The answer is … 3. The answer doesn’t change.

• Discuss how the data could be used and allow students to follow-up suggestions.

Consolidation 6–3

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R.I.C. Publications®

• Students can write their own word problems for other students to give approximate answers to.

New Wave Maths Book E – Teachers Guide • 45 •

Unit 7–1

Student page 19

Outcomes

Indicators

N3.1a, N3.3, S3.2

The student is able to: • talk about what they can and cannot see of an object from different positions and attempt to draw what they see rather than what they know to be there.

Skills • manipulating • drawing • recording • working logically

Memory Masters (N3.1a)

Resources • calculator • Base 10 MAB • 2-cm cubes

Language • multiply • arrangements • full face • cube • draw • sections • cheapest • join • cut

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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 (S3.2) Warm up

• Distribute five 2-cm cubes to each student.

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• Explain to the students they are to use the five cubes to make as many different shapes as possible. When making the shapes the full face of one cube must be in contact with the full face of another cube. • Rotations and flips are to be treated as the same shape in a different orientation. • Cubes may be stacked on each other as well as lying on the desk top. Stacked cubes must also be in full-face contact with another cube. • Draw each shape as it is made so that it is not forgotten. Stacked shapes will need to be drawn in perspective. Students may require assistance to complete these drawings.

Challenge

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• Students will need to show their thinking process by writing explanations and drawing diagrams to show the chain being joined. • Students share their results with the teacher and/or a small group.

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What to do

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Teac he r

• The focus for this unit is the conversion of dollars to cents and cents to dollars.

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• 46 • New Wave Maths Book E – Teachers Guide

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Unit 7–2

Student page 20

Outcomes

Indicators

M3.1, N3.3

The student is able to: • read and right fractional notation to represent unit fractions.

Skills • adding • recording

Resources

Language • multiply • millilitres • litres • fractions • denominators • numerator

• calculator • Base 10 MAB • coloured rods • 1-cm grid paper (see page 199)

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Notes

Memory Masters (M3.1)

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• The focus for this unit is the conversion of litres to millilitres and millilitres to litres.

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) Warm up

• Use the coloured rod whose value (number given to it) corresponds to the denominator as the unit rod; e.g. if the denominator is 8 use the 8 rod as the unit rod—it represents eight eighths. • If adding 3/8 + 2/8 the eight rod may be placed on the desk to represent eight eighths. The three rod and the two rod may then be laid end to end beside the eight rod. The rod that is the same length as the 3 and 2 rods may then be laid along side them to show that 3 + 2 = 5. The answer is 5 of 8 or 5/8. • Alternatively, grid paper may be used to show an array of 8 squares. This may be a row of 8 or two rows of 4. To show 3/8 + 2/8 shade 3 squares to represent 3/8 and then another 2 squares to represent 2/8. A total of 5 squares have been shaded; i.e. 5/8 of the total. • Repeat either or both of these processes until students understand all what is required to add the two numerators and write them over the common denominator. Eight is the common denominator in the above example, therefore 3/8 + 2/8 = 5/8.

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What to do

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• Complete Exercise 3 using one of the methods above to assist if necessary. • In Exercise 4, students must identify that 4/4, 5/5 and so on represents 1 whole.

Challenge

• Students brainstorm as many fractions that equal one whole as they can. • Share results with classmates.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 50–51. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 47 •

Unit 7–3

Student page 21

Outcomes

Indicators The student is able to: • talk about what they can and cannot see of an objects from different positions and attempt to draw what they see rather than what they know to be there.

N3.1b, N3.3, S3.2

Skills • making models • drawing • report writing

Memory Masters (N3.1b)

Resources • calculator • Base 10 MAB • 2-cm cubes (16 per person or group) • pencils

Language • multiply • prisms • model • report • greater than > • less than < • equal to =

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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) Warm up

• Ask students to describe a prism. • Show models of prisms; e.g. duster, box, book. • Organise students into small groups and distribute 2-cm cubes (16 per person or group).

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• Direct students to make as many different models of prisms as possible using the 2-cm cubes. Remind students that rotations, flips and translations all show the same prism. • Direct students to draw the model of each of the cubes when they finish making it. • Note: there are different ways to draw—isometric (3-D), top view, front view or side view. • Ask students to write a short report highlighting any interesting features of the models they have made. • Share these features with the class.

Challenge

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What to do

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• The focus for this unit is ordering of fractions with like denominators.

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• Students are to record their findings, keeping notes of their workings and explaining how they were able to find the triangles. • Share results with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 10–13. • 48 • New Wave Maths Book E – Teachers Guide

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Unit 7—Answers

Student pages 19–21 Unit 7–1 1. (a) 2600 mL (b) 1930 mL (c) 4076 mL (d) 3495 mL (e) 6800 mL (f) 1.725 L (g) 8.7 L (h) 1.043 L (i) 3.259 L (j) 4 L 2. (a) 369 (b) 848 (c) 486 (d) 848 (e) 993 (f) 246 3. (a) 4/5 (g) 2/4 (m) 5/6 (b) 2/6 (h) 6/8 (n) 3/4 6 4 (c) /7 (i) /8 (o) 7/9 (d) 4/10 (j) 3/5 (p) 6/10 (e) 2/3 (k) 5/7 (f) 8/9 (l) 8/10 4 4. (a) /4 = 1 (e) 8/8 = 1 (b) 5/5 = 1 (f) 6/6 = 1 (c) 10/10 = 1 (g) 9/9 = 1 7 (d) /7 = 1 (h) 7/7 = 1 Challenge Answers will vary; possible solutions are: 6 8 19 100/100 50/50 /6 /8 /19

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1. (a) 700c (b) 528c (c) 207c (d) 450c (e) 191c (f) $0.27 (g) $4.60 (h) $2.19 (i) $1.59 (j) $3.08 2. (a) 315 (b) 232 (c) 632 (d) 747 (e) 456 (f) 335 3. Teacher check Challenge Into one long chain $12. (Another possible answer is joining the chains at one central link which costs $1.50)

Unit 7–2

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• Discuss with students which shapes might be able to tile.

Consolidation 7–2 • Provide further opportunities for students to practise adding fractions with common denominators. Some students may find the use of diagrams helpful.

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1. (a) < (b) > (c) > (d) < (e) < (f) < (g) < (h) < (i) < (j) > 2. (a) 494 (b) 676 (c) 378 (d) 892 (e) 951 (f) 276 3. Teacher check The dimensions of the models are related to factors e.g. 16 x 1, 8 x 2, 4 x 4, 4 x 2 x 2. Challenge 8

Consolidation 7–3

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R.I.C. Publications®

• Discuss with students how the shapes change even though there are the same number of cubes.

New Wave Maths Book E – Teachers Guide • 49 •

Unit 8–1

Student page 22

Outcomes

Indicators The student is able to: • use their own methods or a conventional algorithm to multiply whole numbers by single-digit numbers. • explain why the multiplication/ division method used works.

N3.3

Skills • multiplying mentally • dividing mentally

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB

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.

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Teac he r

• divide • mentally • concrete • materials • multiplying • whole numbers • multiple • digit

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

• The focus for this unit is allowing students to explore and discover mental strategies to solve problems. • Students demonstrate facts they know which are related to the fact on the workbook page. They need to show how each calculation is related to each other; e.g. 12 x 20, I can see … 2 x 6 x 20, 2 x 2 x 3 x 20, 2 x 6 x 2 x 10, 2 x 6 x 2 x 5 x 2, 3 x 4 x 20, 3 x 4 x 4 x 5 etc.

Main Activity (N3.3) Warm up

Language

• For this activity have Base 10 MAB available for students to use if required.

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• Discuss with the students the best way to find the answers to each of the multiplication activities—some may choose to use 3 x 10 = 30 and 3 x 2 = 6, the total being 36. Other students may choose a different approach. The important point is to encourage students to think these steps through mentally and write answers only. • To assist students, work through the first four or five examples of Exercise 3 giving students time to find the answers.Then ask students to tell the class how they worked out their answer. Ask the class who worked out the answer the same way. Ask those who chose a different method to explain how they worked out their answer and for those who also chose the same method. • Direct students to complete the rest of Exercise 3. • To assist students, work through the first three or four examples of Exercise 4 giving students time to find the answers.Then ask students to tell the class how they worked out their answer. Ask the class who worked out the answer the same way. Ask those who chose a different method to explain how they worked out their answer and for those who also chose the same method. • Direct students to complete the rest of Exercise 4.

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Challenge • Students may be able to write their explanation immediately. • For those students who wish to try different examples, ask them to show their workings and then write their explanation. • Select a student to explain his/her findings to the class. If there are any different findings, ask for an explanation of these and why they are thought to be correct. • 50 • New Wave Maths Book E – Teachers Guide

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Unit 8–2

Student page 23

Outcomes

Indicators The student is able to: • record frequency data carefully using simple formats based on tallies or organised lists. • use diagrams such as Venn diagrams and two-way tables. • summarise data based on tallying. • summarise data in diagrams and tables which show frequencies for different categories. • report the frequency information provided in a tally produced by a classmate.

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

Skills • recording • collecting data • tallying • thinking logically • working collaboratively

Resources

Language

• calculator • Base 10 MAB • class members

• divide • Carroll diagram • survey • numbers • addition • digit

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

Memory Masters (N3.3)

Notes

Number (N3.3)

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• The focus for this unit is the addition of three numbers 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 (C&D3.2, C&D3.3, C&D3.4) Warm up

• Explain to the class that a Carroll diagram is a table that shows the relationship between two sets of information.

What to do

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Challenge

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• Look at the Carroll diagrams drawn on the page. The first one provides the names of four people; Brett, Rochelle, Coleen and Bob, and the names of four Houses that these people are captains of. This is a logic-type problem. • The second Carroll diagram shows a number of sports and columns to display the number of boys and/or girls who choose each sport as their favourite, as well as the provision for a total of the favourite sports. This one is used to easily record data collected. • The first Carroll diagram requires students to find who is captain of each House. Where a clue gives a definite no, place a cross in the box that shows this information. For example, Bob does not belong to Green—place a cross in the box under Green and along Bob, (the box in the bottom row and third column). Place a tick in a box that gives definite information. • The second Carroll diagram requires students to conduct a survey of their class. In small schools this may be conducted by each student. • The survey may be conducted by each student independently or as part of a small group. Alternatively, you may wish to ask for a show of hands of boys and then of girls to indicate their favourite sport. • Once the survey is completed, total the boys and girls then answer the questions on the page.

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• By using logical thinking and combinations of numbers, the numbers that correspond to the letters shown can be found. • Students are to record all attempts and explain their findings and the reasoning they used. • For those who are unable to find a start—provide the hint that C in the hundreds column can only be one number. Suggest students add 9 + 9 and 5 + 5 to find C.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 110–113. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 51 •

Unit 8–3

Student page 24

Outcomes

Indicators The student is able to: • use the multiplication/division relationship to state division facts from known multiplication facts and to solve missing number problems.

N3.3, N3.2

Skills • finding factors • multiplying • dividing

Memory Masters (N3.3)

Resources • Base 10 MAB • calculator

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)

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• In each example, one factor is missing from the sum. The task is to find the missing factor. (Calculators may be used if necessary.) • Ask students: – Look at 3(a). The sum shows 5 x 7 x _= 70. What do we know of the problem so far? (5 x 7 = 35) – What do we need to find? (The number to multiply 35 by to make 70.) – How can we do this? (By trial and error, or by using our knowledge that multiplication and division are the reverse of each other; i.e. we can divide 70 by 35 to find that 2 is the missing number.) • Students write 2 in the box. • Repeat this as often as required to develop an understanding of the process. • As students feel confident let them proceed at their own rate.

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• Begin with simpler multiplication problems; e.g. 7 x _ = 49. Ask students: What is the missing number? (7) How did you work it out? Provide more examples.Then direct students attention to the examples in the workbook.

Challenge

Notes

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Number (N3.3)

What to do

• divide • numbers • multiplication

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• The focus for this unit is the addition of two numbers less than 10 and the subtraction of a number less than 10.

Warm up

Language

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• Students are to experiment with the theory and then write an explanation. • Share these with the class.

• 52 • New Wave Maths Book E – Teachers Guide

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Unit 8—Answers

Student pages 22–24 Unit 8–2

Unit 8–1 1. (a) 14 (f) 10 2. (a) 9 (f) 8 3.

1. Answers will vary; some possible solutions are 5 x 5 x 2, 5 x 2 x 5, 5 x 1 x 10 2. (a) 82 (b) 71 (c) 43 (d) 81 (e) 91 (f) 81 3. (a) 36 (e) 28 (i) 40 (m) 88 (q) 208 (u) 95 (y) 96

(b) 76 (f) 96 (j) 52 (n) 72 (r) 60 (v) 567 (z) 470

Blue

Gold

(d) 12 (i) 16 (d) 8 Green

(e) 15 (j) 13 (e) 9 Red

Brett Rochelle

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(d) 48 (h) 32 (l) 37 (p) 16 (t) 25 (x) 20

4. Teacher check Challenge A= 9 B= 2 C= 1

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4. (a) 33 (b) 51 (c) 25 (e) 43 (f) 62 (g) 59 (i) 60 (j) 48 (k) 13 (m) 44 (n) 49 (o) 79 (q) 70 (r) 55 (s) 82 (u) 71 (v) 30 (w) 70 Challenge 0 will always be the last digit.

Teac he r

(d) 81 (h) 86 (l) 48 (p) 85 (t) 189 (x) 320

(b) 18 (g) 16 (b) 9

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(d) 5 (i) 5 (d) 23

(e) 4 (j) 13 (e) 25

• Provide students with more examples to further develop their skills.

Consolidation 8–2

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1. (a) 9 (b) 8 (c) 6 (f) 5 (g) 10 (h) 0 2. (a) 36 (b) 33 (c) 33 (f) 32 3. (a) 2 (g) 6 (m) 2 (b) 4 (h) 10 (n) 4 (c) 2 (i) 3 (o) 2 (d) 2 (j) 4 (p) 2 (e) 3 (k) 2 (q) 2 (f) 4 (l) 2 (r) 8 4. (a) 2 (g) 4 (m) 4 (b) 2 (h) 2 (n) 1 (c) 5 (i) 7 (o) 3 (d) 5 (j) 3 (p) 3 (e) 2 (k) 3 (q) 2 (f) 4 (l) 3 (r) 4 Challenge No. Teacher check reasoning.

• Students discuss and devise a survey question in which results can be collected using a Carroll diagram.

Consolidation 8–3

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R.I.C. Publications®

• Provide students with more examples to further develop their skills.

New Wave Maths Book E – Teachers Guide • 53 •

Unit 9–1

Student page 25

Outcomes

Indicators

N3.3, S3.1, S3.2

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 proximity of things in locating key features on maps. • make 3-D models to match their plan.

Skills • modelling • graphing • working collaboratively

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • pencil • paper/light card • glue • recycled material • LEGO®

Language • add • plan • space • models • relative size

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 (S3.1, S3.2) Warm up

• Read through the activity as outlined. • Ask students what it means to provide space for ‘safe passage’ around each site. Will all sites require the same safe passage? Why/Why not?

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• When drawing the plan, remind students they need to have in mind the approximate size of each of the rides or activities. • Also remind students that when planning the location of the rides or activities, they need to take into account possible safety issues such as golf balls and small children near merry-gorounds. • When making models, the models may be made to a larger scale than would fit on the plan, but sizes must be relative to each other. • Models may be drawn and cut from paper or light card, or may be constructed from recycled materials. • Students can work in small groups for this activity.

Challenge

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What to do

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Teac he r

• The focus for this unit is the addition of three addends all less than 10.

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• Provide each student, or small group of students, with six 2-cm cubes. • Explain to the students they are to arrange the six cubes so that 22 surfaces are showing.The surfaces on the bottom are also part of the showing surfaces. • Students draw diagrams of all arrangements made and give an explanation of how they discovered the correct arrangement.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 2–3.

• 54 • New Wave Maths Book E – Teachers Guide

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

Unit 9–2

Student page 26

Outcomes

Indicators

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 • reordering • regrouping • multiplying

Resources

Language • add • commutative (reordering) • associative (regrouping)

• calculator • Base 10 MAB

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

Notes

Memory Masters (N3.3)

Teac he r

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• The focus for this unit is the addition of two numbers followed by the subtraction of another, all 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 (N3.3) Warm up

What to do

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• Discuss the following with the students: In mathematics, there are a number of different terms to explain different processes that are used. It is not necessary for students to remember the names of these processes, but it is important they understand the processes. • The activities in Exercises 3 and 4 use two different processes. One is the commutative process which is the process of reordering. When adding or multiplying, the order in which we add or multiply is not important; e.g. 2 x 3 is the same as 3 x 2; 2 x 6 x 4 is the same as 6 x 2 x 4; 5 + 7 is the same as 7 + 5; and 4 + 3 + 3 is the same as 3 + 4 + 3. • The second process is the associative process or the regrouping process. In addition and multiplication it makes no difference to the answer if we change the grouping of the numbers we add or multiply. For example (3 + 7) + 4 is the same as 3 + (7 + 4) and 2 x (4 x 3) is the same as (2 x 3) x 4. Remember, brackets are dealt with first.

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• Direct students to complete Exercises 3 and 4 in the workbook. Calculators can be used to find the answers if necessary. • Provide assistance if required.

Challenge

• Students are to use brackets appropriately to make this a true statement. • Remind students that the internal set of brackets are worked out first and then the problem worked from left to right.

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New Wave Maths Book E – Teachers Guide • 55 •

Unit 9–3

Student page 27

Outcomes

Indicators

N3.3, M3.4a

The student is able to: • given a perimeter, use grid paper to generate possible shapes where the shapes fit along the grid lines.

Skills • manipulating • drawing • trial and error • problem-solving

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • 1-cm dot grid paper (see page 198) • geoboard • elastic bands

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)

Notes

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Teac he r

• add • fixed perimeter • place value • draw • shapes • geoboard • perimeter • area

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• The ‘Explain how you solved the problem.’ activity involves presenting a calculation to be performed mentally and then asking the students to explain how they went about solving it. • The activity encourages students to explore and discuss mental strategies.

Warm up

Language

• Explain to the class what it means to draw different shapes, each with a fixed perimeter. Using a transparent grid on an overhead or a grid drawn on a blackboard/whiteboard, show different shapes that can be drawn using a fixed perimeter of 12 cm. • Each shape must have each square within it joined to another square by a full side. Part-side joins or diagonal joins are not acceptable.

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• Provide students with a geoboard and elastic bands to experiment with. This will save unnecessary drawing and erasing if the perimeter is incorrect. • Set the class to work to find shapes with a fixed perimeter of 16 cm. • Ask students to show their different shapes by drawing them on the grid in the workbook. • Students count the number of units inside each shape to find the area.

Challenge

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What to do

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• How many different quadrilaterals can you make on a 9-pin geoboard?

• 56 • New Wave Maths Book E – Teachers Guide

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Unit 9—Answers

Student pages 25–27 Unit 9–1

(d) 17 (e) 9 (i) 19 (j) 13 (d) 938 (e) 875

1. (a) 8 (b) 7 (c) 0 (d) 8 (e) 10 (f) 4 (g) 3 (h) 1 (i) 8 (j) 1 2. (a) 1557 (b) 1178 (c) 1779 (d) 1368 (e) 1268 (f) 1757 3. (a) 2; 906 4. (a) (5 x 4) x 7 = 140 (b) 403; 806 (b) (2 x 3) x 9 = 54 (c) 204; 408 (c) (6 x 5) x 4 = 120 (d) 2; 608 (d) (3 x 3) x 8 = 72 (e) 602; 2408 (e) (5 x 2) x 9 = 90 (f) 5; 1505 (f) (4 x 2) x 7 = 56 (g) 201; 1809 (g) (8 x 5) x 3 = 120 (h) 702; 2106 (h) (4 x 5) x 9 = 180 (i) 501; 3507 (i) (5 x 5) x 3 = 75 (j) 4; 3208 (j) (7 x 2) x 10 = 140 5. Teacher check Challenge 2 – (3 – 2) – (2 – 2) =1

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1. (a) 15 (b) 17 (f) 12 (g) 18 2. (a) 977 (b) 657 (f) 833 3. Teacher check Challenge

Unit 9–2

(d) 922 (e) 824

• Students could formulate a fundraising plan: how much money needs to be raised, cost of hiring the equipment, charges for the rides, food stalls etc.

Consolidation 9–2

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1. Teacher check 2. (a) 646 (b) 991 (f) 825 3. 4 x 4 = 16 cm2 Teacher check

• Provide students with further opportunities to practise using the commutative and associative properties.

Consolidation 9–3

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• Students use a fixed perimeter of 25 cm. How many more polygons are they able to make now?

Challenge Teacher check

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New Wave Maths Book E – Teachers Guide • 57 •

Unit 10–1

Student page 28

Outcomes

Indicators

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

The student is able to: • distinguish and order whole numbers. • summarise data based on tallying.

Skills • recording • calculating • analysing • summarising

Memory Masters (N3.1a)

Resources • calculator • Base 10 MAB

Language • round • nearest thousand • subtract • frequency • lowest • highest • range • middle • most common

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, C&D3.3) Warm up

• Discuss the term ‘frequency’ with students. (The rate at which something occurs.) Blackboard/ whiteboard the following set of numbers: 8, 2, 7, 2, 9, 8, 3, 8, 5. Ask students what the frequency or occurrence of each number is: 8 three times or 3 out of 9; 2 twice and so on. • The lowest score and the highest score provide the range of scores. Explain this to the students then ask them to give the range—2 to 9. • To find the middle score, the list needs to be arranged in numerical order—2, 2, 3, 5, 7, 8, 8, 8, 9. The middle score would be the fifth number—7. This is called the median. • The most commonly occurring number is 8. This is called the mode.

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Teac he r

• The focus for this unit is the rounding of whole numbers to the nearest thousand.

What to do

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• Ask students to complete Exercise 3 in the workbook, giving assistance as required.

Challenge

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• Using the information and knowledge gained from the exercises above, students are to write an argument supporting their decision and share this with the class.

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For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 112–113. • 58 • New Wave Maths Book E – Teachers Guide

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

Unit 10–2

Student page 29

Outcomes

Indicators

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

The student is able to: • suggest a suitable way to classify data in order to answer straightforward questions. • tell the time on digital and analog clocks. • read and make straightforward schedules.

Skills • recording • discussing • listening

Resources

Language

• calculator • Base 10 MAB

• > greater than • < less than • = equal • subtract • 24-hour time

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

Memory Masters (N3.1a)

Notes

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Teac he r

• The focus for this unit is ordering whole numbers.

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, M3.2, M3.4a) Warm up

• Organise students into small groups. Ask the groups to discuss what they do on a normal school day. They are to discuss the activities over the full 24 hours. Groups are to include sleeping, eating, after-school activities such as sport as well as regular school activities. • Repeat the same discussion but focus on Saturday activities. • Ask students to return to their desks when the group discussion is finished.

What to do

Challenge

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• Ask students to record as accurately as they can the activities they participate in on a normal school day (any school day may be chosen). • When the above activity is completed, record as accurately as possible the activities they usually participate in on a Saturday.

• In Ancient history there were only 12 hours in a day (i.e. each hour was the same length as two of our current hours). Explain what would happen to your schedule if there were 12 hours in a day.

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For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 90–91. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 59 •

Unit 10–3

Student page 30

Outcomes

Indicators The student is able to: • partition two-digit numbers to assist in adding and subtracting them mentally.

N3.3

Skills • adding • regrouping

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • toothpicks or equivalent

Language • subtract • add • pairs of numbers • regrouping • squares

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) Warm up

• Explain to students that these exercises are aimed at developing their ability to complete sums mentally—that is to work the answers out in their heads and write the answers only on the page.

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• Work with the whole class on Exercise 3(a). Ask students to write the answer in the space provided. • Select a student to explain how he/she worked out the answer. Ask for a show of hands from the rest of the class who used the same method. Ask a student who did not use this method to explain how he/she worked out the answer and ask for a show of hands from students who used this method. • When all different methods have been discussed, remind students that each way is correct. We choose to use a standard method for teaching purposes but this does not preclude other methods being used. • This process may be repeated a number of times before allowing students to continue by themselves. • For students who are having difficulties, direct them to use Base 10 MAB.

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What to do

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Teac he r

• The focus of this unit is the basic facts of multiplication and division.

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• When undertaking this activity, remind students to show and describe each attempt they make in finding a solution.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 62–63.

• 60 • New Wave Maths Book E – Teachers Guide

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Unit 10—Answers

Student pages 28–30 Unit 10–1

1. (a) 6000 (b) 2000 (c) 8000 (d) 5000 (e) 4000 (f) 9000 (g) 3000 (h) 7000 (i) 5000 (j) 7000 2. (a) 80 (b) 62 (c) 92 (d) 80 (e) 62 (f) 72 3.

1. (a) = (b) > (f) < (g) = 2. (a) 68 (b) 87 (f) 46 3. Teacher check Challenge Teacher check

(d) = (i) = (d) 55

(e) > (j) < (e) 78

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(a) 2 (b) 10 (c) 2 to 10 (d) 2, 3, 3, 4, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10 (e) 8 (f) 9 (g) No. Answers will vary (h) No. Answers will vary Challenge mean – 6.2 mode – remains the same (9) median – 7

Teac he r

Unit 10–2

• Use class test results to provide students with further opportunities to develop an understanding of mean, median and mode.

Consolidation 10–2 (c) 578 (g) 75 (k) 243 (o) 642 (c) 478 (g) 72 (k) 332 (o) 433 (c) 35 (g) 44

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(d) 24 (e) 30 (i) 7 (j) 6 (d) 610 (e) 661 (d) 687 (h) 84 (l) 456

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1. (a) 18 (b) 8 (f) 5 (g) 9 2. (a) 821 (b) 473 (f) 491 3. (a) 197 (b) 267 (e) 189 (f) 65 (i) 83 (j) 74 (m) 533 (n) 322 4. (a) 737 (b) 879 (e) 378 (f) 84 (i) 65 (j) 92 (m) 536 (n) 721 5. (a) 33 (b) 45 (e) 57 (f) 54 Challenge Teacher check

• Ask students to select the most significant times and show them on an analog clock.

Consolidation 10–3

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• Provide students with more examples to practise mental addition techniques.

(d) 46 (h) 48

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 61 •

Unit 11–1

Student page 31

Outcomes

Indicators

N3.3, S3.2

The student is able to: • make polyhedra in solid, hollow and skeleton forms and discuss which features of 3-D shapes are emphasised and best represented in each form.

Skills • modelling • describing • observing

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • square pyramid • hexagonal prism • triangular prism • straws • joiners • modelling clay • nets of shapes (see pages 224–230)

Language • multiply • solid shapes • description

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 models of a square pyramid, hexagonal prism and a triangular prism for students to handle and view. • Arrange the students into three groups. • Provide students in group 1 with straws and joiners to make the three models. • Provide students in group 2 with modelling clay to make the three models. • Provide students in group 3 with nets to make the three models. • Allow students time to make their models. • Display models and discuss which features of the models are emphasised in each form.

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Teac he r

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

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What to do

• Allow time for each student to write a description of each of the models focusing on the features of the model. • When a description of each of the models has been written, arrange the students in pairs. One partner chooses one of his/her descriptions and reads it to his/her partner.The partner is to identify which of the three models was described. • Take turns to read the descriptions and see if each partner is able to identify the models.

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• Create a ‘Wanted Poster’ for a cube.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 28–29. • 62 • New Wave Maths Book E – Teachers Guide

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

Unit 11–2

Student page 32

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 • subtracting • using concrete materials

Resources

Language • multiply • Base 10 MAB • difference • divided • shape • size

• calculator • Base 10 MAB

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 division.

• 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

• Organise the class into small groups. • Distribute Base 10 MAB to the groups.

What to do

Challenge

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• The activities in Unit 11-2 are all subtraction sums. • Exercise 3(a) asks to take 36 from 84. Using the Base 10 MAB, show 84. You have four units, can you take six units from four units? (No) What do you need to do? (Trade one ten (a long) for ten units.) You now have how many units? (14) From the 14 units, can you take 6? (Yes) This leaves? (8 units) How many do you have? 7 tens take away 3 tens leaves 4 tens. Your answer is? (48). • It is also important that students see the algorithm written in its standard form; e.g. 84–36 = 48. Ensure students rewrite the algorithm each time. • Allow those students who can to continue. Repeat the process above as often as needed for students who need assistance.

