New Wave Maths - Teachers Guides: Level F - Ages 10-11 by Teacher Superstore

RIC–1089 12.3/591

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

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Robert Dayman 2003

RIC-1089 ISBN 978-1-86311-710-4 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 F – Teachers Guide • i •

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

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

Additional Activities

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Teachers Notes and Answers

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

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

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

Photocopiable Resources

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Grid Paper.........................................................................................................................................198 – 203 Number Charts and Cards......................................................................................................204 – 206 Place Value Charts.........................................................................................................................207 – 208 Fraction Chart and Number Line...................................................................................................... 209 Spinners – Blank............................................................................................................................................ 210 Calendar – Any year..................................................................................................................................... 211 Bingo Cards.......................................................................................................................................212 – 215 3-D Model Attribute Table..................................................................................................................... 216 Venn diagrams – Blank............................................................................................................................... 217 3-D Shapes...................................................................................................................................................... 218 Tangrams.............................................................................................................................................219 – 222 Nets.......................................................................................................................................................223 – 229 Paper Circles.................................................................................................................................................. 230 Curve Stitch and Line Pattern................................................................................................231 – 232 Graph and Table – Blank........................................................................................................................... 233

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

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

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

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

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

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

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

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

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

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

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

(b) its relationship to social and technological changes.

New Wave Maths Book F – Teachers Guide • 1 •

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

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

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

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

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

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

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8. modelling positive attitudes towards mathematics.

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

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• Opportunity to learn

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

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

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• Action and reflection

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

• Motivation and purpose

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

• Inclusivity and difference

Learning experiences should respect and accommodate differences between learners.

• Independence and collaboration

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

• Supportive environment

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

• a supportive environment for learning;

• appropriate mathematical challenge is provided; and

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• fostering processes which enhance learning.

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

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New Wave Maths Book F – 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

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

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

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General Content Outline Goals and Guidelines After completing and understanding Year 5 well, students should then move onto Year 6. 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.

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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. 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 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.

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

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

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

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

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8. estimation of number and measurement activities;

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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.

New Wave Maths Book F – Teachers Guide • 5 •

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

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

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

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

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Working Mathematically

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

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

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

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

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

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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. 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;

r o e t s Bo r e p ok u S 3. making patterns and arrangements; 4. constructing models;

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

8. discussing what they are doing.

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7. inferring, predicting and hypothesising; and

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

9. interpreting results; and

10. communicating findings.

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

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New Wave Maths Book F – Teachers Guide • 7 •

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

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

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

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

Recommended Progression for Algorithms

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

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

• Sharing objects in practical situations.

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

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

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

• Introduction to open number sentences. For example:

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4–1=

3 lots of 4 =

Year 3

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

17 +2

21 23 + 14

372 + 416

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

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43 + 24

54 – 22

469 – 217

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

18 + 19 =

21 + 14 + 37 =

256 + 48 =

329 + 257 =

76 – 25 =

100 – 60 =

700 – 300 =

638 – 73 =

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

40 x 2 =

42 =

34 x 2 =

200 =

6 x 100 =

x 40

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30 x

= 90

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

24 ÷ 4 = 6

Year 4

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

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

54 x 2

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x 21

18 x 3

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

8r1 6 49

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

5 76

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

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

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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

2 – 1 = 3 3

• 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

3 60.24

50 267

New Wave Maths Book F – 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.

Year 7

• Addition and subtraction operations extended to cater for the ability of the student and individual and practical needs in whole number and decimals. Mixed fractions added and subtracted with unlike denominators.

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• Multiplication in whole numbers and decimals limited by ability and needs.

• Division extends to the introduction of division greater than ten with recurring remainders; for example:

Measurement

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•

0.16 6 1.0

Students use direct and indirect measurement and estimation skills in length, area, mass, volume and capacity, time and angles. In the New Wave Maths series, measurement work is practical, concrete and relevant and students are encouraged to make sensible choices as to which units to use. Estimation skills are continuously developed with emphasis on students being able to estimate all measurement activities using standard units.

At this stage, there is a greater use of indirect measurement techniques to find the measurements students need.

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

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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.

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

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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.

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.

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Pre-Algebra

Students should be finding ways to explain generalisations in these early stages of development of algebra. There is little algebra covered in the New Wave Maths series; however, teachers should be aware of this outcome, particularly for talented students who may recognise and describe the nature of variation in situations and are able to read, write and understand the meaning of symbolic expressions. They may also write equations and inequalities to describe situations.

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

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New Wave Maths Book F – Teachers Guide • 11 •

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

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

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

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

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

• improve attitudes towards mathematics through its effective use.

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

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

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

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

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As a computational aid, the calculator can:

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

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

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

• Valid

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

• Educative

Assessment should make a positive contribution to students’ learning.

• Explicit

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

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• Fair

• Comprehensive

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

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

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

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

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It is recommended that students’ mathematics be assessed using the Student Outcome Statements. Commercially prepared assessment packages are available from R.I.C. Publications® as follows: Maths Assessment Level 1 (RIC-0028) Maths Assessment Level 2 (RIC-0029) Maths Assessment Level 3 (RIC-0030) Maths Assessment Level 4 (RIC-0087)

Where possible, links to these pages have been included in the teachers notes, pages 22 – 181. New Wave Maths is not a stand-alone assessment document. Activities may be assessed based on Student Outcome Statements. Teachers will need to be familiar with these to make the appropriate assessments. All activities may be assessed in this way. It is suggested that a random sample of activities only is assessed using Student Outcome Statements to determine progress. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book F – Teachers Guide • 13 •

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

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

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

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

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• 14 • New Wave Maths Book F – Teachers Guide

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

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Teachers Notes and Answers

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Contents

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New Wave Maths Book F – Teachers Guide • 15 •

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

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

Resources have been listed to aid organisation before the lesson.

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

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

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Skills relevant to the main activity have been listed.

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

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

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

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

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Where possible, links to a relevant assessment activity in the R.I.C. Publications® Maths Assessment Level 4 document have been provided.

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Suggested activities for consolidation of the main activity on the workbook page have been provided as a guide only. Feel free to use, modify, extend or disregard these as you feel necessary.

• 16 • New Wave Maths Book F – 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 233. Teachers may photocopy and use them with their class(es). adhesive tape

geometric blocks

reading or library books

analog clock/watch

geostrips

ruler

atlas/street directory

glue

scissors

attribute blocks

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balance

shapes – 2-D

– A4 and A3

balloons balls Base 10 MAB

shapes – 3-D

height measuring stick

small squares

historical reference books or the

spinners

Internet

straws

kitchen scales

string

calculator

large circles • page 230

sundials

calendar • page 211

lead pencil

tangrams • pages 219 – 222

candles

light card – coloured or plain

tape measure

bundles of 10s and 1s

cardboard strips

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interlocking cubes

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bucket

stopwatch

– A4 and A3

tessellating sets

cardboard tubes

magazines

timetables

classmates' clothing

markers (cones)

toothpicks or equivalent

class members

measuring containers (mL/L)

trundle wheel

clock stamp

metre rule

TV program guide

coloured counters

mirror/mira

unit cubes

coloured pencils

modelling clay

water clock

compass

money (coins/notes)

weather maps

containers – various shapes and sizes

nets • pages 223 – 229

wire

cotton

newspapers

wool

cuisenaire rods

number chart 1 – 100 • page 204

world globe

cups

number squares

1-cm grid paper • page 199

curve-stitch sheets • pages 231 – 232

objects for weighing activities

2-cm cubes

diaries

overhead projector

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dice – 6- and 10-sided digital clock/watch dominoes dot paper • page 198 eggtimer elastic bands

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pattern blocks pegboards

Escher drawings

pipe-cleaners

felt-tip pens

place value chart • pages 207 – 208

fraction cake

plastic polygons

fraction/decimal number line • page 209

playing cards

fraction grid • page 209

polyominoes

fraction squares

popsticks

geoboards

protractor

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New Wave Maths Book F – Teachers Guide • 17 •

Term One Week Unit Outcomes

Page

1 1 S4.1—Use grids to follow directions north, south, east and west.

N4.3—Calculate whole numbers mentally.

M4.4a—Understand relationships involving the area and perimeter of regions.

2 N4.1a—Read, write, say, count and compare whole numbers into the thousands and decimals.

M4.2, C&D4.3—Display frequency data and read and make sensible statements about the information provided in tables and line graphs.

N4.1a—Record numbers into the thousands in a place value chart.

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3 S4.1—Locate places according to grid references.

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M4.2, C&D4.3, C&D4.4—Build a model, record details in a table and describe any patterns.

4 N4.3, N4.4—Recognise, describe and use patterns.

C&D4.2, C&D4.3, C&D4.4—Gather, record and analyse data. N4.1a—Record decimal numbers in a place value chart. 5

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N4.2—Record factor pairs of whole numbers.

11 12

5 S4.2—Attend to the shape, size and placement of parts when making nets of 3-D models.

N4.3—Calculate with whole numbers, drawing mostly on mental strategies to add and subtract two-digit numbers related to basic facts.

M4.4a, N4.3—Understand relationships involving the area of regions based on squares and use these for practical purposes.

6 N4.2—Understand and record factors for whole numbers.

C&D4.2, C&D4.3, C&D4.4—Select, collect, tabulate and analyse data.

N4.3, WM4.3, WM4.4—Work logically through a problem to prove or disprove a statement.

7 S4.1—Use scale to draw a map.

N4.3—Calculate whole numbers and money to solve real-life problems.

19 20

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M4.2, S4.4—Understand that while perimeter may remain the same, the area of polygons may change. 8

8 N4.3—Partition numbers to simplify and solve a problem.

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C&D4.3—Summarise data to solve a problem and use the information to answer questions.

WM4.2, C&D4.2, S4.4, S4.2, M4.1, M4.3, N4.2—Pose, ask and contribute mathematical questions prompted by a specific stimulus.

9 S4.4—Compare 2-D shapes and record whether they are congruent or similar in a table.

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N4.3—Partition numbers to simplify and solve and problem.

M4.2—Make models and record details of size and dimension.

10 N4.1a—Use symbols to order whole and decimal numbers.

C&D4.4—Use results provided in a table to answer related questions.

N4.1a—Use diagrams provided to subtract fractions.

• 18 • New Wave Maths Book F – Teachers Guide

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

Term Two Week Unit

Outcomes

Page

1 11 S4.3—Use a compass to repeat a design and a create a pattern.

N4.3—Use mental strategies to add or subtract a constant to solve whole number problems.

M4.2, C&D4.2, C&D4.4—Estimate, order and check mass of items using a balance.

12 N4.1b—Use a fraction grid to locate and list equivalent fractions and use symbols to order fractions.

C&D4.2, C&D4.4—View and record features of 3-D models to find any relationships.

N4.1b—Use diagrams to display fractions.

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13 S4.2—Use mathematical tools to construct an equilateral triangle and 30° angle.

N4.2—Use estimation to complete number sentences.

M4.2, N4.4—Make 3-D models and record details in regard to dimensions to find a pattern.

14 N4.1a, M4.2—Use data from a table to complete a timeline.

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C&D4.3, M4.2—Measure peers, record data and display results as a range, median and mode.

N4.3—Use concrete materials to divide decimal numbers.

5 15 S4.4, C&D4.4—Describe and compare figures and objects on the basis of spatial features, using conventional geometric criteria.

N4.3—Calculate with money to a set criteria.

M4.2—Read and write 12- and 24-hour time. 6

16 N4.1a—Round decimal numbers and write the place value of specified digits.

45 46 47

WM4.2, C&D4.2, N4.2, N4.3—Pose, ask and contribute mathematical questions prompted by a specific stimulus.

7 17 S4.3, C&D4.4, N4.4—Describe and compare 2-D polygons on the basis of spatial features, using conventional geometric criteria.

N4.1a—Read, write, say, count, compare and order whole numbers into the millions.

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M4.2—Measure time reading a calendar. 8

18 N4.3—Calculate with money to a set criteria.

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C&D4.3—Arrange objects to find all possible arrangements and record data.

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C&D4.2, C&D4.3, C&D4.4—Record data and display results as a range, median and mode.

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N4.3—Calculate whole numbers to find rules for divisibility.

51 52 53 54

9 19 S4.3—Describe and compare 2-D figures on the basis of spatial features, using conventional geometric criteria.

N4.1b—Demonstrate understanding of fractions by displaying on diagrams.

M4.2, C&D4.3—Measure peers, record data and tally results.

20 N4.4—Use a number grid to find patterns.

C&D4.2, C&D4.3, C&D4.4—Record data in a Venn diagram and answer relevant questions.

N4.1b—Round and order fractions using symbols.

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New Wave Maths Book F – Teachers Guide • 19 •

Term Three Week Unit Outcomes

Page 61

N4.3—Rearrange numbers to simplify and solve addition problems.

M4.4a—Calculate the perimeter of a region and use the information for practical purposes.

2 22 N4.4—Use and create a code using letters of the alphabet which follow a pattern.

C&D4.2, C&D4.3, C&D4.4—Collect, record and analyse data. Use data to show range, mode and median.

N4.3—Calculate with whole numbers into the millions.

3 23 S4.2—Construct angles using a compass.

N4.1a—Use symbols to order whole and decimal numbers.

M4.4a, N4.3—Calculate the area of a polygon and use the information for practical purposes.

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1 21 S4.3—Use a grid to reduce a figure using scale.

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4 24 N4.1a—Read, write, say, count, compare and order decimals and fractions.

C&D4.2, C&D4.3, C&D4.4, S4.2—Make 3-D models and record data displaying dimensions and summarise any patterns found.

N4.1a—Round whole numbers as required.

5 25 S4.3—Enlarge a diagram and draw it on a grid according to scale.

N4.3—Calculate with whole numbers drawing mostly on mental strategies.

M4.4a, S4.2—Calculate the area of a polygon and use the information for practical purposes. Make 3-D models and record data as required.

6 26 N4.1b—Use diagrams to subtract fractions. Write fraction equivalents.

C&D4.3, N4.2—Use data to create a pie graph.

N4.1b—Calculate with fractions drawing mostly on mental strategies for visualised fractions.

7 27 S4.2, C&D4.3—Locate and describe the purpose of a shape in the school environment.

M4.3—Demonstrate understanding of use of fractions in real-life situations.

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8 28 N4.1b—Order fractions.

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M4.4a—Calculate the perimeter of polygons and use the information for practical purposes.

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C&D4.3, C&D4.4, S4.2—Make models, record data in a table and create a line graph displaying the information.

N4.1b—Use diagrams to demonstrate an understanding of fractions.

9 29 S4.3—Create as many pentominoes as possible on a grid.

N4.3—Solve money problems based on real-life situations.

WM4.2, C&D4.2, C&D4.3—Pose, ask and contribute mathematical questions prompted by a specific stimulus.

10 30 N4.3—Solve money problems based on real-life situations.

C&D4.3—Use an arrow diagram to display possible outcomes.

N4.1b—Use diagrams to demonstrate an understanding of fractions.

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• 20 • New Wave Maths Book F – Teachers Guide

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Term Four Week Unit

Outcomes

Page 91

N4.1a—Round whole and decimal numbers as required.

M4.2—Order areas by counting squares.

2 32 N4.1a, N4.1b—Convert fractions to decimals and decimals to fractions.

C&D4.3, C&D4.4—Collect, tally and analyse data.

N4.3—Estimate and check calculations of whole and decimal numbers.

3 33 S4.3—Use rotations, slides, flips and turns to create a 2-D pattern.

N4.3—Estimate and check calculations of whole numbers into the thousands.

M4.4a, C&D4.3—Locate and record polygons within a design. Calculate perimeter and area.

4 34 N4.3—Estimate and check calculations of whole numbers in the thousands.

100

C&D4.3—Use data to create a pie graph.

101

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N4.3—Use rounding techniques to simplify problems and estimate answers.

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1 31 S4.1, M4.2—Determine order using a map drawn to scale.

102

5 35 WM4.2, C&D4.2, N4.1, N4.2, N4.3—Pose, ask and contribute mathematical questions prompted by a specific stimulus.

103

N4.1b, N4.3—Subtract fractions. Estimate and check calculations of decimal numbers.

104

M4.2—Read a timetable and record relevant answers.

105

6 36 N4.1b, N4.3—Subtract fractions. Estimate and check calculations of whole numbers in the thousands.

106

C&D4.3, C&D4.4—Collect, record and analyse data.

107 108

7 37 S4.1—Use grid coordinates to locate features.

109

N4.1b, N4.1a—Add fractions. Write numbers in the thousands.

110

M4.1—State the correct unit of measure to use and estimate the actual measurement.

111

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8 38 N4.1a, N4.1b—Use symbols to order fractions. C&D4.3, C&D4.4—Represent data in an arrow diagram.

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N4.1b—Subtract fractions. Solve word problems finding totals as required.

C&D4.4, M4.2, M4.3—Answer questions from a timeline. Collect data and record on a chart.

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9 39 S4.1—Use scale and distance on maps and plans.

112 113 114 115

N4.1b—Read, write and order decimal numbers.

116

N4.1a—Convert measures to decimal form.

117

10 40 N4.3—Verify divisibility by four. Prove or disprove its possibility.

118

C&D4.3, C&D4.4—Measure peers, record data in a table and make a scattergraph to show any relationships.

119

N4.3—Estimate and check calculations of decimal numbers.

120

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New Wave Maths Book F – Teachers Guide • 21 •

Unit 1–1

Student page 1

Outcomes

Indicators

N3.3, N4.3, S4.1

The student is able to: • give unambiguous instructions for moving and locating objects in his/ her environment or on models, maps or plans using distance, direction and common map grids.

Skills • writing directions • following directions

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Memory Masters (N3.3)

Resources • calculator

Language • north • south • east • west • direction

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Notes

• The focus for this unit is addition facts to the sum of 20.

• 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 (S4.1) Warm up

• Review directions – north, south, east and west.

What to do

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Number (N4.3)

• Read the question with the class. Explain they may find it easier to draw the pathway in pencil before writing the directions to follow. • Once the directions are completed, erase the pencil pathway. • Students swap workbooks with partners and follow the directions given.

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Challenge • Students study Exercise 2(c) and develop a word problem to match. • Share the word problems with the class.

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For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 4 – 5. • 22 • New Wave Maths Book F – Teachers Guide

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Unit 1–2

Student page 2

Outcomes

Indicators

N3.3, N4.3

The student is able to: • remember most basic multiplication facts (to 10 x 10) and mentally extend to one-digit numbers by multiples of ten.

Skills • calculating mentally • partitioning sums • reasoning • problem solving

Resources

Language

• calculator • Base 10 MAB • pencil

• mental • continuous • network

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Notes

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Memory Masters (N3.3)

• The focus for this unit is subtraction of numbers less than 20.

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Number (N4.3)

• 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 (N4.3) Warm up

© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Revise partitioning of numbers to simplify mental calculation. For example, 56 x 4 may be seen as (50 x 4) + (6 x 4) = 200 + 24 = 224.

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• Work through several examples from Column A with the class to establish the pattern then allow students to complete Column A. • Students who have difficulties in working these problems mentally should be encouraged to write the sums in full then work out the answers. Students should always be encouraged to ‘have a go’. • Check the results from Column A. • Students continue the process with Column B. • Refresh minds before moving on to Column C that the breakdown becomes 124 x 5 = (100 x 5) + (20 x 5) + (4 x 5) = 500 + 100 + 20 = 620. • After completion of Column C ask students how they would attempt Exercise 4. The process is the same as Column C: 390 x 4 – (300 x 4) + (90 x 4) = 1200 + 360 = 1560. • Complete Exercise 4.

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• Students may start at any point on the network. • The pencil must not be lifted off the page. • No line is to be retraced. • There may be several solutions to the problem.

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New Wave Maths Book F – Teachers Guide • 23 •

Unit 1–3

Student page 3

Outcomes

Indicators The student is able to: • investigate rectangles drawn on a square grid and show that a shortcut for finding area is to multiply the number in each row by the number of rows.

N3.3, N4.3, M4.4a

Skills • measuring • reasoning • recording data

Memory Masters (N3.3)

Resources • calculator • overhead of 1-cm grid paper

Language • perimeter • area • shape • grid • table

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Number (N4.3)

• 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 (M4.4a) Warm up

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

• Using an overhead or blackboard/whiteboard-drawn grid, draw a shape similar to the ones in the workbook. • Ask students what perimeter is and how it may be found. For most this will be a case of counting the units to find the total. However, some students will be able to double lengths of two sides or understand that, in rectangles, adding two adjacent sides and doubling will provide the answer. Encourage students to use the method they prefer.

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• Find the perimeter for each given shape. • Ask students what they understand area to be and how they would find it. • Accept all reasonable explanations and use the overhead or blackboard/whiteboard grid used for perimeter; have students explain their reasoning. • Students may understand that for rectangles area is found by multiplying length by width. Students proceed to find the area of the shapes using their own chosen method. • Results of perimeter and area findings are recorded on the table provided. • Once all perimeter and area calculations are completed, ask the students to complete the two sections at the bottom of the page, giving their own descriptions of what they found.

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• Double the length and width of shape H. What happens to the perimeter and area?

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 132 – 133. • 24 • New Wave Maths Book F – Teachers Guide

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Unit 1—Answers

Student pages 1 – 3 Unit 1–1

1. (a) 13 (b) 8 (c) 11 (d) 7 (e) 5 (f) 8 (g) 2 (h) 7 (i) 6 (j) 3 2. (a) 15 156 (b) 13 807 (c) 16 216 (d) 18 339 (e) 16 052 (f) 12 332 3. Column A Column B Column C (a) 156 (a) 504 (a) 620 (b) 96 (b) 185 (b) 472 (c) 201 (c) 392 (c) 1281 (d) 410 (d) 144 (d) 1976 (e) 188 (e) 256 (e) 2172 (f) 225 (f) 126 (f) 1948 (g) 602 (g) 957 (g) 168 (h) 344 (h) 380 (h) 1036 (i) 365 (i) 444 (i) 5607 (j) 141 (j) 468 (j) 2892 3. Column A Column B Column C (a) 1560 (a) 5040 (a) 6200 (b) 960 (b) 1850 (b) 4720 (c) 2010 (c) 3920 (c) 12 810 (d) 4100 (d) 1440 (d) 19 760 (e) 1880 (e) 2560 (e) 21 720 (f) 19 480 Challenge Yes—B, C, D, E, F, D, B, A, G, F

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1. (a) 10 (b) 16 (c) 0 (d) 20 (e) 15 (f) 0 (g) 0 (h) 18 (i) 12 (j) 6 2. (a) 1968 (b) 1179 (c) 1854 (d) 751 (e) 1536 (f) 1561 3. Teacher Check 4. Teacher Check Challenge Teacher Check

Unit 1–2

Perimeter Area

16 16 16 16 16 16 16 16 12 16 13 8 15 7 10 7

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(a) Remains the same (b) Changes each time Challenge Teacher check

• Write a set of directions to move around the school.

Consolidation 1–2 • Brainstorm real-life situations in which multiplication could be used.

Consolidation 1–3

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1. (a) 24 (b) 16 (c) 27 (d) 35 (e) 36 (f) 48 (g) 21 (h) 63 (i) 56 (j) 54 2. (a) 8909 (b) 2661 (c) 981 (d) 4344 (e) 9165 (f) 7577 3. Shape A B C D E F G H

• Select a set perimeter and make various shapes which keep the same perimeter but in which the area changes. Record results.

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New Wave Maths Book F – Teachers Guide • 25 •

Unit 2–1

Student page 4

Outcomes

Indicators The student is able to: • write money amounts in decimal form. • understand the multiplicative nature of the relationship between places for whole numbers.

N3.3, N4.3, N4.1a

Skills • converting decimals

Memory Masters (N3.3)

Resources • calculator • coloured pencils

Language • subtract • decimal form • relationship • adjacent • constant • multiply and divide • calendar • different paths • February

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Notes

Number (N4.3)

• 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 (N4.1a) Warm Up

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

• Ask students to write on a piece of paper, or in pads, one dollar; then separately, fifty cents; and one dollar and twenty cents; again five cents. • Ask a number of students how they have recorded their answers. • Direct students thinking to converting cents to dollars or dollars to cents and how they would record their answers. Would there be any change from how the answers had been recorded? If so, what are the changes? Which is the preferred method of recording?

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• Work through the first two or three examples of Exercise 3 with the students. • Students to complete Exercise 3. • Place value is essential knowledge for students in understanding our number system. Exercise 4 assists in developing the understanding that the relationship between places is constant. • Exercise 4 may be completed as a mental activity or with a calculator. Allow students to choose their preferred method. • After students have completed the activity ask them what happened to the original number when they multiplied by 10 and when they divided by 10. In each case the value of the number increased or decreased by one place.

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What to do

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• It is important that students use a different coloured pencil for each path for ease of reviewing. • Start at 1 and pass through every day of the month once only finishing at 28; find as many different paths as possible.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 60 – 61. • 26 • New Wave Maths Book F – Teachers Guide

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

Unit 2–2

Student page 5

Outcomes

Indicators The student is able to: • express measures of length, capacity and mass using common metric prefixes. • display data in bar graphs in which the frequency axis may be scaled with multiples such as 0, 5, 10, 15, .... and grouped measurement data may be treated as categories.

N3.4, N4.3, M4.2, C&D4.3

Skills • measuring • recording data • reasoning • estimating • graphing

Resources

Language • number patterns • subtract • mass • line graph • relationship • axis • axes

• calculator • twenty 10c coins • kitchen scales or similar • ruler

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Memory Masters (N3.4)

Notes

Number (N4.3)

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• The focus for this unit is completion of number patterns some with one pattern others with a second pattern included.

• 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 (M4.2, C&D4.3) Warm up

• Discuss how money is counted in a bank. (It is weighed.) • Find out how they bag 10c pieces. Is it in $1 or $10 bags? Use this information to give a focus to the activity. How much should $10 weigh?

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• Using a suitably sensitive set of scales, students work in small groups to weigh a 10c coin and record its mass on the table provided in the workbook. • Repeat the process weighing the same coin and one more—recording the mass. • Continue the process by adding one more 10c coin to the scales to be weighed each time. • When 14 coins have been weighed and the mass recorded bring the class together to explain how the graph is drawn. • Remind students that all graphs have a title. What is the title of this graph? All graphs have axes showing the content of the graph. In this graph the axes show what? • In constructing the graph we use the table showing the number of coins weighed and their mass to find the information to be plotted on the graph. • One coin weighs ____ grams. Move up the axis showing the number of coins to find 1 and then across the axis showing the mass recorded to find the figure from your weighing. Draw a small dot at this point. • Continue this for two coins, three coins and so on until you have plotted the mass for all weighings from 1 coin to 14 coins. • Use a ruler to join the dots to make a line graph by starting at the one coin dot and finishing at the 14 coin dot. • From what you have found so far, without weighing twenty 10c coins, can you say how much twenty 10c coins will weigh and how you reached the answer?

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Challenge The Counterfeit Coin: You suspect that you have been given a counterfeit coin (one that weighs slightly less than others of the same value). If you have nine 10c coins, explain how you would find the counterfeit coin using a balance and the last number of weighings. For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 102 – 103. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 27 •

Unit 2–3

Student page 6

Outcomes

Indicators

N3.1a, M3.1, N4.3, N4.1a

The student is able to: • use place value to read, write, say and interpret large whole numbers, oral or written.

Skills • problem solving • reasoning • ordering numbers

Resources • calculator • pencil

Language • kilograms and grams • subtract • place value • tens, units, tenths, hundredths, thousandths, ten thousandths • symbols

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Teac he r

Memory Masters (N3.1a, M3.1)

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

• 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 (N4.1a) Warm up

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Number (N4.3)

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• The numbers displayed in the workbook are to be placed in their respective places. • Work through the first couple with the whole class. How many units? Write this in the units column. How many tens? Write this in the tens column. • Repeat with (b). • Alternatively, ask students for the place value of the first digit in the number. Then ask for its face value and write this in the correct place in the workbook. • Complete the activity. • Students then brainstorm situations in real life where rounding is used.

Challenge

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© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Discuss numbers in general terms – the relationship between places. • Ask students to give the name of the places; e.g. what is the place of the 0 in 607? etc.

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• Students may only use the number 3 and the four operation signs to show three different ways of making 7. Remind students that 3/3 is the same as 3 ÷ 3. • Students may work in groups to find ways of writing 7 in this manner. • Share results with the class.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 56 – 57. • 28 • New Wave Maths Book F – Teachers Guide

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

Unit 2—Answers

Student pages 4 – 6 Unit 2–2

1. (a) 7 (b) 6 (c) 1 (d) 8 (e) 2 (f) 7 (g) 3 (h) 9 (i) 5 (j) 8 2. (a) 732 (b) 245 (c) 411 (d) 622 (e) 213 (f) 323 3. (a) 0.25 (g) 0.75 (m)1.55 (s) 8.30 (b) 0.45 (h) 0.90 (n) 4.10 (t) 2.75 (c) 0.40 (i) 0.65 (o) 8.40 (u) 3.00 (d) 0.50 (j) 0.10 (p) 4.20 (v) 2.10 6.20 (w) 4.20 (e) 0.85 (k) 0.30 (q) (f) 0.15 (l) 0.35 (r) 7.25 (x) 9.45 (f) 80 000 4. (a) 8000 (b) 80 000 (g) 8000 (c) 800 000 (h) 800 (d) 8 000 000 (i) 80 (e) 800 000 (j) 8 Challenge Answers will vary (numerous possibilities)

1. (a) 31 (b) 43 (c) 93 (d) 530 (e) 66 (f) 12 (g) 5 (h) 12 (i) 95 (j) 319 2. (a) 614 (b) 517 (c) 329 (d) 419 (e) 324 (f) 539 3. Number of 10c coins 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mass of coins (grams)

Relationship Between Coins and Their Mass

4. 100 g 5. Answers will vary Challenge Answers will vary

Number of Coins 5 6 7 8 9

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Unit 2–1

Total Mass in Grams

(a) (b) (c) (d)

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(e)

(g) 1

Tens

Units

(i) (j) (k)

• Brainstorm specific situations where money is used in real-life.

Consolidation 2–2 • Complete the activity using $1 coins. Students draw up their own tables and graphs.

Consolidation 2–3

• Ask students one by one to call out a number. Record them on the board. Students write them on the place value chart found on pages 207 – 208.

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(f)

(h)

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1. (a) 4.25 kg (b) 8.17 kg (e) 1.24 kg (f) 6700 g (i) 1790 g (j) 2510 g 2. (a) 457 (b) 179 (e) 176 (f) 449 3. Thousands Hundreds

6 1

(l)

4. Answers will vary Challenge Answers will vary R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book F – Teachers Guide • 29 •

Unit 3–1

Student page 7

Outcomes

Indicators

N3.1a, N4.3, S4.1

The student is able to: • give unambiguous instructions for moving and locating objects in their environment or on models, maps or plans using distance, direction and common map grids.

Skills • mapping • locating • recording • interpreting

Resources • calculator • Base 10 MAB • Atlas

Language • multiply • locate • latitude, longitude • accurate • furthest • east, south • Tropic of Capricorn

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Notes

Teac he r

Memory Masters (N3.1a)

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

• 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 (S4.1) Warm up

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Number (N4.3)

• Have students open their atlas to a map of Australia. Discuss latitude—distance measured in degrees from the Equator—and longitude—distance measured in degrees (meridian at Greenwich in England). Explain the use of latitude and longitude to locate places and, or features on a map, including extension of the grid lines to provide a more accurate location for places that are not right on a grid line. For example, Adelaide lies on the 35 ºS latitude but is between 135 ºE and 140 ºE longitude. Its location is fairly close to 138 ºE.

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• Encourage students to work out the location of each of the 10 cities shown. • After giving the locations, ask students to use their atlas index to find the true locations of the 10 cities and compare the accuracy of their own determinations. • Using the map, complete the questions of page 7 of the workbook.

Challenge

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What to do

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• Use the Internet to find out the relationship between longitude and time zones. How many different time zones are there in Australia?

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 2 – 3. • 30 • New Wave Maths Book F – Teachers Guide

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

Unit 3–2

Student page 8

Outcomes

Indicators

N3.3, N4.3, N4.2

The student is able to: • generate missing numbers which obey a constraint.

Skills

Resources

Language

• calculator • 2-cm cubes • 1-cm grid paper (see page 199)

• multiply • factor • product • factor pairs • network • point

• calculating • recording

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Notes

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Memory Masters (N3.3)

• The focus for this unit is addition of two digits each less than 20.

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Number (N4.3)

• 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 (N4.2) Warm up

e.g. 1 x 10 ,2 x 5 and so on. If an array is not formed then the numbers are not factor pairs. • Students may be set to work to complete the factor pairs.

Challenge

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• Working with the whole class explain what a factor is. Use the example on page 8 of the workbook showing factor pairs of 18. • Ask students to give oral answers to finding factor pairs of 6, 14, 24 and 32. Record their answers on the blackboard/whiteboard for all to see. • Some students may have difficulties finding factors and may find the following method helpful. Use 1-cm grid paper and draw an array, showing factors:

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• Students select a starting point of their own choosing on the network. • The pencil may not be lifted from the network. • No line is to be retraced. • The completed tracing must finish at the starting point.

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New Wave Maths Book F – Teachers Guide • 31 •

Unit 3–3

Student page 9

Outcomes

Indicators

N3.3, N4.3, M4.2, C&D4.3, C&D4.4

The student is able to: • compare and order length, capacity and mass measurements. • represent data in diagrams and tables. • interpret and report on information provided in tables and bar graphs.

Skills • recording • modelling • measuring • reasoning

Memory Masters (N3.3)

Resources • calculator • 2-cm cubes

Language • patterns • measurement • model • length, width, height, volume • table • double, triple, quadruple, quintuple • increasing dimensions

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Number (N4.3)

• 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 (M4.2, C&D4.3, C&D4.4) Warm up

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• The focus for this unit is subtraction of two numbers each less than 20.

• Explain to the whole class that they will be building models using 2-cm cubes following given directions. • Sort the class into small groups and distribute the 2-cm cubes. Allow a few minutes to build their own models with the cubes. • Direct each group to make a basic model with two cubes so that adjoining faces meet full on. See illustrations shown in workbook.

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• Ask students to record the length, width, height and volume of the model on the table on page 9 of their workbook. • Retain this model. Direct students to construct a model with all dimensions doubled. Check with students to ensure they understand what doubling dimensions means; e.g. 4 x 4 x 4. • When the model is made record results on the table in the workbook. • Repeat for a triple scale model of the original model; e.g. 6 x 6 x 6. • Repeat for a model four times the size of the original model; e.g. 8 x 8 x 8. • Repeat for a model five times the size of the original model; e.g. 10 x 10 x 10. • Use the results to see if any patterns can be found in the length or width or height or volume results recorded to date.

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What to do

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• Ask students for ideas as to how a tennis tournament schedule of matches may be displayed. • Without directing students towards a particular schedule, ask them to draw up their own schedule to determine the number of matches played in the tournament to determine the winner.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 146 – 147. • 32 • New Wave Maths Book F – Teachers Guide

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

Unit 3—Answers

Student pages 7 – 9 Unit 3–1

1. (a) 350c (b) 520c (c) 610c (d) 290c (e) 460c (f) $1.60 (g) $5.20 (h) $3.80 (i) $6.10 (j) $4.70 2. (a) 86 (b) 260 (c) 623 (d) 512 (e) 438 (f) 204 3. 35º

139º

24º

134º

Unit 3–2 1. (a) 12 (b) 14 (c) 8 (d) 19 (e) 21 (f) 16 (g) 13 (h) 14 (i) 15 (j) 14 2. (a) 5560 (b) 5292 (c) 1592 (d) 4125 (e) 2778 (f) 3311 3. (a) 10 x 1, 5 x 2 (b) 12 x 1, 6 x 2, 4 x 3 (c) 15 x 1, 5 x 3 (d) 20 x 1, 10 x 2, 5 x 4 (e) 30 x 1, 15 x 2, 10 x 3, 6 x 5 (f) 36 x 1, 18 x 2, 12 x 3, 9 x 4, 6 x 6 (g) 40 x 1, 20 x 2, 10 x 4, 8 x 5 (h) 45 x 1, 15 x 3, 9 x 5 (i) 50 x 1, 25 x 2, 10 x 5 (j) 60 x 1, 30 x 2, 20 x 3, 15 x 4, 12 x 5, 10 x 6 (k) 68 x 1, 32 x 2, 17 x 4 Challenge No

38º

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32º

116º

34º

151º

27º

153º

17º

146º

149º

35º 12º

131º 147º

145º

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43º

4. (a) Brisbane (b) No (c) Brisbane, Perth, Sydney, Adelaide, Canberra, Melbourne, Hobart, Alice Springs

1 2

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• Brainstorm other cities in Australia and locate them on the map in the workbook.

Consolidation 3–2

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1. (a) 3 (b) 2 (c) 7 (d) 9 (e) 3 (f) 8 (g) 5 (h) 7 (i) 9 (j) 2 2. (a) 1920 (b) 1840 (c) 6020 (d) 1850 (e) 3480 (f) 2350 3.

• Brainstorm numbers related to the students such as birthdays, ages, number of family members– write the factor pairs for them.

Consolidation 3–3

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128

250

• Make models of everyday objects, double or triple their original size.

4. Answers will vary Challenge 31 with a knockout tournament

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

New Wave Maths Book F – Teachers Guide • 33 •

Unit 4–1

Student page 10

Outcomes

Indicators

N3.3, N4.3, N4.4

The student is able to: • partition double-digit numbers in order to mentally multiply and divide by single-digit numbers. • follow a rule based on multiplication, division or simple fractions to generate a sequence.

Skills • working mentally • reasoning • following a pattern • problem solving • speaking and listening

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB • pencil • ruler

Language • divide • pattern • sequence • horizontally, vertically

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Notes

Number (N4.3)

• 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 (N4.3, N4.4) Warm Up

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Teac he r

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

• Explain to the class that you are going to show them a simple means of multiplying by 25 but they will need to explain to you how and why it works. • Give students the example 4 x 25. Ask what the answer is. Ask students to divide 4 by 4. What is the answer? (1) Multiply this by 100 and what do you get? (100—the same answer as 4 x 25.) • Try again, divide 8 by 4 and what is your amount? (2) Multiply this by 100 and you now have 200. What is 8 x 25? (200) Use calculators if necessary.

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• Direct students to work through the examples starting with 12 x 25. Remind students to divide the number 25 is being multiplied by, by 4, then multiply their answer by 100. • Point out to students that it only applies to multiples of 4. This method can be extended to find the answer to other numbers multiplied by 25. Still divide by 4, if a remainder of 1 is found place 25 to the right of the whole number answer; e.g. 17 x 25, 4 r1, 425. If the remainder is 2, write 50 to the right of the whole number answer, and if the remainder is 3 write 75 to the right of the whole number answer.This exercise may be left for students to discover for themselves. • How could you use the information you have discovered to multiply 13, 18, 23 and 19 by 25 without using a calculator?

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• Students start at X and traverse squares either horizontally or vertically only to see if they are able to pass through each square once only.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 74 – 75. • 34 • New Wave Maths Book F – Teachers Guide

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

Unit 4–2

Student page 11

Outcomes N3.3, N4.3, C&D4.2, C&D4.3, C&D4.4

Skills

• gathering information • working cooperatively • recording • graphing • reasoning • reading a graph

Indicators

Resources

Language

The student is able to: • suggest what data to collect to help estimate numbers or quantities and represent these data in tables and diagrams. • display data in bar graphs in which the frequency axis may be scaled in multiples of 0, 5, 10 … and interpret and report on information provided in line graphs.

• calculator • classmates’ clothing • coloured pencils

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• divide • gather information • construct • bar graph • tally • total • least • diagram • adjacent

Notes

Memory Masters (N3.3) Number (N4.3)

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Teac he r

• The focus for this unit is basic facts of 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&D4.2, C&D4.3, C&D4.4) Warm up

• Explain to the class that in society many surveys are conducted to collect data for use in marketing, census details and many other purposes. • The activity today will collect information regarding types of clothing worn by students in your class. The information will be collected by me and you will be recording it as a tally, converting the tally to a total then representing the data in a bar graph. You will then interpret the data to determine the type of clothing that is most commonly being worn and give reasons for this.

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• Remind students how to keep a tally – four vertical strokes followed by a diagonal stroke to keep groups of five ( IIII ). • Progressively ask students to stand to show the clothing they are wearing as indicated in the table in the workbook. Students can count and keep a tally to record in their workbooks. Allow sufficient time for recording. • Once all clothing types have been tallied, ask students to write the total for each type of clothing. • Ask students to transfer this total to the correct column on the graph sheet so they can construct a bar graph. Students may need reminding to rule the top line of the bar level with the correct number on the vertical axis. • Finally ask student to complete the interpretation questions on Exercise 4.

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Challenge • Students may use trial and error or educated reasoning to find the least number of colours required to colour the shapes so that no two colours share the same line (are adjacent). They may share a common corner.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 154 – 155. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 35 •

Unit 4–3

Student page 12

Outcomes

Indicators The student is able to: • use place value to read, write, say and interpret large whole numbers, oral or written.

N4.3, N4.1a

Skills • rounding • speaking and listening • reasoning • problem solving

Language

• calculator • Base 10 MAB • place value chart (see pages 207 – 208) • toothpicks or equivalent

• divide • round • approximately equal • tenths, hundredths, thousandths

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• The focus for this unit is to encourage students to find ways to simplify number problems and to support their approach; e.g: If you had one wish and could change one digit in the following question, which one would you change? Explain why. 14 x 6 I would change the 6 to a 5 because it is easier to multiply by fives.

Number (N4.3)

• 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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Memory Masters (N4.3)

Resources

Warm up

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• Direct students to look at a large class place value chart or a small personal chart. • Ask a student or the class to point to various place value columns – ones, hundredths, hundreds, tenths, thousandths, tens and a repeat of these if felt necessary. • Students may be asked to recite the place value column names from smallest to largest as shown on the chart. • Ask students for the rules of rounding. Repeat then to the class – numbers 1 to 4 round down numbers 5 to 9 round up • These rules apply regardless of place value of the number being rounded.

What to do

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Main Activity (N4.1a)

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• Work with the class as a whole on the first two or three numbers. Remind class to focus on the number immediately to the right of the place value to be rounded to. Only when 5 or 0 are in that place value column do students need to check the next column to the right. • Students complete the exercise with careful checking by the teacher.

Challenge • Students should use their mathematical logic to determine which toothpicks to move. • Remind students that the final arrangement must show only three squares – no other shapes are permitted.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 58 – 59. • 36 • New Wave Maths Book F – Teachers Guide

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

Unit 4—Answers

Student pages 10 – 12

Unit 4–1

1. (a) 4 (b) 3 (c) 6 (d) 9 (e) 8 (f) 7 (g) 9 (h) 3 (i) 5 (j) 7 2. (a) 211 (b) 131 (c) 243 (d) 212 (e) 134 (f) 221 3. Teacher check 4. (a) Teacher check (b) Teacher check (c) Teacher check Challenge 2

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1. (a) 36 (b) 32 (c) 72 (d) 45 (e) 42 (f) 81 (g) 48 (h) 54 (i) 49 (j) 36 2. (a) 42 (b) 42 (c) 53 (d) 84 (f) 62 (e) 61 3. (a) 300 Yes (b) 600 (c) 700 (d) 800 (e) 900 (f) 1000 (g) 1200 (h) 1600 (i) 1800 (j) 100 ÷ 25 = 4; therefore dividing by 4 and multiplying by 100 is the same as multiplying by 25. 4. (a)

Unit 4–2

(b) Each row increases by 2 more triangles. Challenge Yes

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• Ask students to develop a method for multiplying by 50.

Consolidation 4–2 • Students gather and record information from classmates about a topic of choice.

Consolidation 4–3

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1. Answers will vary. Some students will see that if they change the 9 to a 10, the problem will be much simpler to calculate. For example: 18 x 9 = ? But: 18 x 10 = 180 2. (a) 25r3 (b) 32r5 (c) 52r2 (d) 65r6 (e) 59r1 (f) 47r2 3. Tenths Hundredths Thousandths (a) 4.6836 ≈ 4.7 ≈ 4.68 ≈ 4.684 (b) 2.1752 ≈ 2.2 ≈ 2.18 ≈ 2.175 (c) 7.8397 ≈ 7.8 ≈ 7.84 ≈ 7.840 (d) 1.4274 ≈ 1.4 ≈ 1.43 ≈ 1.427 (e) 9.5623 ≈ 9.6 ≈ 9.56 ≈ 9.562 (f) 8.8112 ≈ 8.8 ≈ 8.81 ≈ 8.811 (g) 0.3900 ≈ 0.4 ≈ 0.39 ≈ 0.390 (h) 5.0068 ≈ 5.0 ≈ 5.01 ≈ 5.007 (i) 6.0185 ≈ 6.0 ≈ 6.02 ≈ 6.019 (j) 3.5000 ≈ 3.5 ≈ 3.50 ≈ 3.500 Challenge

• Study various symbols used in mathematics – develop a chart for a class reference.