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• When dividing the triangular farm, students should keep a copy of all attempts of the subdivisions. • Jot notes next to the attempts to explain what they were doing. • Share the final results.

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New Wave Maths Book E – Teachers Guide • 63 •

Unit 11–3

Student page 33

Outcomes

Indicators

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

The student is able to: • use a uniform unit consistently and carefully to measure and compare heights. • use the decimal point in representing quantities and money.

Skills • measuring • ordering • calculating • summarising

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • measuring stick • pencil

Language • multiply • centimetre • table • shortest • order • median

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• • • • • •

measure height ascending tallest range mode

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

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Teac he r

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

What to do

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• In pairs, students measure and record their heights in the space provided. • One-by-one students call out their height, while the rest of the class records the heights in a space on the table provided. • Students then use this data to complete Exercise 4.

Challenge

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• Students are to consider alternative methods to measure heights. How accurate would these methods be? • Discuss suggestions.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 92–93. • 64 • New Wave Maths Book E – Teachers Guide

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Unit 11—Answers

Student pages 31–33 Unit 11–1

(d) 16 (e) 6 (i) 3 (j) 4 (d) 759 (e) 728

1. (a) 45 (b) 24 (f) 6 (g) 4 2. (a) 1863 (b) 1828 (f) 2769 3. (a) 48 (b) 36 – 18 = 18 (c) 53 – 27 = 26 (d) 32 – 25 = 7 (e) 41 – 37 = 4 4. (a) 78 – 27 = 51 (b) 79 – 56 = 23 (c) 243 – 172 = 71 (d) 514 – 327 = 187 (e) 56 – 29 = 27 Challenge

(c) 64 (d) 63 (e) 15 (h) 10 (i) 8 (j) 7 (c) 2488 (d) 2139 (e) 3288 (f) 27 – 19 = 8 (g) 82 – 46 = 36 (h) 61 – 35 = 26 (i) 71 – 43 = 28

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

(f) 416 – 223 = 193 (g) 325 – 107 = 218 (h) 124 – 99 = 25 (i) 641 – 273 = 368 (j) 86 – 74 = 12

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1. (a) 20 (b) 8 (f) 4 (g) 9 2. (a) 813 (b) 724 (f) 548 3. Teacher check Challenge Teacher check

Unit 11–2

• Students can select their own 3-D shapes, repeat the activities and write their descriptions.

Consolidation 11–2 • Provide students with more opportunities to develop their own mental strategies to solve subtraction problems.

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(d) 16 (e) 48 (i) 5 (j) 8 (d) 668 (e) 945

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1. (a) 24 (b) 32 (f) 9 (g) 3 2. (a) 795 (b) 958 (f) 740 3. Teacher check 4. Teacher check Challenge Teacher check

Consolidation 11–3

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• Students estimate and measure the heights of objects in their surroundings.

New Wave Maths Book E – Teachers Guide • 65 •

Unit 12–1

Student page 34

Outcomes

Indicators

N3.3, N3.1a

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

Skills • ordering • recording

Memory Masters (N3.3)

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

Language • add • order • smallest • largest • prime numbers • place value • face value • greater than • less than

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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

• Revise the terms greater than and less than.

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• Exercises 3 and 4 require identification of numbers less than or greater than the number provided. Use a place value chart to write numbers in to assist if required. • Work with the whole class revising ordering of numbers for Exercise 5. Look at the number in the highest place value of each of the numbers to be ordered. If all are of the same place value then these numbers can be ordered from here. If, however, the face value of these numbers are also the same then the number in the place value column to the right is to be checked. The numbers may now be ordered using the digits in this column. Should any, or all of these numbers have the same face value then the numbers in the place value column next to the right are to be used.

Challenge

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• Discuss with students what a prime number is. (A number which has two factors, 1 and itself.) • Set students to find the prime numbers between 40 and 50. • Students record their reasoning and any workings they use.

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What to do

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

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

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Unit 12–2

Student page 35

Outcomes

Indicators The student is able to: • use arrow diagrams to summarise data.

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

Skills • recording

Resources

Language

• calculator • Base 10 MAB • restaurant menus • coloured pencils

• add • number, signs, +, –, x, ÷ • arrow diagram • combinations • possible

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Memory Masters (N3.1)

Notes

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• The focus for this unit is the completion of number patterns.

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

• Ask students ‘Who has eaten in a restaurant lately?’. • Discuss menus and selections people make. • Explain to students that sometimes restaurants have a set menu for a certain price. Within the menu, the patron can often make choices. This allows for individual taste.

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Challenge

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• Look at the selections in the workbook. • Students will find it easier to record each contribution using different coloured pencils. • Complete the selections and answer the questions.

• Students are to use only the number 9 and any or all of the four signs and brackets given to make each of the three numbers shown. • Show all workings and provide a written explanation of how the final arrangement was reached.

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New Wave Maths Book E – Teachers Guide • 67 •

Unit 12–3

Student page 36

Outcomes

Indicators The student is able to: • read and write fractional notation (i.e. symbols) to represent unit fractions.

M3.1, N3.3, N3.1b

Skills • recognising fractions • comparing fractions • writing fractions

Memory Masters (M3.1)

Resources • calculator • Base 10 MAB • paper • pencil • coloured pencils • fraction cake • fraction chart (see page 208)

Language • add • equivalence • fraction • diagram • equivalent fraction • number sentence • chart

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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 (N3.1b) Warm up

• Hold up a sheet of paper. Ask students what they see. 1 sheet of paper or 1 whole. Write 1 on the blackboard/whiteboard. • Fold the paper in half and cut or tear along the fold.What do the students see now? (2 halves.) If these halves are rejoined what do you get? (1 whole. )Therefore one whole and two halves represent the same amount—they are equivalent. • Take one half of the paper, fold it in half and cut or tear along the fold. From the half piece of paper there are now two pieces. What are these? If students have difficulty grasping that these smaller pieces represent one-fourth, hold up an equivalent piece of paper to the original. • The one half is now, what? (Two-fourths.) One half is equivalent to two-fourths.

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• The focus for this unit is conversion of centimetres to metres and metres to centimetres.

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What to do

• There are a number of diagrams showing fraction parts. The first diagram is mirrored in the second diagram which shows more divisions. The shaded area in both diagrams is the same; i.e. the shaded area shows equivalent fractions. In Exercise 3(a), the first diagram shows one part out of two shaded. Ask: What fraction is this? (1/2) The second diagram shows how many parts out of four shaded? (two) What fraction is this? (2/4). Therefore, 1/2 is equivalent to 2/4. • Students complete the rest of the equivalent fractions. Assist those students who require more help.

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Challenge

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• Display the fraction chart on page 208. • Discuss the chart and ask students to develop their own fraction chart which would make identification of equivalent fractions easy. • Provide students with a sheet of blank paper and ask them to proceed. • Display those that show equivalent fractions clearly.

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

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Unit 12—Answers

Student pages 34–36 Unit 12–1

1. (a) 15 (b) 11 (c) 27 (d) 16 (e) 36 (f) 12, 12 (g) 40 (h) 41 (i) 54 (j) 35 2. (a) 1835 (b) 1277 (c) 1039 (d) 1058 (e) 1625 (f) 1565 3. (a) TS + FS + CC, TS + FS + SC, TS+ BV + CC, TS + BV + SC, CS + FS + CC, CS + FS + SC, CS + BV + SC, CS + BV + CC VS + BV + SC, VS + BV + CC VS + FS + CC, VS + FS + SC (b) 12 Challenge Answers will vary, some possible answers include: 9 + 9 + 9 = 27 9 + 9 + 9 + 9 + 9 – (9 + 9 ÷ 9 ) = 43 9 + 9 + 9 + 9 + 9 + 9 + 9 – (9 + 9 ÷ 9 ) = 61

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1. (a) 6 (b) 10 (c) 25 (d) 18 (e) 54 (f) 6 (g) 8 (h) 2 (i) 7 (j) 9 2. (a) 1556 (b) 1181 (c) 1077 (d) 1484 (e) 1792 (f) 1345 3. 5, 20, 19, 12 4. 364, 810, 301, 411 5. (a) 409 m, 438 m, 472 m, 490 m (b) 260 kg, 399 kg, 507 kg, 684 kg (c) 8098 L, 8587 L, 8764 L, 8900 L (d) 8569, 8659, 8695, 8965 (e) 400, 587, 652, 697, 699 999, 859 758 (f) 13 569, 13 596, 13 659, 13 695 Challenge 41, 43, 47

Unit 12–2

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• Brainstorm numbers at random. Record on the blackboard/ whiteboard and ask students to write them in order from smallest to largest or largest to smallest.

Consolidation 12–2

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1. (a) 2.36 m (b)8 m (c) 1.08 m (d) 4.79 m (e) 3.12 m (f) 460 cm (g) 249 cm (h) 804 cm (i) 179 cm (j) 400 cm 2. (a) 1615 (b) 1102 (c) 1232 (d) 1541 (e) 1561 (f) 1144 3. (a) 1/2 = 2/4 (e) 3/4 = 6/8 (i) 1/2 = 4/8 1 2 2 4 (b) /3 = /6 (f) /3 = /6 (j) 2/3 = 6/9 (c) 3/5 = 6/10 (g) 1/2 = 3/6 (k) 1/6 = 2/12 (d) 1/6 = 2/12 (h) 3/6 = 6/12 (l) 3/6 = 6/12 Challenge Teacher check

• Students make their own menus and work out the possible combinations.

Consolidation 12–3

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• Use real-life examples to show equivalence; e.g. pizza, cake, apples etc.

New Wave Maths Book E – Teachers Guide • 69 •

Unit 13–1

Student page 37

Outcomes

Indicators

N3.1, N3.3, S3.2

The student is able to: • visualise the folding process to predict which pentominoes can be used as nets for an open box.

Skills • modelling • drawing • constructing

Memory Masters (N3.1)

Resources • calculator • Base 10 MAB • stiff card • scissors • pencil • examples of nets (see pages 224– 230) • boxes

Language • cents • change • dollars • subtract • nets • construct • open base • opposite

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 (S3.2) Warm up

• Revise the word ‘net’. Ask students for their knowledge of the word. • Provide a display of a variety of nets that make different shapes. (See copies provided on pages 224–230 of this book). • Another idea is to cut up a box and lay it out flat to see the net.

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• Explain to the class that their task is to draw different arrangements of a net that can be used to form an open box. The net may be of an open cube or an open rectangular prism. • When the students have drawn different arrangements in the workbook, ask them to copy their designs onto stiff card. Students may need to enlarge their designs and also square off their drawing for greater accuracy in construction. • Once copied onto card, students then cut and fold their designs to see if they were correct in drawing nets that make an open box.

Challenge

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What to do

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

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• Using the drawings in the workbook, students shade (or mark with a X) two sides of the net they think will be opposite each other when the nets are made into open boxes. • Write a brief record of the accuracy of their markings explaining why the two sides were chosen.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 14–15. • 70 • New Wave Maths Book E – Teachers Guide

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Unit 13–2

Student page 38

Outcomes

Indicators 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 reference to place value.

N3.3

Skills • reasoning mathematically • adding • subtracting • multiplying • recording

Resources

Language

• calculator • Base 10 MAB

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

• subtract • add • closed number • sentence • balanced • double • relationship • rule • constant

Notes

Memory Masters (N3.3)

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• The focus for this unit is addition and subtraction of multiples of 10 less than 100.

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) Warm up

• Explain to the class that we can add or subtract a constant to or from both sides of a number sentence and keep the number sentence balanced. That is, the balance or difference remains the same; e.g. 2 + 3 = 5; add 10 to both sides 12 + 3 = 15. Both sides are increased by 10 but the total or balance remains constant—the difference between the two parts of the number sentence that 10 was added to is still 3. A constant is a number that is added to or subtracted from both sides of a number sentence—the number added remains constant—it does not change from side to side. • Use subtraction as an example as well; 24 – 2 = 22; subtract 20 from both sides is 4 –2 = 2. The balance remains constant, or the difference between 24 and 22, and 4 and 2 remains 2.

What to do

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• In the first example of 20 + 7 = 27, a constant of 200 has been added to both sides, keeping the balance of the number sentence. • In the second example a constant of 40 has been subtracted from both sides, keeping the balance of the number sentence. • Students complete Exercise 3 and record the constant that has been added or subtracted in each number sentence. • Open number sentences allow us to find certain information. In Exercise 4, doubling the friend’s age (multiplying by 2) and adding 1 to the total, the student’s age is found; e.g. if my friend is 2 then 2 x 2 + 1 = 5.The student would be five years old. Students are to find other pairs of numbers that follow the rule. • In Exercise 5 the difference between Brett’s and Rochelle’s age is 4 years. By using Rochelle’s age as any age over 4 years we can find Brett’s age by subtracting 4 from Rochelle’s age; e.g. 10 – 4 = 6. Students find three pairs of numbers that fit this rule.

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Challenge • Following the same principles as the open number sentences above, but choosing different relationships, students write a descriptor and the open number sentence that follows the rule. Ask a classmate to provide a solution. This will act as a check on the rule.

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New Wave Maths Book E – Teachers Guide • 71 •

Unit 13–3

Student page 39

Outcomes

Indicators

N3.3, M3.2, S3.2

The student is able to: • construct items using a specified number of cubes to directly compare volume and surface area. • match standard geometric models with realistic drawings and conventional diagrams.

Skills • manipulating • drawing • recording • counting

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • 2-cm cubes

Language • subtract • construct • cube • full face • volume • surface area • relationship

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 (M3.2, S3.2) Warm up

• Students can work in groups or the activity may be done independently. • Distribute twenty 2-cm cubes to each group and allow a couple of minutes free play. • Explain the rules of construction they are to follow. Each cube must be contacting another cube on the full face.

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• Students make a variety of different models using all 20 of the 2-cm cubes in each model. • Draw each model when it is made. • Write the volume and the surface area of each model. The volume is the number of cubes used.The surface area is the exposed surface of each cube after construction. (Remind students about the surfaces on the bottom of the model.) • Students take turns in making the models. • Students explain what they noticed about the volume and the surface area.

Challenge

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What to do

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• The focus for this unit is addition and subtraction of multiples of 10 less than 100.

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• Using the information gained from the model construction, see if a relationship can be found between the volume and the surface area. • Students record their thoughts and workings. Students share their findings with the class.

• 72 • New Wave Maths Book E – Teachers Guide

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Unit 13—Answers

Student pages 37–39 Unit 13–1

1. (a) 50 (b) 100 (c) 120 (d) 60 (e) 110 (f) 70 (g) 20 (h) 50 (i) 10 (j) 10 2. (a) 122 (b) 314 (c) 203 (d) 331 (e) 429 (f) 323 3. (a) 113; +100 (b) 132; +100 (c) 230; +200 (d) 3; -400 (e) 214; +200 4. Answers will vary. One possible answer is: (3, 7), (4, 9), (5, 11) 5. Answers will vary. One possible answer is: (10, 6), (9, 5), (8, 4) Challenge Teacher check

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1. (a) 1020c (b) 1346c (c) 1909c (d) 2000c (e) 1444c (f) $42.75 (g) $31.00 (h) $22.06 (i) $27.40 (j) $34.62 2. (a) 641 (b) 424 (c) 219 (d) 245 (e) 466 (f) 367 3. Teacher check; possible answers include:

Unit 13–2

Challenge Teacher check

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• Students make nets and eventually boxes to store particular items.

Consolidation 13–2

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1. (a) 110 (b) 110 (c) 150 (d) 130 (e) 80 (f) 30 (g) 20 (h) 50 (i) 10 (j) 10 2. (a) 283 (b) 371 (c) 166 (d) 460 (e) 224 (f) 553 3. Teacher check. Volume should remain constant, while surface area will vary. Challenge The more compact the model (e.g. cube) the lower the surface area.

• Provide students with more opportunities to practise solving open number sentences.

Consolidation 13–3

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R.I.C. Publications®

• Use 30 cubes instead of 25 cubes and find out how this affects the number of arrangements that can be made.

New Wave Maths Book E – Teachers Guide • 73 •

Unit 14–1

Student page 40

Outcomes

Indicators

N3.3, N3.1b

The student is able to: • separate objects and collections into equal parts to show unit fractions.

Skills • dividing • sharing • writing decimal fractions

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • streamer • ruler • counters

Language • multiply • solution • problem • lengths

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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 (N3.1b) Warm up

• Explain to the students that the activities involve decimal fractions.They will be asked to break up and/or share different items using decimal fractions. • A decimal fraction is a fraction whose denominator is some power of 10. For example, the decimal fraction 0.5 is actually 5/10. • Distribute 1-m lengths of streamer to each student, or pair of students.

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• Ask students to divide the streamer into lengths of 0.25 m. For those who have difficulties, ask them how long one metre is if given as a length in centimetres. • If 100 cm is 1 metre, how long is 0.25 m? There may need to be some prompting, but an answer of 25 cm should be forthcoming. Direct students to complete the activity. • Exercise 3 (b) asks students to use their ruler to share or mark the strip on the page into 5-cm lengths. • Exercise 3(c) may be completed by using the drawing of a pizza with 0.4 or 4 tenths of the pizza removed. Share the remaining 0.6 among three people to find out how much each have. • In Exercise 3(d), students need to divide 0.9 kg of apples into 0.3 kg lots to work out how many people will get a share.

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Challenge

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• The focus for this unit is addition and subtraction of multiples of 10 less than 100.

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• Supply each student with four counters. • Students are to use the counters to solve the problem. Record each attempt and write an explanation explaining the final result.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 46–47. • 74 • New Wave Maths Book E – Teachers Guide

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

Unit 14–2

Student page 41

Outcomes

Indicators

WM3.2, WM3.4, C&D3.2, C&D3.3

The student is able to: • generate mathematical questions prompted by a specific stimulus. • make lists or tables of data to help solve a problem. • use Venn diagrams and two-way tables to represent data.

Skills • poses questions • analyses data • makes conjectures

Resources

Language

• pencil

• shapes • • different • • colour • • categories • • Venn diagram • two-way table

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Main Activity (WM3.2, WM3.4, C&D3.2, C&D3.3)

Notes

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What to do

similar edges classify sort

• This activity is designed for students working collaboratively in groups. Allow enough time so they can discuss their opinions 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 to investigate the first part of the activity. – What are the shapes? (name them) – What colour are the shapes? – How many sides does each shape have? • Students who are finding this activity challenging can list shapes under these categories and try to find similarities between the categories. • Note: Students will need practise in completing Venn diagrams and creating two-way tables prior to this activity. • Groups may wish to collate and summarise their findings and present them as a poster with a series of graphs, diagrams and information on how they solved the problem. • Allow each group time 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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New Wave Maths Book E – Teachers Guide • 75 •

Unit 14–3

Student page 42

Outcomes

Indicators The student is able to: • estimate sums and products by rounding to double-digit numbers.

N3.3

Skills

Resources • calculator • Base 10 MAB

• estimating • rounding

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.

Notes

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• multiply • round • rectangles • estimate • shape • squares • total • nearest ten • approximate answer • nearest thousand

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Main Activity (N3.3) Warm up

Language

• Ask students who can remember the rules for rounding. Allow discussion then summarise: Round down: 0 – 4, 0–40, 0–400 etc. Round up: 5 – 9, 50–90, 500–900 etc.

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• The first exercise involves rounding to the nearest ten. Rounding to the nearest 10 enables us to quickly find an approximate answer to the multiplication sum. • In the first example shown, after rounding we have 30 x 40 which is … ? (1200) A simple way to multiply multiples of ten is to cross off each zero and write a zero at the right-hand end of the answer for each zero crossed off. If two zeros are crossed off write two zeros in the answer. All that remains is to multiply the remaining two numbers. In this case 3 x 4 = 12 with two zeros added = 1200. • Repeat the process as required, or allow students to continue by themselves. • Exercises 4 and 5 require rounding to the nearest thousand (follow the same rules but focus on the hundreds place). Using the rounded numbers, an estimate of the total can be made. • Demonstrate with the first example in each, provide assistance as required and encourage independent work.

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What to do

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Challenge • Explain why squares are classified as rectangles; i.e. they have all the properties of a rectangle. • When solving the problem given, students will need to provide a record of how they arrived at their conclusion. • Encourage students to find a suitable recording means and use that to support their findings. Notes should also be kept.

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

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Unit 14—Answers

Student pages 40–42 Unit 14–2

Unit 14–1 1. (a) 120 (b) 130 (c) 80 (d) 70 (e) 80 (f) 10 (g) 60 (h) 30 (i) 50 (j) 30 2. (a) 1654 (b) 5496 (c) 2892 (d) 2476 (e) 2481 (f) 2175 3. (a) 4 (b) 3 (c) 2 (d) 3 Challenge

1. Answers will vary. 2.

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four-sided

not four-sided

yellow not yellow

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2. (a) 3582 (b) 7587 (c) 2632 (d) 5992 (e) 4185 (f) 5082

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3. (a) 1200 (b) 40 x 30 = 1200 (c) 50 x 60 = 3000 (d) 80 x 40 = 3200 (e) 70 x 80 = 5600

(f) 50 x 30 = 1500 (g) 40 x 40 = 1600 (h) 70 x 50 = 3500 (i) 90 x 80 = 7200 (j) 20 x 60 = 1200

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Consolidation 14–2 • In groups, students create their own picture (similar to the one on page 41 of the Workbook) using different shapes or objects. Groups can swap their picture with other groups who will decide which categories the objects can be placed in and draw a Venn diagram to represent the data.

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• Continue to use real-life objects to provide further practice with decimal fractions.

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1. Answers will vary; some possible solutions: 2 x 11 20 + 2 44 ÷ 2 30 – 8

Consolidation 14–3

• Provide further opportunities for students to use rounding techniques when estimating answers.

5. (a) 8495 8000 (b) 4986 5000 – 2164 + 2000 + 1851 + 2000 6331 6000 3135 3000 (c) 7281 7000 + 4973 + 5000 2308 2000

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Challenge 96 rectangles Teacher check

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New Wave Maths Book E – Teachers Guide • 77 •

Unit 15–1

Student page 43

Outcomes

Indicators The student is able to: • order angles by direct comparison of the ‘amount of turn’ and by using units such as a quarter turn or an angle or their own making.

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

Skills • measuring angles • comparing angles

Memory Masters (N3.1a)

Resources • calculator • Base 10 MAB • tracing paper (optional)

Language • place value • divide • shape • angles • right angle • less than • superimpose • greater than • acute angle • obtuse angle

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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) Warm up

• Ask students to describe a right angle. From the descriptions, focus on an angle of 90º. • Ask students what common shapes they know of that have a right angle as one or more of its angles. (Square or rectangle.) • Explain that angles larger than 90º are called obtuse angles and angles smaller than 90º are called acute angles.

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• Some students may complete Exercise 3 recognising the angles by ‘look’ alone. • Others can select a shape with a right angle; e.g. the square. • Using the chosen right angle, trace the shape onto tracing paper then superimpose the right angle on each of the angles on the page to find those angles that are right angles, greater than a right angle (obtuse angle) or less than a right angle (acute angle). Write the findings next to each angle in the shape.

Challenge

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What to do

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Teac he r

• The focus for this unit is the identification of place value in a whole number.

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• Write a set of instructions for drawing:

a square; and a rectangle.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 86–87. • 78 • New Wave Maths Book E – Teachers Guide

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

Unit 15–2

Student page 44

Outcomes

Indicators

N3.1a, N3.3

The student is able to: • estimate sums and products by rounding.

Skills • rounding

Resources

Language

• calculator • Base 10 MAB • ruler • pencil

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

• round • nearest hundredth • divide • approximate answer • estimate • nearest ten • nearest hundred • divide • shape

Notes

Memory Masters (N3.1a)

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

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) Warm up

• Revise the rules of rounding with the class. Ask students to provide the rules then reinforce: round down if digit is 4 or less; round up if digit is 5 or greater. • Explain to the class that rounding is used to provide an approximate answer to a sum. This approximation provides a guide for where the actual answer may be or may be used if only an approximation is required. • The rounding allows an estimate to be made of the answer. An estimate is another way of saying an approximate answer.

What to do

Challenge

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• Work through the rounding of the numbers in Exercise 3(a) with the whole class to ensure understanding. Ask students to round the numbers and then record the correct rounding in their workbooks. • As students show confidence, allow them to proceed by themselves to complete Exercises 3–6. • Note: Exercises 5 and 6 involve subtraction.

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• Remind students to keep a record of all attempts to solve the subdivision of the block of land. • Students should also jot notes of their thinking process. • Share final results with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 64–65. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 79 •

Unit 15–3

Student page 45

Outcomes

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

N3.3, M3.2

Skills

Resources • calculator • Base 10 MAB • large clock

• telling the time • writing 12-hour time • writing 24-hour time

Memory Masters (N3.3)

Language • divide • time • clock face • 12-hour • 24-hour • nearest minute

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 (M3.2) Warm up

• To assist students with telling the time, a large clock with readily mobile hands may be useful. • Ask students how many hours there are in a day. • Ask students if they can tell you the two different ways we tell the time. (12-hour time which is divided into morning (a.m.) and afternoon (p.m.) each of 12 hours. 24-hour time where the time for the whole day starts with zero hours and finishes at 24 hours or zero hours, ready to start again.) • Ask students if they can tell you how 24-hour time is read after midday. (Hours continue to be counted from 12 to 23.) • Remind students the 24-hour time is read as a number of hundred hours with the minutes included; e.g. 2.25 p.m. in 24-hour time is read as 14 hundred and 25 hours and written as 1425 hours. Similarly, 3.15 a.m. is read as 03 hundred and 15 hours and written as 0315 hours.

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

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• Work with the class as a whole through the exercise as students may find the conversion confusing and difficult. • The first clock shows what 12-hour time? (6.30 a.m.) Write this in the 12-hour time space. How is this written in 24-hour time? (0630 hours. )Write this in the 24-hour time space. • Continue with each clock encouraging students to read and convert the time.

Challenge

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• Ask students: Have you been alive for a million seconds? • Students will need to provide a record of how they arrived at their conclusion.

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

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

Unit 15—Answers

Student pages 43–45 Unit 15–1

1. (a) ones (e) hundreds (i) tens 2. (a) 21 (e) 31 3.

(b) hundreds (c) tens (d) hundreds (f) ones (g) ones (h) thousands (j) tens (b) 21 (c) 41 (d) 41 (f) 30

Unit 15–2 1. (a) 30 (b) 40 (c) 80 (d) 90 (e) 20 (f) 50 (g) 70 (h) 60 (i) 70 (j) 20 2. (a) 231 (b) 434 (c) 211 (d) 111 (e) 122 (f) 120 3. (a) 59 60 (b) 65 70 (c) 21 20 31 30 87 90 82 80 47 50 49 50 53 50 + 28 + 30 + 58 + 60 + 32 + 30 165 170 259 270 188 180

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5. (a) 76 80 (b) 92 90 (c) 58 60 – 49 – 50 – 27 – 30 – 22 – 20 27 30 65 60 36 40

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4. (a) 524 500 (b) 2 0 (c) 587 600 56 100 619 600 6 0 289 300 382 400 829 800 + 431 + 400 + 74 + 100 + 492 + 500 1300 1300 1077 1100 1914 1900

6. (a) 794 800 (b) 324 300 (c) 882 900 – 239 – 200 – 119 – 100 – 523 – 500 555 600 205 200 359 400 Challenge

Challenge Teacher check

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• Students participate in an angle-hunt in their immediate environment. Locate right angles, acute angles and obtuse angles.

Consolidation 15–2

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1. (a) 0 (b) 12 (c) 4 (d) 36 (e) 35 (f) 3 (g) 9 (h) 4 (i) 1 (j) 7 2. (a) 101 (b) 202 (c) 101 (d) 302 (e) 102 (f) 201 3. (a) 12 hr = 6.30 a.m. (g) 12 hr = 10.52 a.m. 24 hr = 0630 24 hr = 1052 (b) 12 hr = 2.08 p.m. (h) 12 hr = 9.23 p.m. 24 hr = 1408 24 hr = 2123 (c) 12 hr = 2.53 a.m. (i) 12 hr = 3.45 a.m. 24 hr = 0253 24 hr = 0345 (d) 12 hr = 7.35 p.m. 24 hr = 1935 (e) 12 hr = 7.29 a.m. 24 hr = 0729 (f) 12 hr = 5.29 p.m. 24 hr = 1729 Challenge Yes. 60 secs x 60 mins x 24 hrs x 7 days x 52 weeks = 31 449 600 secs/yr

• Provide students with further opportunities to practise their estimating skills.