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New Wave Maths Book F – Teachers Guide • 37 •

Unit 5–1

Student page 13

Outcomes

Indicators The student is able to: • make a model of a cube and pyramid using a net.

N4.1a, N4.3, S4.2

Skills • modelling • drawing nets

Resources • calculator • scissors • glue • tape • pencil • paper • light card • heavy paper • overhead projector

Language • greater • nets • shapes • variations • constructed • cube

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Notes

• The focus for this unit is the identification of the greater of a pair of decimal numbers. • Discuss misconceptions with students; e.g. some students may decide 101 is smaller than 8.3764 since it is made up of fewer numbers.

Number (N4.3)

• 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 (S4.2)

• Explain to students using a 3-D model that the model can be represented in 2-D as a net. To assist, draw the sides of the 3-D model on an overhead or blackboard/whiteboard around the base shape so students can see the net as a graphic representation of the model as a 2-D net. • Students could also cut up a 3-D object to discover the net. • Distribute heavy paper or light card to students and ask them to have scissors and glue or tape ready.

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What to do

• Students either copy the nets from their workbook (page 13) onto the paper/card or use copies provided by the teacher. • Cut around the net. Fold along the dotted lines gluing/taping the tabs under or over the adjoining face of the model. • Set model aside to dry and complete the second model. • Remember that when cut out and folded all faces are to be covered, no face is to be open and no face is to have more than one cover on it. • Ask students to share their nets with their neighbouring students. • Ask for different variations from the class, display these for the class to see and share.

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Warm up

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Memory Masters (N4.1a)

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Challenge • Once both models have been made, draw on a sheet of paper as many variations of nets that can be constructed into a cube as you are able to.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 28 – 29. • 38 • New Wave Maths Book F – Teachers Guide

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

Unit 5–2

Student page 14

Outcomes

Indicators

N3.1a, N4.3

The student is able to: • partition double-digit numbers in order to mentally add by small single-digit numbers.

Skills • working mentally • reasoning

Resources

Language

• calculator • pencil

• round • nearest • hundred • intersection

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Notes

Teac he r

Memory Masters (N3.1a)

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

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Number (N4.3)

• 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 (N4.3) Warm up

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• What to do • Explain to the class that we use different methods to work out sums mentally. Ask students to find the answers to these sums mentally. When answers have been found, ask students how they reached their answers. Ask for different methods used to those given. 24 + 47 86 + 59 79 + 48.

Challenge

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• Work through the examples shown on page 14 of the workbook with the whole class. • Encourage students to apply the methods they used in the previous examples to see if they work with these examples to produce the correct answers. • Acknowledge that there are many ways that answers can be found and that no one method is more correct than another. Explain that for convenience when working out written sums we start addition and subtraction from the right and move across to the left. This allows regrouping to be displayed in written form. Stress it is not the only method of working. • Students work through the exercise in their workbook. Provide assistance as required.

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• This exercise is a different path following activity. There are three concentric circles joined by three intermediary lines.This creates intersection points between the circles and these lines. • The task is to start at point X and follow a path along a circle or across a joining line so that the path passes through each intersection once only and finishes back at point X.The path does not have to follow all of each circle or cross each joining line. • Number each intersection as you pass through it.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 78 – 79. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 39 •

Unit 5–3

Student page 15

Outcomes

Indicators The student is able to: • given a rectangle with a whole number length sides, explain why multiplying the length by the height gives the area. • multiply and divide measurements and amounts of money by a onedigit number.

N3.3, N4.3, M4.4a

Skills • measuring • calculating • reasoning

Memory Masters (N3.3)

Resources

Language • cost • tiling • square • metre • cover • procedure • arrange • hexagon • total

• calculator • pencil

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

Number (N4.3)

• 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 (M4.4a, N3.3) Warm up

• Revise the concept of area. Ask students how they can find the area of a rectangular shape if they know its length and width. • Focus on multiplying length by width. If required, use simple arrays to reinforce this concept: – 2 x 3

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Teac he r

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

etc.

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• Read through the activity on the page. • Ask students what they will need to find and how they will obtain their answers. • First thing to find is the area which is found by ... ? (Student response of multiplying length by width (3.25 m x 2.65 m) is required.) • Tell students that they should use their calculators for all calculations in this activity. • Once the area has been found what next? (Multiply by $23 to find the cost of the tiles.) • Exercise (b) requires students to find the number of tiles required. This is done by ... ? (Multiply area by 16.) • Exercise (c) requires students to multiply the area by $12 to find the cost of laying the tiles.

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Challenge

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What to do

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• Students are to use the numbers 1 – 12 to complete the hexagon so that each side of the hexagon totals 22. • Remind students that there are two numbers on each line that are common to the two lines adjacent to that line. • Show all working out.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 136 – 137. • 40 • New Wave Maths Book F – Teachers Guide

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

Unit 5—Answers

Student pages 13 – 15 Unit 5–1

1. (a) 700 (b) 900 (c) 600 (d) 400 (e) 500 (f) 700 (g) 300 (h) 500 (i) 700 (j) 900 2. (a) 15 609 (b) 20 223 (c) 18 984 (d) 15 637 (e) 27 207 (f) 16 821 3. (a) 30 + 20 + 4 + 7 = 50 + 11 = 61 (b) 60 + 50 + 2 + 6 = 110 + 8 = 118 (c) 30 + 40 + 8 + 9 = 70 + 17 = 87 (d) 20 + 40 + 6 + 7 = 60 + 13 = 73 (e) 50 + 60 + 9 + 6 = 110 + 15 = 125 (f) 40 + 80 + 7 + 1 = 120 + 8 = 128 (g) 20 + 90 + 8 + 8 = 110 + 16 = 126 (h) 70 + 50 + 4 + 8 = 120 + 12 = 132 (i) 60 + 80 + 5 + 9 = 140 + 14 = 154 (j) 50 + 70 + 6 + 6 = 120 + 12 = 132 Challenge

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1. (a) 8.279 (b) 63 (c) 87.21 (d) 47.684 (e) 59 873 (f) 0.0075 (g) 0.0841 (h) 3.8214 (i) 76.1 (j) 101 2. (a) 2306 (b) 2125 (c) 1646 (d) 1909 (e) 2253 (f) 2844 3. Teacher check Challenge Teacher check

Unit 5–2

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• Pull apart various boxes to discover the different nets used. • Design a box to package a kite.

Consolidation 5–2

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1. (a) 12 (b) 15 (c) 11 (d) 17 (e) 10 (f) 13 (g) 14 (h) 13 (i) 10 (j) 9 2. (a) 8645 (b) 9985 (c) 11 233 (d) 861 (e) 15 283 (f) 3151 3. (a) $198.10 (b) 138 tiles 1. Work out the area in square metres. 2. Multiply by the number of tiles (16) per square metre. 3. Round up to the nearest whole tile. (c) $103.35 1. Work out the area in square metres. 2. Multiply by the cost ($12) per square metre. Challenge Answers will vary. Possible solution:

• Discuss and evaluate the various methods which can be used to complete the sums.

Consolidation 5–3

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• Ask students to measure the floor in the girls’ or boys’ toilets, as the school is going to retile these areas. Work out the area and cost of such a project if the tiles cost $45 per metre2 and laying costs $20 per metre2.

New Wave Maths Book F – Teachers Guide • 41 •

Unit 6–1

Student page 16

Outcomes

Indicators

Resources

The student is able to: • generate missing numbers which obey a constraint.

N3.3, N4.3, N4.2

Skills

• calculator • Base 10 MAB

• calculating • recording

Memory Masters (N3.3)

Language • subtract • multiplying • prime numbers • prime factors • product • factor tree • overlapping • enclosed • triangle, square, circle

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

Notes

Number (N4.3)

• 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 (N4.2) Warm Up

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Teac he r

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

What to do

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• Work through the prime factor examples for 24 with the class. Carefully explain each step of finding the factors of 24 such as 4 and 6, then finding the factors of 4 and 6 which are 2, 2, 2 and 3 (note: these are all prime numbers). Rework this example using 3 and 8 and/ or 2 and 12 as the first set of factors and expand the factor tree. This shows students that there are several starting points but they all provide the same answer. 24 24

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• Revise prime numbers and factors. Ask students what a prime number is. • A prime number is a number that has two factors only, 1 and itself. • Ask students what a factor is. • A number when multiplied with another number provides a multiple. • From this, ask students what a prime factor might be. • A prime factor is a prime number that will divide exactly into another number; e.g. 2 and 3 are prime factors of 6.

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4 2 2 6

2 2 2 3

Prime factors 2 x 2 x 2 x 3 Prime factors 2 x 2 x 2 x 3 • Students complete factor trees with the teacher. • Students work through prime factor examples together with teacher direction until the teacher is satisfied that the students can continue by themselves. • Encourage the use of the factor tree.

Challenge • This exercise requires students to follow directions to draw the shapes given.

• 42 • New Wave Maths Book F – Teachers Guide

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

Unit 6–2

Student page 17

Outcomes

Indicators The student is able to: • suggest what data to collect to help estimate numbers or quantities. • represent data in diagrams and tables. • interpret and report on information provided in tables and bar graphs.

N3.3, N4.3, C&D4.2, C&D4.3, C&D4.4

Skills • gathering data • tallying • reasoning • inferring

Resources

Language

• calculator • TV program guide • class members • coloured pencils

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

• subtract • tally • total • inference • data • collected • least • diagram • adjacent

Notes

Memory Masters (N3.3) Number (N4.3)

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Teac he r

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

• 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&D4.2, C&D4.3, C&D4.4) Warm up

• Ask class about their favourite TV programs, and why they like the shows.

What to do

Challenge

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• Direct students to select 10 different TV programs that are screened between 6.00 p.m. and 8.00 p.m. • To save time a consensus opinion of 10 programs selected by the teacher may be listed on the page in the workbook. Alternatively, students select their own and work in a controlled manner around the class questioning every other student in the class.This provides positive interaction and gives a better account of how a survey is conducted. If choosing a generic set of programs, a quick ‘hands up’ tally can be conducted. This saves time but does not expose students to the intricacies of data collection. • Once all data are collected, a total for each program is made. • Students then reflect on the information they have collected and see if they can make some inferences on which programs are watched by class members and which are not. The reasons why programs are watched may require further investigation by questioning a random sample of class members.

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• Students are to find the least number of colours required to colour the diagram so that the same colours are not adjacent.The same colour may have a common intersection but not a shared line.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 156 – 157. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 43 •

Unit 6–3

Student page 18

Outcomes

Indicators

N3.3, N4.3, WM4.2, WM4.3, WM4.4

The student is able to: • make organised lists or tables of data collected to help solve a problem. • collect data to assist in making and testing a conjecture about numbers. • draw on his/her mathematical knowledge to check the reasonableness of an answer.

Skills • reasoning • problem solving • explaining

Teac he r

Memory Masters (N3.3)

Resources • calculator

Language • subtract • sum • prove • individual digits • equals • multiple • original number • divisible • reasoning • conclusion

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

• The focus for this unit is basic facts of 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 (WM4.2, WM4.3, WM4.4) Warm up

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Number (N4.3)

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• Students have been set a problem to find an answer to. They are required to apply their mathematical knowledge, reasoning and cross check to verify their findings. • Read the problem to the students. • Ask students to provide their explanation of what the problem is asking. • Provide an example to show the basis of the problem; e.g. 63, the sum of the digits is 9. Nine is a multiple of 3. Use your calculator to check if 63 is divisible by 3. (Yes) Try 87, the sum of 8 and 7 is 15 (the digits may be totalled again 5 + 1 is 6). 15 is a multiple of 3, use your calculator to check to see if 87 is divisible by 3. (Yes) • Try 1482. Sum of the digits is 15. From the previous example we know that 15 is a multiple of 3. Use your calculator to see if 1482 is divisible by 3. (Yes) • Use your own numbers to check to see if this rule is true for other numbers where digits added total a multiple of 3. • For students unsure on what numbers to use provide them with these – 45, 72, 84, 111, 204, 4701. • Have students share their findings in small groups and a select number share them with the class.

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Challenge • Add the bir th date of students to see if it is a multiple of 3. For example, 14/08/1995 = 1 + 4 + 8 + 1 + 9 + 9 + 5 = 37 = 3 + 7 = 10 = 1 + 0 = 1 – this number is not a multiple of 3.

• 44 • New Wave Maths Book F – Teachers Guide

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

Unit 6—Answers

Student pages 16 – 18 Unit 6–2

Unit 6–1 1. (a) 7 (b) 8 (c) 6 (f) 6 (g) 5 (h) 2 2. (a) 433 (b) 105 (c) 316 (e) 522 (f) 405 16 3. (a) (b)

8 4 2

10 5

3 5

1. (a) 64 (b) 63 (c) 42 (d) 40 (e) 18 (f) 25 (g) 56 (h) 72 (i) 35 (j) 24 347 (c) 271 (d) 432 2. (a) 474 (b) (e) 186 (f) 253 3. Teacher check 4. Teacher check Challenge 2

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4. (a) 2 x 3 (b) 2 x 2 x 2 (e) 3 x 5 (d) 2 x 2 x 3 (g) 2 x 2 x 5 (h) 3 x 7 (j) 2 x 2 x 2 x 2 x 2 Challenge Answers will vary but should resemble this shape.

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(d)

10 2

(d) 4 (e) 8 (i) 6 (j) 1 (d) 254

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• Use real-life numbers which are relevant to the students—e.g. house number, age, birthday etc.—to compare factor trees.

Consolidation 6–2

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1. (a) 8 (b) 4 (c) 6 (d) 4 (e) 5 (g) 9 (h) 4 3 (f) 8 (i) 5 (j) 2. (a) 364 (b) 653 (c) 167 (d) 265 (e) 301 (f) 183 3. Teacher Check

• Complete a tally for weekday and weekend television viewing and compare the results.

Consolidation 6–3

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• Discuss, debate and support any findings.

New Wave Maths Book F – Teachers Guide • 45 •

Unit 7–1

Student page 19

Outcomes

Indicators

N3.3, N4.3, S4.1

The student is able to: • draw maps and plans which show a sense of scale.

Skills • scale drawing

Teac he r

Memory Masters (N3.3)

Resources • calculator • ruler • pencil • tape • metre rule

Language • multiply • scale diagram • bird’s-eye view • position

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

Notes

• The focus for this unit is 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 (S4.1) Warm up

• Ask the class what they would see if they were a bird flying in the sky and looking directly down at the ground. • If the roof were lifted from this classroom and you were a bird flying directly above, what would you see? • You are to pretend that you are the bird and draw a scale plan of this classroom showing where all the furniture is placed in the room. • Scale will need to be discussed.

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Number (N4.3)

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• A grid is provided for your plan of the classroom. The first thing that needs to be done is to make a decision on the scale to be used. How long and wide is the grid on the page? (19 cm x 16 cm). How long and wide is the classroom? (Answers will vary but in the range of 7 m to 9 m as a rule.) • What is the largest scale that can be used to obtain the biggest plan of the class? Most likely 1 cm = 50 cm. • Ask students how they can make an accurate plan. Need to measure furniture and its distance from or along walls and then transfer to the page. • Set class to work. • Display finished plans.

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• Architects sometimes use a special ruler when working with scale drawings. Design a ruler that will help you to easily convert cm to m.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 14 – 15. • 46 • New Wave Maths Book F – Teachers Guide

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

Unit 7–2

Student page 20

Outcomes

Indicators

N3.1a, M3.1, N4.3

The student is able to: • add and subtract money and measures with equal number of decimal places. • multiply and divide measurements and amounts of money by a one-digit number.

Skills • reasoning • problem solving • calculating mentally

Resources

Language • multiply • addition • problems • mentally • tangram • rectangle

• calculator • cuisenaire rods • fraction cake • grid paper

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

Notes

Teac he r

Memory Masters (N3.1a, M3.1)

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

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Number (N4.3)

• 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 (N4.3) Warm up

© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Discuss with the students what would be required to set up a lemonade stand. (Purchase required materials, prepare lemonade, advertising etc.)

Challenge

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• Students use all the tangram pieces to construct a large rectangle. • Draw a diagram to show how the pieces were arranged to make the rectangle.

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• Students can work in groups of four for this activity. • Read through the task step by step. • Set students to work to solve the problems and record their answers on the page. • Answers will vary according to each group’s expectations of the amount of lemonade they expect to sell each day. Remember, people are thirstier when it’s hot. • Share ideas with the whole class.

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New Wave Maths Book F – Teachers Guide • 47 •

Unit 7–3

Student page 21

Outcomes N3.1, N4.3, M4.2, S4.4

Skills • measuring • reasoning

Memory Masters (N3.1)

Indicators

Resources

Language

The student is able to: • find or make different things with one measurement the same but another different. • make figures and objects which meet criteria related to sides, faces, angles and edges.

• calculator • hexagon • L-shaped hexagon • square • rectangle

• divide • straight-sided shapes • perimeter • area • patterns

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

Number (N4.3)

• 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 (M4.2, S4.4) Warm up

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

• Explain to students that we can have regular and irregular shapes. • Regular shape: a polygon is regular if all its sides and angles are congruent. • Irregular shape: a polygon is irregular if all its sides and angles are not congruent. • Irregular shapes may be greatly modified compared to their regular counterparts. For example, a hexagon can become an L shape (six sides). Compare this shape with a regular hexagon. Show examples of both shapes.

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• Your task today is to draw as many straight-sided shapes (polygons – polygon means many angles) that have a fixed perimeter of 12 centimetres as you can. • You will also need to find the area of these shapes. • Record the shapes in your workbook. • Keep a record of the perimeter, area and number of sides each shape has. • See if any patterns develop. • What happens to the area? What happens to the perimeter? • Share your shapes with a small group. • Select some for class display.

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• Keep records of all attempts as you write the numbers 1 – 7 inclusive on the diagram so that each line totals 12. • Share your work with the class or teacher.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 132 – 133. • 48 • New Wave Maths Book F – Teachers Guide

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

Unit 7—Answers

Student pages 19 – 21 Unit 7–1

1. (a) 2.567 L (b) 8.062 L (c) 1.405 L (d) 0.927 L (e) 0.391 L (f) 800 mL (g) 900 mL (h) 30 mL (i) 70 mL (j) 1 mL 2. (a) 2072 (b) 4067 (c) 3551 (d) 4816 (e) 4992 (f) 3384 3. Answers will vary Challenge Teacher check

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1. Teacher check 2. (a) 1350 (b) 4230 (c) 5040 (d) 4760 (e) 2360 (f) 1800 3. Teacher check

Unit 7–2

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• Students draw a bird’s-eye view of their bedroom or school.

Consolidation 7–2 • Discuss and evaluate any problems or variables that may affect the sales of the lemonade stand; e.g. weather.

Consolidation 7–3

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1. (a) $2.96 (b) $5.74 (c) $0.82 (d) $48.00 (e) $7.91 (f) 684c (g) 2700c (h) 47c (i) 6c (j) 10c 2. (a) 4032 (b) 4165 (c) 3848 (d) 7315 (e) 2166 (f) 3600 3. Teaher check 3. Answers will vary Challenge Possible solution

• Discuss: The smaller the fixed perimeter, the fewer the number of shapes which can be made. What would be the ideal perimeter?

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New Wave Maths Book F – Teachers Guide • 49 •

Unit 8–1

Student page 22

Outcomes

Indicators The student is able to: • partition double-digit numbers in order to mentally multiply and divide by single-digit numbers.

N4.3

Skills • working mentally

Memory Masters (N4.3)

Resources

• divide • regrouping • partition strategies • multiples • multiplier • arrange • straight line • extend • multiply

• calculator • counters

Teac he r

Number (N4.3)

• 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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• Ask the class to mentally work out the following; after each example ask students to explain how they reached their answers. 9 x 4 22 x 3 43 x 2 74 x 2 • Focus on one technique, that of partitioning two- or three-digit numbers in preparation for multiplying by a single-digit number. For example, 43 x 2 = (40 x 23 x 2) + (3 x 2) = 6 + 80 = 86 • Repeat this example using 22 x 3 and 74 x 2. • Ask students to demonstrate on the blackboard/whiteboard how they partition then work out the answer.

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Main Activity (N4.3)

What to do

Notes

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• The ‘Today’s number is .... ’ activity asks 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 start on Exercise 3. Check working carefully. If need be work through several more examples with students as a whole or as part of a small group needing assistance. • When Exercise 3 is complete, ask students to think about how they would complete Exercise 4. • Explain that the process is the same except that when multiplying by a multiple of 10 each answer will end in a zero. Perhaps use calculators to complete these sums to assist: 21 x 4 21 x 40 34 x 2 34 x 20 41 x 6 41 x 60 251 x 4 251 x 40 • Students complete Exercise 4 with help as required.

Challenge • Students use their preferred methods to solve the puzzle, which isn’t as straightforward as it appears. For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 82 – 83. • 50 • New Wave Maths Book F – Teachers Guide

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Unit 8–2

Student page 23

Outcomes

Indicators The student is able to: • represent data in diagrams and tables.

N3.3, N4.3, C&D4.3

Skills • working logically • tallying • reasoning • surveying • reflecting

Resources

Language

• calculator • classmates • coloured pencils

• divide • Caroll diagram • summarise • adjacent

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

Notes

Memory Masters (N3.3)

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Teac he r

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

Number (N4.3)

• 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&D4.3) Warm up

• The exercise requires students to solve a logic problem. Students will need to understand how they solve this type of problem.

What to do

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• Explain to the class that a series of clues has been given about the sports that people play. Using the clues, it is possible to show on the Caroll diagram whether the people do or do not participate in a particular sport. • When a clue indicates that one person is involved in a sport, put a dot in the box opposite the correct name and under the identified sport. It is now possible to delete (by putting a cross in the boxes above, below and beside the box with a dot) the rest of the people and sports from this row and column. • If a clue clearly indicates that a person does not participate in a particular sport, a cross is placed in the box opposite the person and under the given sport. • Continue in this way until all boxes are filled and each person has an identified sport next to him/her. • Students are then provided the opportunity to explain how they solved the puzzle, encouraging a reflection of problem-solving strategies.