Consolidation 15–3

o c . che e r o t r s super

R.I.C. Publications®

• Encourage students to read the time at various intervals during the day.

New Wave Maths Book E – Teachers Guide • 81 •

Unit 16–1

Student page 46

Outcomes

Indicators The student is able to: • estimate sums and products by rounding.

N3.3

Skills

Resources • calculator • Base 10 MAB

• estimating • using truncation

Memory Masters (N3.3)

Language • add • truncation • left-most digit • approximate • rounding

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 (N3.3) Warm up

• Explain to the students that there are different ways of finding approximate answers to problems. Previously, rounding has been used. This time they will be using a process of ‘truncation’. • ‘Truncation’ means to cut off or shorten. Examples of truncation are: 825 may be truncated to 800; 894 may be truncated to 800; 362 may be truncated to 300. It is also known as frontend rounding. • Explain the difference between truncation and rounding. (Truncation always goes back to the last place value used at the level of truncation—in the examples above to the last hundred.) • In most cases when truncating, the digit in the highest place value (the digit in the left-most position) is the one to truncate to. • Note: Front-end rounding methods are better for some operations but not others.

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

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• Read the note on truncation and the instructions for the exercise. • In Exercise 3(a) 43 is truncated to 40 and multiplied by 3. The answer is … ? (120) • In 3(b), 52 is truncated to … ? (50) Write 50 in the first space. 50 is multiplied by … ? (7) The answer is … ? (350) • Continue as required gradually easing students into working by themselves. • Check to ensure students are working correctly. Provide assistance as required.

Challenge

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• Students are required to compare rounding and truncating. • Use examples to show comparisons. • Which is more accurate?

• 82 • New Wave Maths Book E – Teachers Guide

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

Unit 16–2

Student page 47

Outcomes

Indicators 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.

N3.3, S3.3, C&D3.3

Skills • classifying • observing • recording

Resources

Language

• calculator • Base 10 MAB • mirror/mira • pencil • ruler

• add • axis of symmetry • classify • horizontal • vertical • symmetry • Venn diagram

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

Memory Masters (N3.3)

Notes

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

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, 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 this 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 students to show all or any axes of symmetry. • Explain to students that there are two axes of symmetry, one horizontal and one vertical. • Rotational symmetry is when the letter can be rotated around a central point, and still look correct; e.g.

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therefore H has rotational symmetry.

What to do

• All the letters of the alphabet are shown. Students are to find those that have only line symmetry and rotational symmetry and those that have both line and rotational symmetry. Some letters will not have any axis of symmetry. • When students have found all the lines of symmetry and rotational symmetry the letters are then to be recorded in the correct place on the Venn diagram. Those that have only line symmetry are placed in the circle segment marked line symmetry. Those with only rotational symmetry are placed in the circle segment marked rotational symmetry; those with both line and rotational symmetry are placed in the overlapping segment of their circles; and, those with no symmetry are placed in the surrounding rectangle.

Challenge • Which numbers have rotational symmetry? For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 110–111. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 83 •

Unit 16–3

Student page 48

Outcomes

Indicators

N3.3, N3.4

The student is able to: • use patterns in sequences of related additions or subtractions to generate new equations.

Skills • following patterns

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • pencil • ruler

Language • add • sequences • straight lines

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

• Explain to the students that mathematics is mostly about logic and patterns. The activities in the workbook will allow them to examine sequences. Sequences are patterns. • Arithmetic sequences are patterns where a constant amount is added or subtracted from a number to create the next term in the sequence.

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• Direct students to look at the sequence in Exercise 3(a). You start with 69 then take 4, take 4, take 4, take 4 and take another 4. What does this leave you with? (49) • Exercise 3(b) starts with 57 then add 6, add 6, add 6, add 6, and add another 6. What does this total? (87) • Can anyone find another way to work out these two sequences? Allow students to explain different ideas. • Repeat the process for the final four examples. • Ask if anyone can find another way to work out the sequences. If no response, suggest grouping the numbers to be added or taken away to make one number; e.g. 5 lots of 4 or 20; 5 lots of 6 or 30; 5 lots of 7 or 35 and so on. Explain this is another way the sequences can be worked out. • Ask students to complete Exercises 4 and 5 by themselves and to explain how they found the answers. Share some of the explanations.

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

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• Make copies of the dots, if required, to show all attempts at finding a solution to the problem. • Write a brief account of how they reached or attempted to reach this solution. Alternatively, jot notes beside each attempt.

• 84 • New Wave Maths Book E – Teachers Guide

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

Unit 16—Answers

Student pages 46–48 Unit 16–1 1. 2. 3.

(a) 15 (f) 3 (a) 888 (f) 785

(b) 6 (g) 7 (b) 993

(d) 36 (e) 12 (i) 6 (j) 3 (d) 696 (e) 975

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1. (a) 4 (b) 12 (c) 12 (d) 21 (e) 42 (f) 5 (g) 4 (h) 10 (i) 10 (j) 2 2. (a) 778 (b) 896 (c) 898 (d) 887 (e) 784 (f) 799 3. (a) 120 (i) 30 x 6 = 180 (b) 50 x 7 = 350 (j) 20 x 9 = 180 (c) 70 x 4 = 280 (k) 300 x 5 = 1500 (d) 600 x 4 = 2400 (l) 200 x 7 = 1400 (e) 500 x 8 = 4000 (m) 700 x 5 = 3500 (f) 20 x 40 = 800 (n) 80 x 30 = 2400 (g) 30 x 30 = 900 (o) 50 x 20 = 1000 (h) 50 x 60 = 3000 (p) 40 x 30 = 1200 Challenge Teacher check (Answer should indicate an understanding that rounding is more accurate.)

Unit 16–2

Challenge 0, 1, 8

• Provide students with further opportunities to practise truncating to approximate answers.

Consolidation 16–2 • Students seek out and record objects that are symmetrical in their immediate environment.

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(d) 48 (e) 45 (i) 9 (j) 6 (d) 939 (e) 709

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1. (a) 32 (b) 27 (f) 7 (g) 5 2. (a) 835 (b) 948 (f) 889 3. (a) 49 (d) 83 (b) 87 (e) 95 (c) 36 (f) 13 4. (a) 80c (b) Answers will vary. 5. (a) $6 (b) Answers will vary. Challenge

Consolidation 16–3

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R.I.C. Publications®

• Students work with a partner to make up their own arithmetic sequences.

New Wave Maths Book E – Teachers Guide • 85 •

Unit 17–1

Student page 49

Outcomes

Indicators

N3.1a, N3.3, S3.4, S3.2

The student is able to: • visualise and describe crosssections of familiar objects. • imagine and draw different crosssections of simple 3-D shapes and then check and improve the drawings by observing the crosssection.

Skills • modelling • manipulating • visualising

Memory Masters (N3.1a)

Resources • calculator • Base 10 MAB • items to cut • knife • modelling clay

Language • > greater than • < less than • = equal • subtract • cross-section

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 (S3.4, S3.2) Warm up

• Have a selection of items which can easily be cut to make a cross-section. • Display the item and ask the class—’What shape will be made if I cut the –––(item) across here (showing students the position of the cut)? • Allow students to respond—circle, triangle, square etc. • Repeat this process with several items. • Distribute modelling clay to students. Direct them to make various 3-D shapes. Students then make cross-section cuts to find the 2-D shape created.

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• The focus for this unit is the expression of equality or inequality of number statements.

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• Students can then look at the photographs in their workbook. Consider the point of the cross-section and write the 2-D shape they think will be made. • Students complete all questions for Exercise 3. • Check answers and ask students to explain how they came to each conclusion.

Challenge

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What to do

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• Students are to use their knowledge of Magic Squares to complete the puzzle. • If the students do not know the clue for completing the Magic Square encourage them to investigate means of their own. • Students are to record all attempts and to keep notes explaining their workings.

• 86 • New Wave Maths Book E – Teachers Guide

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

Unit 17–2

Student page 50

Outcomes

Indicators

N3.1, N3.3, N3.2, N3.4

The student is able to: • understand the terms ‘multiple’, ‘factor’ and ‘prime’ and use them appropriately. • recognise, describe and use patterns to define prime numbers.

Skills • following instructions

Resources

Language

• calculator • Base 10 MAB • coloured pencils • ruler

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

• change • 24-hour time • subtract • Eratosthene’s Sieve • prime number • composite number • horizontal • equal • multiples • vertical • diagonal • added

Notes

Memory Masters (N3.1)

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• The focus for this unit is the conversion of 12 hour-time to 24-hour time and 24-hour time to 12-hour time.

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, N3.4) Warm up

• Provide students with some background information about Eratosthene. He was a Greek astronomer and mathematician who lived in 276–196 BC in Alexandria. He used geometry to calculate the size of our planet. He estimated the circumference of the Earth to be about 40 233.6 km. This was amazingly accurate. (40 076.5 km is the current known equatorial circumference of the Earth.) Eratosthene was also a mathematician who used a grid to discover prime numbers.

What to do

Challenge

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• Read the full set of instructions to the class before directing them to take out their ruler and coloured pencils. • Instruct the class to follow the directions given carefully. Any error in following the directions will lead to an error in finding the prime numbers. • When the students have finished, ask them to list the prime numbers on the page. • Ask the students what is special about this set of numbers. (They have two factors only—one and themselves.)

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• Ask the students to find a pair of prime numbers (two prime numbers that are next to each other in the sequence of prime numbers) that when added together will make the next prime number. • Show all pairings and workings.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 87 •

Unit 17–3

Student page 51

Outcomes

Indicators

N3.1, N3.3, M3.2

The student is able to: • use a standard calendar to locate and calculate particular days.

Skills

Resources • calculator • coloured pencils

• reading a calendar • counting

Memory Masters (N3.1)

Language • cents • subtract • before • months • leap year

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

• • • •

dollars calendar after year

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 (M3.2) Warm up

• Before starting work, ask each student to write his/her birthday on the page in the space provided. • Hold an open discussion on the calendar—How many months? Why do some months have 30 days and others 31? How many weeks? Which months have 31 days? Which months have 30 days? Are there any months with a different number of days? The year shown starts on a Tuesday. What day will the next year start?

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• The focus for this unit is the conversion of dollars to cents and cents to dollars.

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• Ask students to mark on the calendar in coloured pencil—New Years’ Day, Australia Day, Anzac Day, Foundation Day (first Monday in June), Christmas Day, Boxing Day and their own birthday. • Read through each question separately with the students and ask them to complete the answer. • Discuss each answer briefly, with students providing answers and how they found the answer.

Challenge

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• Research to find the reason why July and August have 31 days.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 104–105. • 88 • New Wave Maths Book E – Teachers Guide

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

Unit 17—Answers

Student pages 49–51 Unit 17–1

1. (a) 1130 (b) 1615 (c) 2245 (d) 0654 (e) 1200 (f) 3.00 p.m. (g) 11.20 p.m. (h) 2.15 a.m. (i) 10.54 a.m. (j) 7.42 p.m. 2. (a) 273 (b) 325 (c) 518 (d) 113 (e) 175 (f) 224 3. Teacher check 4. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 Challenge 2+3=5

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1. (a) < (b) > (c) < (d) = (e) > (g) > (h) = (j) = (f) < (i) < 2. (a) 315 (b) 237 (c) 229 (d) 118 (e) 516 (f) 328 3. (a) circle (d) rectangle (g) oval (b) circle (e) rectangle (h) square (c) triangle (f) square (i) circle Challenge

Unit 17–2

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Consolidation 17–2

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1. (a) 4000c (b) 657c (c) 64c (d) 2727c (e) 1090c (f) $0.14 (g) $6.84 (h) $36.47 (i) $20.02 (j) $0.98 2. (a) 5718 (b) 4533 (c) 2458 (d) 4819 (e) 2056 (f) 4561 3. (a) Teacher check (d) (i) Teacher check check (b) Teacher check (ii) Teacher 2 (iii) /5 (c) Teacher check (iv) 11/

• Provide students with further opportunity to discover prime numbers into the hundreds and thousands.

Consolidation 17–3

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(e) March, June, September, December (f) (i) 365 (ii) 366

• Practise reading the calendar at regular intervals for special events, planning etc.

Challenge Julius Caesar reformed the Roman calendar, which is the basis of the one we use today. In the Julian calendar the months alternated between 30 and 31 days, with February having 29 but 30 in a leap year. For this he was honoured with having the fifth month (Quintilis) named after him (Julius/ July). His grand-nephew Augustus was later honoured by the senate by having a month named after him (August). They chose the sixth month of the year, but as this was only a 30 day month they rearranged the calendar so that he should also have 31 days in his month, preventing anyone claiming that Emperor Augustus was saddled with an inferior month.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 89 •

Unit 18–1

Student page 52

Outcomes

Indicators

N3.3

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

Skills • subtracting

Memory Masters (N3.3)

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

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.

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Number (N3.3)

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• Organise students into small groups. Distribute Base 10 MAB between the groups. • Ask students to show their knowledge of Base 10 MAB by asking what each piece of wood represents. • Ask students to make 2156 with the Base 10 MAB. Leave the Base 10 MAB as made up and select more pieces of wood to show 1138. • Ask students to find the difference between the two sets of wood by matching the two sets. • Two large cubes and one large cube – one large cube the difference. One flat and one flat – no flats difference. Five longs and three longs – two longs difference. Six units and eight units – two units too many. What do you need to do?( Trade.) Trade one long for ten units. Match the two units with the ten large cubes – eight units the difference. You are left with 1018. • The order of matching does not matter. Always start with the wood from the larger number and match the wood from the smaller number to it.

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Main Activity (N3.3)

What to do

• multiply • matching blocks • difference between • subtract • regrouping • tetromino • squares • edge

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

Warm up

Language

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• Repeat for Exercise 3 (a–e), or allow students to explore by themselves. • Students should be able to complete Exercise 4 mentally. If not, encourage them to use the Base 10 MAB as for Exercise 3.

Challenge • Explain that a tetromino is a shape made from four squares joined at the edges. • Ask students to find as many tetrominos as they can, showing all drawings. • Ask whether flips and rotations represent the same or different shapes. • Display final results. For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 68–69. • 90 • New Wave Maths Book E – Teachers Guide

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

Unit 18–2

Student page 53

Outcomes

Indicators The student is able to: • order a few easily-understood situations from least likely to most likely. • record frequency data carefully using simple formats based on tallies or organised lists and take care with their measurements. • summarise data in diagrams and tables which show frequencies for different categories.

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

Skills • recording • collecting data • inferring • rationalising • summarising • justifying decisions

Resources

Language

• calculator • Bass 10 MAB • class surnames • library book • reading book

• multiply • table • frequency • most common • rank • most likely • least likely

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

Memory Masters (N3.3)

Notes

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• The focus for this unit is the addition of a multiple of 100 less than 1000 to a whole number 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 (C&D3.1, C&D3.2, C&D3.3) Warm up

• List all surnames of students in the class on the blackboard/whiteboard.

What to do

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Challenge

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• Exercise 3 is looking at gathering, recording and analysing data. • Use the frequency table to make a tally of the number of times each letter is the first letter of a student’s surname. • When the frequency table is completed, students may write the total in each box. Once this is completed, they can write the letter that is most frequently the initial letter of a surname in the space provided. • Exercise 4 is looking at chance outcomes. When selecting pairs for the imaginary tennis tournament, if class discussion follows, ensure that discussion involves rational capabilities and not personal put downs. • Rank student pairs according to the likelihood of a pair to win or lose to other pairs. • This exercise may be completed in groups of eight and rankings made within the groupings. Alternatively, the teacher may place rankings of students or world players on the blackboard, and have the students rank these.

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• Students are to select a library book or a reading book and make their decision on which letter of the alphabet is the most frequent. • Students may find it necessary to write each letter of the alphabet and record tally marks each time it occurs. • Show all recordings and provide a written explanation for final decisions.

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

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 91 •

Unit 18–3

Student page 54

Outcomes

Indicators

N3.3, N3.2

The student is able to: • understand the terms ‘multiple’, ‘factor’, and ‘prime’ and use them appropriately. • remember basic addition facts and many multiplication facts and calculate mentally basic multiplication facts they don’t recall.

Skills • finding factors • problem-solving

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • coloured pencils

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)

Notes

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Number (N3.3)

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• Ask students to try this with Exercises 3 (a) and (b). • In Exercise 4, students are to find all the factor pairs and write them in the space provided. If they think of the factor pairs of 12 as 1 x 12, 2 x 6, 3 x 4 they can follow that pattern for 16 and 30. Help as required. • Exercise 5 asks for factors that are prime numbers. Find the factors for 10 and 18 and write the prime numbers that make these numbers when multiplied together. Using the number 12 as an example, prime factors can come from 4 x 3, which may be written as 2 x 2 x 3. Students may be required to use a prime factor more than once. More examples may be needed. • Exercise 6 is a problem-solving exercise to complete the number sentences. In each activity the symbol represents the same digit for that number sentence. Symbols may represent different numbers in different examples. Remind students that zero is a possible choice.

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• What is a factor? Give an example—2 x 3 is a factor of 6. • All numbers, including prime numbers (except for 1) have at least one pair of factors. • Ask students to help find the factor pairs for 12; i.e. any number that will divide into 12 without a remainder is a factor. • List the factors on the blackboard/whiteboard. • Find factors as pairs and start with the number 1. Use 12 as the example—1 x 12; 2 x 6; 3 x 4; 4 x 3 (already have as 3 x 4) 5 x – no pair; 6 x 2 (already have as 2 x 6). What can you tell about 9, 8, 9, 10 and 11 as factors of 12?( They can’t be because multiplied by 1 they remain the same number and multiplied by 2 they are larger than 12.) The last factor of 12 is 12 x 1 (already have as 1 x 12).

What to do

• multiply • factor pairs • numbers • product • prime numbers • symbol • sum • least • shape • neighbouring

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• The focus for this unit is the addition of a multiple of 10 less than 1000 to a whole number less than 10.

Warm up

Language

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Challenge • Students use the least number of colours possible to colour the diagram. No two adjoining areas can have the same colour.

• 92 • New Wave Maths Book E – Teachers Guide

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Unit 18—Answers

Student pages 52–54 Unit 18–1

1. (a) 305 (b) 508 (c) 202 (d) 607 (e) 401 (f) 906 (g) 109 (h) 703 (i) 202 (j) 404 2. (a) 606 (b) 609 (c) 804 (d) 408 (e) 505 (f) 808 3. Teacher check 4. 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.

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1. Answers will vary; some possible solutions: 4 x 4 x 4, 2 x 8 x 4, 2 x 2 x 2 x 2 x 4, 2 x 2 x 2 x 2 x 2 etc. 2. (a) 690 (b) 860 (c) 480 (d) 390 (e) 880 (f) 960 3. (a) 1223 (b) 3241 (c) 2223 (d) 2332 (e) 630 4. (a) 14 (i) 21 (q) 17 (b) 46 (j) 37 (r) 4 (c) 32 (k) 2 (s) 28 (d) 68 (l) 18 (t) 12 (e) 23 (m) 3 (u) 29 (f) 79 (n) 9 (v) 36 (g) 54 (o) 15 (w) 21 (h) 5 (p) 36 (x) 27 Challenge Teacher check

Unit 18–2

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• Provide students with further opportunity to practise their own methods when comparing subtraction problems.

Consolidation 18–2

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1. (a) 523 (b) 386 (c) 436 (d) 679 (e) 581 (f) 916 (g) 657 (h) 242 (i) 194 (j) 863 2. (a) 400 (b) 800 (c) 800 (d) 900 (e) 600 (f) 600 3. (a) 20 x 1, 10 x 2, 5 x 4 (b) 24 x 1, 12 x 2, 8 x 3, 6 x 4 4. (a) 16 x 1 = 8 x 2 = 4 x 4 (b) 30 x 1 = 15 x 2 = 10 x 3 = 5 x 6 5. (a) 5 x 2 (b) 2 x 3 x 3 6. (a) 3 (g) 2, 2 (b) 4, 4 (h) 1, 1 (c) 0, 0 (i) 0, 0, 0 or 6, 6, 6 (d) 4, 4 (e) 0, 0, 0 or 5, 5, 5 (f) 1 Challenge 3

• Students survey class members for selected data and record as a tally in two-way tables.

Consolidation 18–3

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R.I.C. Publications®

• Develop class charts which can be used as a reference by students. Charts could include prime numbers, factor pairs, composite numbers etc.

New Wave Maths Book E – Teachers Guide • 93 •

Unit 19–1

Student page 55

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.

Skills • symmetrical drawing • scale drawing

Memory Masters (N3.3)

Resources • calculator • 1-cm grid paper (see page 199) • mirror/mira

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• 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)

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• Explain that half of a drawing has been shown on the page.The task is to complete the other half of the drawing so that the completed drawing is symmetrical—the top half is a mirror image of the bottom half. • To assist, the outside of the top half has been shown with a dotted line. Five of the internal lines have also been shown as dotted lines. Draw these lines in fully and find the rest of the lines. Using a mirror/mira may help. • When the drawing is complete, ask the students to draw the outline on a separate sheet of paper so that it is half the length and width of the shape in the workbook. Now complete the diagram so that it is a 1:2 (half dimension) replica of the original. 1-cm grid paper may assist.

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• Review the notion of symmetry. A shape, picture or object is symmetrical if it is identical on both sides of a line dividing it into two parts. • Brainstorm items students consider to be symmetrical.

Challenge

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Number (N3.3)

What to do

• add • symmetrical shape • reduce • half the dimensions • original size • squares

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• The focus for this unit is the addition of a whole number less than 1000 to a whole number less than 10 with no regrouping.

Warm up

Language

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• Students use the drawing in the workbook to find the number of squares in the picture. Remind them that all squares will not necessarily be the same size or on the same angle. • Students show how they counted the squares by writing a report of what they did or the use of any other means to assist in counting. • Share results with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 20–21. • 94 • New Wave Maths Book E – Teachers Guide

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

Unit 19–2

Student page 56

Outcomes

Indicators

N3.3, N3.1b, N3.1a

The student is able to: • read and write fractional notation (i.e. symbols) to represent units fractions. • read and write any whole number into the thousands.

Skills • recording fractions • calculating

Resources

Language

• calculator • fraction squares

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• add • fraction • whole • shaded • diagrams • calculator • largest number • smallest number • three different digits • four different digits • five different digits • second last • fourth

Notes

Memory Masters (N3.3)

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Number (N3.3)

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• The focus for this unit is the addition of a whole number less than 1000 to a whole number less than 10 with no regrouping.

• 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, N3.1a) Warm up

© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Explain to students they will be participating in a number of different activities, requiring them to write fractions and whole numbers.

Challenge

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• Ask how many parts are shown in the diagram in Exercise 3 (a). (8) How many parts are shaded? (3) This represents 3 out of 8 or 3/8. Write the answer in the space provided. • Repeat for (b), (c) and (d) or allow students to find the fraction which is shaded. • You may need to discuss the changes incurred by adding 1 in Exercise 4.The number created is formed using the next place value. • Exercise 5 asks the students to use 3, 4 or 5 different digits to make whole numbers. Ask students to give examples of numbers with 3, 4 or 5 different digits. • If you are to make the largest number possible which different digits would you use? (Those closest to 10.) Which three are closest to 10? (7, 8 and 9) Which of these three digits will be in the highest place value? (9) • Either repeat for (b) and (c) or allow students to find the numbers.

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• Students must show all drawings and make notes to explain. • The final explanation should have notes to allow others to understand how the answer was reached. • Share solutions with the class and steps taken to reach the answer.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 46–47 and 34–35. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 95 •

Unit 19–3

Student page 57

Outcomes

Indicators

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

The student is able to: • use a uniform unit of length to measure and order heights. • remember basic facts of addition.

Skills • measuring • ordering

Memory Masters (N3.1a)

Resources • calculator • paper streamer • metre rule • measuring tape • pencil • ruler

Language • circle • number • hundreds • add • measure • height • centimetres • metres • tallest • shortest • points • closest • furthest • total • distance • path • difference between • traversable

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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 (M3.2, N3.3) Warm up

• Organise students into groups of four. • Provide groups with paper streamers and measuring tapes. • Measure items around the classroom and compare heights.

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• Ask the groups to measure each other’s height with the paper streamer. Once this has been finished, use the measuring tape to find out the actual height of each group member. • Record the names of each group member and the measured height on the table provided. The table should show group members with the tallest first through to the shortest at the bottom of the table. • Exercise 4 shows a series of points representing towns that are the distances apart shown on the page. Note: This map is not drawn to scale. • Use the distances shown to answer the questions.

Challenge

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What to do

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• The focus for this unit is the identification of place value.

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• Using a pencil, students are to check to see if the path is traversable. • To be traversable, each path segment can only be travelled along once. Path segments may be crossed over and points may be reached more than once.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 78–79. • 96 • New Wave Maths Book E – Teachers Guide

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

Unit 19—Answers

Student pages 55–57 Unit 19–1 1. (a) 607 (f) 407 2. (a) 906 (f) 895

(b) 305 (g) 482 (b) 912

3. (a) 3/8 (b) 15/20 4. (a) 1000 (b) 10 000 (c) 100 000 (d) 1 000 000 (e) 10 000 000 5. (a) 987 (b) 98 765 (c) 0123 Challenge

(d) 306 (e) 725 (i) 509 (j) 704 (d) 959 (e) 653

(d) 8/8

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4. Teacher check Challenge 31

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1. (a) 807 (b) 509 (c) 107 (d) 908 (e) 309 (f) 608 (g) 409 (h) 709 (i) 208 (j) 909 2. (a) 2288 (b) 1598 (c) 1679 (d) 1998 (e) 1689 (f) 2169 3.

Unit 19–2

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• Students create their own symmetrical designs.

Consolidation 19-2 • Ask students to write the numbers made on the page in word form.

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1. (a) 8 (b) 4 (c) 9 (d) 7 (e) 8 (f) 4 (g) 8 (h) 9 (i) 1 (j) 5 2. (a) 1972 (b) 2358 (c) 1394 (d) 1392 (e) 1344 (f) 2079 3. Teacher check 4. (a) C and D (b) B and E (c) 224 km (d) 84 km Challenge Yes. E to A to B to D to D to C to B to E to D to A to C to E

Consolidation 19–3

• Students can use a street directory to devise a path. Ask other students to work out the distance of the path using scale.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 97 •

Unit 20–1

Student page 58

Outcomes

Indicators

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

The student is able to: • understand that multiplication can be used for repeated addition situations. • remember basic addition facts and many multiplication facts and calculate basic multiplication facts they don’t recall.

Skills • reasoning mathematically

Memory Masters (N3.1a)

Resources • calculator

Language • round • numbers • nearest ten • subtract • investigate • addition • mathematical properties • multiplication • odd numbers • even numbers • continuous addition

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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 (N3.2, N3.3) Warm up

• Ask the students to explain to you, what they understand addition and multiplication to be. • Ask if they know of any relationship between multiplication and addition. • Ask the following questions to promote discussion or gain specific answers: When you add two even numbers together you get an … number? (Even) When you add two odd numbers together you get an … number? (Even) When you add an odd and an even number together you get an … number? (Odd) Who can explain why this is so? When you multiply two even numbers you get an … number? (Even) When you multiply two odd numbers you get an … number? (Odd) When you multiply an odd and an even number you get an … number? (Even) If multiplication is repeated addition why is this so? Discuss. • Demonstrate examples on the board to support each question; e.g. 2 + 2 = 4; 3 + 3 = 6; 2 x 2 = 4; 3 x 3 = 9; 2 x 5 = 10. • Show repeated addition as multiplication: 3 + 3 + 3 = 3 x 3 = 9, 8 + 8 + 8 + 8 = 4 x 8 = 32.

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

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What to do

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• Students write their own explanation of why they think that when adding two odd numbers the answer is an even number, but, when multiplying two odd numbers the answer is an odd number. • Share a selection of explanations.

Challenge

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• Record all attempts and keep notes to explain the thoughts and reasoning of each attempt. • Share results with the class.

• 98 • New Wave Maths Book E – Teachers Guide

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

Unit 20–2

Student page 59

Outcomes

Indicators

N3.3, C&D3.3

The student is able to: • use diagrams such as Venn diagrams and two-way tables to represent a two-way classification. • add and subtract whole numbers using their own written method or a conventional algorithm, explaining the method by reference to place value.

Skills • using Venn diagrams • recording • problem-solving

Resources

Language

• calculator • Base 10 MAB

• number sentences • share • Venn diagram

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Memory Masters (N3.3)

Notes

Number (N3.3)

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

• 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, N3.3) Warm up

• Discuss Venn diagrams as a means of showing information. For example, show prime and even numbers inclusive of 1–10. The numbers 1, 3, 5 and 7 will be included in one circle. The numbers 4, 6, 8 and 10 will be included in the other circle. The number 2 is prime and even so it will be shown in an overlap of the two circles.

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What to do

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• Rochelle and Taylor were given 18 Easter eggs between them. Rochelle was given six herself. How can this be shown on the Venn diagram? (Draw six Easter eggs, or write the number 6 in Rochelle’s circle.) • Are we told how many eggs Taylor was given? (No.)We have been told that they were given seven eggs to share. How can this be shown? (By drawing seven eggs or writing the number 7 in the overlap of the two circles.) These eggs belong here because they are both Rochelle and Taylor’s eggs. • How many eggs have been shown? (6 for Rochelle and 7 shared, a total of 13. )How many eggs did Taylor have? (18 – 13 = 5.) • Students complete Exercise 4 by themselves.

Challenge • Students are to read the question carefully then write an explanation to describe their answer. • Share solutions. For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 116–117.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 99 •

Unit 20–3

Student page 60

Outcomes

Indicators The student is able to: • add decimals with like denominators using diagrams and concrete materials for support.

N3.3

Skills • working with fractions • using concrete materials • manipulation • sharing

Memory Masters (N3.3)

Resources • calculator • bar of chocolate • coloured pencils • 2-cm cubes

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)

Notes

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• subtract • diagrams • fractions

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• The focus for this unit is the multiplication of a whole number less than 10 by a multiple of 100 less than 1000.

Warm up

Language

• Show students eight 2-cm cubes. Place them together and then separate them into eight individual pieces.

What to do

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• The exercises in the workbook all involve sharing parts of a whole or fractions. • In Exercise 3 (a) Bob and Brett each give 2/5 of their chocolate bar to Coleen. Students are to work out how much chocolate Coleen received as a fraction of a whole bar. She received 2 /5 and another 2/5 for a total of 4/5. • Students work through Anne and Claire’s sharing of the pizzas in Exercise 3 (b) and Kelly and Michelle’s sharing of their marbles in Exercise 3 (c).

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Challenge

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Select two students. Give one student 3 8 of the 2-cm cubes. Give the second student 1 8 of the 2-cm cubes. How many of the 2-cm cubes did the two students receive? (4 8) Repeat several times using different fractions. • A similar activity using a bar of chocolate may be done to reinforce the idea of sharing items. Separating a whole item into pieces creates a fraction.

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• Students are to show all their working and keep notes to explain how they found out the cost of the eraser. • Remind students that the pen costs $2.00 more than the eraser, therefore the eraser can not cost 10 cents.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 50–51. • 100 • New Wave Maths Book E – Teachers Guide

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

Unit 20—Answers

Student pages 58–60 Unit 20–1

1. (a) 260 (b) 520 (c) 850 (d) 390 (e) 470 (f) 710 (g) 490 (h) 240 (i) 590 (j) 170 2. (a) 7047 (b) 6127 (c) 2227 (d) 8027 (e) 6302 (f) 8261 3. An odd number of odd numbers will always add to an odd number; hence, when you multiply two odd numbers the number will be odd. For example; 3 x 3 = 3 + 3 + 3 = 9 5 x 3 = 3 + 3 + 3 + 3 + 3 = 15 5 x 7 = 7 + 7 + 7 + 7 + 7 = 35 Challenge Answers will vary; one possible solution

Unit 20–2 1. (a) 6 (b) 5 (c) 4 (f) 6 (g) 19 (h) 5 2. (a) 1146 (b) 197 (c) 274 (f) 7513 3. Taylor was given 5 eggs.

(d) 9,9 (e) 27 (i) 2 (j) 5 (d) 6253 (e) 7653

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4. Coleen bought 7 cassettes for herself.

Challenge The ladder’s position will remain the same, as the ship rises on the tide.

3. (a) 4/5, teacher check drawing. (b) 4/6 (c) 5, teacher check drawing. Challenge 5c

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• Can students find any relationship between subtraction and division?

Consolidation 20–2

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1. (a) 200 (b) 800 (c) 600 (d) 900 (e) 400 (f) 300 (g) 700 (h) 500 (i) 100 (j) 400 2. (a) 3.7 L (b) 1.4 m (c) 1.2 cm (d) 5.7 km (e) 1.8 kg (f) 2.5 mm

• Provide students with opportunities to collect and record data in Venn diagrams.

Consolidation 20–3

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R.I.C. Publications®

• Regularly use classroom situations to encourage students to share items and consider the fractions involved.

New Wave Maths Book E – 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.

Skills • symmetrical drawing

Memory Masters (N3.3)

Resources • calculator • mirror/mira • pencil • symmetrical patterns/drawings

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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Teac he r

• multiply • diagram • symmetrical pattern • large rectangle • small triangle • inside

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• The focus for this unit is the addition or subtraction of whole numbers less than 10 to or from a whole number less than 100 with no regrouping.

Warm up

Language

• Display a number of symmetrical diagrams or drawings. Ask students where the line(s) of symmetry are. Discuss symmetry—ensure students understand that each side of the line of symmetry is a mirror image of the other. • If using a mirror, explain how it is used and what it shows. The mirror is placed on the line of symmetry. By looking into the mirror the mirror image is displayed. It is possible to ‘trace over’ this image by holding the mirror in place and following the image lines.

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• On the page is part of a symmetrical pattern. Students are to complete the pattern around each of the lines of symmetry. Use mirrors/miras if available. • Parts of the pattern overlap the lines of symmetry to assist with the drawing. • Students may colour their finished pattern.

Challenge

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What to do

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• This activity requires students to follow directions. • Show all attempts and jot notes beside drawing(s) to explain what was done. • Share finished drawings.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 20–21. • 102 • New Wave Maths Book E – Teachers Guide

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

Unit 21–2

Student page 62

Outcomes

Indicators

N3.3, M3.3

The student is able to: • with a physical model of the unit available for comparison, estimate which regions they can see have an area of about the same as, less than or more than a square metre, which containers have capacity of about the same as, less than or more than a litre, which objects have a mass of about the same as, less than or more than a kilogram.

Skills • estimating • comparing • recording • measuring

Resources

Language • multiply • less than • same as • more than

• calculator • newspaper made into 1 m2 • container holding 1 L of liquid • various containers • item weighing 1 kg • various items • balance scales • classroom/school environment

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Notes

Memory Masters (N3.3)

Teac he r

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• The focus for this unit is the addition or subtraction of a whole number less than 10 to or from a whole number less than 100.

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 (M3.3) Warm up

• Display the newspaper which has an area of 1 m2. Invite students (in small groups) to come forward and have a closer look. Encourage students to count the number of their feet it takes to step along one side. • Display a container holding 1 L of liquid. Discuss its shape, size etc. Pour the liquid into a different container (of a different shape and size but which still holds 1 L). Talk about the changes. Does this mean there is more or less liquid? (No.) • Display an object which weighs exactly 1 kg; e.g. sugar, flour, Weetbix® etc. Invite students to heft the object to get a feel for the weight of 1 kg.

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What to do

• Read and explain the tasks involved in Exercises 3, 4 and 5. • Students are required to estimate and list three items for each category. Students may find it helpful to refer back to the items presented in the warm up activity. Ensure they are still available as a ready reference. • Students may complete the activities in small groups or independently.

Challenge

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• The number 60 is to be made using just the four operation signs and the number 6. • Students are to record all attempts with explanations of what they were doing. • Final answers to be shared with the class. • If finished early, find a different combination to make 60 using the same rules.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 92–97. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 103 •

Unit 21–3

Student page 63

Outcomes

Indicators

N3.3, M3.4a

The student is able to: • measure the perimeter of an irregular shape, ensuring that they follow the edge closely and begin and end at the same place. • given a perimeter, use grid paper to generate possible shapes where the shapes fit along the grid lines.

Skills • measuring • estimating

Memory Masters (N3.3)

Resources • calculator • ruler • pencil • 1-cm grid or dot paper (see pages 198–199)

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 (M3.4a)

estimate perimeter longest greatest vertical

Notes

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• Before finding the distance around the outside of each of the shapes shown they are to estimate the distance first. Write the estimate in the appropriate space. • Students now measure the perimeter of the shape. Ask how they will do this. (Count the number of lines between individual dots or use a ruler). Write the actual measure of the perimeter beside the shape. • When all perimeters have been measured, write which shape has the longest perimeter. • Exercise 4 asks students to use 12 squares to find the shape with the longest perimeter.When drawing different shapes, each square must have one full edge in contact with a full edge of at least one other square. • Students see how many different shapes they can find with 12 squares to find the one with the longest perimeter. • Use 1-cm grid or dot paper to record shapes. Direct students to number each shape they draw.

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• Direct students to look at the workbook. • They will see seven different shapes drawn on the page. Ask them to find the shape with the longest perimeter. • Who can tell me what a perimeter is? (peri meter—to measure around)

Challenge

• • • • •

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Number (N3.3)

What to do

• multiply • measure • shapes • arc • diagonal • horizontal

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• The focus for this unit is the addition or subtraction of a whole number less than 10 to or from a whole number less than 100 with no regrouping.

Warm up

Language

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• To some students the answer will be obvious—as all shapes have 12 squares then all areas are the same. • For others the notes that are kept as they think and work through the problem are a valuable guide to the development of the concept of area.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 100–101. • 104 • New Wave Maths Book E – Teachers Guide

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

Unit 21—Answers

Student pages 61–63 Unit 21–1

1. (a) 45 (f) 61 2. (a) 515 (f) 621 3.

(b) 56 (g) 73 (b) 816

(d) 69 (e) 86 (i) 85 (j) 62 (d) 927 (e) 618

1. (a) 96 (b) 46 (c) 38 (d) 57 (e) 67 (f) 31 (g) 44 (h) 61 (i) 53 (j) 81 2. (a) 640 (b) 660 (c) 690 (d) 840 (e) 690 (f) 460 3. Teacher check 4. Teacher check 5. Teacher check Challenge Answers will vary; some possible solutions: 6+6+6+6+6+6+6+6+6+6 6x6+6+6+6+6 etc.

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Unit 21–2

Challenge

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• Students can make their own symmetrical patterns.

Consolidation 21–2 • Provide students with everyday opportunities to estimate area, capacity and mass.

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1. (a) 78 (b) 87 (c) 49 (d) 69 (e) 79 (f) 92 (g) 24 (h) 10 (i) 33 (j) 45 2. (a) 760 (b) 760 (c) 840 (d) 540 (e) 520 (f) 870 3. Teacher check estimates (a) 20 units (b) 16 units (c) 22 units (d) 20 units (e) 20 units (f) 26 units (g) 22 units 4. 26 units. Challenge No, all areas are the same—12 squares.

Consolidation 21–3

• Keep the perimeter the same and ask students to compare the area of the shapes they create.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 105 •

Unit 22–1

Student page 64

Outcomes

Indicators

N3.3, N3.4

The student is able to: • use a number-letter code to write a message.

Skills

Resources • calculator

• encoding/decoding

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 (N3.4)

Notes

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• add • substitution • code • numbers • line • total

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• The focus for this unit is the addition or subtraction of a whole number less than 10 to or from a whole number less than 100.

Warm up

Language

• Using codes has been a means of getting messages from one place to another. Explain how morse code was once widely used as a means of sending messages. It used a system of dots and dashes corresponding to letters of the alphabet, which were signalled by telegraph. • There are many different ways of making codes. One of the simplest is to match the letters of the alphabet with numbers; such as A = 1, B = 2 and so on to Z = 26. When a message is written using this code you would see something like this: 3 15 18 18 5 3 20. • When decoded, what does this say? (CORRECT)

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• Exercise 3 uses a number-letter substitution code starting with A = 26 to Z = 1. • Students make up a simple message of their own using this code. • Exercise 4 asks students to make up their own number-letter substitution code to write a message to a friend. The friend can decode the message using the code or see if he/she is able to break the code.

Challenge

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What to do

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• The numbers 1–9 are to be used once only when placed in the circles. • Each line is to add to a total of 15. • Keep a record of all attempts. Students should make notes to explain how they solved the problem.

• 106 • New Wave Maths Book E – Teachers Guide

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Unit 22–2

Student page 65

Outcomes

Indicators

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

The student is able to: • report the frequency information provided in a tally produced by a classmate. • suggest information to collect to answer particular questions.

Skills • tallying • ordering • recording

Resources

Language

• calculator • birth dates of class members

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• patterns • add • years • months • range • common • most • least • tree diagram

Memory Masters (N3.4)

Notes

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• The focus for this unit is the completion of patterns, including number patterns.

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&D3.4, C&D3.2) Warm up

• Compile a list showing ages of students in the class in years and months on the blackboard/ whiteboard. Next to the age, write the month the student was born in.

What to do

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• Using the information on the board, ask students which is the earliest birth date and which is the latest birth date. Explain that these two dates cover the range or spread of the birth dates of all members of the class. Write these two dates in the workbook in the space provided. • Ask the class to help you tally the birth months and record these as a tally and total on the board; e.g. January 111–3. • From the tally sheet, find the month with the most birthdays—the most common birth month. Write the answer in the space provided. • Do the same for the least common birth month. • A tree diagram has branches or arms like a tree leading to or coming from one trunk. • In a tennis tournament, players are paired. The winner of the pair is written on the next line that is joined by a bracket to the playing pair; e.g.

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• The top two winners on the tree diagram play each other. The next pair play and so on. A bracket joins the pairs with the winner written on the line, as shown above. Further explanation may be needed. • Ask students to complete the tree diagram. No names are required, just the diagram.

Challenge • Students are to find the greatest possible range of birth dates over one calendar year. • Record findings and share with class.

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New Wave Maths Book E – Teachers Guide • 107 •

Unit 22–3

Student page 66

Outcomes

Indicators

N3.1a, N3.3

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.

Skills • recording place value • partitioning numbers

Memory Masters (N3.1a)

Resources • calculator • place value chart (see pages 205– 206)

Language • > greater than • < less than • = equal to • add • total value • digit • place value • number • partition

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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 place value chart for all students to see. Ask students to name the place value on the chart as you point to it. Ask students for the relationship between places. (Always a factor of 10.) • Use a number of examples and record in the correct columns on the place value chart.

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• The task is to write the total value of each digit of each given number in its correct place on the chart. • Exercise 3 (a) has been done as an example to follow. Work through each of these examples as follows: What is the number in (a)? (785 964) The four belongs in which place? (Ones. Its total value is 4 ones or 4.) Repeat for each digit of the examples given. • If necessary, continue to work with the whole class or small groups as required. Students who are able may continue by themselves. • Exercise 4 requires students to follow the example to write the given numbers in expanded form using partitioning techniques.

Challenge

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What to do

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• The focus for this unit is the completion of expressions of equality or inequality.

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• Students record all working out in the space provided. • Share results with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 36–37.

• 108 • New Wave Maths Book E – Teachers Guide

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Unit 22—Answers

Student pages 64–66 Unit 22–1

1. (a) 111 (b) • (c) (d) c (e) 8 (f) 4 (g) 12 (h) 30 (i) (j) 2. (a) 1654 (b) 2092 (c) 1194 (d) 1453 (e) 1761 (f) 1758 3. Teacher check 4.

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Challenge 365 or 366 days if a leap year.

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1. (a) 97 (b) 68 (c) 96 (d) 49 (e) 59 (f) 94 (g) 52 (h) 73 (i) 13 (j) 22 2. (a) 2648 (b) 2009 (c) 1249 (d) 1578 (e) 2059 (f) 2539 3. Teacher check 4. Teacher check Challenge

Unit 22–2

(d) > (i) > (d) 59

(e) > (j) > (e) 86

• Students make up their own codes using symbols. Make messages and swap to decode.

Consolidation 22–2

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(b) > (g) < (b) 59

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1. (a) < (f) = 2. (a) 78 (f) 29 3.

• Develop a list of data that could be collected. Brainstorm the best way to collect, record and analyse that information.

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Consolidation 22–3

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• Provide opportunities for students to define place value in daily situations.

4. (a) 100 + 70 + 2 (g) 800 + 7 (b) 200 + 80 + 1 (h) 900 + 90 (c) 400 + 10 + 9 (i) 300 + 10 + 3 (d) 600 + 3 (j) 400 + 40 + 4 (e) 300 + 20 (k) 500 + 50 + 7 (f) 1000 + 80 + 9 (l) 700 + 10 + 8 Challenge Be paid $2 and have your salary double as you end up with more money. $20 x 7 = $140 1 2 3 4 5 6 7 2 + 4 + 8 + 16 + 32 + 64 + 128 = $254

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New Wave Maths Book E – Teachers Guide • 109 •

Unit 23–1

Student page 67

Outcomes

Indicators

N3.1, N3.3, S3.3

The student is able to: • identify repetitions of component parts in symmetrical objects/ arrangements.

Skills • symmetrical drawing • enlarging • drawing

Memory Masters (N3.1)

Resources • calculator • mirror/mira • pencil • 1 cm grid paper (see page 199)

Language • dollars • change • cents • diagram • symmetrical shape • squared paper • broken line • line of symmetry • enlarge • number • signs

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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

• View and discuss symmetrical shapes. What makes them symmetrical?

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• On the page is a partial drawing with a dotted line. The dotted line is the line of symmetry. Complete the drawing by drawing the mirror image of the partial drawing on the page. Use a mirror/mira if one is available. • When the drawing is completed, an enlargement of the drawing is to be made on 1-cm grid paper. The enlargement is to be twice the dimensions of the original. If a line is two squares long on the original drawing, it will be four squares long in the enlargement. • Exercise 4 allows students to draw three shapes of their own choosing. Suggest they use simple shapes. When the shapes have been drawn, all lines of symmetry, if any, are to be shown on each shape.

Challenge

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What to do

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Teac he r

• The focus for this unit is the conversion of cents to dollars and dollars to cents.

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• Using only the number 2 and any or all of the signs, students are to make the numbers 21, 17 and 32. • All attempts to make each number are to be shown. • Share results with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 20–21. • 110 • New Wave Maths Book E – Teachers Guide

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

Unit 23–2

Student page 68

Outcomes

Indicators

N3.3, N3.1a

The student is able to: • read and write whole numbers into the thousands. • distinguish and order whole numbers.

Skills • ordering • manipulating concrete materials

Resources

Language • number pairs • Base 10 MAB • least • most • order • greatest

• calculator • Base 10 MAB • counters • place value chart (see pages 205– 206)

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Notes

Memory Masters (N3.3)

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

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

• Organise students into small groups and distribute Base 10 MAB. • Ask students to use the Base 10 MAB to show 238. Leave the wood on the table and take more Base 10 MAB to show 283. • Ask students to count how many pieces of wood they used to make both numbers.

Challenge

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• Students now have the answer to Exercise 3 (a). • Ask students to write the total number of pieces of wood used. • Students are to continue using the Base 10 MAB to show each number pair and the total amount of wood used. • Exercises 4 and 5 both require ordering of prices from least to most expensive. Guide students to look at the highest place value digit first, then the next if required. If need be, ask students to write the numbers in the correct place value columns on a chart and check least to most expensive this way.

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• Use the exercises just completed to help make a final decision. • Record any reasoning to share with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 34–37. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 111 •

Unit 23–3

Student page 69

Outcomes

Indicators

N3.3, M3.1

The student is able to: • choose shapes that can cover a region with no gaps or overlaps to use as units of area.

Skills • tessellating • colouring

Memory Masters (N3.3)

Resources • calculator • pencil • ruler • coloured pencils • square tiles • 1-cm grid paper (see page 199)

Language • multiply • tessellate • divide • pentomino

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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 (M3.1) Warm up

• Ask students to explain what they understand about ‘tessellations’. • If students are unsure, explain that a ‘tessellation’ is tiling. For shapes to tessellate they must not leave space between tiles.

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• A grid has been provided with a pentomino shape shown. A pentomino is a shape made from five squares all joined by at least one edge to another edge. • Use the same pentomino shape to make a continuous tessellating pattern that fits inside the grid. To make it easier to see each pentomino shape, colour the pentominos in a variety of different colours. • When reaching the edge and the pentomino shape does not fit exactly, what might you do? Arrive at a consensus from the class discussion. • Cover the grid neatly with pentominos.

Challenge

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What to do

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Teac he r

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

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• Using 5 square tiles, or by drawing directly onto 1-cm grid paper, students make as many different pentomino shapes as they can. • Flips and rotations count as the same shape. • Remind students that at least one full side of a square must be in contact with one full side of another square. • Show all attempts. Make notes on any shapes rejected to explain why it was rejected.

• 112 • New Wave Maths Book E – Teachers Guide

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Unit 23—Answers

Student pages 67–69 Unit 23–1

Teac he r

1. (a) $6.27 (b) $5.00 (c) $7.90 (d) $4.69 (e) $5.20 (f) 700c (g) 291c (h) 506c (i) 1000c (j) 352c 2. (a) 311 (b) 221 (c) 321 (d) 110 (e) 211 (f) 232 3.

Unit 23–2 1. (a) 24 (b) 40 (c) 24 (d) 8 (f) 2 (g) 3 (h) 2 (i) 4 2. (a) 76 (b) 74 (c) 69 (d) 38 (f) 84 3. (a) 8 (d) 16 (b) 8 (e) 11 (c) 8 4. $5.55, $5.60, $5.70, $5.85 5. $5000, $12 000, $40 000, $68 000 Challenge The value of the left-most digit.

(e) 63 (j) 2 (e) 92

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4. Teacher check Challenge Answers will vary; some possible solutions: 2 x 2 x 2 x 2 x 2 = 32 2 x 2 x 2 x 2 x 2 + 2 ÷ 2 = 17 2 x 2 x 2 x 2 x 2 + 2 + 2 + 2 + 2 ÷ 2 = 21

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(e) 4 (j) 3 (e) 10

• Students create their own symmetrical designs.

Consolidation 23–2 • Students order items from least to most expensive from a shopping catalogue.

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1. (a) 18 (b) 54 (c) 10 (d) 49 (f) 3 (g) 8 (h) 2 (i) 7 2. (a) 20 (b) 20 (c) 30 (d) 40 (f) 20 3. Teacher check Challenge Teacher check; some possible answers are:

Consolidation 23–3

• Create a tessellating pattern using a different shape.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 113 •

Unit 24–1

Student page 70

Outcomes

Indicators

N3.3, S3.1

The student is able to: • attempt to provide a bird’s-eye view of familiar locations such as their classroom.

Skills • drawing a plan to scale • measuring

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • square tiles or 1-cm grid paper (see page 199)

Language • subtract • plan • rearrange

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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 students how the classroom would look if they were looking down from the ceiling. • Together with the class, draw a floor plan of the classroom on the blackboard/whiteboard. • Point out the size of objects; e.g. the teacher’s desk is larger than a student’s desk which is larger than a bookshelf.

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• Explain to students that you are considering rearranging the furniture. • Tell students they are to draw a classroom plan they think will work best. All of the current furniture must be used in the new layout. • Share designs with the class. Choose some of the plans that will suit best and rearrange the furniture over the next few weeks to trial the designs.

Challenge

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What to do

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Teac he r

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

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• Students may use square tiles or 1-cm grid paper for this activity. • Explore arrangements of four squares so the shape has a perimeter of 12 units. • Tiles or squares may join full edge along full edge or corner to corner.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 2–3. • 114 • New Wave Maths Book E – Teachers Guide

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

Unit 24–2

Student page 71

Outcomes

Indicators

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. • summarise data in diagrams and tables which show frequencies for different categories.

Skills • surveying • analysing data

Resources

Language

• calculator • class members

• subtract • survey • Carroll diagram

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Memory Masters (N3.3)

Notes

Number (N3.3)

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

• 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 planning a three-day menu for a school camp. To assist with the planning, a survey of the class will be made to find out favourite foods. From the survey, (remembering that being on camp does not allow for a full range of foods to be prepared) a menu for the three days is to be planned. • Students may work in small groups for this activity.

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• A survey of the whole class, organised by the teacher and recording the results on the blackboard/whiteboard, may assist students in selecting the preferred options or most practical options. These can be recorded as tallies and totals in the workbook. • Remind students when surveying dinner that soup, main meal and dessert are three parts of the meal. When preparing the menu a decision on one, two or all three of these choices needs to be made. • Survey the class for the three meals. • Students now choose the favourite or most logical options to record on the Carroll diagrams in their workbook. • Plan the menus. Note: Only supply one option of soup, main and dessert for each class. • If working in groups, a menu may be prepared using art materials or the computer to display to the class.

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• How many different meal combinations may be served if there are three soups, four main meals and three desserts to choose from? Assume a person orders a soup, a main meal and a dessert.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 115 •

Unit 24–3

Student page 72

Outcomes

Indicators

N3.3, N3.1a

Skills • reading and writing numbers • estimating • multiplying

Memory Masters (N3.3)

The student is able to: • read and write any whole number into the thousands. • estimate sums and products by rounding.

Resources • calculator • Base 10 MAB • tangram (see page 221)

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.

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Teac he r

• subtract • round • nearest • approximate • tangram • triangle

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

Language

• Organise students into small groups. • Explain to the students they will be writing the numerals beside the word numbers to show the numbers.

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• Read the first number to the class. Ask a student to write the number on the blackboard/ whiteboard. Check with the class to see if they agree. Students write the correct number in their workbook. • This process may be repeated for the rest of the exercise or allow students to work in their groups to find the answers. • Exercise 4 involves rounding. Revise the rules for rounding up and rounding down. Round down: 0–4, 0–40, 0–400 etc. Round up: 5–9, 50–90, 500–900 etc. • Students write the rounded number and its multiple beside each multiplication problem. Complete each problem to provide an approximate answer.

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Challenge

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What to do

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• Provide students with a tangram or a sheet of paper with the tangram pieces marked on it for them to cut out. • Students manipulate the shapes to make a large triangle using all seven tangram pieces. • Draw the final arrangement on the page in the space provided.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 34–35. • 116 • New Wave Maths Book E – Teachers Guide

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

Unit 24—Answers

Student pages 70–72 Unit 24–1 1. (a) 56 (b) 6 (c) 32 (f) 9 (g) 9 (h) 7 2. (a) 280 (b) 280 (c) 560 (f) 450 3. Teacher check estimates Challenge 36 combinations

(d) 54 (e) 35 (i) 8 (j) 6 (d) 370 (e) 290

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1. (a) 64 (b) 16 (c) 81 (d) 9 (e) 25 (f) 9 (g) 7 (h) 6 (i) 8 (j) 4 2. (a) $2.32 (b) $5.32 (c) $4.21 (d) $3.24 (e) $3.44 (f) $4.44 3. Teacher check Challenge Answers will vary; one possible solution:

Unit 24–2

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• Students could draw a plan of their bedroom.

Consolidation 24–2 • Use the same approach to design and present a menu for a new ‘family’ restaurant.

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1. Answers will vary; some possible solutions: 5 x 5, 20 + 5, 30 – 5, 50 ÷ 2 2. (a) 215 (b) 429 (c) 276 (d) 457 (e) 538 (f) 335 3. (a) 172 (f) 70 005 (b) 6600 (g) 362 (c) 32 (h) 200 000 (d) 3060 (i) 98 (e) 615 (j) 36 018 4. (a) 180 (h) 400 x 5 = 2000 (b) 50 x 6 = 300 (i) 200 x 7 = 1400 (c) 80 x 4 = 320 (j) 800 x 5 = 4000 (d) 700 x 4 = 2800 (e) 500 x 8 = 4000 (f) 30 x 5 = 150 (g) 80 x 3 = 240 Challenge

Consolidation 24–3

• Offer students regular opportunities to read and write numbers and estimate through use of rounding.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 117 •

Unit 25–1

Student page 73

Outcomes

Indicators

WM3.1, WM3.2, M3.2

The student is able to: • identify devices which have a mathematical basis. • generate mathematical questions.