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• Find the least number of colours required to colour the shape following the given rules. • Share results with the class.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 162 – 163. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 51 •

Unit 8–3

Student page 24

Outcomes

Indicators

WM4.2, C&D4.2, S4.4, S4.2, M4.1, M4.3, N4.2

The student is able to: • pose mathematical questions prompted by a specific stimulus or familiar contexts. • ask organising questions to get him or her started. • contribute questions in a brainstorming situation.

Skills • reasoning • taking risks • problem-solving • estimating • recording data • working mentally • speaking and listening • collaborative learning and working

Resources • calculator • pencil and paper • the Internet (if available)

Language • data • plan • classify • organise • questions • brainstorm • estimate • round • category

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What to do

• This activity is designed for students working collaboratively in groups. As students will need to discuss their opinions and ideas, allow enough time so they do not feel rushed and for ideas to evolve. Open-ended tasks such as these are a good opportunity for students to ‘take a risk’ with maths. • The activity is designed to be open-ended and investigative. Students may request resources such as books from the resource centre, Internet access or magazines and catalogues. Others may wish to ask adults from the building profession or adults they know who have recently had a house built. • When completing open-ended tasks, some students may be more successful in mixedability groups rather than same-ability groups. • Some groups will be able to work independently while others may need guidance. The stimulus questions below may prompt such groups. – How do you think a house is built? – In what order do you think a house is built? – What materials do you think a house is made of? – What groups of people build a house? What do they do? – Have you ever seen house plans? – How many bricks do you think a house is made of? (Take the students to a wall and calculate the number of bricks on that wall.) – Why are windows and doors the shape they are? – How do painters know how to calculate the amount of paint needed for a house? – Why are roofs the shapes they are? • Groups may wish to collate their findings and present them as a poster with diagrams and information or as a series of graphs and calculations. • 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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Main Activity (WM4.2, C&D4.2, S4.4, S4.2, M4.1, M4.3, N4.2)

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• 52 • New Wave Maths Book F – Teachers Guide

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Unit 8—Answers

Student pages 22 – 24 Unit 8–1

1. (a) 18 (b) 5 (c) 7 (d) 9 (e) 10 (f) 13 (g) 8 (h) 9 (i) 6 (j) 15 2. (a) 28 (b) 49 (c) 68 (d) 43 (f) 32 (e) 36 3. (a)

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(b) hockey, football, netball, golf (b) Answers will vary Challenge 3

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1. Teacher check 2. (a) 90.5 (b) 70.75 (c) 70.55 (d) 60.625 (e) 80.57 (f) 60.75 3. Set one Set two (a) 159 (b) 148 (a) 753 (b) 948 (c) 164 (d) 246 (c) 648 (d) 786 (e) 126 (f) 184 (e) 566 (f) 426 (g) 305 (h) 426 (g) 1119 (h) 562 (i) 168 (j) 219 (i) 1048 (j) 908 4. (a) 880 (f) 2050 (k) 7680 (p) 7800 (b) 660 (g) 3280 (l) 9460 (c) 2080 (h) 4970 (m)4560 (d) 990 (i) 1830 (n) 10 840 (e) 1240 (j) 2550 (o) 7840 Challenge One counter on top of another.

Unit 8–2

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• Discuss the strategies used with the students. Extend to discover if it still work with numbers into the thousands.

Consolidation 8–2

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1. Answers will vary; e.g. measuring, counting, ratios, angles, scale, time, planning, organising, timetabling, collecting, estimating etc. 2. Surveying of the block Site works Plumbing pipes, formwork Concrete pad Bricks Roof Electrical wiring Internal plumbing Rendering Plastering Windows and doors Fit kitchen, bathroom, laundry cupboards and fixtures Tiling Fit internal doors Complete electrical Painting Driveway Cleanup etc.

• Students write their own logic problems.

Consolidation 8–3

• Collect house plans and ask students to work out materials required and estimate the amount of each type of material required.

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New Wave Maths Book F – Teachers Guide • 53 •

Unit 9–1

Student page 25

Outcomes

Indicators

N3.3, N4.3, S4.4

The student is able to: • choose geometric language with care in order to describe things clearly.

Skills

Resources

• subtract • protractor • shapes • congruent • similar • comparison

• calculator • ruler • protractor

• measuring • working geometrically • comparing

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Memory Masters (N3.3)

Language

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Notes

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

• 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 (S4.4) Warm up

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Number (N4.3)

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sizes; for example. • Use 2-D shapes to reinforce these terms with students: congruent – the same in all features; similar – some features the same—e.g. angle size—others may be different—e.g. side lengths. • Revise the use of a protractor demonstrating correct placement of the intersection of the 0º – 180º base line and the vertical 90º line over the intersection of the two lines that make the angle to be read. Stress that the base of the protractor is not the measuring line unless it is also the 0º – 180º line. • Note: Students need to understand right, obtuse, acute angles etc. before using a protractor.

What to do

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• Ask students to check a mathematics dictionary for the meanings of congruent and similar. – Congruent: Shapes that are the same shape and size in all respects; e.g. – Similar: Shapes that are the same shape, have the same size angles but are different

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• Ask students to practice, as a whole class, measuring the angles in shapes 6, 7 and 11. • When satisfied that the protractor is being used correctly, revise definitions of congruent and similar shapes then set class to work to identify the shapes that are congruent or similar to shapes 1, 2 and 3. • Students record their findings on the table provided on the page.

Challenge • Draw a triangle with at least one acute angle in it. Now draw a triangle with a right angle in it. Then draw a triangle with an obtuse angle in it. • Try drawing a triangle with three acute angles; a triangle with two right angles; and a triangle with two obtuse angles. Which one can be drawn? Which can not be drawn? Why/Why not?

• 54 • New Wave Maths Book F – Teachers Guide

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Unit 9–2

Student page 26

Outcomes

Indicators

N3.3, N4.3

The student is able to: • partition double-digit numbers in order to mentally multiply and divide small single-digit numbers.

Skills

Resources

Language

• calculator • counters to represent pills

• subtract • simple terms • multiply • multiples • partition

• working mentally • reasoning

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Notes

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Memory Masters (N3.3)

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

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Number (N4.3)

• 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 (N4.3) Warm up

• Revise activities on pages 22 and 24 of the workbook. • Partitioning of multiples to work out answers provides a convenient method of completing these sums mentally. Note:This is also known as the distributive property of multiplication over addition. It does involve partitioning, which means splitting up or breaking up. For example, partitioning a two- or three-digit number in preparation for multiplying by a single digit number: 67 x 3 = (60 x 3) + (7 x 3) = 180 + 21 + 201

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• Students are asked to view the examples there and work through them with the teacher before completing the exercises on the page.

Challenge

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• Students are reminded to show their working, including taking notes on their thought processes when working this problem out.

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New Wave Maths Book F – Teachers Guide • 55 •

Unit 9–3

Student page 27

Outcomes

Indicators The student is able to: • make items which meet straightforward measurement specifications in standard units.

N4.1a, N4.3, M4.2

Skills • modelling • measuring • speaking and listening • reasoning

Resources • calculator • 2-cm cubes • pencil • ruler

Language • subtract • prism • model • base area • dimensions • total • maximum

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Memory Masters (N4.1a) Number (N4.3)

• 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 (M4.2) Warm up

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• The focus for this unit is identification of place value of decimals to four decimal places.

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What to do

• Give the directions for the construction of the first cube as detailed on page 27 of the workbook: three cubes wide, two cubes high and four cubes long. Students are then to determine how many cubes altogether in the model. Ask students what they are actually finding. (Volume) • Direct students to draw their model in the workbook. • Continue with the remainder of the exercise.

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• Distribute 2-cm cubes to small groups of students and allow a brief time for free play. • Ask students to explain what a prism is. • Ask students to build a simple prism with no more than 10 cubes; e.g. 10 x 1 or 5 x 2 – no other possibilities. • Check prisms. • Explain to the class that they will be asked to build prisms according to set criteria. The criteria will include some or all of the following – base area (tells how many cubes are on the base), width, height and length.

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• Students are to use logical thinking to find the answer to the problem. Students use trial and error and work systematically. • Remind students that there are both bicycles and tricycles in the shop. • Students are to record their working to show their reasoning. • Share findings with the class.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 146 – 147. • 56 • New Wave Maths Book F – Teachers Guide

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Unit 9—Answers

Student pages 25 – 27 Unit 9–1

1. (a) 45 (b) 18 (c) 16 (d) 30 (e) 49 (f) 21 (g) 32 (h) 27 (i) 64 (j) 40 2. (a) 219 (b) 454 (c) 327 (d) 148 (e) 659 (f) 228 3. Shape

Congruent

Similar

1. (a) 8 (b) 9 (c) 3 (d) 7 (e) 2 (f) 5 (g) 3 (h) 1 (i) 6 (j) 4 2. (a) 413 (b) 471 (c) 219 (d) 514 (e) 146 (f) 375 3. (a) (7 x 20) + (7 x 8) = 140 + 56 = 196 (b) (6 x 30) + (6 x 4) = 180 + 24 = 204 (c) (8 x 40) + (8 x 3) = 320 + 24 = 344 (d) (9 x 20) + (9 x 9) = 180 + 81 = 261 (e) (5 x 30) + (5 x 7) = 150 + 35 = 185 (f) (7 x 50) + (7 x 3) = 350 + 21 = 371 (g) (8 x 20) + (8 x 4) = 160 + 32 = 192 (h) (6 x 30) + (6 x 2) = 180 + 12 = 192 (i) (8 x 40) + (8 x 6) = 320 + 48 = 368 (j) (9 x 20) + (9 x 7) = 180 + 63 = 243 4. (a) (80 x 40) + (80 x 3) = 3200 + 240 = 3440 (b) (60 x 70) + (60 x 6) = 4200 + 360 = 4560 (c) (40 x 90) + (40 x 2) = 3600 + 80 = 3680 (d) (70 x 50) + (70 x 9) = 3500 + 630 = 4130 (e) (90 x 60) + (90 x 5) = 5400 + 450 = 5850 (f) (50 x 60) + (50 x 2) = 3000 + 100 = 3100 Challenge 2 hours

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Unit 9–2

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• Study and discuss the total of angles in a triangle, square and rectangle. What do you notice?

Consolidation 9–2

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1. (a) thousandths (b) units or ones (g) (c) ten thousands (h) (d) ten thousandths (e) tenths 2. (a) 4117 (b) 5215 (f) 2123 (e) 3141 3. 24 cubes

• Discuss how breaking problems into simple terms and using brackets helps to make them easier to solve.

Consolidation 9–3

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• Use 81 cubes to make a prism. What are the possible dimensions? Make it and check.

6. 4 length, 4 width, 4 height

Challenge 32

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New Wave Maths Book F – Teachers Guide • 57 •

Unit 10–1

Student page 28

Outcomes

Indicators The student is able to: • use the symbols =. < and > to state comparisons.

N4.1a, N4.3

Skills • reasoning

Resources • calculator • reading book/library book • place value charts (see pages 207 – 208)

Language • multiply • symbols • greater than, equal to, less than • order • relationship • frequently • round • nearest

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Notes

Memory Masters (N4.1a) Number (N4.3)

• 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 (N4.1a) Warm Up

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

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• Revise place value using a class chart or individual student chart. Ask students to point to various place value columns from both whole number and decimal places; e.g. tens, hundredths, thousands, ones, tenths etc. • Ask students how they would decide which number in a pair is the greater. • Look at the place value of the digit on the left-hand side of the number. The number in the higher place value will be the larger number. If digits from the number are of the same place value, then the bigger digit indicates the larger number. Should both digits be the same place value and the same face value, then you move to the digit immediately to the right and repeat the process. This continues until you are able to find one digit larger than its comparative place value digit, thus determining the larger number; e.g. 5601 is larger than 5061 because of the number in the hundreds place. • Remind students of the symbols < for less than and > for greater than. Easy ways to remember this are, the smallest part of the symbols indicates the smallest number or the large open crocodile mouth is about to eat the larger number.

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• Set students to work to complete the activities.

Challenge

• Students are to investigate a page in their reading or library book to find which letter occurs most frequently on the page. • Remind students that there are many ways to do this, but they are to choose a method for themselves then apply this method to find the most frequently occurring letter. • All working out, methods and strategies used must be recorded. • Share findings with the class.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 54 – 55. • 58 • New Wave Maths Book F – Teachers Guide

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Unit 10–2

Student page 29

Outcomes

Indicators The student is able to: • interpret and report on information provided in tables and bar graphs where data are grouped into simple intervals which can be regarded as categories.

N3.3, N4.3, C&D4.4

Skills • ordering data • inferring • reasoning

Resources

Language

• calculator

• multiply • results • chronological order • previous • greatest • decrease • improvement

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Notes

Memory Masters (N3.3)

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

Number (N4.3)

• 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&D4.4) Warm up

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• What to do • Talk to the class about census collections and surveys and how the data are used for future planning, marketing and resource allocation. • If possible, obtain some census data from the last available census and ask students for their interpretations of the data. If possible, compare their interpretations with actual results.

Challenge

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• Ask students to examine the information provided on the results of the men’s 100-metre freestyle final from the Olympic Games. • Explain to students they will be asked to work with this information in a number of ways to answer the questions on the page. • The first task is to place the winning times in chronological order – the order in which they occurred from the earliest year to the latest year noted. • The next exercise requires specific information that can be obtained by checking the data provided.

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• If one swimming stroke takes me forward 11/2 m, how many swimming strokes will it take to swim 200 m?

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 164 – 165. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 59 •

Unit 10–3

Student page 30

Outcomes N3.1, N4.3, N4.1b

Indicators

Resources

The student is able to: • use models to represent decimals.

• calculator • toothpicks or equivalent • coloured pencils or felt-pens

Skills • subtracting fractions • reasoning

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Memory Masters (N3.1)

Language • multiply • diagrams • fraction • difference • wholes • squares

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• The focus for this unit is the commutative property of addition.

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

Main Activity (N4.1b) Warm up

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• Ask students to follow your instructions attentively. • Exercise 4(a) shows 4 parts of 5 shaded.That is four-fifths of a whole. By outlining one unit of the four-fifths shaded it is possible to show 4/5 – 1/5. How many fifths are left that are shaded but not outlined? (3 is correct.) This shows an answer of 3/5. • Repeat with (b). • Exercise 4(c) shows 13/4 wholes shaded.You are to shade 3/4 of a whole by outline 3/4 using the shaded area. How many units will be outlined? (3 is correct.) Where would you outline these three units? Does it matter where? (No.) For simplicity the three shaded units by themselves make it easier to see the answer. By outlining the 3 units we have 4 units or one whole left shaded but not outlined. We have shown that 13/4 – 3/4 = 1. • Repeat for Exercise 4(d) through to (g). Work with all students together.

Challenge

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• Discuss what a fraction is and what fractions represent.

What to do

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Number (N4.3)

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• Students need to remember that there must be only two squares left, other shapes may be made in final arrangements after the eight toothpicks have been removed.These shapes may not be squares. • Remind students to record their workings so that strategies and attempts can be viewed. • Share final results with class if desired or leave for further trials by those who are unable to solve the problem.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 64 – 65. • 60 • New Wave Maths Book F – Teachers Guide

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Unit 10—Answers

Student pages 28 – 30

Unit 10–1 1. Answers will vary 2. (a) 24 900 (b) 29 400 (d) 25 600 (e) 47 200 3. (a) 1952 57.40 (b) 1956 55.40 (c) 1960 55.20 (d) 2964 53.30 (e) 1968 52.20 (f) 1972 52.22 (g) 1976 49.99 (h) 1980 50.40 (i) 1984 49.80 (j) 1988 48.63 3. (a) 1972, 1980 (b) 1972 and 1976—2.23 secs (c) 1972 and 1976—2.23 secs (d) 1976, Montreal Games (e) USA (6 wins) (f) twice (g) it was not a record

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1. (a) 80 000 (b) 10 000 (c) 30 000 (d) 70 000 (e) 10 000 (f) 60 000 (g) 70 000 (h) 90 000 (i) 70 000 (j) 40 000 2. (a) 32 240 (b) 56 320 (c) 28 630 (d) 30 150 (e) 30 300 (f) 20 120 3. (a) 423 < 500 < 536 (b) 600 < 694 < 700 (c) 327 < 350 < 369 (d) 6937 < 7000 < 7468 (e) 80 < 86 < 90 (f) 4.7 < 5.0 < 5.3 (g) 6.1 < 6.5 < 6.9 (h) 4.01 < 4.06 < 4.80 (i) 2.003 < 2.005 < 2.008 (j) 5.007 < 5.3 < 5.82 (k) 6.07 < 6.27 < 6.72 (l) 8088 < 8808 < 8880 (m) 6.1 < 61 < 610 (n) 0.39 < 39 < 390 4. >, <, <, >, >, <, > Challenge Answers will vary

Unit 10–2

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(d) 6, 15 (e) 3, 11 (i) 7, 10 (j) 4, 12 (c) 39 200 (f) 43 000

• Use symbols >, < to order students in the class by age.

Consolidation 10–2 • Students run 100 m, record finishing times. Write in order from fastest to slowest.

Consolidation 10–3

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1. (a) 7, 8 (b) 9, 11 (c) 5, 9 (f) 7, 15 (g) 4, 13 (h) 8, 13 2. (a) 31 500 (b) 29 600 (d) 66 600 (e) 28 800 3 3 3. (a) /5 (e) /5 4 (f) 14/10 (b) /8 (c) 1 (g) 3/6 (d) 21/3 Challenge

• Brainstorm where students would use fractions.

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New Wave Maths Book F – Teachers Guide • 61 •

Unit 11–1

Student page 31

Outcomes

Indicators

N3.3, N4.3, S4.3

The student is able to: • use appropriate language of transformation in describing how one shape can be superimposed on another.

Skills • technical drawing • measuring • compass use

• calculator • compass • pencil • coloured pencils • paper

Number (N4.3)

• 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 (S4.3)

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What to do

• By placing the compass point on each of the dots on the circumference of the circle in turn and drawing a full circle, a pattern can be made. • Once the pattern has been drawn, colour it. • After completing your pattern you may experiment with your compass on a separate sheet of paper and make different patterns of your own choosing.

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• Before starting this activity ensure that students are familiar with the correct setting up and use of their compass. The compass point and pencil point should be level when the compass is closed and the two points beside each other. • When drawing with the compass, the compass point should remain in contact with the point designated on the page. The compass is held and turned from the top point so that a light but visible line, arc, or circle is drawn on the page. • Show students how to set the radius by placing the compass point on the centre point of the circle and opening the compass so that the pencil point is placed on the circle.

Challenge

• divide • compass • radius • circle • pattern • location • overlap

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• The focus for this unit is the addition of a whole number less than 10 to a whole number less than 100.

Warm up

Language

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Memory Masters (N3.3)

Resources

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• The TARGET™ logo is made up of concentric circles (that is circles within each other that have the same centre but different radiuses. Try to make your own TARGET™ logo.

• 62 • New Wave Maths Book F – Teachers Guide

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Unit 11–2

Student page 32

Outcomes

Indicators

N3.3, N4.3

The student is able to: • add or subtract a constant to solve a problem.

Skills

Resources

Language • divide • add • subtract • quantity • numbers • mental

• calculator • 2-cm cubes or counters

• working mentally • reasoning

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Notes

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Memory Masters (N3.3)

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• The focus for this unit is the subtraction of a whole number less than 10 from a whole number less than 100.

Number (N4.3)

• 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 (N4.3) Warm up

What to do

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• Use concrete materials such as 2-cm cubes or counters to demonstrate that the addition or subtraction of a constant to or from both numbers in a subtraction sum does not alter the answer. • Distribute the concrete materials among small groups of students. • Ask students to take out 6 cubes (‘cube’ here represents the concrete material chosen) and another set of 4 cubes. Ask what the difference is between the two sets. (2) Ask students to add 3 cubes to each set. Students now have a set of 9 and a set of 7 cubes. The difference between both sets is still ... 2. • Take 5 cubes from each set. You now have a set of 4 and a set of 2 cubes. The difference is still ... 2. • Repeat starting with 12 cubes and 7 cubes. Take 4 from each set then add 9 to each set. The difference remains at 5.

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• Explain to students using example (a) how this may be represented in written form. Use 2-cm cubes with students to work through the examples. • Students complete examples on the page using concrete materials if required. • Check answers.

Challenge

• Use the examples to provide an explanation of what happened in each sum and why this happens. • Keep a written record of your findings to share with the class.

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New Wave Maths Book F – Teachers Guide • 63 •

Unit 11–3

Student page 33

Outcomes

Indicators

N3.3, N4.3, M4.2, C&D4.2, C&D4.4

The student is able to: • order objects according to mass. • suggest what data to collect to help estimate numbers or quantities. • interpret and report on information provided in tables and bar graphs.

Skills • measuring • estimating • weighing • ordering • reasoning

Memory Masters (N3.3)

Resources • calculator • ten different sized and shaped objects • balance

Number (N4.3)

• 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 (M4.2, C&D4.2, C&D4.4)

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• divide • variety • different • size • shape • estimate • actual • mass • heaviest, lightest

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• The focus for this unit is basic facts of multiplication with addition of a whole number less than 10. • Even though these examples do not involve rule of order, it should be discussed as the questions involve more than one operation.

Warm up

Language

• Allow students a few minutes to collect ten different sized and shaped objects. This may be done with students working in small groups.

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• Students record their items in the table on page 33 of their workbook. • Ask the group to reach an agreement on the estimated order of the mass of their objects and write the order next to each object in the table. • Provide each group with a balance and ask them to check the actual order of the items. Ask students how they intend to use the equipment to find the actual order. Share this information with the class then set the groups to work to find the actual order using the balance as they best see fit. • Possible methods – place an object in the scale pan and check the mass of other objects against it, repeating until an order is found – weigh two objects against each other then order them in pairs, continue until all pairs are ordered.Then repeat, crosschecking objects from each pair. • Complete the activity by writing responses to (b) and (c).

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What to do

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• Students are to use their intuitive thinking to answer this question. • Reasoning behind the answer must be given in writing.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 102 – 103. • 64 • New Wave Maths Book F – Teachers Guide

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Unit 11—Answers

Student pages 31 – 33

Unit 11–1 1. 2. 3.

(a) 87 (f) 98 (a) 71 (e) 92

(b) 53 (g) 37 (b) 71 (f) 61

(d) 27 (i) 50 (d) 81

(e) 88 (j) 93

1. (a) 81 (b) 41 (c) 73 (d) 23 (e) 61 (f) 92 (g) 43 (h) 81 (i) 20 (j) 64 2. (a) 201 (b) 301 (c) 702 (d) 701 (e) 701 (f) 501 3. (b) 3 14 – 11 = 3 (c) 7 17 – 10 = 7 (d) 2 13 – 11 = 2 (e) 2 15 – 3 = 2 (f) 8 10 – 2 = 8 (g) 8 11 – 3 = 8 (h) 8 10 – 2 = 8 (i) 3 6–3=3 (j) 8 16 – 8 = 8 (k) 12 18 – 6 = 12 (l) 5 25 – 20 = 5 (m)12 22 – 10 = 12 (n) 10 30 – 20 = 10 (o) 6 28 – 22 = 6 Challenge The answers remained the same.

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4. Answers will vary

Unit 11–2

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(e) 21 (j) 21

• Use a compass to create other circular patterns.

Consolidation 11–2 • Extend the activity to add or subtract constant numbers into tens and hundreds.

Consolidation 11–3

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(a) 85 (b) 50 (c) 37 (f) 52 (g) 26 (h) 49 (a) 206 (b) 206 (c) 108 (e) 107 (f) 104 Teacher check N; Explanations will vary

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1. 2. 3. 4.

• Complete the activity again using objects found in nature; e.g. rocks, sticks, leaves.

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New Wave Maths Book F – Teachers Guide • 65 •

Unit 12–1

Student page 34

Outcomes

Indicators The student is able to: • state fractional equivalents in words and symbols.

N3.3, N4.3, N4.1b

Skills • decimal/fraction conversion • ordering • reasoning • modelling

Memory Masters (N3.3)

Resources • calculator • toothpicks or equivalent

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 (N4.1b)

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Number (N4.3)

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• Ask students to look at the fraction grid on the page. • Explain to the class that by using the grid it is possible to find fraction equivalents and to order fractions. • An equivalent fraction is one fraction that is equal in representation to another. For example, if you find 1/2 on the grid, equivalent fractions may be found by finding other fractions that have a line on the grid immediately below the 1/2 line. What fraction equivalents are these? • Students should find fourths (2/4), sixths (3/6), eighths (4/8) and tenths (5/10). • Try a more difficult equivalent such as 2/3. Students should find 4/6 and 6/9 are equivalent. If students are having difficulty reading the grid, suggest that they use their ruler to align equivalent fractions.

What to do

• subtract • fraction grid • order • fractions • greater than > • less than < • triangles • model • equivalent

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• The focus for this unit is subtraction of a whole number less than 10 from basic facts of multiplication.

Warm Up

Language

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• Ask students to complete Exercise 3 using the grid to find equivalent fractions. • Exercise 4 may be completed using the grid by finding whether the second fraction pair is larger or smaller than the first. Again, by using a ruler students can see whether the second fraction is bigger or smaller. For example, if the first fraction is 2/5, by placing the ruler along the 2/5 line this may be compared with 3/7; 3/7 is larger than 2/5. • Remind students of the use of > and < signs with the larger, open side closer to the larger number. • Students complete the exercise.

Challenge • Students use all the sticks and arrange them to make two triangles only. • Record each attempt and write any necessary explanations so that this may be shared with the class or the teacher.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 62 – 63. • 66 • New Wave Maths Book F – Teachers Guide

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

Unit 12–2

Student page 35

Outcomes

Indicators The student is able to: • construct and use his/her own strategies to answer specific questions.

N3.4, N4.3, C&D4.2, C&D4.4

Skills • recording data • comparing • measuring • reasoning

Resources

Language

• calculator • cube • rectangular, triangular, square, pentagonal and hexagonal prisms or geoshapes • trundle wheel • stopwatch • markers

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

• subtract • models • three-dimensional • relationship • shapes • prism • distance • time • metre • wheel • measure • table • actual • estimate

Notes

Memory Masters (N3.4)

Teac he r

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• The focus for this unit is completion of number patterns including some with two patterns included.

Number (N4.3)

• 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&D4.2, C&D4.4) Warm up

• With small groups of students, distribute the 3-D models or use geoshapes to build models. • Discuss with students what a prism is. (A shape named by its end shapes which are congruent and flat.) • Ensure that students know what edges, vertices (corners) and faces are by having them point to each as asked. • Euler was the first to discover the relationship between edges, vertices and faces.

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What to do

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• Direct students to complete the table on page 35 of the workbook by examining each shape in turn then writing their findings in the table. • When all models have been examined, students should explore the table and explain any relationships that they find in writing. • Exercise 4 requires students to look at the pyramids, develop a table of their own and discuss any relationships they can find.

Challenge

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• Encourage students to think of strategies that they will need to solve the problem. • Remind students that each number to be placed is used in more than one total. • Record all strategies and attempts to share with the teacher.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 52 – 53. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 67 •

Unit 12–3

Student page 36

Outcomes

Indicators

Resources

N3.1, M4.1, N4.3, N4.1b

The student is able to: • use materials and diagrams to represent fractional amounts where the ‘whole’ may be an object, quantity or collection.

• calculator • pencil • ruler • overhead projector

Skills • reasoning

Language • metres • subtract • shapes • squares • cubes • fourths • eighths • circles • tenths

• centimetres • divide • fractions • halves • thirds • rectangles • triangles • sixths • symbols

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

Teac he r

Memory Masters (N3.1, M4.1)

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

• 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 (N4.1b) Warm up

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Number (N4.3)

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What to do

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• Direct students to follow the directions on the page and to divide the shapes provided into the fractional parts as requested. • Write the total fraction parts for each group of shapes. • Check work carefully.

Challenge

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• Draw a square on an overhead, blackboard/whiteboard and ask students how this may be divided into fourths. • Allow a student to complete the task or do it yourself following student directions. • Draw another square beside the first and repeat the process of dividing this square into halves. • Ask students how many halves in each square, and how many halves altogether. • Draw a rectangle on the overhead, blackboard/whiteboard. Ask how this may be divided into thirds. Allow students to do so or do yourself following student directions. • Draw another two rectangles beside the first and divide each into thirds as for the first. • Ask students how many thirds in each rectangle, and how many thirds altogether.

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• Students will need to experiment with adding, subtracting, multiplying and dividing to find ways of making other numbers using a single number and the four operations. • As a hint, ask students how they might make the number 1 from the number 4 using one or all of the operations; for example, 4 ÷ 4 = 1. • Leave students to experiment either by themselves or in small groups. • Share answers with the class and encourage the finding of different combinations. Praise efforts but don’t give answers that students can not find.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 62 – 63. • 68 • New Wave Maths Book F – Teachers Guide

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Unit 12—Answers

Student pages 34 – 36

Unit 12–1

1. (a) 16 (b) 30 (c) 34 (d) 440 (e) 200 (f) 9 (g) 40 (h) 24 (i) 5 (j) 35 2. (a) 6085 (b) 1563 (c) 6142 (d) 3091 (e) 1183 (f) 5192 3. Shape Edges Vertices Faces Cube

Rectangular prism

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

Square prism

Pentagonal prism

Hexagonal prism

Total number of edges = edges of base shape x 3 Total number of vertices = vertices of base shape x 2 Total number of faces = edges of bases shape + 2 1 4. Teacher check Challenge 4 10 14 2

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Teac he r

1. (a) 22 (b) 59 (c) 17 (d) 7 (e) 50 (f) 27 (g) 3 (h) 36 (i) 20 (j) 9 2. (a) 3107 (b) 3238 (c) 6209 (d) 6409 (e) 2318 (f) 4164 3 4 3. (a) 2/4 /6 /8 5/10 3 (b) 2/6 /9 2 (c) /8 (d) 2/10 4. (a) = (f) > (k) > < (b) > (g) < (l) (c) < (h) > (m)> < (n) < (d) < (i) (e) > (j) > (o) > Challenge

Unit 12–2

• Brainstorm situations where fractions need to be used; e.g. birthday parties, cooking recipes etc.

Consolidation 12–2

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1. (a) 0.67 m (b) 0.03 m (c) 47.56 m (d) 8.92 m (e) 0.11 (f) 894 cm (g) 46 310 cm (h) 8 cm (i) 75 cm (j) 1 cm 2. (a) 2423 (b) 1921 (c) 6961 (d) 3670 (e) 4732 (f) 4934 3. (a) 8

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(b) 6

• Complete the information for other prisms; e.g. octagonal prism, nonagonal prism etc.

Consolidation 12–3

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• Students make shapes out of modelling clay to cut into fractions.

(e) 12 (f) 24 (g) 30 Challenge Answers will vary R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book F – Teachers Guide • 69 •

Unit 13–1

Student page 37

Outcomes

Indicators

N3.1a, N4.3, S4.2

The student is able to: • use some mathematical conventions in drawings.

Skills • reasoning • using a compass • drawing technically

Resources • calculator • compass • ruler • pencil

Language • multiply • compass • ruler • construct • point • arc • extend • intersection • base line • total • inclusive • equilateral triangle

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

Memory Masters (N3.1a)

Notes

Number (N4.3)

• 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 (S4.2) Warm up

• The purpose of this activity is to familiarise students with the skills and language related to using a compass. • Remind students how to use the compass. • When drawing with the compass, the compass point should remain in contact with the point designated on the page and be held and turned from the top so that a light but visible line, arc, or circle is drawn on the page. • Remind students construction work should be light.

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Teac he r

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

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• Ask students to take out their compass and ruler. • Explain to students that they will be constructing two items – an equilateral triangle and a 30º angle using a pencil, compass and ruler only. • Work through the activity with the students repeating the instructions as detailed in the workbook. Check student work while giving the instructions for both activities.

Challenge

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What to do

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• Share the following story with the students. Frederick Gauss developed a simple way to add the numbers 1 to 100 while at school. His teacher would punish students by making them add the numbers 1 to 100. Gauss discovered ... There are 50 pairs of numbers, so he multiplied 101 by 50 to find the answer of 5050, astounding the teacher with his brilliance!

• Students are to think about different ways that they can complete this activity. • Record findings and share with the class. Specifically look for simple or short methods to find the answer. • 70 • New Wave Maths Book F – Teachers Guide

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Unit 13–2

Student page 38

Outcomes

Indicators

N3.3, N4.3, N4.2

The student is able to: • complete numerical statements involving brackets.

Skills

Resources

Language

• calculator • pencil

• estimating • reasoning

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

• multiply • estimation • number sentences • brackets • algorithm • draw • shape • trace • continuous

Memory Masters (N3.3)

Notes

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Teac he r

• The focus for this unit is addition of a whole number less than 10 to a whole number less than 100.

Number (N4.3)

• 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 (N4.2) Warm up

What to do

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• Provide students with two calculators—one basic, one scientific. Ask students to input the following in both calculators: 3+4x5 The basic calculator will give the answer of 35 where as the scientific calculator will show 23. Explanation: Some operations are more powerful than others so when operations are mixed in the same number sentence, the order in which they are done varies. – multiply (x) and divide (÷) are calculated first – add (+) and subtract (–) are calculated last We work left to right. Brackets are used here to highlight which part to do first for the students. • In working through these examples the answer is greater than the number found by solving the number sentence. Because of this, answers will vary. It is normal to use the next largest number but this does not have to be the case.

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• Work through the first few examples with the whole class reminding them to work out the bracket then solve the number sentence. Once solved, find a number greater than the answer to the number sentence. • Exercise 4 is similar to Exercise 3 except that the final number is subtracted and the answer is now less than the solution to the number sentence. Again, the next smallest number is normal but not essential. • Students complete the activity.

Challenge • Students select their own starting point which does not have to be one of the intersection points named on the shape. • Record all attempts and write a brief description of what was done to share with the teacher. For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 74 – 75. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 71 •

Unit 13–3

Student page 39

Outcomes

Indicators

N3.3, N4.3, M4.2, N4.4

The student is able to: • compare and order length, capacity and mass measurements provided in common standard units. • identify and describe several different patterns in an array of numbers such as the 100 chart or a similar 6 by 6 chart.

Skills • modelling • recording • measuring • reasoning

Memory Masters (N3.3)

Resources • calculator • 2-cm cubes

Number (N4.3)

• 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 (M4.2, N4.4)

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Teac he r

• multiply • models • table • information • height • width, length • pattern • sum • inclusive

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• The focus for this unit is subtraction of a whole number less than 10 from a whole number less than 100.

Warm up

Language

• Distribute 2-cm cubes to small groups of students. • Explain to the class that they will be building models, to given directions, to find the volume of the models. • Ask students how they think they might determine the volume of a model built using 2-cm cubes. (By counting each cube used.)

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• Direct students to make the first model and record the length, width and height in unit cube measures.Then find the total number of cubes used. Record the information gathered in the spaces provided. • Students are then to move on to construct each of the other models shown and record details in the spaces provided. • Once this has been completed, students are to use the information collected to identify any patterns they see emerging from the data.

Challenge

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What to do

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• Students are to think about different ways they can complete this activity. • Record findings and share with the class. Specifically look for simple or short methods to find the answer.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 146 – 147. • 72 • New Wave Maths Book F – Teachers Guide

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Unit 13—Answers

Student pages 37 – 39

Unit 13–1 1. (a) $47.82 (b) $30.06 (c) $0.49 (d) $0.65 (e) $3.82 (f) 7c (g) 15c (h) 680c (i) 7203c (j) 8109c 2. (a) 17 940 (b) 27 930 (c) 48 000 (d) 47 040 (e) 31 450 (f) 45 360 3. C

1. (a) 29 (b) 75 (c) 89 (d) 37 (e) 56 (f) 30 (g) 71 (h) 66 (i) 18 (j) 94 2. (a) 23 829 (b) 45 192 (c) 53 756 (d) 37 157 (e) 26 390 (f) 45 248 3. (a) ≈16▲ (g) ≈54▲ (m) ≈14▲ (b) ≈5▲ (h) ≈12▲ (n) ≈21▲ (c) ≈13▲ (i) ≈33▲ (o) ≈20▲ (d) ≈14▲ (j) ≈13▲ (p) ≈22▲ (e) ≈26▲ (k) ≈22▲ (q) ≈15▲ (f) ≈10▲ (l) ≈13▲ (r) ≈17▲ 4. (a) ≈3▼ (g) ≈5▼ (m) ≈0 (b) ≈13▼ (h) ≈6▼ (n) ≈9▼ (c) ≈4▼ (i) ≈6▼ (o) ≈2▼ (d) ≈2▼ (j) ≈6▼ (p) 0 (e) ≈4▼ (k) ≈1, 0 (q) ≈11▼ (f) ≈0 (l) ≈2▼ (r) ≈9▼ Note: ▲ denotes any number larger ▼ denotes any number smaller Challenge Yes; A, E, C, A, B, C, D, E, B

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Unit 13–2

Challenge 45 Find combinations that add to 10; e.g. 1 + 9, 2 + 8 etc.

Length 1

Width 1

Height 1

Number of Cubes 1

Length

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3 Width 3

Width 4

Height

Width 2

Height 2

Number of Cubes 8

• Use a compass to create a design.

Consolidation 13–2 • Develop a class chart explaining how to estimate.

Consolidation 13–3

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1. (a) 78 (b) 47 (c) 25 (d) 56 (e) 42 (f) 44 (g) 7 (h) 44 (i) 87 (j) 41 25 272 (c) 62 604 (d) 52 472 2. (a) 38 793 (b) (e) 40 205 (f) 29 516 3. Block 1 Block 2 Block 3 Block 4

• Extend the activity to 81 and 100 cubes. Does the pattern continue?

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Number of Cubes 64

4. L x W x H = Number of cubes Challenge 190

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New Wave Maths Book F – Teachers Guide • 73 •

Unit 14–1

Student page 40

Outcomes

Indicators The student is able to: • count forwards and backwards from any whole number. • read scales to the nearest graduation, including instances where the graduations are not labelled.

N3.3, N4.3, N4.1a, M4.2

Skills • researching • ordering data

Memory Masters (N3.3)

Resources • calculator • historical reference books or the Internet

Language • divide • information • time line

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

Notes

Number (N4.3)

• 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 (N4.1a, M4.2) Warm Up

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Teac he r

• The focus for this unit is addition of a whole number less than 10 to basic facts of division.

• Ask students to undertake some research of their own prior to attempting this activity to find as many early explorers of Australian shores from 1600 to 1700 as they are able. • This information may be brought in and presented as a research activity, including a time line and a description of the section of coast discovered, who the explorers were, the names of their ships and the countries of origin of the vessels. • Alternatively, the information collected may be used to complete the time line provided on page 40 of the workbook.

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• By sharing all found information students should be able to complete the missing information on the table provided. • Students may wish to extend beyond the time period specified and gather a wider range of information than that requested. Encourage them to do so and to present their information for assessment and sharing. • This activity may be completed as part of the Society and Environment Learning area or as an adjunct to it.

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Challenge

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• Students will require access to the Internet. Working in small groups may alleviate pressure on resources.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 122 – 123. • 74 • New Wave Maths Book F – Teachers Guide

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

Unit 14–2

Student page 41

Outcomes

Indicators The student is able to: • find the mean where there is sufficient data to make summarising sensible and use it to get an estimate number. • put data in order and describe the highest, lowest and middle scores. • read scales to the nearest graduation.

N3.3, N4.3, C&D4.3, M4.2

Skills • recording data • gathering data • measuring • reasoning • interpreting

Resources

Language • divide • measure • height • tally • range • table • total • median • mode • subtract • consecutive number • original number

• calculator • height measuring stick

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Notes

Memory Masters (N3.3)

Teac he r Number (N4.3)

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• The focus for this unit is the addition of a whole number less than 10 to basic facts of 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&D4.3, M4.2) Warm up

What to do

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• Discuss with the whole class the use of statistical information in providing support for planning, introduction of change, representing points of view etc. • Within the statistics, certain types of information may be used: Range: is the spread of scores from lowest to highest or a spread in which a category may be placed. Median: is the middle score when all the scores are given in ascending/descending order. If two middle scores, then add the two and divide by two. Mean: is the average of the scores. Mode: is the most frequently occurring score and there may be more than one mode. • The exercise today is to determine what the outcomes of some of these terms will be based on recording the height of all class members.

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• Direct students to record, using a tally, the height of each student as it is measured under the correct height range. • When all class members have been measured, students are to write the total for each group on the table. • Use this information to answer the questions on the page. This may be completed as a whole-class activity directed by the teacher to ensure all students gain the required understanding.

Challenge • Follow the instructions, recording your actions as you go. • Repeat several times, starting with different numbers. • Explain why you think you obtain the answers you are getting.

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New Wave Maths Book F – Teachers Guide • 75 •

Unit 14–3

Student page 42

Outcomes

Indicators The student is able to: • multiply and divide measurements and amounts of money by a onedigit number.

N4.3

Skills

Resources • calculator • Base 10 MAB • counters

Language • divide • division • rectangles

• reasoning

Memory Masters (N4.3)

Teac he r

• 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.

Number (N4.3)

• 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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• Organise the class into small groups. Distribute the Base 10 MAB and counters. • Ask students to show 25 with their Base 10 MAB (two longs, 5 units.) • Share this into 5 equal groups. What do you need to do? (Trade) How much wood is in each group? (This represents 5 ones.) • Repeat the process for 42 ÷ 2; 63 ÷ 3; 36 ÷ 4; and 51 ÷ 3. • Explain to the students that the pieces of wood may represent whatever place value we give to them. With this in mind, how might we show 2.1 using the Base 10 MAB? Accept all suggestions. Clear use of a counter for a decimal point gives the best representation. • Ask students to show 5.36, 4.38, 1.59.

What to do

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Main Activity (N4.3) Warm up

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• Ask students to show 4.32 using the Base 10 MAB. • Ask students to share this into 3 equal groups applying the same principles that they used for earlier sharing. For clarity, suggest that the shared wood is placed immediately above or below the original placement of the wood – this includes use of the counter as a decimal point. • What answer was obtained? (1.44) • Repeat the process with the rest of the examples. If need be, work with whole class or any groups having difficulties directing their actions step by step.

Challenge • Students are to record their findings as they work through the problem. • Findings are to be presented to the teacher and perhaps shared with the class. • Remember that results are as the student sees it.There is a finite number of rectangles but not all students will find them all. • 76 • New Wave Maths Book F – Teachers Guide

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Unit 14—Answers

Student pages 40 – 42

Unit 14–1

1. (a) 13 (b) 15 (c) 7 (d) 16 (e) 9 (f) 13 (g) 7 (h) 13 (i) 13 (j) 7 2. (a) 112r2 (b) 111r2 (c) 222r1 (d) 221r2 (e) 222r1 (f) 332r2 3. Teacher check 4. Teacher check Challenge 5

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Teac he r

1. (a) 10 (b) 17 (c) 6 (d) 10 (e) 11 (f) 9 (g) 7 (h) 10 (i) 12 (j) 14 2. (a) 707 (b) 707 (c) 404 (d) 707 (e) 909 (f) 707 3. 1606: Jansz – Cape York – ‘Duyfken’ 1606: Torres – Torres Strait – ‘San Pedrico’ 1616: Hartog – Shark Bay – ‘Eendracht’ 1618: Jansz – Exmouth Gulf – ‘Mauritius’ 1619: Houtman – Abrolhos Islands – ‘Amsterdam’ 1622: Unknown – Cape Leeuwin – ‘Leeuwin’ Pelsaert – Abrolhos Islands – ‘Batavia’ 1629: 1642: Tasman – Tasmania – ‘Zeehaen’ and ‘Heemskerck’ 1644: Tasman – Northern Coast – ‘Limmen’, ‘Zeemeuw’ and ‘Bracq’ 1688: Dampier – Cygnet Bay – ‘Cygnet’ De Vlamingh – Rottnest – ‘Geeluink’, 1696: ‘Nyptangh’ and ‘Weeseltje’ Challenge Teacher check

Unit 14–2

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• Develop a time line of development in your local area if possible.

Consolidation 14–2 • Complete a table recording daily temperatures for one month and calculate the mean, mode and median.

Consolidation 14–3

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1. Teacher check 2. (a) 202r1 (b) 201r2 (c) 101r2 (d) 401r1 (e) 101r2 (f) 302r1 3. (a) 1.44 (b) 1.79 (c) 0.87 (d) 0.69 (e) 0.69 Challenge 20

• Discuss how the division problems would be affected if we used Base 5 MAB instead of Base 10 MAB.

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New Wave Maths Book F – Teachers Guide • 77 •

Unit 15–1

Student page 43

Outcomes

Indicators

N4.1a, N4.3, S4.4, C&D4.4

Skills • measuring • recording • comparing • drawing conclusions

The student is able to: • choose geometric language with care in order to describe things clearly. • interpret and report on information provided in tables and bar graphs where data are grouped in intervals which can be regarded as categories.