Skills • observing • analysing • posing questions

Resources • the Internet (optional) • various clocks

Language • clock • analog • digital • hands • face • time • minutes • seconds • watch • number • Roman numerals • marks • dots

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Notes

What to do

• This activity is designed for students working collaboratively in groups. Allow enough time so students can discuss their opinions 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 to investigate the first part of the activity. – What is the difference between a digital and an analog clock? – What does the face of an analog clock look like? Are they all the same? – What are Roman numerals? – Do all watches have numbers? – What is different between a stopwatch and a digital watch? (Show seconds.) • The second part of the activity may require further research for most groups. Students may request to use the resource centre or the Internet to find out how a sundial works. Sundials are probably the oldest clocks built. Egyptians built obelisks as early as 300 BC. Over the centuries sundials have become smaller and more portable. • Groups may wish to collate and summarise their findings and present them as a poster with a series of graphs, diagrams and information on how they solved the problem. • Allow each group time 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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Teac he r

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

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• 118 • New Wave Maths Book E – Teachers Guide

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

Unit 25–2

Student page 74

Outcomes

Indicators

N3.1a, N3.3

The student is able to: • use their own methods or a conventional algorithm to multiply whole numbers by single-digit numbers.

Skills • partitioning • multiplying • adding • subtracting

Resources

Language

• calculator • coloured pencils

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• round, • multiply • expanded form • mental • method • least • space • shape

Notes

Memory Masters (N3.1a)

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Teac he r

• The focus for this unit is the rounding of whole numbers to the nearest hundred.

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) Warm up

• Write 43 x 6 on the blackboard/whiteboard. • Explain that expanded notation means breaking a number into its place value components and writing the total value for each digit. In the case above, 43 is 4 tens and 3 ones or 40 + 3. The answer to the problem 43 x 6 can be found by multiplying 40 x 6 and adding 3 x 6. This gives 240 + 18 or 258. • Another way is to use the next highest ten to 40; that is 50 and multiply this by 6. As the original number was 43 and I have used 50, I now need to take off the difference between 50 and 43, which is 7, and multiply this by 6. The sum now reads (50 x 6) – (7 x 6) or 300 – 42 = 258. (The same answer as was reached before.)

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What to do

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Challenge

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• Students are to colour the shape using as few colours as possible. No area may be coloured the same as another area with which it shares a common boundary. • Students write a brief explanation of their findings.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 119 •

Unit 25–3

Student page 75

Outcomes

Indicators

N3.3, M3.2

The student is able to: • use a unit consistently and carefully to measure and compare lengths.

Skills • measuring with nonstandard units • recording data • analysing data

Memory Masters (N3.3)

Resources • calculator • streamers • tape measure • measuring stick

• • • •

measure length greatest difference

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)

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Teac he r

• multiply • distance • height • smallest

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

• The focus for this unit is the addition of whole numbers less than 10 to a multiple of 100 and the subtraction of a number less than 10 from a multiple of 10.

Warm up

Language

• Open a general discussion on the use of body parts for measuring. One foot length was close to a ‘foot’ (30 cm); nose to finger tips approximately a metre; middle bone in finger approximately 2.5 cm; one large pace equals nearly a metre and so on. Students may be asked to research other measures particularly Imperial measurements; e.g. ‘foot’, ‘inch’, ‘yard’. These measures were based on adult bodies.

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• Organise students into groups of five. Ask each group to measure the distance from their nose to the end of their outstretched right arm and record the measure for each individual in their workbook. • Repeat for measuring their arm span and their height. • Once all measurements have been recorded, answer the questions in the workbook. • As a follow-up activity, students could use the data collected to draw a scattergraph.

Challenge

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What to do

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• Use the information gained from the measurements to determine whether arm span is a good indicator of a person’s height. This may be completed as a group activity to allow different views to be expressed before a reason is recorded.

• 120 • New Wave Maths Book E – Teachers Guide

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

Unit 25—Answers

Student pages 73–75 Unit 25–1

1. (a) 3700 (b) 6000 (c) 8500 (d) 600 (e) 74 900 (f) 69 700 (g) 200 (h) 700 (i) 8900 (j) 2000 2. (a) 3000 (b) 6880 (c) 5700 (d) 5920 (e) 4830 (f) 3450 3. (a) (8 x 50) + (8 x 4) = 432 or (8 x 60) – (8 x 6) = 432 (b) (6 x 70) + (6 x 3) = 438 or (6 x 80) – (6 x 7) = 438 (c) (9 x 40) + (9 x 6) = 414 or (9 x 50) – (9 x 4) = 414 (d) (5 x 60 ) + (5 x 7) = 335 or (5 x 70) – (5 x 3) = 335 (e) (6 x 80) + (6 x 6 ) = 516 or (6 x 90) – (6 x 4) = 516 (f) (4 x 90) + (4 x 7) = 388 or (4 x 100) – (4 x 3) = 388 (g) (8 x 60) + (8 x 8) = 544 or (8 x 70) – (8 x 2) = 544 Challenge 4

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Teac he r

1. Answers will vary. Students should reflect on the difference between a digital and analog clock. Discuss that some digital clocks are in 24-hour time. Students may also comment on the different faces of analog clocks and watches with some having numbers, Roman numerals, dashes or dots to read. Other observations may include stopwatches showing the seconds, some analog clocks having second hands etc. 2. As the Earth turns on its axis, the sun appears to move across our sky. The shadows cast by the sun move in a clockwise direction for objects in the Northern Hemisphere and anticlockwise in the Southern Hemisphere. If the sun rose and set at the same time and spot on the horizon every day, sundials would be quite accurate clocks. However, the sun’s path through the sky changes every day because the Earth’s axis is tilted. The North Pole is tilted toward the sun for half the year and away from the sun for the other half. This means the shadows cast by the sun change from day to day. Also, because the Earth’s surface is curved, the ground at the base of the shadow stick is not at the same angle to the sun’s rays as at the equator. This means the shadow does not move at a uniform rate during the day.

Unit 25–2

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• Students can create their own sundial using a small plastic soft drink bottle, dry sand, 5-mm dowel, newspaper, art paper, marker pens and four little rocks (weights) to create their own sundial.

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1. (a) 709 (b) 306 (c) 808 (d) 202 (e) 507 (f) 67 (g) 74 (h) 26 (i) 33 (j) 88 2. (a) 800 (b) 2400 (c) 800 (d) 5600 (e) 3600 (f) 3500 3. Teacher check. 4. Teacher check. Challenge Yes. Taller people generally have longer arms and vice versa.

Consolidation 25–2

o c . che e r o t r s super

• Provide students with regular opportunities to practise the skill of partitioning.

Consolidation 25–3

• Students can measure and compare other body parts, strides etc.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 121 •

Unit 26–1

Student page 76

Outcomes

Indicators

N3.3, S3.3

The student is able to: • informally explain why they think a figure won’t tile.

Skills

Resources • calculator • Base 10 MAB

• analysing shapes • supporting conclusions

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 (S3.3)

Notes

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Teac he r

• shapes • tile • explain • larger • conclusion

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

• The focus for this unit is the addition of a whole number less than 100 to a multiple of 100 and the subtraction of a whole number less than 10 from a multiple of 10.

Warm up

Language

• Discuss with students what it means to ‘tile’. (To cover an area without leaving gaps.) Also called ‘tessellate’. • Brainstorm examples of situations where tiling is used (not just the wet areas of a home)— brickwork, tiles, paving, material designs, artwork etc. • Brainstorm the shapes students see in these situations; e.g. bricks are rectangular, paving bricks can be square or rectangular, tiles can be square or rectangular.

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• Look at the four shapes shown in the workbook. Students are to decide which shapes will tile and those that won’t. Students may like to trace over the shapes in a tiling pattern to check, if that method is easier. • Students are then required to offer an explanation for their answer; e.g. this shape will leave a gap, therefore it won’t tile. • Check students’ work and explanations.

Challenge

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What to do

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• Students can use concrete aids, diagrams or any other means to help decide which fraction in the diagram is larger. • Show all working out and keep notes to explain thought processes. • Share answers with classmates.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 22–23. • 122 • New Wave Maths Book E – Teachers Guide

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

Unit 26–2

Student page 77

Outcomes

Indicators

N3.3, C&D3.1

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.

Skills • considering outcomes • ordering events

Resources

Language

• calculator

• likely • not likely • order • events • explain

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

Memory Masters (N3.3)

Notes

Teac he r

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• The focus for this unit is the addition of a whole number less than 100 to a multiple of 100 and the subtraction of a multiple of 10 from a multiple of 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 (C&D3.1) Warm up

• Discuss the difference between events that are likely to happen and events that are not likely to happen. • Ask students to offer examples of each type of event. Record some on the blackboard/ whiteboard. • Invite students to help you order the events recorded on the board from most likely to least likely.

What to do

Challenge

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• Direct students to read the statements in Exercise 3 or read them as a whole group. • Students tick, circle or cross their selection for each event. • Exercise 4 requires students to consider the likelihood of the event occurring and to order the events. Students may find it easier to record ‘likely’ or ‘not likely’ next to each statement and then order them accordingly. • Share results with the class.

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• Students are responsible for devising their own list of events and ordering them according to likelihood of it happening. This activity is a direct extension of Exercises 3 and 4. • Explanations must be clear and support their chosen order. • Share with the class.

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

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 123 •

Unit 26–3

Student page 78

Outcomes

Indicators

N3.3, N3.1b

The student is able to: • read and write fractional notation (i.e. symbols) to represent unit fractions.

Skills • shading • writing fractions

Memory Masters (N3.3)

Resources • calculator • coloured pencils

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 (N3.1b)

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Number (N3.3)

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• Exercise 3 (a) shows a diagram of five blocks or five-fifths. Ask students to colour three fifths or three blocks and then another fifth or block (suggest a second colour is used for the second colouring). How many blocks have been coloured? (3 and 1 or a total of 4) This shows 3/5 + 1 /5 = 4/5. Record this on the board for students to view but do not expect students to record in this way. • Repeat this process for each of the activities on the page, working with the whole class.

Challenge

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• Students show all working and keep notes of reasoning/thinking when working out how to divide the block of chocolate. • Share possible solutions with classmates.

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• Diagrams have been drawn on the page to represent fractions. By colouring the diagrams as directed, it is possible to show addition of the fractions named.

What to do

• diagrams • divide

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

• The focus for this unit is the addition of a multiple of 10 to a multiple of 100 and the subtraction of 15 from a multiple of 10.

Warm up

Language

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For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 50–51.

• 124 • New Wave Maths Book E – Teachers Guide

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

Unit 26—Answers

Student pages 76–78 Unit 26–1 1. (a) 505 (b) 239 (f) 70 (g) 10 2. (a) 697 (b) 366 (f) 338 3. (a) Not likely (b) Not likely (c) Likely (d) Not likely (e) Likely (f) Not likely 4. Teacher check Challenge Teacher check

(d) 207 (e) 862 (i) 20 (j) 20 (d) 787 (e) 478

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Teac he r

1. (a) 115 (b) 924 (c) 213 (d) 786 (e) 447 (f) 51 (g) 84 (h) 5 (i) 32 (j) 18 2. (a) 88 (b) 29 (c) 99 (d) 48 (e) 88 (f) 99 3. (a) Won’t tile. Teacher check explanation. (b) Will tile. Teacher check explanation. (c) Will tile. Teacher check explanation. (d) Won’t tile. Teacher check explanation. Challenge is larger. Teacher check explanation.

Unit 26–2

(a) 4/5

(e) 5/6

(b) 8/10

(f) 8/8

(g) 6/7

(d) 8/10 Challenge

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(d) 390 (e) 670 (i) 55 (j) 35 (d) 67 (e) 82

• Students select a 2-D shape, explore its tiling properties and develop a design.

Consolidation 26–2

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1. (a) 120 (b) 450 (c) 940 (f) 45 (g) 25 (h) 65 2. (a) 76 (b) 81 (c) 73 (f) 53 3. Teacher check shading.

• Provide students with regular opportunities to evaluate the likelihood of chance events.

Consolidation 26–3

o c . che e r o t r s super

R.I.C. Publications®

• Provide further opportunities for students to develop the concept of adding fractions with like denominators.

New Wave Maths Book E – Teachers Guide • 125 •

Unit 27–1

Student page 79

Outcomes

Indicators

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

The student is able to: • understand that units are needed, when direct comparison is not possible or we wish to know ‘how big …’ or ‘how much bigger …’. • 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 • counting • recording • comparing • calculating

Memory Masters (M3.1)

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

Language • grams • kilograms • subtract • enlarge • shape • twice the dimensions • perimeter • area • doubling

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 (M3.1, M3.2, M3.4a) Warm up

• Explain to the students the shape shown is to be enlarged so that its dimensions are twice those of the original shape. If part of the shape is three squares long in the original, it will be six squares long in the enlarged shape.

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• Set the class to work to draw the enlarged shape. At regular intervals, check to ensure the enlargement is accurate. Inaccurate enlargements will adversely affect the rest of the activity. • When the students have completed their enlargements, ask the class to find the perimeter of the original shape and the enlargement. Remind students that the perimeter is the distance around the outside. The grid is not to scale—ask students to work out the perimeter using one square as 1 centimetre. • When students have found the perimeter, ask them to find the area. Remind students the area is the number of squares inside the perimeter. • One square represents 1 cm2.

Challenge

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What to do

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Teac he r

• The focus for this unit is the conversion of kilograms to grams and grams to kilograms.

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• Students will need to draw other shapes, possibly using 1-cm grid paper, to find out whether there is a consistent effect on perimeter and area relationships between the original shape and the double dimension shape. • Show drawings of originals and enlargements. • Show perimeters and areas. • Show any workings used to find a relationship. Keep notes and use these to explain any findings. HINT: Use simple and small shapes.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 18–19. • 126 • New Wave Maths Book E – Teachers Guide

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

Unit 27–2

Student page 80

Outcomes

Indicators

M3.1, N3.3, N3.1b

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 (i.e. symbols) to represent unit fractions. • describe and record simple fractional equivalences in words.

Skills • ordering fractions • reading fractions • writing fractions • matching equivalent fractions

Resources

Language • change • millilitres • smallest • order • draw • model

• calculator • 2-cm cubes

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

• • • •

litres fractions largest equivalent

Notes

Memory Masters (M3.1)

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Teac he r

• The focus for this unit is the conversion of millilitres to litres and litres to millilitres.

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

• Revise that a fraction is made when a whole is divided into equal partitions. • Discuss the parts that make up a fraction.

What to do

Challenge

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• Work as a whole class to order the fractions in Exercise 3 (a). Note:The denominators are the same so students need only focus on the numerator for the purpose of this activity. Students then complete the activity independently. • Exercise 4 requires students to read the fraction as a word and write it in numeric form. Complete Exercise 4 (a) as a whole class. Students then complete the activity independently. • Exercise 5 is focusing on two skills. Students need to match equivalent fractions across numbers and words. Complete Exercise 5 (a) as a whole class. Students then complete the activity independently.

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• Provide students with 2-cm cubes to make the model shown. • There are three views of the model to be drawn; from directly above; from one end; and from in front. • Students draw the model as they would see it from each of these positions.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 48–49. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 127 •

Unit 27–3

Student page 81

Outcomes

Indicators The student is able to: • directly compare the volume of two structures.

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

Skills

Resources • calculator • 2-cm cubes

• measuring • understanding volume

Memory Masters (N3.1a)

Language • cents • dollars • subtract • cubes • volume • double dimensions • triple • quadruples

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

• Organise students into small groups and distribute 2-cm cubes to each group. • Explain to students the concept of volume. Volume is the amount of space taken up by an object. It is measured in cubic units.

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• Ask each group to take one cube and say what the volume of the cube is. (One cube.) • Ask each group to make a model that has double the dimensions of the original cube. Remind students, or ask for an explanation, that all dimensions are to be doubled – length, width and height. What is the volume of the new cube? (8 cubes.) • Make a model that has triple the dimensions of the original cube. Again remind the class that length, width and height are all to be tripled. What is the volume of this model? (27 cubes.) • Repeat this for a model that quadruples the dimensions of the original cube. The volume is now 64 cubes. • Ask students to discuss the volume of a cube five times the volume of the original cube in their groups. Groups may check their answers by building the model.

Challenge

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What to do

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Teac he r

• The focus for this unit is the conversion of dollars to cents and cents to dollars.

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• The alphabet code uses matching pairs of letters in the alphabet. Students see if they can find the matching pairs to decode the message.

• 128 • New Wave Maths Book E – Teachers Guide

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

Unit 27—Answers

Student pages 79–81 Unit 27–1

1. (a) 7.248 L (b) 4.097 L (c) 3.5 L (d) 6.275 L (e) 1.006 L (f) 8200 mL (g) 6000 mL (h) 5493 mL (i) 1690 mL (j) 2538 mL 2. (a) $2.37 (b) $7.15 (c) $3.57 (d) $5.27 (e) $4.39 (f) $4.54 3. (a) 1/4, 1/2, 3/4, 1

(d) 1/10, 5/10, 8/10, 9/10

(b) 1/6, 2/6, 4/6, 5/6

(e) 1/5, 2/5, 3/5, 4/5

(f)

r o e t s Bo r e p ok u S 4. (a) 1/3

/12, 3/12, 6/12, 8/12

(d) 7/10

(b) 3/4

(d) 4/5

(d) 1/2

5. (a) 2/4 •

• three-fourths

(b) 6/8 •

• three-fifths

• five-sixths

(d) 10/12 •

• one-half

/10 • (f) 2/8 •

(e)

Challenge

Side view

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Teac he r

1. (a) 4800 g (b) 2076 g (c) 8743 g (d) 60 g (e) 4275 g (f) 0.068 kg (g) 0.827 kg (h) 4.936 kg (i) 1.2 kg (j) 2.745 kg 2. (a) 6.14 m (b) 4.18 m (c) 2.47 m (d) 6.26 m (e) 4.36m (f) 3.38 m 3. Teacher check. 4. (a) ≈42 cm (b) ≈84 cm 2 4. (a) 25 cm (b) 100 cm2 Challenge Yes

Unit 27–2

• one-fourth • two-fifths

Top view

End view

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www.ricpublications.com.au

• Students create their own geometric designs. Enlarge to a given scale and compare the perimeter and area.

Consolidation 27–2

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1. (a) 4300c (b) 3800 c (c) 2160 c (d) 1840 c (e) 2090 c (f) $60.00 (g) $80.00 (h) $47.80 (i) $69.10 (j) $24.70 2. (a) $3.25 (b) $3.23 (c) $3.44 (d) $3.22 (e) $5.14 (f) $6.11 3. (a) one cube (b) eight cubes (c) teacher check (d) 27 cubes (e) teacher check (f) 64 cubes (g) 125 cubes Challenge I think maths is awesome.

• Provide students with opportunities for cooking experiences using fractional measures as part of the process.

Consolidation 27–3

o c . che e r o t r s super

R.I.C. Publications®

• Students can explore the volume of various 3-D shapes. What is the best way to work it out?

New Wave Maths Book E – Teachers Guide • 129 •

Unit 28–1

Student page 82

Outcomes

Indicators The student is able to: • partition two-digit numbers to assist in adding and subtracting them mentally.

N3.3

Skills

Resources • calculator

• partitioning • rounding

Memory Masters (N3.3)

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)

Notes

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Number (N3.3)

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• Write 42 + 76 on the board. • Instruct the students how to add these numbers using partitioning techniques. Firstly, add the two tens together—40 + 70. Then add the two ones together—2 + 6 . This gives 110 and 8. Add these numbers together to get the answer of 118. • Work through a second example, 69 + 54, with the class. Invite them to verbalise the process for you as you record on the board. Repeat for several examples as necessary. • Instruct the students how to add two numbers using rounding techniques. Use the same numbers as before (42 + 76). • First of all round 42 to 50 and 76 to 80. This gives 50 + 80. Always round up for the purpose of addition. Then add the number used in the rounding of the ones (8 + 4). Eight because 42 went up 8 to 50 and 4 because 76 went up 4 to 80. This gives 130 and 12. Then subtract the two numbers 130 – 12, giving the answer of 118. • Work through the same examples as before, using the rounding technique.

What to do

• multiply • add • partition • round • signs

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

• The ‘Today’s number is …’ activity ask students to list all they know about a particular number; e.g. Today’s number is … 12 2 + 2 + 2 + 2 + 2 + 2 = 12, 3 x 4 = 12, 24 ÷ 2 = 12, 120 ÷ 10 = 1

Warm up

Language

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• Set students to work independently to complete Exercises 3 and 4. Assist those students who still require help. • Exercise 5 encourages students to use their favoured technique; however, they can use a combination of the two to complete the exercise.

Challenge • Students are to follow the instructions given, record all attempts and keep notes of working thoughts. • Share final results with the class.

• 130 • New Wave Maths Book E – Teachers Guide

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

Unit 28–2

Student page 83

Outcomes

Indicators

The student is able to: • describe outcomes as having an equal chance or being equally likely. • clarify questions in order to decide what data to collect. • summarise data based on tallying. • summarise data in tables and diagrams which show frequencies for different categories. • interpret straightforward one- and two-way tables. • read frequencies from a bar graph.

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

Skills • collecting data • recording data • analysing data

Resources

Language

• calculator • dice • coloured pencils

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

• multiply • tally • total • dice • record • results • graph • most frequent

Memory Masters (N3.3)

Notes

Teac he r

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• The focus for this unit is multiplication of a multiple of 10 by a whole number less than 10 and basic facts of division.

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

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• Explain to the class that each student, in turn, is to have 36 throws of the two dice. Each time the dice are thrown the total of the two dice are to be found by adding the number shown on each die. The total is then to be recorded in the tally column of the table on the left of the page. • Together with students, discuss the possible totals available when throwing two dice. Once the students offer the numbers 2 to 12, record them in the table under ‘Total of Throw’ and on the horizontal axis of the graph. • After 36 throws find the total or the number of times each number was thrown. • Use this information to complete the graph on the right of the page. One square represents a recorded total. If one total is greater than 13 either add squares to the top of the column or write the total on top of the column. • Colour the columns in the graph using different colours for each total. • Compare graphs with those in the group. Explain any differences that may have occurred. Think widely for reasons. Discuss chance, bias of die, small simple error and so on. Students write their own explanation.

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Challenge • Students examine the two dice and use their results to assist in giving an explanation as to why seven should be the most frequently occurring total. • Show any working, or recordings made. • Discuss results as a whole class. For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 112–113. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 131 •

Unit 28–3

Student page 84

Outcomes

Indicators

N3.3, N3.1b

Skills • ordering fractions • fraction equivalences

Memory Masters (N3.3)

Resources • calculator • pencil • coloured pencils • 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 (N3.1b)

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• Alongside the diagram are sets of five fractions. Each set of five fractions is to be arranged from largest to smallest. Use the diagram to assist in arranging the fractions. Students may find a ruler helpful. • Find 1/6. Is it larger or smaller than 1/8? (Larger) Find 1/3. Is it larger or smaller than 1/6 and 1/8? (Larger) Find 1/5. Is it larger or smaller than 1/3, 1/6 and 1/8? (Smaller than 1/3 but larger than 1/6 and 1/8.) Find 1/2. Is it smaller or larger than the other fractions? (Larger than all.) From largest to smallest in order is: 1/2, 1/3, 1/5, 1/6, 1/8. • Work through 3 (b) in the same way. If necessary, work through the rest of the exercises with the students. Allow students to proceed by themselves as they are able to. • Exercises 4, 5 and 6 are asking students to colour equivalent fractions. For example, students colour 1/2, then the equivalent fractions of 2/4, 3/6, 4/8 and 5/10.

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• On the page is a diagram showing one whole and the equivalent fractions ranging from halves to tenths inclusive. • Ask students to choose ten different colours to colour one block in each fraction bar—1/2, 1 /3, 1/4, … 1/10. The coloured portions may assist in reading the diagram.

Challenge

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Number (N3.3)

What to do

• multiply • ordering • diagram • fractions • largest • smallest • equivalent

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

• The focus for this unit is multiplication of a multiple of 10 by a whole number less than 10 and basic facts of division.

Warm up

Language

o c . che e r o t r s super

• In this activity students are required to select a starting point, then trace over all lines once only without lifting the pencil from the page. • It may be useful to copy the diagram two or three times on another sheet of paper for any extra attempts rather than erasing too often. • The use of a different coloured pencil for second and more attempts on one diagram will assist in being able to see the path followed.

• 132 • New Wave Maths Book E – Teachers Guide

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

Unit 28—Answers

Student pages 82–84 Unit 28–1

1. (a) 240 (b) 400 (c) 240 (d) 80 (e) 630 (f) 8 (g) 6 (h) 8 (i) 5 (j) 3 2. (a) 400 (b) 1088 (c) 364 (d) 525 (e) 774 (f) 532 3. Teacher check. 4. Teacher check. Challenge 1 + 6, 2 + 5, 3 + 4, 6 + 1, 5 +2, 4 + 3. It can occur six different ways.

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Teac he r

1. Answers will vary; some possible solutions: 5 x 3 x 6, 5 x 3 x 2 x 3 2. (a) 372 (b) 882 (c) 168 (d) 572 (e) 484 (f) 693 3. (b) (80 + 90) + (3 + 8) = 170 + 11 = 181 (c) (50 + 30) + (1 + 7) = 80 + 8 = 88 (d) (40 + 90) + (8 + 3) = 130 + 11 = 141 (e) (30 + 40) + (5 + 2) = 70 + 7 = 77 4. (b) (70 + 20) – (8 + 3) = 90 – 11 = 79 (c) (80 + 70) – (5 + 2) = 150 – 7 = 143 (d) (50 + 100) – (0 + 8) = 150 – 8 = 142 (e) (60 + 70) – (3 + 5) = 130 – 8 = 122 5. Teacher check, technique will vary. (a) 95 (b) 143 (c) 86 Challenge Answers will vary; possible solution: 3 ÷ 3 + 3 + 3 + 3 + 3 + 3 + 3 = 19 3 x 3 x 3 + 3 + 3 + 3 x 3 + 3 ÷ 3 = 37

Unit 28–2

3. (a) 1/2, 1/3, 1/5, 1/6, 1/8

(b) 2/3, 5/8, 3/5, 5/10, 3/10

(d) 1/3, 1/4, 1/6, 1/8, 1/10

(e) 5/6, 3/4, 2/3, 1/2, 3/9

(f) 4/5, 7/10, 5/8, 1/2, 1/3

(g) 7/8, 6/7, 3/5, 5/10, 4/9

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• Provide students with further opportunities to practise the techniques with addition problems.

Consolidation 28–2

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1. (a) 180 (b) 540 (c) 100 (d) 490 (e) 40 (f) 8 (g) 7 (h) 3 (i) 5 (j) 6 2. (a) 2318 (b) 1968 (c) 2666 (d) 3224 (e) 2592 (f) 3096

• Repeat the activity—use spinners with numbers 1–6. Do the results differ?

Consolidation 28–3

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• Brainstorm where students might see and use fractions. Discuss the importance of knowing equivalent fractions; e.g. when cooking—it is good to know how to mix and match measuring cups to make a certain fraction.

4. 1/2, 2/4, 3/6, 4/8, 5/10 should be coloured blue. 5. 1/3, 2/6, 3/9 should be coloured green. 6. 2/5, 4/10 should be coloured yellow. Challenge Yes

start

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 133 •

Unit 29–1

Student page 85

Outcomes

Indicators

Resources

N3.3, S3.2, M3.2

The student is able to: • draw 2-D shapes to prove or disprove their tessellating qualities.

• calculator • scissors • light card • pencil • regular 2-D shapes

Skills • tessellating • reasoning mathematically

Memory Masters (N3.3)

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, M3.2)

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• Either use the shapes distributed to students, or ask them to copy the four regular shapes on the page onto card and cut them out to use to make tessellating patterns on the page. • Select a section of the page to show whether each of the four shapes tessellates. • If students have access to other regular shapes, ask them to also test these to see if they tessellate. A separate sheet of paper will be required. • Students are to write a brief report on what was found out in the space provided at the bottom of the page. • Note: A shape will tessellate when the sum of the angles of the shapes at any given point will total 360º.

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• Ask students if they know what a regular shape is. (One whose sides are all equal in length and each of the internal angles are all the same size.) • Distribute regular 2-D shapes to the class to examine.

Challenge

Notes

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Number (N3.3)

What to do

• add • regular shapes • tessellate • square • equilateral triangle • pentagon • hexagon

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• The focus for this unit is the multiplication of a multiple of 10 by a whole number less than 10 and the basic facts of division.

Warm up

Language

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• What do you notice about regular shapes that tessellate? How many squares can you see around a point? How many triangles? How many hexagons?

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 22–23. • 134 • New Wave Maths Book E – Teachers Guide

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

Unit 29–2

Student page 86

Outcomes

Indicators

N3.3

The student is able to: • subtract amounts of money.

Skills • adding money

Resources

Language

• calculator • Base 10 MAB • coins • shopping catalogues

• add • subtract • sums • money

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Notes

Memory Masters (N3.3)

Teac he r

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• The focus for this unit is the multiplication of a multiple of 10 by a whole number less than 10 and the basic facts of division.

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) Warm up

• Using items out of a shopping catalogue, ask students to help you calculate the difference in price between two items. • Demonstrate all working on the blackboard/whiteboard. • Students may use Base 10 MAB or coins to help them.

What to do

Challenge

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• Following the first example, students complete the activities in Exercise 3. Record the sum horizontally first then the answer. • Exercise 4 is still working with subtracting amounts of money, but shows it written as an algorithm. • Assist students as required.

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• Distribute a range of plastic coins to individuals or small groups. • Ask students to find what combination of coins they require to use 6 coins to make 50c. • Show working and keep notes.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 40–41. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 135 •

Unit 29–3

Student page 87

Outcomes

Indicators

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

The student is able to: • use a unit consistently and carefully to measure and compare containers.

Skills • measuring accurately and carefully • counting • recording

Memory Masters (N3.1a)

Resources • calculator • four different sized containers (not too large) for each group • beads (not too small) • several different containers for teacher use

Language • digit • tens • add • subtract • capacity • least • most • difference • order

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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) Warm up

• Discuss, explain and clarify the concept of capacity. Capacity is the amount a container can hold. • Hold up several different containers; e.g. 2 L ice-cream container, and ask students the capacity. (2 L) Repeat with containers of different capacities.

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• Organise students into groups of four. Provide each group with four containers and a pile of beads. • Draw each container in the workbook. Ensure each group draws the containers in the same space; e.g. the ice-cream container needs to be drawn in the Container 1 on each student’s workbook. • Students use the beads to fill Container 1. Pat the container to ensure all the beads settle and the container is as full as possible. • Students count out the beads from the container. They may find it easier to count them into piles of 10 then add these together. Record the total under the correct container. • Repeat this process until all containers are measured. • Students can then work independently to complete Exercises 4 and 5.

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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 the identification of place value to three decimal places.

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• Students are to use any resources they feel necessary to find the answer. Suggest Base 10 MAB and/or a calculator. • Record all working out including notes of thinking processes.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 94–95. • 136 • New Wave Maths Book E – Teachers Guide

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

Unit 29—Answers

Student pages 85–87 Unit 29–1

1. (a) 360 (b) 560 (c) 60 (d) 320 (e) 540 (f) 7 (g) 8 (h) 6 (i) 5 (j) 3 2. (a) 513 (b) 565 (c) 1510 (d) 1482 (e) 1518 (f) 381 3. (b) $2.19 (h) $1.36 (c) $1.03 (i) $1.21 (d) $2.42 (j) $1.04 (e) $1.18 (k) $0.26 (f) $1.18 (l) $1.23 (g) $0.72 (m) $2.03 (n) $2.14 4. (a) $1.48 (c) $2.36 (b) $3.62 (d) $1.51 Challenge 4 x 10c + 2 x 5c = 50c or 20 c + 10 c + 4 x 5c = 50c

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1. (a) 640 (b) 160 (c) 810 (d) 90 (e) 250 (f) 9 (g) 7 (h) 3 (i) 8 (j) 5 2. (a) 118 (b) 90 (c) 136 (d) 104 (e) 158 (f) 119 3. Not all regular shapes tessellate. Challenge Teacher check. Answers should indicate an understanding that the angles around any given point need to add up to 360º in order for the shapes to tessellate. 4 squares 6 triangles 3 hexagons

Unit 29–2

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• Students select a regular 2-D shape which tessellates and make a design.

Consolidation 29–2

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1. (a) 8 (b) 6 (c) 7 (d) 8 (e) 9 (f) 3 (g) 5 (h) 9 (i) 0 (j) 9 2. (a) 8924 (b) 12 807 (c) 8813 (d) 6949 (e) 7555 (f) 6231 3. Teacher check 4. Teacher check 5. Teacher check Challenge 100

• Students can have a set amount of money; e.g. $5.00, and work out the change for items in a catalogue.

Consolidation 29–3

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R.I.C. Publications®

• Evaluate the accuracy of using beads to find capacity. What would be better to use?

New Wave Maths Book E – Teachers Guide • 137 •

Unit 30–1

Student page 88

Outcomes

Indicators The student is able to: • add money amounts.

N3.1a, N3.3

Resources • calculator • shopping catalogues

Skills • adding of money

Memory Masters (N3.1a)

Language • round • nearest • ten • subtract

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) Warm up

• Using items out of a shopping catalogue, ask students to help you calculate the total of two items. • Demonstrate all working on the blackboard/whiteboard. • Students may use Base 10 MAB or coins to help them.

• Students complete the activities in Exercise 3. • Exercise 4 requires students to add more numbers, horizontally. Some students may choose to write them vertically and then work it out. • Assist students as required.

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Challenge

• Students are to use the clues given to find who is the tallest and who is the shortest. • Guide students by suggesting they use a table or chart to record the answers to each question. • Suggest students are to work in a logical manner.

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What to do

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Teac he r

• The focus for this unit is rounding to the nearest 10.

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• 138 • New Wave Maths Book E – Teachers Guide

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

Unit 30–2

Student page 89

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 as a tally or list. • display and interpret frequency data in a two-way table and bar graph.

Skills • collecting data • recording data • analysing data • interpreting data

Resources

Language

• calculator • two-colour counters • cup or jar or can • 5c coins

• number sentence • true • subtract • record • totals

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Memory Masters (N3.3)

Notes

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

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

• Organise students into small groups. • Provide each group with ten two-colour counters and a cup or similar container in which to place the counters. • Students record the two-colour counters in the two-way table; e.g. a = red, b = blue.

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• Ask the students to place all counters in the cup and give it a shake.Tip the cup upside down, remove from the counters. Spread the counters out carefully without turning any over. Count the number of faces showing each of the two colours. • Explain to the class they will be repeating this exercise ten times, counting the total of each colour and recording the totals in their workbook. • When the ten counts have been made and recorded, ask students to look at the graph below the table. Explain that each student is to now show on the graph the total of colour (a) and (b) for each turn. All ten turns are to be shown. • When the graph is completed, write a summary of the findings shown on the graph. Look for closeness of totals of both colours for each turn and for the total number of turns. Explain why these are close or far apart.

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• Attempt the same experiment using 5c coins. Compare heads and tails.

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

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 139 •

Unit 30–3

Student page 90

Outcomes

Indicators The student is able to: • select an object from a collection given a description of its spatial features.

N3.3, S3.4

Skills

Resources • calculator • 3-D shapes as shown in the workbook

• naming 3-D shapes • reading descriptions • matching shapes

Memory Masters (N3.3)

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 (NS3.4)

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Number (N3.3)

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• Organise students into small groups. • Distribute 3-D shapes to each group. • Direct students to observe and feel each item. • Name each shape. • Instruct students to find the ‘corners’ of the objects—these are called ‘vertices’ in mathematics when describing 3-D shapes. Record the word ‘vertices’ on the board. • Instruct students to find a ‘side’ of an object—these are called ‘faces’. Record the word ‘face’ on the board. • Direct students to find where two faces meet.This is called an ‘edge’. Record the word ‘edge’ on the board.

What to do

• subtract • three-dimensional • hexagonal prism • cone • cube • square pyramid • cylinder • face • edge • vertices

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

• The focus for this unit is the addition of a whole number less than 100 to a multiple of 100 and the subtraction of a whole number less than 10 from a whole number less than 100.

Warm up

Language

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• Ask students to look at the shapes in their workbook. Record the correct name under each shape. • Discuss the features of the shapes. Some have round faces, some square etc. • Students then complete Exercise 4 independently—recording the shapes next to the correct descriptions. • Share results with the class.

Challenge • Students are to show all their working and to keep notes to explain how they reached their answer. • Share results with other students or the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 32–33. • 140 • New Wave Maths Book E – Teachers Guide

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

Unit 30—Answers

Student pages 88–90 Unit 30–1

(c) 2200 (d) 1080 (e) 7220 (h) 4200 (i) 510 (j) 730 (c) 243 (d) 353 (e) 392 (g) $8.46 (j) $6.50 (h) $9.85 (k) $5.99 (i) $9.92 (l) $8.64

1. (a) 2 (b) 2 (c) 50 (d) 8 (e) 20 (f) 10 (g) 8 (h) 2 (i) 3, 3 (j) 4 2. (a) 2.89 km (b) 3.86 L (c) 1.89 m (d) 3.87 kg (e) 5.57 m (f) 3.57 L 3. Teacher check Challenge Teacher check

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1. (a) 6270 (b) 8440 (f) 6880 (g) 5380 2. (a) 231 (b) 272 (f) 416 3. (a) $3.27 (d) $6.45 (b) $6.34 (e) $6.16 (c) $4.64 (f) $7.21 4. (a) $9.90 (b) $11.55 (c) $13.05 (d) $16.00 (e) $14.40 Challenge Shortest Rochelle Brett Coleen Bob Tallest

Unit 30–2

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• Students can organise a shopping list for their next birthday party. Add the total cost.

Consolidation 30–2

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1. (a) 382 (b) 773 (c) 926 (d) 683 (e) 191 (f) 31 (g) 44 (h) 61 (i) 53 (j) 81 2. (a) $4.02 (b) $2.43 (c) $2.47 (d) $2.61 (e) $1.24 (f) $1.44 3. hexagonal prism, square pyramid, cylinder, cone, cube 4. (a) cone (b) hexagonal prism (c) cylinder (d) cube Challenge 5 times

• Discuss and explore how data may change with more counters or repeating the activity more times.

Consolidation 30–3

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R.I.C. Publications®

• Students write their own descriptions of 3-D shapes. Ask a classmate to work out which 3-D shapes are being described.

New Wave Maths Book E – Teachers Guide • 141 •

Unit 31–1

Student page 91

Outcomes

Indicators

N3.3, S3.1

The student is able to: • attempt to provide a bird’s-eye view of familiar locations such as their classroom.

Skills • scale drawing

Memory Masters (N3.3)

Resources • calculator • pencil • ruler • house plan

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 (S3.1)

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• Read the instructions given. Remind students that the size of each part and the position of each part in the plan is to be relative to the actual size of each part of the house and block or farm. For example, the house will be longer than the shed, which will be longer than the letterbox. • Use as much of the space on the page as possible to draw the plan. Use a ruler for straight lines. Use lead pencil for ease of correcting the drawing. • Students imagine themselves above the block and house or farm, then draw the plan as they would see it.

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• Show students a house plan, a plan of a block and house and/or a farm plan. Explain, or ask for an explanation of the view taken for a plan. (Bird’s-eye view.) • Before drawing a plan the person drawing the plan must imagine they are directly above the block and house or farm. From this position, they can draw the outline of what they see.

Challenge

Notes

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Number (N3.3)

What to do

• divide • plan • relative size • position • main features

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• The focus for this unit is the addition of a whole number less than 100 to a multiple of 100 and the subtraction of a whole number less than 10 from a whole number less than 100.

Warm up

Language

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• Students will need to find the perimeter of the nine tiles as they are shown. • Remove one tile from the nine and keep the perimeter the same. • Show how this was done. Record all attempts and keep notes on each attempt.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 2–3. • 142 • New Wave Maths Book E – Teachers Guide

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

Unit 31–2

Student page 92

Outcomes

Indicators

N3.3, N3.1a

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.

Skills • working with decimals • calculating

Resources

Language • divide • grid • lots of • triangles • shape

• calculator • coloured pencils • 1-cm grid paper (see page 199)

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

Notes

Memory Masters (N3.3)

Teac he r

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• The focus for this unit is the addition of a multiple of 100 less than 1000 to a multiple of 10 less than 100 and the subtraction of a multiple of 10 from a whole number less than 100.

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

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• What to do • This activity is a good introduction to calculating decimal numbers. Using the grid to represent one whole, students are able to show the relationship between one whole and the decimal. • On the page there is a 10 x 10 grid. The grid represents one whole. One row or column represents one-tenth (0.1).

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• What represents 0.3 or three tenths?( Three rows or three columns.) If 3 rows represents 0.3, how can you show three lots of 0.3? (Colour in nine rows.) Using their blue-coloured pencil, colour nine rows. What is the total? (three lots of 0.3 or 0.9.) Remember one row is not coloured. One row is one-tenth or 0.1. There are nine rows coloured or nine lots of 0.1; i.e. 0.9 is coloured. • Repeat this process for showing 2 lots of 0.2. Outline rows or columns in red. • Using this information, tracing over the grid if required, solve the multiplication activities in Exercise 4. Prepare more grids for students or use grid paper so students can draw 10 x 10 grids if needed. • Exercise 5 has examples that are the same as Exercise 4, except they are shown as vertical multiplication not horizontal. Using grids, if needed, complete these activities. Ask students how they would show 25 tenths. (2 wholes and five-tenths or 2.5) • Students may also use a calculator if necessary.

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• Students are to show all recordings with notes to explain how they reached their total. • Share results with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 44–45. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 143 •

Unit 31–3

Student page 93

Outcomes

Indicators

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

The student is able to: • choose to make numerical measurements of objects to order the objects. • use the decimal point in representing quantities and money. • remember basic addition facts and many multiplication facts and calculate mentally basic multiplication facts they don’t recall.

Skills • analysing data • grouping • measuring • recording • rounding • ordering

Memory Masters (N3.3)

Resources • calculator • ruler • paper • jugs and water

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, N3.1a, N3.3)

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Teac he r

• divide • measure • length • record • results • nearest centimetre • shortest • longest • range • arrange • rearrange

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

• The focus for this unit is the addition of a multiple of 100 less than 1000 to a multiple of 10 less than 100 and the subtraction of a multiple of 10 less than 100 from a whole number less than 100.

Warm up

Language

• Ask students to use their ruler to measure the length of the middle finger of their left hand and the length of their left shoe. • Record all measurements on the blackboard/whiteboard. Students are to copy student names and measurements onto a separate sheet of paper. This part may be deleted to save time, as results may be read from the board.

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• From the list of measures, ask students to find the shortest shoe and write the student’s name in their workbook. Find and write who has the longest shoe. Look at these measurements carefully as they provide the range of measurements. • Repeat this for finger measures. • Arrange ten measurements of fingers and shoes, separately, from shortest to longest in Exercise 4. • Exercise 5 requires students to think of different groupings of the numbers shown to make addition easier. Show these groupings. • Invite students to share their reasoning for the grouping they have chosen being easier to add.

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• Students may find it easier to solve this problem using real materials. • A logic problem, such as this, encourages students to think outside the square. • Share and discuss results with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 78–79. • 144 • New Wave Maths Book E – Teachers Guide

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

Unit 31—Answers

Student pages 91–93 Unit 31–1

(d) 865 (e) 547 (i) 13 (j) 22 (d) 6 (e) 7

1. (a) 520 (b) 260 (c) 810 (f) 36 (g) 34 (h) 22 2. (a) 17 (b) 12 (c) 25 (f) 14 3. Teacher check grid shading. 0.90 and 0.40 4. (a) 0.8 (b) 0.6 (c) 1.2 (d) 0.6 (e) 1.0 5. (a) 0.4 (d) 1.8 (g) 1.8 (b) 01.6 (e) 1.8 (h) 1.4 (c) 0.9 (f) 2.7 Challenge 16

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(d) 150 (e) 430 (i) 26 (j) 13 (d) 14 (e) 19

(i) 2.1 (j) 1.2

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1. (a) 458 (b) 234 (f) 94 (g) 52 2. (a) 8 (b) 6 (f) 7 3. Teacher check. Challenge

Unit 31–2

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• Design a space of your own. Include all key features to make it everything you want it to be.

Consolidation 31–2

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1. (a) 680 (b) 570 (c) 380 (d) 640 (e) 760 (f) 51 (g) 32 (h) 24 (i) 25 (j) 36 2. (a) 32 (b) 21 (c) 22 (d) 42 (e) 32 (f) 11 3. Teacher check. 4. Teacher check. 5. (a) 27 (b) 29 (c) 22 (d) 29 (e) 21 Challenge • Fill the jug that holds 3 cups. • Pour into the jug that holds 5 cups. • Fill the jug that holds 3 cups again. • Pour into the jug that holds 5 cups. • This jug only has room for 2 more cups so 1 cup will be left in the other jug.

• Provide students with further opportunities to multiply with decimals.

Consolidation 31–3

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R.I.C. Publications®

• Measure and compare other body parts. Ensure sensitivity is applied if using body mass as a measure.

New Wave Maths Book E – Teachers Guide • 145 •

Unit 32–1

Student page 94

Outcomes

Indicators

Resources

N3.3, N3.1a

The student is able to: • use the decimal point in representing quantities and money. • regroup money to the fewest number of notes or coins. • add and subtract money for a purpose.

• calculator • canteen product and price list • coins

Skills • deduction • addition

Memory Masters (N3.3)

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 (N3.1a, N3.3)

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• Read through Exercise 3 (a). Students are to select their favourite items for each, keeping in mind they only have $5.00 to spend.This will require a degree of estimation on students’ part. • Once the total is reached, students then subtract it from $5.00 to find the change. • In order to complete the table in Exercise 3 (b), students must select an item for each category and record its cost.The final part of the table requires students to consider the least number of coins they would use to purchase the item. Some students may find it easier to have coins in front of them to work this out, while others can work this out mentally. • Students will need to concentrate only on the drinks available at the canteen for Exercise 3(c). Note: Some drinks can be grouped together to form one category; e.g. chocolate milk, milk and strawberry milk may all be the same price and could be listed as milk.

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• Display a copy of the products available in the school canteen along with each price. • As a group, categorise the items available into lunchtime foods, drinks, foods, morning tea foods and snacks. • Record these groups on the board for all to see.

Challenge

Notes

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Number (N3.3)

What to do

• multiply • list • items • greatest number • more than • items • add

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• The focus for this unit is the addition of a multiple of 100 to a whole number less than 100 and the subtraction of a multiple of 10 less than 100 from a whole number less than 100.

Warm up

Language

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• Students’ choices should not vary here, so personal taste is not involved. Students must be directed by cost alone. • Students should list each most expensive item and its cost. Add these to find the total cost. • Repeat to find the least expensive total.

• 146 • New Wave Maths Book E – Teachers Guide

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Unit 32–2

Student page 95

Outcomes

Indicators 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.

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

Skills • collecting data • recording data • analysing data

Resources

Language

• calculator • ten-sided dice

• patterns • multiply • table • possible totals • adding

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

Memory Masters (N3.4)

Notes

Number (N3.3)

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Teac he r

• The focus for this unit is the completion of patterns, including 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) Warm up

• Organise the class into small groups. Provide each group with two ten-sided dice. • Ask the students what is the smallest total they could get if they rolled two dice and added the two numbers showing on the top surface. (0) • What would be the largest number? (18)

Challenge

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• There is a table shown that will give the possible totals if two dice are thrown and the numbers on the top surface are added together. • Complete the table. • Make a list of all the different possible numbers that can be made by adding the totals of the two dice together. • From the table, which total do you think will occur most often? (9) • From the table, which total do you think will occur least often? (0 and 18) • Were you correct? Can you explain your result?

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• Using the information from the table and the graph, students write how many times they think the total 2 might occur in 36 throws. • Write an explanation for their answer.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 147 •

Unit 32–3

Student page 96

Outcomes

Indicators

M3.1, N3.3, N3.4

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. • fill in number sequences involving addition or subtraction by a constant amount. • identify particular terms in a sequence. • identify patterns in the multiplication tables and use to make predictions.

Skills • solving number patterns • searching for number patterns • explaining number patterns

Teac he r

Memory Masters (M3.1)

Resources • calculator • Base 10 MAB

Language • centimetres • metres • sums • patterns • rule • divisible

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

• The focus for this unit is the conversion of metres to centimetres and centimetres to metres.

• 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

• Display 2, 4, 6, 8, 10, 12 on the board. • Ask students to read the numbers out loud. • Invite students to look carefully at the numbers. Ask students to share their observations about the numbers on the board. (Even numbers, increasing by two each time.) • Display 3, 6, 9, 12, 15, 18 on the board. • Invite students to share their observations with the class. (Increasing by three each time, three times table, odd-even-odd-even.) • Discuss the idea of patterns. Mathematics is all about patterns.

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Number (N3.3)

What to do

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• Direct students to look at the patterns provided in Exercise 3. • In pairs, encourage students to discuss any patterns they can see in the numbers. • Record the patterns discussed and write the next three numbers in the pattern. • Discuss and share ideas with the whole class. • Now direct students to look at the pattern in the tables. Students are to look at the digits in the units column of the answers. 0, 6, 2, 8, 4, 0, 6, 2, 8, 4, 0. Can students see a pattern occurring? • From this, students should be able to determine which numbers are divisible by six, without having to work out the tables.

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Challenge

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• Students need to understand reversal of digits; e.g. 15 – 51. Apply this knowledge to find the answer to the problem. • All attempts are to be recorded and notes kept explaining the process used in each attempt. • Share results with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 70–73. • 148 • New Wave Maths Book E – Teachers Guide

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

Unit 32—Answers

Student pages 94–96 Unit 32–1

1. (a) 535 (b) 348 (c) 437 (d) 293 (e) 242 (f) 74 (g) 15 (h) 28 (i) 22 (j) 27 2. (a) 3886 (b) 6384 (c) 5394 (d) 3312 (e) 3315 (f) 3886 3. Teacher check. Challenge Teacher check

Unit 32–2 1. (a) (b) (c) (f) 21 (g) 62 (h) 15 2. (a) 420 (b) 540 (c) 840 (f) 850 3. (a) 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (b) Teacher check (c) Teacher check (d) Teacher check Challenge

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1; there is only a 1/36 chance of 2 being thrown.

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(d) (e) (i) 21 (j) 33 (d) 1350 (e) 840

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• Invite students to devise their own school canteen menu.

Consolidation 32–2 • Throw two 10-sided dice 36 times and record the total each time. Check against answers at Exercise 3 (b) and (c).

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1. (a) 527 cm (b) 600 cm (c) 804 cm (d) 289 cm (e) 47 cm (f) 27.68 m (g) 0.36 m (h) 8 m (i) 4.2 m (j) 87.46 m 2. (a) 3840 (b) 3120 (c) 4450 (d) 6090 (e) 2860 (f) 4060 3. Rule Next three digits (a) start at 100 –1, –2, –3 etc. 90, 85, 79 (b) start at 13 +2 etc. 21, 23, 25 (c) start at 150 –2, –4, –6 etc. 128, 118, 106 (d) start at 5 +5, +10, +15 etc. 55, 80, 110 4. (a) 0, 6, 2, 8, 4 pattern repeats. (b) 624 and 848 are divisible by 6; teacher check explanation Challenge 93 – 39 = 54

Consolidation 32–3

• Explore other tables to find any patterns.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 149 •

Unit 33–1

Student page 97

Outcomes

Indicators

N3.3, S3.3

The student is able to: • find repetitions of figures and objects within decorative patterns.

Skills • verifying • testing theories

Memory Masters (N3.3)

Resources • calculator • ruler • pencil • coloured pencils

Language • change • dollars • cent • add • verify • pattern • square tile

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

• Explain to the class, or ask the class to check the meaning of ‘verify’ in a dictionary. • To ‘verify’ a mathematical statement will require students to check on all possibilities.

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• Students are asked to verify that the pattern provided can only be placed on a tile in four ways. • As part of the verification, space is provided for the drawing of alternative placements of the pattern. • Direct the class to show all possible placements of the pattern. • Discuss with students the findings they reached after their work in verifying the statement. • While answers may vary, two key results should arise: 1. There are four orientations with the dot moved to an alternative location. 2. There is only one orientation of the pattern; the tile may be rotated through 90º on three separate occasions to show a different position of the same pattern. • Students should have their solutions accepted if they have a reasonable argument for their decision.

Challenge

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What to do

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Teac he r

• The focus for this unit is the conversion of cents to dollars and dollars to cents.

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• Students design their own tile that may be oriented in four different ways.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 24–25. • 150 • New Wave Maths Book E – Teachers Guide

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

Unit 33–2

Student page 98

Outcomes

Indicators

N3.3, N3.2

The student is able to: • match word problems with particular calculations. • remember basic addition facts and many multiplication facts and calculate mentally basic multiplication facts they don’t recall.

Skills • estimating • rounding

Resources

Language • add • estimate • approximate • sums • surfaces • combination

• calculator • 2-cm cubes

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

Notes

Memory Masters (N3.3)

Teac he r

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• The focus for this unit is the multiplication of a whole number less than 10 by a multiple of 100 and the division of a multiple of 100 by a whole number 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 (N3.2, N3.3) Warm up

• These activities involve estimating to give an approximate answer. • Ask students what they might do to assist in finding the approximate answer. (Rounding, truncating or guessing. )Which of these three methods would be the most reliable? (Rounding.) • Revise the rules of rounding: numbers ending in 1– 4 round down numbers ending in 5 – 9 round up.

What to do

Challenge

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• Ask students to provide the approximate totals for the five examples given. Assist those who need help with reading. • Remind students to show any working they are doing in the space provided.

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• Provide each student with three 2-cm cubes to make the model shown. • In finding the number of surfaces there are in the combination shown, keep notes and any other means used to find the total. • Share findings with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 64–65. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 151 •

Unit 33–3

Student page 99

Outcomes

Indicators

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

The student is able to: • find the perimeter of a polygon by measuring each side and adding the lengths. • use regular concrete units to measure area of polygons. • choose to make numerical measurements of objects to order the objects.

Skills • measuring • using a ruler • counting • adding

Memory Masters (N3.3)

Resources • calculator • ruler • 1-cm tiles • 1-cm grid paper (see page 199) or transparency

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, M3.4a)

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• The activity in the workbook asks students to measure the perimeter of the shapes shown using a ruler. Students can measure each side and add the measurements together, or continue around the shape by measuring the next side using the measured length of the previous side as the starting point. If the first side is 4-cm then start measuring the second side from the 4-cm mark. • The area is the measure of the inside part of the shape. 1-cm tiles, 1-cm grid paper or 1-cm transparency may be used to cover each shape and count the squares required to cover the shape. • Complete Exercise 4 after students have found the perimeter and area of each shape.

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• Explain the use of a ruler as required. The measuring point is from the 0 mark on the ruler, not the end of the ruler.

Challenge

Notes

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Number (N3.3)

What to do

• add • ruler • measure • perimeter • shapes • nearest centimetre • area • record • findings

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

• The focus for this unit is multiplication of a whole number less than 10 by a multiple of 100 and the division of a multiple of 100 by a whole number less than 10.

Warm up

Language

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• Students are to read the question carefully then write an explanation to describe their answer. • Share solutions.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 80–81, 100–101. • 152 • New Wave Maths Book E – Teachers Guide

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Unit 33—Answers

Student pages 97–99 Unit 33–1

1. (a) 200 (b) 800 (c) 400 (d) 900 (e) 300 (f) 100 (g) 100 (h) 100 (i) 100 (j) 100 2. (a) 2322 (b) 2312 (c) 2720 (d) 15 589 (e) 9158 (f) 2148 3. (a) 1300 km (b) 18 000 people (c) $50 (d) 140 seconds (e) 1900 students Challenge 18

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1. (a) $0.82 (b) $7.56 (c) $8.00 (d) $47.00 (e) $68.27 (f) 428c (g) 92c (h) 600c (i) 840c (j) 2764c 2. (a) 8151 (b) 12 044 (c) 19 831 (d) 7490 (e) 16 954 (f) 1695 3. (a) Teacher check. (b)

Unit 33–2

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• Find decorative patterns which contain repetitious designs.