Resources • calculator • ruler • protractor

Language • subtract • shapes • equilateral • isosceles • scalene • triangle • angles • sides • degrees

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

Notes

Teac he r

Memory Masters (N4.1a)

• The focus for this unit is determining equality and inequality of pairs of numbers.

• 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 (S4.4, C&D4.4) Warm up

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Number (N4.3)

What to do

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• In this activity you are comparing three types of triangles – equilateral, isosceles and scalene. – equilateral: all sides and angles are the same – isosceles: two sides and two angles are the same – scalene: all sides and angles are different Comparison is made by measuring and recording the length of all sides of the triangles and measuring and recording all angles of each triangle. • When this has been completed you are to use this information to describe what you found out about each pair of triangles and record this in your workbook.

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Challenge

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• Distribute a number of different shaped triangles among small groups of students. • Ask students to observe the triangles and record similarities and differences among the triangles. • Select one person from each group to share the group’s findings with the rest of the class. • Ask students to draw or sketch triangles. Note: There are seven different kinds. • Remind students how to accurately measure angle sizes using the protractor— intersection point of 0º–180º line and 90º line is placed over the intersection point of the two arms of the angle with the lower arm along the 0º–180º line on the protractor.

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• Use the information from the Main Activity to calculate the total number of degrees in the sum of the three angles in a triangle to find out if this is the same for all triangles. • Students may wish to explore this further with other triangles. • Rip the corners out of a triangle and line up along a straight edge.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 48 – 49. • 78 • New Wave Maths Book F – Teachers Guide

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

Unit 15–2

Student page 44

Outcomes

Indicators

N4.1a, N4.3

The student is able to: • add and subtract money and measures with equal numbers of decimal places.

Skills

Resources

Language

• calculator

• reasoning • adding • budgeting

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• subtract • round • thousand • nearest • budget • approximate • total • equal

Notes

Teac he r

Memory Masters (N4.1a)

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

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Number (N4.3)

• 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 (N4.3) Warm up

© R. I . C.Publ i cat i ons orr evi ew pur posesonl y• What to• dof • Ask the students how many of them have dined at a restaurant. • Select some students to explain what they had to do to get their meal served. • Focus on the use of a menu. Ask how it is set out, what are its key points?

Challenge

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• A simple menu has been provided on the page for students to use. • The activity requires students to make three orders each to a maximum of $20. One order is to have three courses and the other two orders are to have two courses. Students are free to make their own choices but must not go over $20 for any single order. • Share orders within small groups or select some students to share their orders with the class. • Students can make their own menus as directed. If pricing the meal, suggest reasonable limits for prices.

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• Students are to use the numbers 1 – 10 inclusive to complete the lines on the shape so that each line totals 22. • Remind students that the corner numbers are common to two lines. • Students are to record all their attempts and keep notes on their working and thinking strategies.

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New Wave Maths Book F – Teachers Guide • 79 •

Unit 15–3

Student page 45

Outcomes

Indicators The student is able to: • read scales to the nearest graduation, including instances where the graduations are not labelled.

N3.3, N4.3, M4.2

Skills • time conversion • reasoning

Memory Masters (N3.3)

Resources • calculator • analog clock • digital watch

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 (M4.2)

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Number (N4.3)

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• Ask which students can tell the time. Ask them if this is on both an analog clock and a digital clock. • Show the analog clock and move the hands to a number of different settings, asking students to give you the time. • Ask students how they know whether it is morning or afternoon. Use a.m. or p.m. when writing times or speaking about the time. – a.m. - ante meridiem - Latin for before noon – p.m. - post meridiem - Latin for after noon • Do timetables and armed forces use the same time? (No) They use 24-hour time not 12-hour time. Video recorders often use 24-hour time. • Explain that a.m. or morning time is the same but p.m. or afternoon time is different. After 12 noon in 24-hour time, the p.m. (12-hour) time is added to 12 to give 24-hour time. For example, 3 p.m. is 12 noon + 3 = 15. This is written as – 1500 hours; 3.20 p.m. is written as 1520 hours.

What to do

• subtract • 24-hour time • 12-hour time • clock faces • sum • product • consecutive numbers

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• The focus for this unit is addition of a whole number less than 10 to a whole number less than 100.

Warm up

Language

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• When students show understanding of conversion of 12-hour to 24-hour time and vice versa, ask them to complete Exercise 3 in their workbook. Some students will need special guidance to understand the concept and to read analog clocks. Work extensively, in a practical manner, with these students. • Complete Exercise 4 as directed.

Challenge • Ensure students understand what consecutive numbers are (numbers that run in order; e.g. 5, 6, 7). • Encourage students to explore the possibilities, recording their attempts. • If one set of numbers is found, ask students to try to find others. For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 116 – 117. • 80 • New Wave Maths Book F – Teachers Guide

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Unit 15—Answers

Student pages 43 – 45

Unit 15–1 1. (a) > (b) < (c) < (d) < (e) > (f) > (g) > (h) = (i) < (j) = 2. (a) 6279 (b) 4438 (c) 3048 (d) 4147 (e) 3298 (f) 5485 3. (a) Triangle 1 2 3 4 5 6

Sides (cm)

Angles (degrees)

three equal sides

three equal angles

two equal sides

two equal angles

Unit 15–2 1. (a) 66 000 (b) 29 000 (c) 57 000 (d) 24 000 (e) 91 000 (f) 37 000 (g) 85 000 (h) 66 000 (i) 18 000 (j) 42 000 2. (a) 3569 (b) 5573 (c) 4687 (d) 4779 (e) 3586 (f) 3529 3. Answers will vary 4. Teacher check Challenge Possible solution

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sides all different lengths angles all different sizes

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(b) They are similar. Challenge Yes

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• Explore the angles to find that the total of angles in a triangle equals 180º.

Consolidation 15–2

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1. (a) 50 (b) 42 (c) 70 (d) 60 (e) 93 (f) 45 (g) 35 (h) 86 (i) 48 (j) 59 3818 (c) 2247 (d) 2465 2. (a) 4806 (b) (e) 3559 (f) 2616 3. (a) 0602, 6.02 a.m. (b) 2227, 10.27 p.m. (c) 0327 (d) 17.12 (e) 0358, 3.58 a.m. (f) 8.39 p.m. (g) 8.39 p.m. (h) 12.24 a.m. 4. (a) 12.45 p.m. (b) 0849 (c) 1928 (d) 7.16 a.m. (e) 0658 (f) 7.04 p.m. Challenge 1 + 2 + 3 = 6 1 x 2 x 3 = 6

• Use a real menu to complete the activity again with a $30 budget.

Consolidation 15–3

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

• Record what you do in a day, recording the times as you go.

New Wave Maths Book F – Teachers Guide • 81 •

Unit 16–1

Student page 46

Outcomes

Indicators The student is able to: • use place value to read, write, say and interpret large whole numbers, oral or written.

N3.3, N4.3, N4.1a

Skills

Resources

• divide • place • round • value • approximately • digit • equal to • decimal place • tenth • hundredth • thousandth

• calculator

• rounding • reasoning

Memory Masters (N3.3)

Teac he r

Number (N4.3)

• 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 (N4.1a)

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• Work with the class as a whole on the first two or three numbers in Exercise 3. Remind the class to focus on the number immediately to the right of the place value to be rounded to. Only when 5 or 0 are in that place value column do students need to check the next column to the right. • Students complete the exercise with careful checking by the teacher. • Exercises 4 and 5 may be completed as soon as students finish the first exercise.

Challenge

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• Direct students to look at a large class place value chart or a small personal chart. • Ask a student or the class to point to various place value columns – ones, hundredths, hundreds, tenths, thousandths, tens, thousands and repeat if felt necessary. • Students may be asked to recite the place value column names from smallest to largest as shown on chart. • Ask students for the rules of rounding. Repeat them to the class: numbers 1 to 4 round down numbers 5 to 9 round up These rules apply regardless of the place value of the number being rounded.

What to do

Notes

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• The focus for this unit is subtraction of a whole number less than 10 from a whole number less than 100.

Warm Up

Language

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• Students use logic and trial and error to solve this problem.

• 82 • New Wave Maths Book F – Teachers Guide

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Unit 16–2

Student page 47

Outcomes

Indicators The student is able to: • realise that different classifications may tell different things and suggest alternative questions to answer new questions. • find the mean, order data to find the middle scores and use a mean to get an estimate of a number. • interpret tables and graphs.

N3.3, N4.3, C&D4.2, C&D4.3, C&D4.4

Skills • recording data • interpreting data

Resources

Language

• calculator

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

• divide • table range • median • mode • scores • tally • total • triangles

Notes

Memory Masters (N3.3)

Teac he r Number (N4.3)

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• The focus for this unit is addition or subtraction of a whole number less than 10 to basic facts of multiplication.

• 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&D4.2, C&D4.3, C&D4.4) Warm up

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What to do

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• Refresh student knowledge of statistical terminology – range, median and mode. • Within the statistics, certain types of information may be used: Range: is the spread of scores from lowest to highest or a spread in which a category may be placed. Median: is the middle score when all the scores are given in ascending/descending order. If two middle scores, then add them and divide by two. Mean: is the average of the scores. Mode: is the most frequently occurring score; there may be more than one mode.

• Explain that the table shows a range of scores that are to be used to tally actual test scores as given on the page. • Ask students to tally the scores on the table provided then write the total of each score range in the space provided. • Using the information from the table, complete the questions in Exercise 4. • Complete Exercises 5 and 6 and share explanations.

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Challenge

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• Students are to record their working and findings so these may be shared with the class and/or teacher. • Not all students will find all the triangles. • Remind students that using diagrams and tables may assist in their search for the triangles.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 160 – 161. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 83 •

Unit 16–3

Student page 48

Outcomes WM4.2, C&D4.2, N4.2, N4.3

Skills • reasoning • estimating • problem solving • recording data • working mentally • speaking and listening • taking risks • collaborative learning and working

Indicators

Resources

• calculator • pencil and paper • jars (optional) • jellybeans (optional)

Language • data • classify • questions • estimate • category

• plan • organise • brainstorm • round

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

What to do

• This activity is designed for students working collaboratively in groups. As students will need to discuss their opinions and ideas, allow enough time so they do not feel rushed and for ideas to evolve. Open-ended tasks such as these are a good opportunity for students to ‘take a risk’ with maths. • The activity is designed to be open-ended and investigative. Students may request resources such as calculators, jellybeans, jars etc. • When completing open-ended tasks, some students may be more successful in mixedability groups rather than same-ability groups. • Some groups will be able to work independently while others may need guidance. The stimulus questions below may prompt such groups. – What are the dimensions of a jellybean? – What resources do you think you could use to solve the problem? – How much does a jellybean weigh? – Can an exact answer be calculated? Why/Why not? – What can we estimate to solve the problem? For example, can we estimate the size of the jar, how many rows of jellybeans in the jar and so on?. – Do you think the colour affects the jellybean in any way? – Is there a better shape for a jellybean that would make it easier to guess the number of jellybeans? • Groups may wish to collate their findings and present them as a poster with diagrams, graphs and calculations. Discuss the reason for the range of answers from the groups.Which groups had answers that were similar (perhaps within 50 jellybeans of each other)? Did these groups use similar estimation and calculation strategies? • Allow each group to discuss and evaluate their ability to problem solve and 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. • Now that students have considered strategies for solving a problem such as this, it may be fun to hold a similar ‘competition’ such as Smarties™ or gobstoppers in a jar.

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Main Activity (WM4.2, C&D4.2, N4.2, N4.3)

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• 84 • New Wave Maths Book F – Teachers Guide

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Unit 16—Answers

Student pages 46 – 48

Unit 16–1

1. (a) 52 (b) 19 (c) 14 (d) 30 (e) 54 (f) 31 (g) 22 (h) 1 (i) 3 (j) 7 2. (a) 400 (b) 500 (c) 200 (d) 500 (e) 600 (f) 600 3. Score Range

56 – 60

61 – 65

66 – 70

71 – 75

Tally

Total

r o e t s Bo r e p ok u S 4. (a) 57, 100 (b) 91 – 95 (c) 96 – 100 5. (a) 57 – 97 (b) 91 – 95 (c) 96 – 100 (d) Teacher check 6. (a) 68 – 100 (b) 96 – 100 (c) 96 – 100 (d) Teacher check Challenge 27

76 – 80

81 – 85

86 – 90

91 – 95

96 – 100

IIII III

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1. (a) 20 (b) 76 (c) 27 (d) 38 (e) 67 (f) 95 (g) 63 (h) 77 (i) 16 (j) 46 2. (a) 119r4 (b) 119r1 (c) 117r2 (d) 147r3 (e) 123r1 (f) 132r2 3. (a) 5.5 5.48 5.483 (b) 2.2 2.21 2.207 (c) 7.4 7.35 7.353 (d) 8.4 8.41 8.411 (e) 4.3 4.27 4.272 (f) 9.6 9.61 9.614 (g) 3.6 3.59 3.588 (h) 6.2 6.21 6.210 (i) 1.9 1.90 1.900 (j) 6.7 6.73 6.729 (d) units (g) units 4. (a) tenth (b) thousandth (e) hundredth (h) hundredth (c) hundredth (f) tens (i) units 5. (a) 1258 (d) 1.5 (g) 3.06 (b) 0.50 (e) 200.03 Challenge A = 2, B = 1, C = 7, D = 8 2178 x 4 = 8712

Unit 16–2

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• Students write a set of rules to follow when rounding.

Consolidation 16–2 • Brainstorm and discuss situations which may use mean, mode and median.

Consolidation 16–3

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1. Answers will vary. Students may come to the conclusion that to estimate a reasonable answer, a jar and ‘jellybean-shaped’ objects are required.

• Try using other lollies or items for students to guess and test their ideas.

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New Wave Maths Book F – Teachers Guide • 85 •

Unit 17–1

Student page 49

Outcomes

Indicators

Resources

N4.2, N4.3, N4.4, S4.3, C&D4.4

The student is able to: • recognise and describe patterns. • decide which of rotation, reflection or translation is involved in producing a symmetrical arrangement and describe it. • interpret information in tables and bar graphs.

• calculator • coloured pencils • ruler • 2 dice • large cutout shape (one to be used by students) • mirror/mira

Skills • recording data • working geometrically

Memory Masters (N4.2)

Language • lines of symmetry • diagonals • shapes • pentagon • triangle • square • octagon • hexagon • sides • relationship • score • multiply

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Notes

Number (N4.3)

• 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 (S4.3, C&D4.4, N4.4) Warm up

• Hold up a large cutout square. Ask students about the following. – How many sides? – How many diagonals? (Ask a student to show them either by drawing the diagonals or folding the shape to show the diagonals.) – How many lines of symmetry? (Remind students that a line of symmetry exists when each side of the line is an exact mirror image of the other. This may be confirmed by using a mirror/mira.) It may assist students if the shape is folded along the lines of symmetry as they are offered by the class. This shows each half as being a mirror image of the other half as they overlap each other exactly.

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

What to do

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• Ask students to use the shapes on the page to complete the table. • When all the information has been gathered, students are to use this to see if they are able to find any relationships between the number of side, diagonals and lines of symmetry. • Share findings in small groups or have selected students share with the class.

Challenge

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• Students are to record their reasoning and findings. • It may be necessary for students to have 2 dice to assist them with their research. Part of this may include throwing the dice. • To answer this question, students may wish to construct a table. For example:

• Answer is 7 (with a 6/36 or 1/6 chance.) Share results with small groups and/or teacher. For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 34 – 35. • 86 • New Wave Maths Book F – Teachers Guide

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Unit 17–2

Student page 50

Outcomes

Indicators

N3.1a, M3.1, N4.3, N4.1a

Skills

The student is able to: • use place value to read, write, say and interpret large whole numbers, oral or written.

Resources

Language

• calculator

• add • city • population • total • order • most • least

• ordering • calculating

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

Memory Masters (N3.1a, M3.1)

Notes

Number (N4.3)

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Teac he r

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

• 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 (N4.1a) Warm up

• Discuss and name the capital cities of Australia. • Discuss which ones are classed as eastern states’ cities (Brisbane, Sydney, Canberra, Melbourne.)

• Read through the activity to ensure students understand what the task is. • Discuss the best way to work out the order of the populations. • Allow students to complete the activity using their preferred method. • Discuss answers as a whole class.

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• Encourage students to think widely about how they can prove the statement. • It may be beneficial for students to work collaboratively in pairs or small groups, recording their workings and sharing their findings with the class.

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For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 56 – 57. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 87 •

Unit 17–3

Student page 51

Outcomes

Indicators The student is able to: • read a calendar and answer relevant questions.

N3.1a, N4.3, M4.2

Skills

Resources • calculator

Language • add • information • calendar • how many • longest • difference • between

• calculating dates

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

Memory Masters (N3.1a, M3.1) Number (N4.3)

• 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 (M4.2) Warm up

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

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• Look at the calendar. • Work with the whole class to identify individual birthdays and mark them on the calendar with a star. Draw a square around Christmas Day and Boxing Day – which dates? • When is New Year’s Day? Mark it with a circle. Students may need to be told when Easter is and to mark the two days with circles. • When is Anzac day? Mark it with a circle. • Tell students when the school holidays are. • Students should follow through the rest of the activities with the teacher to ensure that each activity is understood. • Set students to work to complete activities. • Check student work – remember several answers will vary due to different dates being used.

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© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Begin with a general discussion about calendars, why we have them, months of the year, days in each month, weeks in a year, special days (Easter, Anzac, Christmas, Birthdays etc.).

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• Anzac Day is always on 25 April but the date for Easter changes each year.Why? Investigate. It may help you to research information on lunar calendars.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 120 – 121. • 88 • New Wave Maths Book F – Teachers Guide

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

Unit 17—Answers

Student pages 49 – 51

Unit 17–1

1. (a) 1930 (b) 0215 (c) 2345 (d) 0600 (e) 1200 (f) 11.20 a.m (g) 9.40 p.m. (h) 5.50 a.m. (i) 10.10 a.m. (j) 5.15 p.m. 2. (a) 196.35 (b) 147.44 cm (c) $208.23 (d) 18.232 km (e) $149.64 3. (a) Sydney, Melbourne, Brisbane, Perth, Adelaide, Canberra, Hobart, Darwin (b) 12 250 400 (c) 9 683 200 (Brisbane, Sydney, Canberra, Melbourne, Hobart) (d) 2 567 200 Challenge Answers will vary 7/12 < 2/3 7/12 < 2/3 x 4/4 = 8/12 7/12 < 8/12

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4. The number of lines of symmetry equals the number of sides. The number of diagonals = Triangles = 1 + 2 - 3 = 0 Square = 1 + 2 + 3 - 4 = 2 Hexagon = 1 + 2 + 3 + 4 + 5 - 6 = 9 Pentagon = 1 + 2 + 3 + 4 - 5 = 5 Octagon = 1 + 2 + 3 + 4 + 5 + 6 + 7 - 8 = 20 Challenge 2178 x 4 = 8712 A = 2, B = 1, C = 7. D = 8

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1. Answers will vary 2. (a) 108 (b) 87.8 m (c) $125.90 (d) 161.6 (e) 83.0 L (e) 166.8 3.

Unit 17–2

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• Continue the activity with a nonagon, decagon etc.

Consolidation 17–2 • Record the populations on a place value chart found on pages 207 and 208.

Consolidation 17–3

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1. (a) $32.84 (b) $0.14 (c) $0.92 (d) $32.84 (e) $91.76 (f) 491c (g) 1853c (h) 1c (i) 87c (j) 503c 2. (a) 171.142 (b) 106.131 (c) 551.87 (d) 235.141 (e) 184.385 (f) 83.446 3. Teacher check (a) – (j) (k) 34 weeks and 6 days Challenge Answers will vary

• Highlight on the calendar special events and days which may be celebrated by the class.

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New Wave Maths Book F – Teachers Guide • 89 •

Unit 18–1

Student page 52

Outcomes

Indicators The student is able to: • add and subtract money measures with equal numbers of decimal places.

N4.3

Skills • reasoning

Memory Masters (N4.3)

Resources • calculator

• subtract • cost • difference • charged • half price • numbers • row • column • diagonal

Teac he r

Number (N4.3)

• 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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• The focus for this unit is to encourage students to find ways to simplify number problems and to support their approach. For example: If you had one wish and could change one digit in the following question, which one would you change? Explain why. 14 x 6 I would change the 6 to a 5 because it is easier to multiply by fives.

Main Activity (N4.3) Warm Up

Language

What to do

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• The menu in this case is set with the option of ordering tea or coffee. Each of the three courses is charged against the patrons regardless of whether they order the course or not; that is to say both adults and children will be charged for the entree, main meal and desserts. • Students are required to find the cost of an adult meal including tea or coffee and the cost of a child’s meal excluding tea or coffee – remember tea and coffee are not included. • Having found the cost of an adult meal and a child meal it now becomes necessary to find the cost of two adult and two child meals so that the total meal cost can be found. Use space on the page for working if needed. • Students are then to find the difference between the cost of an adult and a child meal.

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• Remind students that they recently completed an activity using a menu to make up sums to find the costs of meals.

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• Students will need to arrange the four digits in the rows and columns provided so that each digit appears in each row and column once. • Students are to record their working and describe, in writing, their thinking processes. • Display finished activity.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 80 – 81. • 90 • New Wave Maths Book F – Teachers Guide

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

Unit 18–2

Student page 53

Outcomes

Indicators The student is able to: • represent data in diagrams and tables which may include arrow diagrams, Venn diagrams and twoway tables.

N3.3, N4.3, C&D4.3

Skills • reasoning

Resources

Language

• calculator • ruler • pencil • markers in four different colours • coloured counters

• subtract • different ways • arrange • arrangements • congruent shapes

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Notes

Memory Masters (N3.3)

Teac he r

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• The focus for this unit is addition of a whole number less than 10 to a whole number less than 100.

Number (N4.3)

• 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&D4.3) Warm up

What to do

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• Take one marker and ask the class in how many different arrangements it can be placed in a line. (1) • Take two different markers and ask students in how many different arrangements they can be placed in a line. (2) • Take three different coloured markers and ask students to find how many different ways, they can arrange them in a line. (6) It may help to record the arrangements as they are found. • Take four different coloured markers and tell students that this is the activity they will be completing—finding how many different arrangements can be made using the four markers in a line. • Students could also use coloured blocks. Discuss the arrangements of triple-decker icecreams.