Consolidation 33–2 • Invite students to formulate their own word problems to swap with a classmate to solve.

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1. (a) 2400 (b) 4000 (c) 2400 (d) 800 (e) 6300 (f) 200 (g) 300 (h) 200 (i) 400 (j) 200 2. (a) 14.66 L (b) 21.63 cm (c) $26.20 (d) 20.59 m (e) 22.39 m (f) 19.86 kg 3. (a) P = 16 cm A = 15 cm2 (b) P = 32 cm A = 60 cm2 (c) P = 22 cm A = 28 cm2 (d) P = 16 cm A = 16 cm2 (e) P = 20 cm A = 25 cm2 4. (a) 16, 16, 20, 22, 32 (b) 15, 16, 25, 28, 60 Challenge The ladder’s position will remain the same as the ship rises on the tide.

Consolidation 33–3

• Students find the perimeter and area of larger items found in the classroom environment.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 153 •

Unit 34–1

Student page 100

Outcomes

Indicators The student is able to: • use their own methods or a conventional algorithm to multiply whole numbers by single-digit numbers. • match word problems with particular calculations.

N3.3, N3.2

Skills • mental multiplication • partitioning

Memory Masters (N3.3)

Resources • calculator • coloured pencils

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, N3.2)

Notes

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Number (N3.3)

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What to do

• Parts of the first three problems have been provided to assist. Ask students to ‘have a go’ at solving these. • Exercise 4 can be solved mentally, in written form or by using a calculator.

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• Write 67 x 9 on the blackboard/whiteboard. Ask a student if he/she can work out the answer mentally? in written form? or do they need a calculator? • Show how it might be solved mentally by expanding the number. Multiply the parts of the number then add the answer together on the blackboard/whiteboard; e.g. 67 x 9 = (60 + 7) x 9 = (60 x 9) + (7 x 9) = 540 + 63 = 603. • Display another example—26 x 4 and ask a student to work through the problem in the same manner. This may be repeated several times with a student or the whole class working through the steps together to solve the problem.

Challenge

• subtract • distributive • solve • problems • different • draw • single line • calendar month

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• The focus for this unit is multiplication of a whole number less than 10 by a multiple of 100 and the division of a multiple of 10 by a whole number less than 10.

Warm up

Language

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• Students will need to press lightly with coloured pencils to show the different paths they were able to make on the calendar following the directions given. • Share work with the class.

• 154 • New Wave Maths Book E – Teachers Guide

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Unit 34–2

Student page 101

Outcomes

Indicators The student is able to: • describe outcomes as having an equal chance or being equally likely.

N3.3, C&D3.1

Skills

Resources

Language

• calculator • 6-sided dice • 10-sided dice

• patterns • multiply • table • possible totals • adding

• investigating probability

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

Memory Masters (N3.3)

Notes

Teac he r

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• The focus for this unit is the multiplication of a whole number less than 10 by a multiple of 100 and the division of a multiple of ten by a whole number 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 (C&D3.1) Warm up

What to do

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• Probability is the likelihood of an event happening. On a coin, one side is known as a head and the other side is known as tails. As there are two sides, the probability of the coin landing, if thrown, tails up is 1 chance in 2, or 1/2. • Put three different coloured counters—red, blue and yellow—into a bag, then, without looking take one out. The probability of it being red is 1 chance in 3 or 1/3. • Explain to students they will be using a six-sided die and later a 10-sided die to determine the probability of a given number being rolled. • How many sides on a six-sided die? (6.) The numbers are? 1, 2, 3, 4, 5 and 6. If you want to roll a 3; what is the probability of rolling a 3? It is one chance in ...? (6.) • What is the probability of not rolling a 3? There are now 5 chances in 6 because there are 5 other numbers that might be rolled.

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• Distribute 6- and 10-sided dice. Using the six-sided die, determine the chance of rolling a 6. The die does not need to be rolled. This chance is found by knowing the number of sides on the die and the number of 6s shown on the die. There is one 6 and six sides. The chances of rolling a 6 are 1 in 6 or 1/6. • Repeat for the rest of Exercise 3, then Exercises 4 and 5. Students who are able to work by themselves should do so.

Challenge

• Show all attempts of how numbers were arranged and keep notes of thinking/ reasoning when solving this problem. • Final solutions may be shared with the class.

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

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 155 •

Unit 34–3

Student page 102

Outcomes

Indicators

N3.3, M3.3

The student is able to: • estimate time of day, week or year using ‘clues’ such as shadows, weather, clothing, fullness of car parks, shop signs, plant behaviour.

Skills • observation

Memory Masters (N3.3)

Resources • 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.

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Teac he r

• multiply • decimals • seasons

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

Main Activity (M3.3) Warm up

Language

What to do

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• Direct students to look at the photographs in their workbook. • Ask students to look at each photo carefully and decide the time of day and season they think the photographs were taken. Give reasons for answers. • Share as a whole class.

Challenge

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• Discuss the different times of the year. Talk about the types of clothes we wear and how the environment changes. • Now discuss the time of day. Morning, noon, late afternoon, evening and night. How can we tell the approximate time of day without a clock? (Shadows.)

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• Students are to show and describe how a person can be third and third-last in a line. • Record workings on paper with explanations to give understanding to the workings.

• 156 • New Wave Maths Book E – Teachers Guide

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Unit 34—Answers

Student pages 100–102 Unit 34–1

1. (a) 6400 (b) 1600 (c) 8100 (d) 900 (e) 2500 (f) 20 (g) 50 (h) 70 (i) 60 (j) 70 2. (a) 4.31 kg (b) 3.23 km (c) 5.24 L (d) 3.32 m (e) 5.21 kg (f) 4.53 kg 3. (a) 1/6

(b) 1/6

(d) 5/6

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(b) 1/10

(d) 9/10

5. (a) Teacher check

Challenge

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1. (a) 1800 (b) 5400 (c) 1000 (d) 4900 (e) 400 (f) 30 (g) 80 (h) 20 (i) 70 (j) 30 2. (a) $2.92 (b) $5.94 (c) $1.98 (d) $3.93 (e) $1.92 (f) $2.98 3. (a) (30 + 5) x 3 = (30 x 3) + (5 x 3) = 105 (b) (20 + 7) x 8 = (20 x 8) + (7 x 8) = 216 (c) (40 + 6) x 7 = (40 x 7) + (6 x 7) = 322 (d) (30 + 2) x 4 = (30 x 4) + (2 x 4) = 128 (e) (80 + 4) x 6 = (80 x 6) + (4 x 6) = 504 (f) (70 + 5) x 8 = (70 x 8) + (5 x 8) = 600 (g) (50 + 3) x 5 = (50 x 5) + (3 x 5) = 265 (h) (90 + 6) x 7 = (90 x 7) + (6 x 7) = 672 (i) (60 + 2) x 9 = (60 x 9) + (2 x 9) = 558 4. (a) 42 marbles Challenge (b) $30 One possible solution: (c) $20 (d) 20 tickets

Unit 34–2

(b) Teacher check

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• Provide students with further opportunities to practise techniques learned.

Consolidation 34–2

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1. Answers will vary; some possible solutions: 4 x 12, 40 + 8, 96 ÷ 2, 50 – 2 2. (a) 0.8 (b) 0.9 (c) 0.8 (d) 0.6 (e) 0.4 (f) 0.6 3. (a) Time: evening Season: spring/autumn Reason: Teacher check. (b) Time: afternoon Season: autumn Reason: Teacher check. (c) Time: morning Season: summer Reason: Teacher check. (d) Time: afternoon Season: winter Reason: Teacher check. Challenge

• Work out probability for a number of daily events.

Consolidation 34–3

• Create collages of photographs which display different seasons.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 157 •

Unit 35–1

Student page 103

Outcomes

Indicators

N3.1a, N3.3, S3.4

The student is able to: • describe and compare features of 2-D shapes including the number of sides.

Skills

Resources • calculator • pencil • ruler

• geometry

Memory Masters (N3.1a)

Language • place value • digit • multiply • shapes • hexagons • pentagons • quadrilaterals • common feature

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 (S3.4) Warm up

• Ask students to explain what a hexagonal shape is. Encourage in-depth discussion. Focus of the discussion should involve a six-sided shape. Ask if all sides are the same length. Are all angles the same size? Encourage students to justify their answers. Allow them to draw the shapes they are describing on the blackboard/whiteboard. • Once the students should come to the conclusion that the criteria for a 2-D shape to be classified as a hexagonal shape is for it to have six sides and six angles. The sides and angles may be different in size.

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• The focus for this unit is the identification of place value to three decimal places.

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• There are four different hexagonal shapes shown in the workbook. • From the information gained when talking about hexagonal shapes, the task is now to apply this knowledge to draw pentagonal and quadrilateral shapes. • How many sides in a pentagon? (5.) How many angles in a pentagon? (5.) Do all sides and angles have to be equal to each other? (No.) • How many sides and angles in a quadrilateral? (4.) Do all sides and angles have to be equal to each other? (No.) • Draw a number of pentagonal and quadrilateral shapes in the space provided. • Describe the common feature of each set of pentagons and quadrilaterals.

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• Draw a square. Mark the midpoints of each side. Join the midpoints to each other using a straight line. Repeat the process on the new shape. What do you find?

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 26–27. • 158 • New Wave Maths Book E – Teachers Guide

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

Unit 35–2

Student page 104

Outcomes

Indicators

N3.1, N3.3, N3.1b

The student is able to: • use basic division facts to find a unit fraction of a whole number multiple.

Skills • finding unit fractions • reading

Resources

Language

• calculator • 2-cm or 1-cm blocks or counters. • 1-cm grid paper (see page 199) • fraction cake

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

• round • nearest ten • multiply • word problems • one-fourth • one -fifth • two-fifths • half

Notes

Memory Masters (N3.1)

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Teac he r

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

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 10 blocks to each student.Tell students the blocks represent apples.Therefore they each have 10 apples. • Direct students to pretend to eat one-tenth of the apples. How many apples did you eat? (One.) • Ask students to explain how they worked this out. • Distribute another two blocks to each student, so each person has 12 blocks.Tell the students the blocks represent party pies. • Direct students to pretend to eat one-fourth of the party pies. How many party pies did you eat? (Three.) • Ask students to explain how they worked this out. • Students need to divide the whole amount into equal groups according to the given fraction. In this case, 12 pies should be shared into four equal groups, to find there are three pies in each group.

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What to do

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• Direct students to read through Exercise 3(a) altogether. Think carefully about what is being asked. • Some students may need 28 blocks to help them work out this problem. Support as required. • Students should use the strategies acquired in the warm up to solve all problems in Exercise 3. • Assist students as required.

Challenge • Use a fraction chart or grid paper to assist in finding the answer to the question. • Show working out and notes to assist with the explanation of findings.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 159 •

Unit 35–3

Student page 105

Outcomes

Indicators The student is able to: • read and make straightforward schedules.

N3.3, M3.4a

Skills

Resources • calculator • various timetables

Language • multiply • timetable

• reading time tables • recording daily events

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 (M3.4a) Warm up

• Display timetables of rail or bus schedules, television programs, class activities or other programs as are available. • Discuss the reason for timetables—provides knowledge of what is happening and when.

• A timetable of Carl’s School Day and the other activities he participates in during the day is shown in the workbook. Alongside, is a blank timetable for students to fill in showing what activities they participate in during a day. • Students may choose any school day and show what they do on that day. • When the timetable is complete, read and answer the questions comparing timetables.

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Challenge

• Record what happens to a shadow over the period of a school day. Explain why this occurs.

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What to do

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• The focus for this unit is the addition and subtraction of multiples of 100 less than 1000.

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For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 102–103. • 160 • New Wave Maths Book E – Teachers Guide

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

Unit 35—Answers

Student pages 103–105 Unit 35–1

1. (a) tens (b) thousands (c) ones (d) hundreds (e) tens (f) ten thousands (g) tens (h) ones (i) ones (j) thousands 2. (a) 3.6 (b) 4.0 (c) 5.6 (d) 5.4 (e) 5.6 (f) 4.2 3. (a) Teacher check. (b) The number of sides. Challenge

Unit 35–2 1. (a) 6980 (b) 3470 (c) 7490 (d) 2090 (e) 1500 (f) 7440 (g) 6920 (h) 4760 (i) 8220 (j) 4290 2. (a) 24.5 m (b) 28 m (c) 37 m (d) 27 m (e) 25.6 m (f) 36.8 m 3. (a) 7 (b) 3 (c) 4 (d) 10 (e) 50 m (f) $20 Challenge

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/6 is bigger than 8/10. 50/60 is bigger than 48/60

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• Discuss which hexagon, pentagon and quadrilateral would be best suited for use in a structure or for a storage container.

Consolidation 35–2

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1. (a) 900 (b) 1400 (c) 1200 (d) 1100 (e) 1800 (f) 300 (g) 100 (h) 300 (i) 200 (j) 600 2. (a) 73.6 (b) 90.2 (c) 71.4 (d) 40.8 (e) 72.6 (f) 39.6 3. (a) Teacher check. (b) Teacher check. (c) Teacher check. (d) 2 hours. (e) Teacher check. (f) Teacher check. Challenge Teacher check. Students should note that the shadow shortens then lengthens over the course of the day.

• Provide students with further opportunities to devise and solve their own word problems.

Consolidation 35–3

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R.I.C. Publications®

• Devise a timetable of their ‘perfect’ day.

New Wave Maths Book E – Teachers Guide • 161 •

Unit 36–1

Student page 106

Outcomes

Indicators

N3.3, S3.1

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

Skills • drawing • locating • logical thinking • working collaboratively

Memory Masters (N3.3)

Resources • calculator • coloured pencils

Language • add • path • horizontally • vertically • cell • grid • sums • difference 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 (S3.1) Warm up

• Ask students to imagine their local shopping centre or park. • Think about the location of such things as telephones, lights, bus stops, seats, toilets, postboxes, fences etc. • Discuss why certain items are situated where they are; e.g. there is usually a seat at the bus stop, so people can sit down while waiting for the bus.

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• Direct students to look at the map in the workbook. • In small groups, discuss the best location for the items listed.Working in small groups encourages students to justify their placement and reasoning behind it. • Students then draw the items on the map. • Share placement of items with the class. Discuss and justify ideas.

Challenge

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What to do

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• The focus for this unit is the addition and subtraction of multiples of 100.

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• Students use coloured pencils to show the different attempts and different paths. Draw lightly and carefully. More diagrams may be drawn on the page or on grid paper if required. • Share final results with the class.

• 162 • New Wave Maths Book E – Teachers Guide

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Unit 36–2

Student page 107

Outcomes

Indicators

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. • summarise data based on tallying.

Skills • analysing data • recording data

Resources

Language

• calculator • students in the school or student records (be careful of confidentiality of information)

• add • tally • total • order • reasons • most • fewest

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

Memory Masters (N3.3)

Notes

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• The focus for this unit is the addition and subtraction of multiples of 10 and multiples of 100.

Number (N3.3)

• The focus for this unit is the addition of decimals.

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

• Write the seasons of the year on the board. Asks students to tell you which months belong where. • Revise how to record data using a tally. 1111

What to do

Challenge

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• Make a tally of students in the class with birth dates in the different months. To complete the tally, send two or three students to other classes to collect the months of birth of students in those classes (organise with the teachers beforehand). Alternatively, use school records and call out the months of birth for class members to record on their tally sheet. Try to collect close to 100 samples at least. • Find the total births for each month then write the order from one for the largest number of births through to 12 for least number of births per month. Ask students what they will do if more than one month has the same total. Write as the same order then miss the next number before continuing; e.g.. 1, 2, 3, 3, 3, 6, 7 etc. • When all details are complete, answer the questions using the information from the table.

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• A new reception centre is going to open and the owners would like to know which month(s) are the most popular for weddings.You task is to find out. Explain how you would determine the answer.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 112–113. www.ricpublications.com.au

R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 163 •

Unit 36–3

Student page 108

Outcomes

Indicators The student is able to: • interpret pictographs produced by others where each picture represents more than one unit. • remember basic addition facts and many multiplication facts and calculate mentally basic multiplication facts they don’t recall.

N3.3, C&D3.4

Skills • reading pictographs

Memory Masters (N3.3)

Resources • calculator

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 (C&D3.4, N3.3)

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• Look at the pictograph displayed in the workbook.The graph is showing the number of children who play team sports. • Each person on the graph represents 200 children while the half-person represents half that amount—100 children. • Firstly, students are to work out the total number of children who play each sport. These results will be used to complete the rest of the questions.

Challenge

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• Students are to find three different ways to write the number one. • Make notes to justify his/her choice. • Share answers with the class.

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• Discuss what a pictograph is. A pictograph is a graph in which data is represented by pictures. One picture could represent one unit or many.

What to do

• add • money • pictograph • total • difference

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• The focus for this unit is the addition and subtraction of multiples of then and multiples of 100.

Warm up

Language

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• 164 • New Wave Maths Book E – Teachers Guide

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Unit 36—Answers

Student pages 106–108 Unit 36–1

1. (a) 430 (b) 650 (c) 820 (d) 760 (e) 350 (f) 170 (g) 730 (h) 380 (i) 10 (j) 640 2. (a) 95.67 (b) 68.23 (c) 73.94 (d) 59.63 (e) 106.07 (f) 106.72 3. Teacher check Challenge Teacher check

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1. (a) 900 (b) 1200 (c) 1000 (d) 1600 (e) 700 (f) 100 (g) 400 (h) 400 (i) 100 (j) 300 2. (a) 102.60 (b) 92.13 (c) 76.54 (d) 68.83 (e) 101.18 (f) 140.90 3. Teacher check Challenge Not possible.

Unit 36–2

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• Students design their own shopping centre, considering locations of various features.

Consolidation 36–2

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1. (a) 190 (b) 520 (c) 270 (d) 910 (e) 860 (f) 420 (g) 850 (h) 560 (i) 290 (j) 440 2. (a) $60.67 (b) $60.43 (c) $43.01 (d) $57.54 (e) $48.88 (f) $147.23 3. (a) Aussie rules: 1100, T-ball: 800, Soccer: 1000, Cricket: 700 (b) 1100 – 700 = 400 (c) 3600 Challenge Answers will vary; possible answers one, 1, 5 – 5, 1 x 1, 11 – 10

• Draw a graph to show the results of the tally completed on page 107 of the workbook.

Consolidation 36–3

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R.I.C. Publications®

• Students create their own pictograph to represent data collected of their own choosing.

New Wave Maths Book E – Teachers Guide • 165 •

Unit 37–1

Student page 109

Outcomes

Indicators The student is able to: • informally describe the symmetry of a figure or arrangement. • draw informal maps and plans which show a sense of scale, that is, look ‘roughly right’.

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

Skills • measuring • drawing

Memory Masters (N3.3)

Resources • calculator • playing fields – netball court, basketball court, square ball, soccer, tennis court etc. • trundle wheel • measuring tape

Language • number sentences • divide • draw • describe • spatial features • markings • regularity • symmetry • measurements

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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, M3.4b) Warm up

• The activity asks for the examination of a range of playing field markings. Students can identify playing fields in the school for the teacher to record on the blackboard/ whiteboard.

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• Students are to draw one of these playing fields roughly to scale. All markings relevant to the playing field should be included. • Students are to look for regularity (evenness of spacing of markings), symmetry and measure dimensions. All of these points are to be noted on the drawing of the playing field. Measuring wheels (trundle wheel) and long (20–50 m) measuring tapes should be provided. • Organise students into small groups and take them out to examine the available playing fields. • Share the findings when they have finished the activity.

Challenge

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What to do

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

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• 166 • New Wave Maths Book E – Teachers Guide

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

Unit 37–2

Student page 110

Outcomes

Indicators

M3.1, 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 • adding • reading • problem-solving

Resources

Language • change • 24-hour time • add • subtract • expand • multiply

• calculator • six-sided dice

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Notes

Memory Masters (M3.1)

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• The focus for this unit is the conversion of 12-hour time to 24-hour time and 24-hour time to 12-hour time.

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) Warm up

• Students work in pairs to play a dice game. • Each student rolls the die five times, recording the number that lands on top each time. • Students then add the five numbers to obtain a total.The student with the highest total wins. Keep playing to find the best of five. • Discuss with students their strategy for adding their score each time.

What to do

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• Exercise 3 is designed to encourage students to look at lists of numbers and rearrange them to make adding easier; e.g. 9 + 2 + 1 + 8, it would be easier to add 9 + 1 then 2 + 8 to get 10 + 10, which equals 20. • Set students to work. • Exercise 4 are word problems. This type of activity encourages students to read a problem, then to work out whether they need to divide or multiply, add or subtract. • Once students work out the type of algorithm required, the problems can then be solved using traditional methods. • Support students who need assistance and allow the rest of the class to work independently.

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• Students are to use the clues to find the value of each letter of their name (first name only, surname only, or whole name). • Students write the value of their name. • Share with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 66–67.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 167 •

Unit 37–3

Student page 111

Outcomes

Indicators

Resources

N3.1, N3.3, M3.2

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

• calculator • clocks – analog and digital

Skills • writing time

Memory Masters (N3.1)

Language • how many • cents • change • dollars • time

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

• Display analog and digital clocks. • Discuss the features of an analog clock. It displays each minute, with main points at 5-minute intervals.There is a long hand to show the minutes and a short hand to show the hour. Some clocks also have a second hand. An analog clock only shows the time in 12-hour format, which means the hour hand goes around the clock twice in one day. • Work through a number of examples, allowing students to display a range of times on analog clocks. For example; 12.20, 1.40, 3.10, 5.15, 6.30, 7.45, 10.05, 11.25, 2.35, 4.50, 8.55 and 9.00. • Digital clocks can show the time in 12-hour or 24-hour format. Usually, clocks on video recorders show 24-hour time, while alarm clocks might use 12-hour time, with a light showing a.m. or p.m.

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• The focus for this unit is the conversion of dollars to cents and cents to dollars.

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What to do

• Students look at the analog clocks in Exercise 3. Read the time and write it in the space provided. Assist those students having difficulty. • Exercise 4 uses 24-hour time in digital format and requires students to write the time displayed in words. Assist those students having difficulty.

Challenge

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• Students use their knowledge about time to answer this. • It would be best for students to break down the problem; e.g. work out how many minutes in one hour, one day, then one week.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 90–91.

• 168 • New Wave Maths Book E – Teachers Guide

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Unit 37—Answers

Student pages 108–111 Unit 37–1

1. (a) 0600 (b) 2200 (c) 0920 (d) 0145 (e) 1515 (f) 2.25 p.m. (g) 7.10 p.m. (h) 12.30 a.m. (i) 12.45 p.m. (j) 7.20 a.m. 2. (a) 30 (b) 20 (c) 14 (d) 25 (e) 30 (f) 20 3. (a) 2 + 5 + 7 = 14 (b) 4 + 5 + 9 = 18 (c) 3 + 3 + 5 = 11 (d) 7 + 3 + 2 + 5 = 17 (e) 4 + 2 + 9 = 15 (f) 2 + 2 + 2 + 6 = 12 4. (a) 1500 pages (b) $700 (c) $950 (d) $5829 (e) $324 (f) $54 Challenge Teacher check

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1. (a) 17 (b) 7 (c) 9 (d) 13 (e) 5 (f) 6 (g) 6 (h) 9 (i) 7 (j) 4 2. (a) 14 (b) 12 (c) 17 (d) 19 (e) 12 (f) 13 3. Teacher check Challenge Teacher check. Possible reasons are human error, one student starting from the 0 on the ruler and the other starting from the end, or different sized rulers being used by the students.

Unit 37–2

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• Design a new playing field, complete with markings for an invented sport.

Consolidation 37–2

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1. (a) 899c (b) 2415c (c) 3005c (d) 1827c (e) 2140c (f) $0.16 (g) $47.28 (h) $50.00 (i) $4.25 (j) $0.03 2. (a) 19 (b) 14 (c) 19 (d) 18 (e) 24 (f) 13 3. (a) 12.45 (b) 10.00 (c) 7.35 (d) 11.10 (e) 8.40 (f) 4.20 4. (a) half past six in the morning or six thirty (b) twenty-five past one in the afternoon or one twenty-five (c) quarter to six in the afternoon or five forty-five (d) ten to eleven in the morning or ten fifty Challenge 60 mins = 1 hr 1440 mins = 1 day 10 080 mins = 1 week

• Invite students to create their own word problems for a partner to solve.

Consolidation 37–3

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• Make a timing device to measure a minute.

New Wave Maths Book E – Teachers Guide • 169 •

Unit 38–1

Student page 112

Outcomes

Indicators

WM3.2, WM3.4, N3.1a, N3.2, N3.3

The student is able to: • generate mathematical questions. • decide which information in a problem needs to be represented. • make lists or tables of data to help solve a problem. • check their answers with their estimates.

Skills • estimating • calculating • posing questions • surveying • collecting data • analysing data • makes conjectures

Resources • catalogues • calculators • Internet (optional)

data average estimate standard cost family tallying

Notes

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• problem • open-ended • • calculations • • population • • survey • • money • • multiply • • pet •

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Main Activity (WM3.2, WM3.4, N3.1a, N3.2, N3.3) What to do

Language

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• Groups may decide they would like to survey the number of people in the class or school who have dogs and use the information to estimate the number of dog owners in Australia. This surveyed information can be shared with the whole class. • Students will be working with large numbers to complete this activity. Calculators are essential. Note: For some groups, estimating and calculating with such large numbers may be very challenging. The task could be scaled down from ‘in one year’ to ‘in one day’ or ‘in one week’. The task could also be changed it from ‘in Australia’ to ‘in your school’ or ‘ in 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. How much do they vary? Why do the students think this is? • Students may wish to use the Internet or board of statistics to find an answer to the amount of money spent on dog food each year. • Allow each group time 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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• 170 • New Wave Maths Book E – Teachers Guide

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

Unit 38–2

Student page 113

Outcomes

Indicators

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. • suggest a suitable way to classify data.

Skills • analysing data • recording data

Resources

Language

• calculator • students • toothpicks or similar

• multiply • table • tally • total • classify • combinations

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

Memory Masters (N3.3)

Notes

Teac he r

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• The focus for this unit is the multiplication of a multiple of 10 by 10 and the division of a multiple of ten less than 1000 by a whole number 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 (C&D3.2, C&D3.3) Warm up

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• What to do • Discuss different forms of transport used by the students to travel to and from school. Discuss that forms of transport may change according to the weather. • The table requires information to be collected on the means by which each student came to school on this day.

Challenge

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• Information is collected for boys and girls. Ask girls first and then boys to stand to indicate the means they used to come to school. Girls – walk; Boys – walk; Girls – car; Boys – car; Girls – bus; and so on. Students record totals in the correct space in their workbook. When all means have been covered, students find the combined total of how many walk, came by car and so on. • Exercise 4 asks for the students in the class to be classified in three different ways. Classifying in two ways is easy—girls and boys. Discuss the following classifications—closed shoes, open shoes (include thongs, sandals) and no shoes; blonde, brown and other hair colour; by eye colour; dressed in school uniform, same school uniform, no school uniform. Students may provide other examples.

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• Provide toothpicks or equivalent for students to use. • Students draw each attempt and make notes to explain their thinking and reasoning. • Share final results with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 112–113.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 171 •

Unit 38–3

Student page 114

Outcomes

Indicators

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 and express results with remainders or fractions. • use the decimal point in representing quantities and money.