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• Students are asked to record the arrangements as they find them. It may assist them to use four different coloured counters to represent the four letters. Alternatively, by recording the letters in the places provided in their workbook on page 53, students may be able to find all the possible combinations. • Remind students that working systematically through the activity assists in ensuring that all combinations have been found; e.g. start with A and arrange the other letters in an orderly fashion until all combinations beginning with A have been used, then move to B and so on.

Challenge • Students will need to show all the possibilities they explored before finding the solution to this problem. • Share results with the class/teacher.

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New Wave Maths Book F – Teachers Guide • 91 •

Unit 18–3

Student page 54

Outcomes

Indicators The student is able to: • enter divisions into the calculator. • use properties of operations to complete and justify number sentences without completing the calculations.

N3.3, N4.3, N4.2

Skills • reasoning • evaluating

• subtract • circle • divisible • calculator • rule • magic square • row • column • diagonal

• calculator

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

• The focus for this unit is subtraction of a whole number less than 10 from a whole number less than 100.

Number (N4.3)

• 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 (N4.2, N4.3) Warm up

Language

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Memory Masters (N3.3)

Resources

• Use a tables chart to assist students to identify the multiples of 3. • Where a multiple of 3 has more than one digit in it ask students to add the digits and tell you what they find. (All the added digits totalled 3, 6 or 9.) • Remind students that these answers (3, 6 and 9) were obtained by adding the digits of the multiples of 3. Therefore they are all divisible by 3. Extend this knowledge and apply it to Exercise 3 on page 54 of the workbook.

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• When students have completed Exercise 3 ask them to develop their own rules for numbers that are divisible by 5 and for 4. • Exercise 5 requires studnets to evaluate the activity and support their reasoning.

Challenge

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What to do

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• Remind students that magic squares must have all columns, rows and diagonals adding to the same total. • Note: There are nine different ways you can arrange the numbers 1 – 9 in a magic square. • Students are required to record all their working to share with the teacher and perhaps the class.

• 92 • New Wave Maths Book F – Teachers Guide

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

Unit 18—Answers

Student pages 52 – 54

Unit 18–1

1. (a) 27 (b) 97 (c) 96 (d) 41 (e) 82 (f) 45 (g) 31 (h) 73 (i) 61 (j) 80 2. (a) 0.6 (b) 0.5 (c) 0.1 (d) 0.1 (e) 0.1 (f) 0.1 3. 24 different ways A B C D B D A C D B A C A B D C B D C A D B C A A C B D C A B D D C B A A C D B C A D B D C A B A D B C C B A D A D C B C B D A B A C D C D A B B A D C C D B A B C A D D A B C B C D A D A C B (Short cut: 4 x 3 x 2 x 1 = 24. For the first letter there are 4 possibilities, second there are 3, third there are 2 and fourth there is only 1 possible letter.) Challenge

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1. Answers will vary 2. (a) 1997 (b) 2999 (c) 5998 (d) 2999 (e) 2994 (f) 5997 3. (a) Adult Child $ 6.90 $15.80 $ 6.90 $ 5.95 $ 7.90 $ 2.00 $ 5.95 $30.65 $20.75 (double) $ 30.65 (b) Adults Children (double) $ 41.50 $102.80 $ 30.65 (b) Adult Meal Child Meal $ 20.75 $ 9.90 Challenge

Unit 18–2

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• Select items to make a meal from a grocery store catalogue and add to find the total.

Consolidation 18–2

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1. (a) 26 (b) 66 (c) 37 (d) 24 (e) 51 (f) 58 (g) 41 (h) 68 (i) 87 (j) 28 8.2 (c) 4.3 (d) 3.2 (e) 2.4 2. (a) 6.6 (b) (f) 9.1 3. (a) 678 555 849 (b) If the sum of the digits is a multiple of 3, then the number is divisible by 3. 4. (c) 4795 9460 (d) If the last two digits are divisible by 5, then the number is divisible by 5. 5. (e) 5236 6152 3796 (f) If the last two digits are divisible by 4, then the number is divisible by 4. 6. Answers will vary. Challenge 15

• Have students in groups of five. Rearrange them to find the possible orders available. Record.

Consolidation 18–3

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

• Find numbers divisible by 6 and state an appropriate rule.

New Wave Maths Book F – Teachers Guide • 93 •

Unit 19–1

Student page 55

Outcomes

Indicators

N3.3, N4.3, S4.3

Resources

The student is able to: • decide which of rotation, reflection or translation is involved in producing a pattern or formation.

Skills

Memory Masters (N3.3)

• multiply • polygons • rotational symmetry • lines of symmetry • sides • points of symmetry • shapes

• calculator • mirror/mira • ruler • pencil • plastic polygons • overhead

• symmetry • measuring

Language

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

Number (N4.3)

• 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 (S4.3) Warm up

• Display a set of polygons from a central location as models or drawings on an overhead. • Ask students to show lines of symmetry – explaining why they chose those particular lines. Ask the class for their comments on the choice of lines of symmetry. • This activity may also be completed by students in small groups with the teacher circulating to check their findings. • Working from a central location, or in small groups, ask students to find points or lines of rotational symmetry. Students may need to be given an understanding of rotational symmetry – a point or a line about which a shape may be rotated. An order of rotational symmetry being the line(s) and/or point(s) of rotational symmetry. For example;

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• The focus for this unit is addition or subtraction of a whole number less than 10 to or from basic facts of multiplication.

2 3

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4 3

3 4

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point of origin for rotational symmetry

What to do

• Students are to apply the knowledge to answer the questions in Exercise 2. • Use the shapes to show working. • Share findings with the class.

Challenge • All the letters with straight lines are above the line, while all the curved letters are below the line. For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 36 – 37. • 94 • New Wave Maths Book F – Teachers Guide

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

Unit 19–2

Student page 56

Outcomes

Indicators

N3.3, N4.3, N4.1b

The student is able to: • use materials and diagrams to represent fractional amounts where the ‘whole’ may be an object, quantity or collection.

Skills • interpreting diagrams

Resources

Language

• calculator • 2-cm cubes • fraction cake • fraction grid (see page 200) • various objects/ counters

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• multiply • diagram • fraction • arrange • construction • individual • surfaces • block

Memory Masters (N3.3)

Notes

Number (N4.3)

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• The focus for this unit is addition or subtraction of a whole number less than 10 from basic facts of 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 (N4.1b) Warm up

• Distribute a copy of a fraction grid, showing up to tenths, to each student. • Ask students to point to various fractions – 1/5, 1/8, 1/2, 1/3, 2/6, 3/8, 9/10, 4/5 etc. Check by circulating around the room that students are able to identify these fractions. • Distribute counters, 2-cm cubes or other objects that may have been chosen for this activity. • Ask students to take three of the same object and place together as a group on their desk. Ask students to show 2/3 of this group. Circulate to check. Repeat this activity for 5/8, 3/10, 3 /4, 2/5.

What to do

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• Explain that they are to colour each of the three diagrams for each question to show the fraction equivalent shown at the beginning of the question. • Work through Exercise 3 with the whole class. Colour 3/4 of the first diagram. How many columns will be coloured? Colour 3/4 of the second diagram. How many small squares will be coloured? Is there more than one way to do this? Students tell these different ways. In 3(a), how many sectors of the circle will be coloured? Colour them. • Direct students to complete the rest of the exercise.

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Challenge

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• Students will require six 2-cm cubes for this activity.They are required to make a construction that has 24 individual surfaces exposed on the faces of the construction. • The final construction should be shown to the teacher. • All attempts should be recorded as diagrams or in written descriptions.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 62 – 63. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 95 •

Unit 19–3

Student page 57

Outcomes

Indicators The student is able to: • express measures of length, capacity and mass using common metric prefixes. • represent data in diagrams and tables which may include arrow diagrams, Venn diagrams and twoway tables.

N4.1a, N4.3, M4.2, C&D4.3

Skills • recording • measuring

Resources • calculator • tape measure • height measuring stick

Language • multiply • table • mass • height • tally • Carroll diagram • hand span

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

Memory Masters (N4.1a) Number (N4.3)

• 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 (M4.2, C&D4.3) Warm up

• Divide the class into two groups, one to have its members’ heights measured and the other to have members’ hand spans measured. Depending on available measuring sticks and tape measures, further divide these two groups into smaller groups.

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• The focus for this unit is ordering numbers.

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• As one measure is taken, students record it on page 57 of their workbook, then move to the other group to have the second measure taken. • When all measurements have been taken, students return to their seats. • Students in order as directed by the class teacher call the height and hand span that they have written down for themselves so that the rest of the class can record these in their own workbooks. Actual recording order is irrelevant. • When all student measures have been recorded, students then tally hand span against height. For example, all students who are between 135 – 139 centimetres in height are tallied under hand span range given, such as 18 – 19 cm. • Students will need to double check to ensure that all measurements have been transferred to the Carroll diagram.

Challenge

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• Students can be directed to make a scatter graph to show the details recorded on the Carroll diagram as an extension of this activity. The scatter graph would display height against hand span as shown on the Carroll diagram.

• 96 • New Wave Maths Book F – Teachers Guide

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

Unit 19—Answers

Student pages 55 – 57

Unit 19–1

1. (a) 12 (b) 13 (c) 11 (d) 11 (e) 10 (f) 0 (g) 0 (h) 1 (i) 1 (j) 1 2. (a) $201.60 (b) $153.50 (c) $301.80 (d) $ 2 0 3 . 0 0 (e) $425.60 (f) $152.10 3. 3 4

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7 10

3 8

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1. (a) 23 (b) 40 (c) 37 (d) 23 (e) 25 (f) 10 (g) 47 (h) 24 (i) 65 (j) 0 2. (a) 30.4 (b) 25.5 (c) 27.2 (d) 25.2 (e) 34.2 (f) 25.5 3. (a) A, B, C, D, F, I, K (b) A, C, F, K (c) A, C, F, K Challenge The letters above the line are straight-line letters. The letters below the line have curves.

Unit 19–2

Challenge

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• Find objects in nature which have symmetry; e.g. butterflies.

Consolidation 19–2 • Use real-life objects—e.g. apples, oranges, cake—to cut into fractions.

Consolidation 19–3

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1. (a) 0.0899 (b) 0.9482 (c) 3.946 (d) 4.729 (e) 6.87 (f) 6.94 (g) 30.0879 (h) 57 (i) 84 (j) 86.21 2. (a) $54.00 (b) $111.00 (c) $92.00 (d) $72.00 (e) $282.00 (f) $210.00 3. Teacher check

• Measure arm span and compare results with hand span.

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New Wave Maths Book F – Teachers Guide • 97 •

Unit 20–1

Student page 58

Outcomes N4.1a, N4.3, N4.4

Skills • speaking and listening • understanding patterns

Indicators

Resources

Language

The student is able to: • identify and describe several different patterns in an array of numbers such as the 100 chart or a similar 6 by 6 chart.

• calculator • counters • overhead projector

• divide • patterns • numbers • properties • arithmetic • column • represented • grid • triangle • twice • row • square numbers

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

Memory Masters (N4.1a) Number (N4.3)

• 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 (N4.4) Warm Up

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• The focus for this unit is rounding of decimals to the nearest whole number.

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What to do

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• Talk to the class about early mathematical discoveries, particularly by the Greeks. They discovered many interesting patterns as they explored the number system. Two such patterns are square and triangular numbers. • Ask the class if they could explain why these patterns were described as square numbers and triangular numbers. • Show on an overhead or blackboard/whiteboard how these numbers are arranged. The number 1 is always the starting point. Square numbers may be arranged in the form of a square. The next square number is? (Students may be asked to try to make their arrangements themselves to discover the patterns without first seeing them demonstrated. In this case, encourage them to work in small groups to explore the patterns then share their findings with the class.) • Ask students how they might build on to this square to find the next number in the pattern and so on. • Repeat the process for triangular numbers. (Another approach may be to demonstrate for square numbers and allow students to explore for themselves the patterns in triangular numbers.)

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• Complete activities on page 58 of workbook.

Challenge

• Students are to use the numbers 1, 2 and 3 to arrange them on the grid so that each number occurs in each row and column only once. • All attempts and thinking processes are to be recorded.

• 98 • New Wave Maths Book F – Teachers Guide

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

Unit 20–2

Student page 59

Outcomes

Indicators

N4.2, N4.3 ,C&D4.2, C&D4.3, C&D4.4

The student is able to: • construct and use their own categories to answer specific questions. • represent data in diagrams and tables. • describe information from diagrams such as Venn diagrams.

Skills • interpreting diagrams

Resources

Language

• calculator • overhead of a Venn diagram

• divide • survey • Venn diagram • how • many

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

Notes

Memory Masters (N4.2) Number (N4.3)

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

• 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&D4.2, C&D4.3, C&D4.4) Warm up

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What to do

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• Ask students what a Venn diagram is. Show a two overlapping circle Venn diagram. Explain that the diagram is used to show characteristics of data collected. • Give some examples; e.g. boys and girls in the class shown on the Venn diagram would have no recording in the overlap segment. • Focus on language associated with Venn diagrams such as ‘and’ and ‘or’. • If we record the students who have only brothers in one segment, sisters in the other segment, then those with both brothers and sisters would be recorded in the overlapping segment. (Those with no siblings would be recorded outside the two overlapping circles.)

• A survey of a class of students indicates the means by which they travel to school. This data may be transferred to the Venn diagram. • Suggest to students that it may be best to record the multiple options first – those who at some stage walk, ride or are driven; those who ride or are driven; those who ride or walk; those who walk or are driven. Once these are noted it is possible to then determine how many only walk, only ride or only travel by car. • Remind students that of the 23 who walk, some also ride or are driven; therefore, the walk only segment will be less than 23. • The total of all segments of the Venn diagram will be the class total. • Students use the information that they recorded to answer the questions on the page in their workbook. • Share the results with the class.

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Challenge • Make up a Venn diagram for a friend to solve. Begin by completing the Venn diagram and then write the explanation and questions to go with it.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 162 – 163. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 99 •

Unit 20–3

Student page 60

Outcomes

Indicators The student is able to: • use equivalences that are readily visualised to compare and order fractions.

N3.3, N4.3, N4.1b

Skills

Resources • calculator • cuisenaire rods

• reasoning

Teac he r

Memory Masters (N3.3)

Language • divide • round • nearest • arrange • order • smallest • largest • fractions • decimals • comparison • fraction grid • whole number

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

• The focus for this unit is the commutative property of addition.

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

Main Activity (N4.1b) Warm up

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Number (N4.3)

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What to do

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• Set class to complete Exercise 3 or work through each activity with the class as a whole. • Exercise 4 requires fractions to be changed to decimals or vice versa. Use this as a whole class exercise to find the equivalent fraction and/or decimal for each given. • Working with the class as a whole order the fractions and decimals for each group.

Challenge

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• Distribute cuisenaire rods and allow students a brief time to play freely with the materials. • Using the cuisenaire rods, or the fraction grid on page 60 of your workbook, make up the following fractions; 5/6, 1/4, 2/3, 1/10 and 3/4. • If using cuisenaire rods place the equivalent 1 rod next to the fraction. • Using the comparison shown with the rods or the fraction grid, look at each fraction in turn and tell me whether it is closer to one whole or zero. Explain how you came to this answer. Ask the class if they have any other ideas. • When rounding fractions the same rules apply in principle as for whole numbers. If the numerator (top number), is less than half of the denominator (bottom number), you round down. If the numerator is greater than half the denominator you round up. If the numerator is equal to the denominator you may round up or down.

• Students are required to use their creativity to come up with an answer to this problem. • Encourage them to experiment, discuss and record their ideas.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 62 – 63. • 100 • New Wave Maths Book F – Teachers Guide

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

Unit 20—Answers

Student pages 58 – 60

Unit 20–1 1. (a) 8 (b) 24 (c) 64 (d) 47 (e) 3 (f) 9 (g) 153 (h) 77 (i) 924 (j) 4 2. (a) 1184r6 (b) 1220r5 (c) 1456r3 (d) 1342r3 (e) 1228r4 (f) 1349r3 3. (a) 1, 4, 9, 16, 25, 36, 49 (b) 1, 4, 9, 16, 25, 36, 49 (c) 64, 81, 100, 121, 144, 169, 196 (d) 1, 3, 6, 10, 15, 21, 28, 36, 45, 55 (e) 66, 78, 91 Challenge

3 1 2

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1. Teacher check 2. (a) 2 (b) 3 (c) 2 (d) 4 (e) 2 (f) 1 3. Walk 13 2

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4. (a) 23 (b) 11 (c) 3 (d) 13 (e) 19 (f) 29 (g) 29 (h) 32 (i) 3 (j) 32

5 3 Bike

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Unit 20–2

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Consolidation 20–2

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1. (a) 5, 13 (b) 6, 9 (c) 2, 10 (d) 7, 11 (e) 3, 5 (f) 5, 11 (g) 7, 16 (h) 5, 14 (i) 8, 14 (j) 3, 7 2. (a) 2 (b) 2 (c) 4 (d) 2 (e) 3 (f) 2 3. (a) 0 (i) 1 (b) 1 (j) 1 (c) 1 (k) 0 (d) 1 (l) 1 (e) 1 (m)0 (f) 1 (n) 0 (g) 0 (o) 1 (h) 1 (p) 1 4. (a) > (d) > (g) > (b) < (e) > (h) < (c) > (f) > (i) < Challenge Answers will vary

• Develop a question and follow up with data collection to complete a Venn diagram.

Consolidation 20–3

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

• Develop further examples for students to complete.

New Wave Maths Book F – Teachers Guide • 101 •

Unit 21–1

Student page 61

Outcomes

Indicators

N3.3, N4.3, S4.3

The student is able to: • reduce a diagram using scale as a reference.

Skills

Resources

• divide • redraw • shape • grid • one-third of the dimensions • total • equal

• calculator • pencil • ruler

• scaling

Teac he r

Memory Masters (N3.3)

Language

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

Notes

• The focus for this unit is the addition of two multiples of 10 each less than 100.

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

Main Activity (S4.3) Warm up

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• Ask students to look at the shape drawn on the page. How many grid units wide is the top of the head? (3) You will be asked to redraw this shape one-third the size of the original. How many grid units wide will the head be in the redraw? (1) • Ask how tall (long) is the original drawing. (21) How tall will the redraw shape be? (7) • Ask where the ideal place to draw would be to draw the smaller shape. There are many possibilities – but the best to ensure everything will fit is to draw the head from the top of the page in the centre square. • Instruct students to follow the same process – count the grid units, located as previously suggested, then work on the lines that are attached to the head and then the lines attached to these lines and so on until the redraw is completed. • Work out the perimeter of each of the shapes and record on the table provided. • Check your results to see if there are any relationships between the results. • Ask students to design a shape/model to be enlarged.

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• Discuss scale etc. such as scale models. Where is scale used in everyday situations?

What to do

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Number (N4.3)

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• Students are to place the digits 1 – 6 in the rows so that each row totals 10. • Show all attempts and record your thinking processes to share with the class or your teacher.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 144 – 145. • 102 • New Wave Maths Book F – Teachers Guide

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

Unit 21–2

Student page 62

Outcomes

Indicators

N3.3, N4.3

The student is able to: • order numbers to make addition easier.

Skills

Resources

Language • divide • rearrange • lists • numbers • addition • group • symbols

• calculator

• working mentally

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

Notes

Teac he r

Memory Masters (N3.3)

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• The focus for this unit is subtraction of a multiple of 10 less than 100 from a multiple of 10 less than 100.

Number (N4.3)

• 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 (N4.3) Warm up

• Explain to students that your ultimate aim is to be able to work all problems mentally. To do this it helps to look for combinations that are easy to add. Such combinations are those that add to 10; e.g. 6 + 4, 3 + 7 etc. (compatible numbers), doubling and adding on 9 by adding 10 and taking 1. • Where there are sets of several numbers to add, finding these combinations within the groupings will usually mean reorganising the order of the numbers.

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What to do

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• Ask students what groupings they can use in (a) that may help make the addition of the ones easier; e.g. 2 + 8, 5 + 5 and 9. The total is 29. • Ask students to find groupings in the tens, including the 2 from the regrouping of the ones total. For example, 2 + 8, 4 + 6 , 2 + 3. Total is 25. • Students are to rewrite their columns so that the pairs of numbers are grouped. • Complete the exercise. • For Exercise 4 you may use the regrouping strategy from Exercise 3 or round the numbers to gain an appropriate answer to check whether the sums are correct. • Work through the first two: (a) looks correct; (b) by checking the thousands it shows the answer should be greater than 9; therefore, it needs to be recalculated.

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Challenge

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• Students will need to experiment with adding, subtracting, multiplying and dividing to find ways of making other numbers using a single number and the four operations. (Brackets may also be used.) • As a hint, ask students how they might make the number 1 from the number 5 using one or all of the operations: 5 ÷ 5 = 1. • Leave students to experiment either by themselves or in small groups. • Share answers with the class and encourage the finding of different combinations. Praise efforts but don’t give answers that students can not find. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 103 •

Unit 21–3

Student page 63

Outcomes

Indicators The student is able to: • use perimeter to solve real-life problems.

N3.3, N4.3, M4.4a

Skills

Resources • calculator

• divide • calculate • length • total • digits • correct • counting • order • number sentence • equals

• measurement

Memory Masters (N3.3)

Number (N4.3)

• 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 (M4.4a) Warm up

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

Teac he r

Language

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• Explain to the class that they are required to find the distance around the inside of each of the rooms so that the length of cornice required to complete the task is known. • Ask students how they would find the distance around each room (perimeter). Students may offer several solutions. The two most preferred methods are to add the lengths of all sides, or to add the length and width and double the result. • Ask students to find the lengths of the walls that have not been noted on the page. The walls of the inside largest shape will create the most discussion. For those who are unable to see how to find these lengths, remind students that the internal walls are the same length as the external walls. In the case of the bottom side walls, these are each the same length as that shown on the outside right wall. • Once individual shape perimeters have been found, those totals are to be added.