Skills • following instructions • problem-solving • writing decimals • sharing decimal fractions

Memory Masters (N3.3)

Resources • calculator • paper streamer • string • 2-cm blocks • paper • 1-cm grid paper (see page 199) • fraction cake • metre 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 (N3.3, N3.1a)

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• Read through each activity in turn providing students with ideas to help solve the problems; e.g. (a) Measure 80 cm or 0.8 m of paper streamer then divide it into 20-cm or 0.2-m lengths. (b) Use nine 2-cm cubes to show nine-tenths (0.9) of a kilogram of apples. Share the nine cubes into 0.3-kilogram lots (1 cube = 0.1 kilograms). (c) Use six 2-cm cubes to represent 0.6 of a block of chocolate. Share these into two groups, or use 1-cm grid paper to draw 0.6 of a block of chocolate then share it. (d) Measure 2 m of string then cut it into 50-cm or 0.5-m lengths. (e) Measure 0.5 m (50 cm) of string, or use a piece of string from (d). Cut the string into 0.1-m or 10-cm lengths. (f) Use a fraction cake to show 0.8 or 8/10. Share the 0.8 or 8/10 among four people. • Write answers in the spaces provided.

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• Distribute paper streamers, 2-cm blocks, 1-cm grid paper, string, fraction cake and any other resources available among small groups of students within the class.

Challenge

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Number (N3.3)

What to do

• multiply • solutions • problems • divided • lengths • share • weigh

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

• The focus for this unit is the multiplication of a multiple of 10 by 10 and the division of a multiple of 10 less than 100 by a whole number less than 10.

Warm up

Language

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• Students are to explain the answer they give to justify their findings. • A discussion about density and volume may arise.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 6–7. • 172 • New Wave Maths Book E – Teachers Guide

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Unit 38—Answers

Student pages 112–114 Unit 38–1

1. (a) 300 (b) 200 (c) 700 (d) 500 (e) 800 (f) 60 (g) 50 (h) 80 (i) 70 (j) 40 2. (a) 1.61 (b) 3.42 (c) 3.84 (d) 5.64 (e) 3.75 (f) 3.78 3. Teacher check 4. Teacher check Challenge or

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1. In 2001, it was reported by the United States Department of Agriculture that Australians spend US$217.7 million on dog food each year which roughly converts to AUD$400 000 000 (400 million Australian dollars).

Unit 38–2

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• Students can use the same thought processes to estimate other open-ended problems such as: – the number of tissues used in one year. – the number (and cost) of disposable nappies used in one year. – the amount of Milo™ eaten in one month.

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1. (a) 600 (b) 900 (c) 400 (d) 100 (e) 0 (f) 30 (g) 30 (h) 20 (i) 30 (j) 40 2. (a) 114 (b) 112 (c) 144 (d) 153 (e) 90 (f) 84 3. (a) 4 lengths of ribbon (b) 3 people (c) 12 squares or 2 rows of chocolate (d) 4 lengths of rod (e) 5 lengths of string (f) 0.2 of a whole pizza Challenge Both will be the same weight.

Consolidation 38–2

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R.I.C. Publications®

• Brainstorm other data that could be collected and recorded in a table.

Consolidation 38–3

• Use class parties, art activities and cooking to reinforce the sharing of decimal fractions.

New Wave Maths Book E – Teachers Guide • 173 •

Unit 39–1

Student page 115

Outcomes

Indicators

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

The student is able to: • order and show a sense of proximity of things in locating key features on maps. • attend to the order and proximity of things in giving directions. • interpret order and proximity from maps. • draw informal maps and plans which show a sense of scale. • use their informal understanding that maps and plans are drawn ‘to scale’ to make simple comparisons.

Skills • mapping • drawing • using scale

Memory Masters (N3.3)

Resources • calculator • pencil • coloured pencils

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 (S3.1, M3.4b)

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• In the space on the page students draw a rough outline of their suburb, town or location. • Note: Communities vary greatly and may not have all the facilities listed. Students can add in any which are particularly relevant to their community. • Students draw the route they follow from home to school. Show main points of interest on the way. These are called landmarks. • Choose a different coloured pencil and draw the route they follow to go to the park. Mark in all important landmarks. • Choose another colour to show the route they take to go to the shop. Mark in all important landmarks. • Choose another colour to show the route they follow to go to the sports ground. Mark in all important landmarks on the way.

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• Discuss the various facilities in your local community. Talk about distance between them and locations in relation to home or school. • Talk about the particular facilities used by the students.

Challenge

Notes

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Number (N3.3)

What to do

• add • space • draw • routes • landmarks • path

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

• The focus for this unit is the multiplication of a whole number less than 100 by 10 and the division of a multiple of 10 less than 1000 by a whole number less than 10.

Warm up

Language

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• Test the traversibility options for the map shown. Show all workings and notes of their working out.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 8–9.

• 174 • New Wave Maths Book E – Teachers Guide

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

Unit 39–2

Student page 116

Outcomes

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

N3.3, N3.1a

Skills • multiplying • dividing • rearranging numbers • recognising place value

Resources

Language

• calculator • toothpicks or equivalent • place value chart (see pages 205– 206)

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• add • digits • different • whole numbers • circle • largest • smallest • total value • squares

Notes

Memory Masters (N3.3)

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• The focus for this unit is the multiplication of a whole number less than 100 by 10 and the division of a multiple of 10 less than 1000 by a whole number 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 (N3.1a) Warm up

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• What to do • The first task is to rearrange the numbers shown to create as many different combinations as possible. With two numbers such as 2 and 1, they can be written as 12 or 21. With three numbers such as 3, 4 and 7, how many different numbers can be made? (347, 374, 437, 473, 734, 743) Ask students to share combinations while you record them on the board.

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• Using the four digits shown on the page, rearrange them to make as many different numbers as possible.When writing the new combinations, it is worthwhile to think in terms of patterns; e.g. start with 2 and write all the possible arrangements using 2 in the thousands, then 3 in the thousands, then 7 in the thousands and finally 9 in the thousands. In this way, numbers should be in order from smallest to largest. • Record the number of combinations found. (There are 24 possible combinations.) • Once all numbers have been found circle the largest and the smallest. • Find the difference between the largest and smallest combinations. • Exercise 4 asks for the total value of the digit underlined in each number. To find the total value of the digit, the place value is multiplied by the face value; e.g. a number in the tens place with a face value of 3 is 10 x 3 or 30. • Students may write the numbers in their correct places on a place value chart to assist in working out the answer.The total value is then easier to work out. In the above example, the 3 is in the tens place, hence total value is 3 tens or 30. • Work through several examples with the class.

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Challenge • Provide toothpicks or equivalent for students to use. • Students draw each attempt and make notes to explain their thinking and reasoning. • Share final results with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 36–37.

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New Wave Maths Book E – Teachers Guide • 175 •

Unit 39–3

Student page 117

Outcomes

Indicators

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

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. • measure time intervals using natural units of time, artificial non-standard units, timers or standard units.

Skills • estimating time

Memory Masters (N3.1a)

Resources • calculator • stopwatch • eggtimer • skipping rope • netball/basketball

Language • place value • digit • add • measure • estimate • different • combinations

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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 (M3.3, M3.2) Warm up

• Explain to students that we frequently estimate time rather than specify, particularly for the length of time to complete a task. • Organise students into small groups or pairs.

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• Outline the process for the class to follow—estimate the time they anticipate will take to complete each activity as outlined in the workbook. Write the estimate in the space provided. • Discuss estimates with students to encourage them to make statements about how well they think their estimates will match the actual time taken. • Complete the first two tasks before allowing students to complete their estimations for the rest of the tasks. This will help students to refine their estimation skills as they go. • Take turns in the group, or with a partner, to complete each activity. Another group member or partner times to see how long was actually taken to complete the activity. Record this time in the space provided. • Continue until all activities are completed and each person has completed each activity.

Challenge

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• The focus for this unit is the identification of place value.

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• Students are to find the various combinations possible using the photographs as shown. • Recordings are to be kept of attempts and findings.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 6–7.

• 176 • New Wave Maths Book E – Teachers Guide

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Unit 39—Answers

Student pages 115–117 Unit 39–1

1. (a) 230 (b) 490 (c) 960 (d) 580 (e) 660 (f) $60 (g) $40 (h) $40 (i) $80 (j) $50 2. (a) 724.4 L (b) 837.0 L (c) 866.7 L (d) 905.0 L (e) 885.4 L (f) 890 .8 L 3. Teacher check 4. (a) 600 (k) 6000 (b) 70 (l) 3000 (c) 9000 (m) 700 (d) 800 000 (n) 400 (e) 30 (f) 300 (g) 4 (h) 6 (i) 1 (j) 6 000 000 Challenge

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1. (a) 870 (b) 390 (c) 520 (d) 340 (e) 470 (f) 50 (g) 30 (h) 40 (i) 20 (j) 20 2. (a) 938.3 m (b) 558.5 m (c) 775.2 m (d) 889.4 m (e) 778.5 m (f) 688.3 m 3. Teacher check Challenge Yes.

Unit 39–2

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• Design a community to include all facilities students feel are necessary. Discuss locations and proximity of facilities.

Consolidation 39–2

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1. (a) thousands (b) ones (d) thousands (e) tens (g) thousands (h) ones (j) thousands 2. (a) 8.355 km (b) 6.782 L (e) 8.942 km (f) 8.936 kg 3. Teacher check Challenge 6

• Provide students with further examples of digits to rearrange in order to make new combinations of numbers.

Consolidation 39–3

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• Encourage students to estimate the time that will be taken to complete various activities every day.

New Wave Maths Book E – Teachers Guide • 177 •

Unit 40–1

Student page 118

Outcomes

Indicators

N3.1a, N3.3

The student is able to: • use the decimal point in representing quantities and money. • subtract measures and money.

Skills • subtracting decimals

Memory Masters (N3.1a)

Resources • calculator • Base 10 MAB • counters

Language • round • nearest tenth • Base 10 MAB • differences

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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, N3.3) Warm up

• Organise the students into small groups. • Distribute Base 10 MAB and allow a brief time for free play. • Ask students to show 3.25 using the Base 10 MAB. A counter may be used to indicate the decimal point. • Take 2.6 from the 3.25 shown using Base 10 MAB. Ask a student to explain how they found the answer. Ask other students if they used another method. Remind students that there are different methods that may be used. The important thing is to start with the wood showing the number then take the other number away from this number. Where there is not enough wood what must they do? (Trade, then proceed with the subtraction.) • Work through several examples, asking students to show the number on the left using the Base 10 MAB, then take away the number on the right.

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

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• Allow students to proceed by completing Exercise 3 themselves.When students reach Exercise 4 ask students to try to work the sums without using the Base 10 MAB. If they have difficulties, allow them to use the Base 10 MAB. • Exercise 5 consists of straightforward subtraction algorithms. Students can work independently. Assisting those who need further help.

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• Remind students that magic squares have all rows, columns and diagonals each having the same total. In this case the total is 15. • Show all attempts at solving the problem. Keep notes to show reasoning. • Share the final answer.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 50–51.

• 178 • New Wave Maths Book E – Teachers Guide

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Unit 40–2

Student page 119

Outcomes

Indicators

Resources

N3.3, C&D3.4

The student is able to: • read frequencies from a bar graph and hence describe the data.

• calculator • toothpicks or similar

Skills

Language

• analysing graphs

• number sentences • multiply • graph • hottest • coldest • wettest • amount • change • triangles • arrangement

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Memory Masters (N3.3)

Notes

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

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&D3.4) Warm up

• The page shows a graph combining rainfall and average temperature for a 12-month period. The rainfall is shown using a bar graph and the temperature using a line graph. The vertical axis shows a zig-zag line which indicates that the line has been broken to allow the scale to fit on the page and to accommodate two different scales on the same graph.

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• Provide toothpicks or equivalent for students to use. • Students draw each attempt and make notes to explain their thinking and reasoning. • Share final results with the class.

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For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 116–117.

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New Wave Maths Book E – Teachers Guide • 179 •

Unit 40–3

Student page 120

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 • calculator • 1-cm grid paper (See page 199)

• using a calculator

Memory Masters (N3.3)

Language • multiply • solutions • problems • divided • lengths • share • weigh

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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 calculator and explore its functions for a few minutes. • Ask students if they know what the constant function is and what it does.The constant function allows you to press a function; e.g. x, – , +, ÷, followed by = and the calculator will continue to work the function at the rate asked; e.g. press 2 + =, =, =, the calculator will provide the following answers; 2, 4, 6, 8. • Students try these with their calculator: 6 + 4 = = = =; 14 – 2 = = = =; 2 x 2 = = = =. • What did they discover? Allow students to describe what happened.

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

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• In all the examples given, the constant function will be as per the example. Although the problems show multiplication, we use subtraction as the means of finding how many times the number shown is to be added to itself (repeated addition is multiplication). In the example shown: __ x 3 = 9, entering 9 into the calculator followed by – and 3 then = shows 6, then = shows 3, then = shows 0. The equal sign was pushed 3 times ,therefore 3 lots of 3 equal 9. • Work through several with the class to see they understand the process, then let them explore by themselves.

Challenge

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• It may be necessary to draw the grid a number of times on 1-cm grid paper to be able to show all workings. • Students show all attempts and keep notes to provide an explanation of how they were trying to reach the answer. • Share the final result with the class.

For a relevant assessment activity refer to RIC-0030 Maths Assessment – Level 3 pages 72–73.

• 180 • New Wave Maths Book E – Teachers Guide

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Unit 40—Answers

Student pages 118–120 Unit 40–1

Unit 40–2 1. (a) 2 (b) 6 (c) 2 (d) 10 (e) 3 (f) 4 (g) 2 (h) 5 (i) 5 (j) 2 2. (a) 2.76 m (b) 4.83 m (c) 10.56 m (d) 5.16 m (e) 4.84 m (f) 7.26 m 3. (a) December (b) June and July (c) 200 mL (d) 25 mL each (50 mL altogether) Challenge

1. (a) 3 (b) 4 (c) 2 (d) 7 (e) 2 (f) 3 (g) 2 (h) 2 (i) 2 (j) 3 2. (a) 17.28 m (b) 35.25 m (c) 44.52 m (d) 46.23 m (e) 26.22 m (f) 39.22 m 3. (a) 3 (h) 8 (o) 19 (b) 12 (i) 6 (p) 12 (c) 4 (j) 9 (q) 11 (d) 8 (k) 14 (r) 9 (e) 6 (l) 7 (s) 4 (f) 16 (m) 14 (t) 6 (g) 9 (n) 7 (u) 7 Challenge

• Use shopping catalogues to give students practice adding and subtracting money.

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1. (a) 5690 (b) 8750 (c) 6310 (d) 8920 (e) 7240 (f) 4790 (g) 6270 (h) 1070 (i) 2730 (j) 1680 2. (a) $25.92 (b) $22.74 (c) $66.88 (d) $34.51 (e) $28.35 (f) $39.41 3. (a) 0.65 4. (a) 1.65 m 5. (a) $2.64 (h) 1.763 (b) 1.413 m (b) $5.97 (i) 1.75 (c) 6.61 m (c) $2.15 (j) 2.269 (d) 2.733 m (d) $3.34 (b) 2.67 (e) 1.64 m (c) 1.126 (f) 2.78 m (d) 1.752 (e) 1.164 (f) 1.748 Challenge (g) 0.965 (k) 1.299 (l) 3.527 (m) 2.802 (n) 2.716 (o) 2.211 (p) 2.887

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Consolidation 40–2 • Search through newspapers to find graphs students can use to analyse and glean information.

Consolidation 40–3

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• Use calculators on a regular basis to study patterns in numbers.

New Wave Maths Book E – Teachers Guide • 181 •

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Additional Activities

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

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New Wave Maths Book E – Teachers Guide • 183 •

Space Activities S3.1

1. Students give directions to each other in order to successfully reach a particular place in

the school grounds. 2. Students use directional language when giving each other directions to successfully reach a particular place in the school grounds. S3.2, S3.4 1. A lump of modelling clay and fishing line (approximately 60 cm) is required for this activity.

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(a) Make a cube using the modelling clay. (b) Use the fishing line to cut the cube of modelling clay to give the following cross-sectional shapes: • a square • a rectangle • a triangle • an equilateral triangle • a hexagon • a regular hexagon (c) Is each cross-section congruent after the modelling clay is cut? How can students be sure? (d) Students trace around each cross-section on a page in their pad and label it with the name of the shape made. S3.3

© R. I . C.Publ i cat i ons •f oInvestigate rr ethe vi ew p ur p ose so nl y •packaging. merits of using cylindrical packaging against rectangular or cube Study various shapes and ask students to explain why some shapes will tile while other shapes

won't tile.

S3.4

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(a) Use an A4-size sheet of paper to make a cylindrical package without enclosing the ends. (b) Use an A4-size sheet of paper to make a rectangular package the same height as the cylinder, again without the ends being enclosed. (c) Which package holds the most? Use peas, beans or similar to fill the containers. Students may need to tape the bottom end to a piece of card to help with stability. (d) Try the same task but use the A4-size sheet of paper to make the ends as well. Does this have any effect on the final comparison of the two shapes?

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Measurement Activities M3.1

Measure and compare objects using arbitrary units. Students should come to realise that units must be kept consistent to ensure accuracy and allow for comparisons.

M3.2, M3.3 Note: Estimate prior to measuring in all activities. 1. Boxes, such as cereal packets, may be cut and laid out flat. The surface area can then be checked by laying a transparent grid over the box or by drawing squares on the surface of the box and counting them. 2. Using straight-sided, transparent containers, marked in standard units, measure the volume of a variety of different objects in the following ways. Note: Use some containers to catch the overflow. (a) Mark off the level the water is raised to when placing an object in the water container. Alternatively, pour off the water above the original water mark after immersing the object and measure the displacement in standard units. (b) Place an object in an empty container, partly filling the container with water. Remove the object and note how much the water level drops. (c) Fill a straight-sided container with water, place an object in the water, collect the overflow and measure it in standard units. 3. (a) Select a variety of objects and order them by lifting. Each object should be compared by changing hands prior to making a decision. Repeat this activity a number of times using different objects. (b) Using a simple balance scale (a simple balance scale may be made from a wire coat hanger with plastic cups suspended from opposite ends), or a ruler balanced across a pencil, compare a variety of objects to find those that have the same weight or rank objects from lightest to heaviest. (c) Select a range of objects and see how many are needed to balance a set weight or a heavier object. For example, the number or paper clips to balance a 50-g or 100-g weight. Repeat each of these activities a number of times for students to develop the concepts. 4. Collect a variety of boxes, some the same size and some different. Fill the boxes with different materials so the boxes of the same size may have the same weight, or different; and the small boxes may be the same mass as larger boxes, heavier than them or lighter. Cover the boxes with contact or something similar, so contents cannot be viewed or spilt. Label each box with either a number or a letter . Order the boxes by: (a) height (b) length (c) surface area (d) mass Record the order in sequence from smallest to greatest. Encourage students to talk and/ or write about their discoveries.

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Have students collect a number of objects less than 10 centimetres. Ask the students to estimate in centimetres and millimetres the length of each object. Students record the estimate and then measure, as accurately as they can, the actual length of each object and record their measurements. Total accuracy to the nearest millimetre is not required.

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Measurement Activities cont. M3.4a 1. Students use a fixed perimeter (16 cm) and draw as many shapes as they can on 1-cm

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grid paper on page 199 of the New Wave Maths for WA Teachers Guide. This activity may also be completed using a geoboard. (a) Which shape has the longest perimeter? (b) Which shape has the largest area? 2. Students give the length of time taken to do certain activities. Use a timer, stopwatch, watch/clock with sweep second hand, eggtimer or similar, to time the activity. Such activities could include: • tie shoelaces • run 100 m • to call the roll • run around the oval • line up in two even lines • walk around the oval • find a given word in the dictionary • hop the length of the netball court • to complete a given task – solve a mathematics problem – draw a set picture, etc. • run until you need to walk • skip without stopping the rope • to throw a netball goal • length of time a paper aeroplane remains in the air Use similar lists to determine which activity would take longer than another.

M3.4b 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

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Number Activities N3.1a

Use shopping activities to provide students with opportunities to regroup money to the fewest number of notes or coins. N3.1b

Use cooking activities to read and write fractional notation. For example 1/2 cup, 3/4 tablespoon. N3.2

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Use day-to-day situations to encourage students 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? (4 counters). N3.

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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

Search for patterns in all areas of mathematics and daily routines, classroom/school environment, at home and in the community.

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Assessment

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

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New Wave Maths Book E – Teachers Guide • 189 •

Student Outcomes Working Mathematically WM3.1 The student identifies familiar mathematical features inherent in the activities and products of own and other communities. WM3.2 The student poses mathematical questions prompted by a specific stimulus or familiar context and uses problem-solving strategies which include those based on representing key information in models, diagrams and lists.

Chance and Data C&D3.1 The student distinguishes certain from uncertain things and describes familiar, easilyunderstood events as having equal chances of happening or being more or less likely.

Space

The student understands a map or plan a 'bird's-eye view' and uses order, proximity and directional language associated with quarter and half turns on maps and in descriptions of locations and paths.

C&D3.2 The student contributes to discussions to clarify what data would help answer particular questions and take care in collecting, classifying, sequencing and tabulating data in order to answer those questions. C&D3.3 The student displays and summarises data using frequencies, measurements and manyto-one correspondences between data and representation.

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WM3.4 The student uses alternative ways, when prompted, to check working and choice of method.

S3.2

M3.4b The student attends informally to scale when making and using plans, maps and models.

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WM3.3 The student understands mathematical conjectures as more than simply a guess, makes straightforward tests of conjectures and discards those that fail the test.

S3.1

measurements which cannot be obtained directly.

C&D3.4 The student reads and makes sensible statements about the information provided in tallies and in simple tables, diagrams, pictographs and bar graphs.

The student attends to the shape and placement of parts when matching, making and drawing things, including matching 3-D models which can be seen and handled with conventional drawings of them and with their nets.

N3.1a The student reads, writes, says, counts with and compares whole numbers into the thousands, money and familiar measurements.

S3.3

The student recognises repetitions of the same shape within arrangements and patterns and uses repetitions of figures and objects systematically to produce arrangements and patterns.

N3.1b The student reads, writes, says and understands the meaning of fractions, flexibly partitioning and rearranging quantities to show equal parts.

S3.4

The student interprets common spatial language and uses it to describe and compare features of things.

N3.2

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The student understands the meaning, use and connections between the four operations on whole numbers, and uses this understanding to choose appropriate operations and construct and complete equivalent statements.

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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.

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 and uses standard scales to measure length and time.

M3.3

Number

The student makes 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 • 190 • New Wave Maths Book E – Teachers Guide

N3.3

The student adds and subtracts whole numbers and amounts of money and multiplies and divides by one-digit whole numbers, drawing mostly on mental strategies for doubling, halving, adding to 100, and additions and subtractions readily derived from basic facts.

N3.4

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 subtractionbased strategy.

Extracted from Mathematics Outcomes and Standards Framework – Student Outcome Statements, Education Department of Western Australia 1998. R.I.C. Publications® www.ricpublications.com.au

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Working Mathematically—Record Sheet

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Space—Record Sheet

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Measurement—Record Sheet

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Chance and Data—Record Sheet

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Number—Record Sheet

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New Wave Maths Book E – 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 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. ✄

Date

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Demonstrated

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Photocopiable Resources

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Grid Paper ............................................................................................................................... 198–201 Number Charts and Cards ............................................................................................ 202–204 Place Value Charts .............................................................................................................. 205–206 Number Lines and Fraction Chart ............................................................................. 207–208 Spinners ..................................................................................................................................... 209–210 Calendar – Any year ..................................................................................................................... 211 Clocks – Blank ....................................................................................................................................212 Bingo Cards ............................................................................................................................. 213–216 3-D Model Attribute Table ...................................................................................................... 217 Venn diagrams – Blank ................................................................................................................ 218 3-D Shapes ......................................................................................................................................... 219 Tangrams .................................................................................................................................. 220–223 Nets ............................................................................................................................................. 224–230 Paper Circles ..................................................................................................................................... 231 Curve Stitch and Line Pattern ...................................................................................... 232–233 Graph and Table – Blank ........................................................................................................... 234

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2-cm grid paper.

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1-cm triangle grid paper.

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0–99 Chart

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Basic Facts

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Number Cards

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1000

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New Wave Maths Book E – Teachers Guide • 205 •

Tens

Ones

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Tenths Hundredths

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Ten thousands Thousands Hundreds

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Ten thousands Thousands Hundreds

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Number Lines 0

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Spinners – Blank

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New Wave Maths Book E – Teachers Guide • 209 •

Spinners

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Calendar January

March

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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.

February

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Clocks 12

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Note: These cards can be cut along the dark, solid lines to make bingo cards.

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Note: These cards can be cut along the dark, solid lines to make bingo cards.

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Note: These cards can be cut along the dark, solid lines to make bingo cards.

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Attributes

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Note: This table can be used to explore any attributes of 3-D shapes.

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Note: These Venn diagrams can be used as required by the students to sort items into categories.

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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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Enlarge to A3. Use this tangram puzzle to make various shapes.

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Enlarge to A3. Use this Pythagorean puzzle to make various shapes.

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Enlarge to A3. Use this circle puzzle to make various shapes.

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Circle Puzzle

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Enlarge to A3. Use this broken heart puzzle to make various shapes.

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The Magic Egg Puzzle

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Enlarge to A3. Use this magic egg puzzle to make various shapes.

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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.

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Cone Net

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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.

Cylinder 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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Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

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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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Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

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Curve Stitch and Line Pattern 10 9 8 7 6 5

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Curve Stitch and Line Pattern 10 9 8 7 6 5

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20 19 18 17 16 15 14 13

Units of Measure:

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Tally

Total

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Parent Information

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Expectations of Knowledge of Basic Facts ....................................................................... 236 Primary School Mathematics ......................................................................................... 237–238 Problem-solving Strategies ........................................................................................................ 239 Concrete to Mental ....................................................................................................................... 240 Mathematical Learning Areas ................................................................................................... 241 Homework Policy ........................................................................................................................... 242

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New Wave Maths Book E – Teachers Guide • 235 •

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

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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.

Year 5

Recall basic addition, subtraction, multiplication and division facts.

Years 6 and 7

Automatic response is desirable.

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.

• 236 • New Wave Maths Book E – 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.

6 + 7 + 6 = 19 Write the ones (9) under the ones column and add the tens (1) to the tens column.

111

1756 2837 + 4276 8869

7.51 + 21.08 28.59

2+ 1= 3 5 5 5

3 – 7 is not possible. So we exchange one 10 for ten ones to make 13 – 7, which we can do.

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Year 5

Add the numerators 2 + 1 = 3 and write the denominator as it appears.

6.80 – 2.93 3.87

783 – 217 566

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Year 7

1+ 1= 2 3 3 3

11 + 22 = 33 5

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4– 2= 2 5 5 5

Subtract the numerators 4 – 2 = 2 and write the denominator as it appears.

Add or subtract the numerator. Add or subtract the whole number. Write the denominator as it appears.

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24 – 13 = 11 5

Consolidation of above.

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New Wave Maths Book E – Teachers Guide • 237 •

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 5 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.

4 3

576 x 6 3456

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1 468 –300 168

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:

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This is recorded as:

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)

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$5 4 . 2 7 x 46 3 2 5 .6 2 2 1 7 0 .8 0 $2 4 9 6 . 4 2 Year 7

$4 7 . 6 9 x 64 1 9 0 .7 6 2 8 6 1 .4 0 $3 0 5 2 . 1 6

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This is recorded as:

156 468 –300 168 –150 18 – 18 0

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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)

76 x 58 608 + 3800 4408

0 .5 8 x 69 5 .2 2 + 3 4 .8 0 4 0 .0 2

468

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10 = 4

3 x 8

7 = 21 = 2 5

1 x 6

50 = 8 2 = 8 13 6

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.

• 238 • New Wave Maths Book E – 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.

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.

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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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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 239 •

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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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.

Teac he r

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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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m . u

Concrete to Mental – Including the Calculator

o c . che e r o t r s super

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). 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

• 240 • New Wave Maths Book E – Teachers Guide

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Parent Information Mathematical Learning Areas Mathematics is comprises 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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Teac he r

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.

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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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o c . che e r o t r s super

Understand symbols and graphs. Represent variation. Solve equations and inequalities.

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R.I.C. Publications®

New Wave Maths Book E – Teachers Guide • 241 •

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'.

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

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

ew i ev Pr

Teac he r

Should you encounter any problems, please contact me.

Classroom Teacher

✄

w ww

Dear Parent(s)

m . u

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'.

. te

o c . che e r o t r s super

Classroom Teacher

• 242 • New Wave Maths Book E – Teachers Guide