Challenge

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• Students can use their workings from above, deleting the doorways to calculate the length of the skirting required.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 136 – 137. • 104 • New Wave Maths Book F – Teachers Guide

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

Unit 21—Answers

Student pages 61 – 63

Unit 21–1 (d) 70 (e) 90 (i) 80 (j) 80 (d) 3r1 (e) 3r2

1. (a) 20 (b) 40 (c) 70 (d) 30 (e) 10 (f) 10 (g) 50 (h) 40 (i) 0 (j) 40 2. (a) 6 (b) 4 (c) 4 (d) 7 (e) 8 (f) 4 285 (c) 315 (d) 275 (e) 255 3. (a) 259 (b) (f) 238 (g) 247 (h) 167 (i) 168 (j) 163 4. (a) Correct (b) Incorrect – 21 246 (c) Incorrect – 13 945 (d) Incorrect – 64 083 (e) 24 445 (f) 41 819 (g) 221 855 (h) 103 954 Challenge Answers will vary. Possible answer: 5 – (5 ÷ 5) = 4

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4. (a)

(b) Answers will vary; e.g. the reduced shape has a perimeter 1/3 and area 1/9 of the original shape.

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1. (a) 60 (b) 80 (c) 60 (f) 100 (g) 90 (h) 80 2. (a) 2r4 (b) 2r5 (c) 4r3 (f) 2r8 3.

Unit 21–2

Challenge

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• Use the 1-cm grid on page 199. Students design their own picture with a total area of 10 squares. Give to another student to redraw at twice the size.

Consolidation 21–2

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1. (a) 120 (b) 100 (c) 270 (d) 150 (e) 210 (f) 160 (g) 160 (h) 100 (i) 80 (j) 120 7r15 (c) 5r27 (d) 8r17 (e) 8r11 2. (a) 4r1 (b) (f) 7r4 3. (a) 94.5 m (b) $1011.15 Challenge 86.85 m, $390.85

• Develop further examples for students to practise their estimation skills.

Consolidation 21–3

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• Students can complete the same task for cornicing the classroom or another room in the school grounds.

New Wave Maths Book F – Teachers Guide • 105 •

Unit 22–1

Student page 64

Outcomes

Indicators

Resources

The student is able to: • identify, describe and continue patterns, linking pairs of numbers on a coordinate grid or in a table by a single digit operation.

N3.3, N4.3, N4.4

Skills • encoding • decoding • reasoning

Memory Masters (N3.3)

• calculator

Language • subtract • code • message • number • group

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Notes

Number (N4.3)

• 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 (N4.4) Warm Up

• Talk to students about codes, using the example of morse code as an original means of transmitting messages via wire to distant locations. • During times of war, or if countries and businesses see the need to send confidential messages to the members of their organisations, they use codes. In war, the enemy can be desperate to break these codes to find out what is happening. People known as ‘codebreakers’ played an important part in WWII. For example, here is morse code:

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

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Letter Morse Letter Morse Letter Morse A .- H .... O B -... I .. P C -.-. J .--- Q D -.. K -.- R E . L .-.. S M -- T F ..-. G --. N -. U

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What to do

Letter Morse --- .--. --.- .-. ...

V ...W .-X -..- Y -.-- Z - - . .

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• A simple code can be developed using the alphabet. Look at the code given in Exercise 3(a). Use this code to make a message for your partner to look at and read. • Use the alphabet in Exercise 4(a) to make your own code. Use your code write a message on the page.

Challenge Technology Opportunity! • Look a the numbers given and use logical reasoning to work out which number does not belong to the group. ☛ Investigate codes and ciphers on the Internet. • Show all working for sharing with class or teacher. • 106 • New Wave Maths Book F – Teachers Guide R.I.C. Publications® www.ricpublications.com.au

Unit 22–2

Student page 65

Outcomes

Indicators The student is able to: • suggest what data to collect to help estimate numbers or quantities. • order data from highest to lowest. • find the mean and use it to estimate. • interpret and report on information provided in tables.

N3.4, N4.3, C&D4.2, C&D4.3, C&D4.4

Skills • organising data • reasoning

Resources

Language

• calculator • class members’ ages

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• number patterns • subtract • record • tally • total • youngest • eldest • range • median • middle • common • Venn diagram • mode

Memory Masters (N3.4)

Notes

Number (N4.3)

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• The focus for this unit is completion of number patterns including some with two combined 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&D4.2, C&D4.3, C&D4.4) Warm up

• Refresh students’ memory of range, median and mode. • Tell students that they will be recording the ages of all class members on the table provided in the workbook.

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Challenge

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• Ask students to stand when their age range is called out. Call the age ranges as shown in the workbook, starting at 10 years through to 10 years 3 months for the first group. It may be worthwhile to have students work their ages out prior to this by writing the number 0 – 12 on the blackboard/whiteboard and writing the corresponding months below or beside. For example, if this activity is completed in May, students born in May will be 0, students born in June 1 and so on. • When the tally of all students has been recorded, write the total below. • Use the information to answer the questions in Exercise 3. • For Exercise 4 ask students who have only brothers to stand in one group, those who have only sisters to stand in another group, those who have both brothers and sisters to stand in a third group, and those who are only children to stand in a fourth group. Students in each group are then recorded on the Venn diagram. Write the only child group outside the circles.

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• Refresh students’ memory of range, median and mode. • The mean of three numbers is 60. What might the numbers be?

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 160 – 161. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 107 •

Unit 22–3

Student page 66

Outcomes

Indicators The student is able to: • use a calculator to multiply and divide a sequence of calculations.

N4.1a, M4.1, N4.3

Skills

Resources • calculator • place value chart (see pages 207 and 208)

• reasoning • calculating

Language • kilograms • subtract • numbers • calculator • multiplying • grams • dividing • place value

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Memory Masters (N4.1a, M4.1)

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

• 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 (N4.3) Warm up

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Number (N4.3)

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What to do

• Students are to complete the activity by following the directions on the page. Start by entering 10 000 into the calculator then x and 10. Write the result in the space provided. Continue to follow the instructions, dividing or multiplying by multiples of 10. • In Exercise 4 students will need to find whether they are to multiply or divide the starting number by a multiple of 10, 100 or 1000 to obtain the next answer. • To assist, ask students what they need to do if the next number is smaller (÷) or larger (x).

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Challenge

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• Ask all students to take out their calculator and a place value chart or look at a large class place value chart. • Ask students to enter 24 into their calculator and then have students suggest what will happen if x and 10 are then entered into the calculator. Repeat this process with 16 x 100, 47 x 10, 83 x 100. • Using the place value chart, show where each number would be on the chart; e.g. 24 = 2 tens and 4 ones, then 24 x 10 becomes 2 hundreds, 4 tens and zero ones. • Multiplying by 10 moves the digits one place to the left. Multiplying by 100 moves the digits two places to the left.

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• Using the information from the main activity, students are to give a sound mathematical shortcut to multiplying or dividing by 100. Note: Decimal points do not move. Numbers increase and decrease in size. • Record all thinking processes and workings for sharing with class/teacher.

• 108 • New Wave Maths Book F – Teachers Guide

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Unit 22—Answers

Student pages 64 – 66

Unit 22–1 (d) 10 (e) 20 (i) 10 (j) 10 (d) 2.1 (e) 4.3

1. (a) 76 (b) 92 (c) 52 (f) 10 (g) 60 (h) 20 2. (a) 0.9 (b) 3.8 (c) 6.4 (f) 2.6 3. Teacher check 4. Teacher check 5. Teacher check

(d) 74 (e) 75 (i) 400 (j) 67 (d) 4.9 (e) 0.7

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1. (a) 20 (b) 20 (c) 30 (f) 40 (g) 30 (h) 10 2. (a) 6.1 (b) 6.2 (c) 4.7 (f) 6.2 3. Teacher check 4. Teacher check Challenge 13 (not a square number)

Unit 22–2

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• Students can develop their own code using symbols, letters or numbers.

Consolidation 22–2

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1. (a) 0.528 kg (b) 2.464 kg (c) 0.3 kg (d) 0.09 kg (e) 0.256 kg (f) 4800 g (g) 900 g (h) 1820 g (i) 50 g (j) 61 084 g 2. (a) 0.42 (b) 0.41 (c) 0.33 (d) 0.12 (e) 0.12 (f) 0.22 3. (i) 1 000 000 (j) 10 000 (a) 100 000 (h) 100 000 (k) 10 (b) 1000 (g) 10 000 (l) 100 000 (c) 100 (f) 10 (m)1000 (d) 100 000 (e) 1000 4. (a) ÷100 (b) ÷10 (c) x1000 (f) ÷10 (e) ÷10 (d) x10 Challenge Answers will vary

• Students develop a research question as a class; collect and record data to find the mode, median and range.

Consolidation 22–3

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• Students develop their own number chains. Swap to solve.

New Wave Maths Book F – Teachers Guide • 109 •

Unit 23–1

Student page 67

Outcomes

Indicators

N4.1a, N4.3, S4.2

The student is able to: • use some mathematical conventions in drawings.

Skills • technical drawing • observation • recording • discussing • comparing

Resources • calculator • compass • protractor • pencil • ruler

Language • divide • construct • right angle • compass • angle • triangle • observe • environment

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Notes

Teac he r

Memory Masters (N4.1a)

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

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

Main Activity (S4.2) Warm up

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Number (N4.3)

• Remind students of correct set up for compass with compass point and pencil both level. • Remind students that construction lines are light. • Remind students that the compass should be operated by holding the top to turn the pencil end and that the radius needs to be a reasonable size, 5 – 8 cm, for more accurate construction.

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• Walk through the construction requirements for drawing a right angle. Place compass point at M and draw a small arc across the base line to the right of M and another to the left of M. • Open the compass wider and mark two arcs above M with the compass point placed alternatively at each arc on the base line. Construct a 45º by bisecting the right angle. This may be done by placing the compass point alternately on the intersection of the two arcs at the top of the 90º line and at one of the areas on the base line and drawing arcs. Join the intersection of these two arcs to the starting point on the base. • Check accuracy of construction using a protractor. • To construct a triangle with an angle of 45º follow the above construction on the base line. Draw a third line from the 45º arc intersections to a point on the base line below the arc intersection to make a triangle with a 45º angle. Alternatively, repeat above to the right and left of M and complete a triangle with two 45º angles and one right angle. • Encourage students to observe their own environment by seeking out 45º angles and right angles. Discuss where they may be found. Which angle is more common?

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What to do

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Challenge • Students follow given directions using written methods and record all findings for sharing later. Using written methods provides the students with remainders which will aid the students in finding the pattern. • Write an explanation of the discovery.

• 110 • New Wave Maths Book F – Teachers Guide

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Unit 23–2

Student page 68

Outcomes

Indicators

N3.3, N4.3, N4.1a

The student is able to: • use the symbols =, < and > to state comparisons.

Skills

Resources

Language

• calculator

• comparing • ordering

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• divide • symbols • greater than • less than • different • numbers • order • place value

Notes

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Memory Masters (N3.3)

• The focus for this unit is addition of two multiples of 10 each less than 100.

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Number (N4.3)

• 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 (N4.1a) Warm up

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• What to do • Write these symbols on the blackboard/whiteboard: >,<. • Ask students what they represent and how they know which symbol is which. • Tell students we use these symbols to represent pairs or groups of numbers showing their relationship. Give an example like 64 > 27 or 2.6 < 4.03. Use further examples if required.

Challenge

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• Work through the first few examples as a whole class. • Before setting students to work, remind them that the place value of the number on the left is the first reference point for determining largest numbers. If these are equal, move to the number in the place immediately to the right and so on. • Exercise 4 is the same process. Although the numbers are in a line it is only the two numbers immediately adjacent to the space for the symbol that are of concern in determining the order.

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• Students are to use their information to come up with five different ways of writing the number 5. Leave students by themselves to think of ways that this may be done. • All attempts are to be recorded and any necessary explanations are to be written down for future sharing

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 58 – 59. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 111 •

Unit 23–3

Student page 69

Outcomes

Indicators The student is able to: • decide whether a shape is sufficiently close to rectangular that adding adjacent sides and doubling will be a ‘good enough’ estimate of perimeter for the task. • partition double digit numbers in order to mentally multiply and divide small single-digit numbers.

N3.3, N4.3, M4.4a

Skills • reasoning • measuring • problem solving

Memory Masters (N3.3)

Resources • calculator • 2-cm cubes

Language • divide • calculate • area • square metre • arrange • tallest • equal

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Number (N4.3)

• 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 (M4.4a, N4.3) Warm up

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• The focus for this unit is subtraction of a multiple of 10 less than 100 from 100.

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• Students may be set to work finding their own answers; or: • Work with the class to find out how they will work out the area of the lawn. Without drawing a scaled grid pattern, students will need to focus on calculation of area from multiplication of length and width. This may be done using a calculator or in written form using the space on the page. • Having found the area, students are now required to find the cost of fertilising the lawn at the given rate. Encourage students to offer suggestions as to how this might be done then let them find the answer by using one of these suggestions or a method of their own choosing. • Students are again required to explain how they reached their answer. Calculations may be completed in the space provided or be done using a calculator. • Share procedures with the class by asking several students to explain different procedures they may have used.

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Challenge

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© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Ask students what they would need to do to find the cost of fertilising a large grass area similar to an oval. Once the students are focused on finding the area of the lawn, find the cost of fertiliser application.

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• Students use ten 2-cm cubes to arrange them in three stacks as directed. • Record the construction and an explanation of the reasoning behind the work done to reach the final construction.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 136 – 137. • 112 • New Wave Maths Book F – Teachers Guide

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Unit 23—Answers

Student pages 67 – 69 Unit 23–2

Unit 23–1 1. (a) $0.81 (b) $0.04 (c) $7.50 (e) $10.00 (f) 400c (g) 5c (i) 608c (j) 1501c 2. (a) 5r37 (b) 8r33 (c) 8r16 (e) 4r66 (f) 5r52 3. (a)

(d) 3r44

1. (a) 100 (b) 100 (c) 100 (d) 70 (e) 70 (f) 120 (g) 150 (h) 130 (i) 50 (j) 90 2. (a) 30r2 (b) 20r4 (c) 30r6 (d) 20r5 (e) 20r7 (f) 30r6 < (o) < 3. (a) < (h) (i) > (p) < (b) > (c) < (j) < (q) < (d) < (k) < (r) > (e) > > (l) > (s) (m)> (t) (f) > < (g) < (n) < (u) > 4. (a) >, <, >, >, < (b) >, <, <, >, < (c) >, >, >, <, > (d) >, >, <, >, < (e) >, <, <, >, < (f) >, <, >, >, < Challenge Answers will vary; e.g. 2 x 3 – 1 = 5

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4. Answers will vary Challenge 361÷2 = 180r1 361÷6 = 60r1 361÷3 = 120r1 361÷7 = 51r1 361÷4 = 90r1 361÷8 = 45r1 361÷5 = 72r1 361÷9 = 40r1 All leave a remainder of 1 except for 7.

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(b) Possible answers

(d) $8.00 (h) 64c

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• Students use their knowledge to construct other angles; e.g. 60º, 75º, 110º etc.

Consolidation 23–2

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1. (a) 30 (b) 50 (c) 20 (d) 10 (e) 90 (f) 40 (g) 60 (h) 70 (i) 80 (j) 0 2. (a) 18r32 (b) 16r16 (c) 16r1 (d) 14r23 (e) 14r60 (f) 14r4 3. (a) $649.40 (b) Answers will vary; e.g. find the area by multiplying length by width, divide by 100 and multiply by $10.85 (c) 299.25 kg (d) Answers will vary; e.g. find the area in square metres then multiply by 50 to give the total number of grams required Challenge

• Students create five pairs of numbers. Swap with another student and use the symbols >, < to show the relationship.

Consolidation Unit 23–3

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• Repeat the activity with students measuring a grassed area in the school.

New Wave Maths Book F – Teachers Guide • 113 •

Unit 24–1

Student page 70

Outcomes

Indicators The student is able to: • rewrite a decimal as a fraction.

N3.3, N4.3, N4.1a

Skills • decimals

Memory Masters (N3.3)

Resources • calculator • fraction grid • fraction/decimal number line

Teac he r

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

Main Activity (N4.1a)

Notes

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Number (N4.3)

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What to do

• Use the number line and the information learnt so far to express the decimals as fractions. Where possible ask students to simplify fractions. • Work through the first six examples with the whole class before directing students to complete the task. • Using the same number line, order the decimals and fraction in Exercise 3 from smallest to largest. It is recommended that fractions be changed to decimals first to assist in ordering. • Work through the first example with the class as a whole before directing them to complete the remainder.

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• Ask students if they are able to write these decimals as fractions – 0.1, 0.6, 0.4, 0.9. • Explain that all may be written as tenths – 1/10, 6/10, 4/10, and 9/10 without simplifying them. • Similarly, decimals that finish in the hundredths column may be written as fractions shown as hundredths – 0.27, 0.93 as 27/100, 93/100. • Either use a large class display number line showing decimal and fraction or individual number lines.The lines should show main equivalents between 0 and 1.0; e.g. 0.1, 0.2, 0.25, 0.3, 0.4, 0.5, 0.6, 0.7, 0.75, 0.8, 0.9 and 1.0 and their fraction equivalents.

Challenge

• divide • decimals • fractions • simply • arrange • order • smallest • largest • comparison • symbols

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

Warm Up

Language

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• Students will need to experiment with adding, subtracting, multiplying and dividing to find ways of making other numbers using a single number and the four operations. (Brackets may also be used.) • As a hint, ask students how they might make the number 1 from the number 2 using one or all of the operations: 2 ÷ 2 = 1. • Leave students to experiment either by themselves or in small groups. • Share answers with the class and encourage the finding of different combinations. Praise efforts but don’t give answers that students can not find. For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 66 – 67. • 114 • New Wave Maths Book F – Teachers Guide

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Unit 24–2

Student page 71

Outcomes

Indicators The student is able to: • use a calculator to multiply and divide a sequence of calculations.

N4.3, S4.2, C&D4.2, C&D4.3, C&D4.4

Skills • scaling • modelling • recording data • measuring

Resources

Language

• calculator • ruler • pencil • 2-cm cubes

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• divide • scaling • base model • volume • surface area • ratio • double • triple • line graph • relationship

Notes

Memory Masters (N4.3)

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

Number (N4.3)

• 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 (S4.2, C&D4.2, C&D4.3, C&D4.4) Warm up

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• Distribute 2-cm cubes to small groups of students. • Ask students to make a rectangular prism that is two cubes wide, three cubes long and one cube high. • Ask students to find the volume – that is, the number of cubes used. • Ask students to find the base area – that is, the number of faces on the base. • Ask students to find the base perimeter – that is, the number of cube edges or unit edges around the base.

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What to do

• Ask students to make a double scale model of their original prism. How long will this one be? How wide will the new prism be? How high will this prism be? • Having made the prism, find the volume, base area and base perimeter as you did for the original prism. Record your results on the Carroll diagram. • Repeat this for the triple and quadruple scale models. • When all recordings have been made, look at the Carroll diagram and write about any patterns that you are able to find. • Share pattern discoveries with the whole class.

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Challenge

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• Write a story/word question to match one of the division problems in Exercise 2.

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New Wave Maths Book F – Teachers Guide • 115 •

Unit 24–3

Student page 72

Outcomes

Indicators The student is able to: • count forwards and backwards from any whole number.

N4.3, N4.1a

Skills

Resources • calculator • number line (1 – 100)

• rounding

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• The ‘Today’s number is ...’ activity asks 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.

Number (N4.3)

• 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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• Use a number line (1 – 100) to explain rounding of numbers to the class. Numbers that end in 1 – 4 are rounded down to the nearest ten. Numbers that end in 5 – 9 are rounded up to the nearest ten. • Numbers 10 – 40 are rounded down, numbers 50 – 90 are rounded up to the nearest hundred. • Numbers 100 – 400 are rounded down, numbers 500 – 900 are rounded up to the nearest thousand.

What to do

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• Use this information to round the numbers shown on the chart in Exercise 3. • Work through the first two examples, 3959 and 4113, with the whole class before allowing the students to proceed. • Apply the same process for Exercise 4, following the instructions for each example.

Challenge

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Main Activity (N4.1a) Warm up

• round • nearest • ten • hundred • thousand

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Memory Masters (N4.3)

Language

• Students are to list some numbers that would be rounded to 700. • Any numbers from 650 onward would be suitable solutions.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 56 – 57. • 116 • New Wave Maths Book F – Teachers Guide

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Unit 24—Answers

Student pages 70 – 72 Unit 24–2

1. (a) 180 (b) 560 (c) 240 (d) 420 (e) 350 (f) 540 (g) 480 (h) 360 (i) 400 (j) 480 2. (a) $0.62 (b) $0.52 (c) $0.31 (d) $0.31 (e) $0.21 (f) $0.42 1 /10 3. (a) 3/10 (g) 7/10 (m)9/10 (s) 1 4 1 3 (b) /5 (h) /5 (n) /2 (t) /5 1 1 3 4 (c) /4 (i) /2 (o) /4 (u) /5 11 (d) 79/100 (j) 43/100 (p) 67/100 (v) /100 7 931 471 819 (e) /40 (k) /1000 (q) /1000 (w) /1000 (f) 1/4 (l) 2/5 (r) 3/4 (x) 19/20 1 3 3. (a) 1/10 0.2 /2 /5 0.75 7 3 (b) 0.25 0.3 0.6 /10 /4 2 9 (c) 0.2 /5 0.5 0.75 /10 1 4 (d) 0.1 /4 0.5 /5 0.9 1 2 (e) / 0.3 / 0.75 0.8 5 3

1. (a) 90 (b) 80 (c) 80 (d) 90 (e) 60 (f) 90 (g) 70 (h) 90 (i) 80 (j) 90 2. (a) $0.25 (b) $0.24 (c) $0.58 (d) $0.37 (e) $0.23 (f) $0.28 3.

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5x Volume = 750 Base Area = 150 Base Perimeter = 50 10 x Volume = 6000 Base Area = 600 Base Perimeter = 100

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Unit 24–1

Challenge Answers will vary; e.g. 2 x 2 + 2 + 2 = 8

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• Develop further examples for the students to complete.

Consolidation 24–2 • Discuss the students’ responses to Exercise 5.

Consolidation 24–3

4. (a) 3600 (f) 280 km (g) 370 km (b) 9300 (c) 73 000 (h) 1400 km (d) 80 (i) 27 000 (e) 15 400 (j) 4830 Challenge Any numbers that fall between 650 and 749.

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1. Teacher check 2. (a) $0.61 (b) $0.51 (c) $0.61 (d) $0.72 (e) $0.51 (f) $0.92 3.

• Students select 5 numbers at random and round them accordingly.

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New Wave Maths Book F – Teachers Guide • 117 •

Unit 25–1

Student page 73

Outcomes

Indicators

N4.1a, N4.3, S4.3

The student is able to: • enlarge a diagram using scale as a reference.

Skills • scale drawing

Resources • calculator • pencil • ruler • small squares (4) or 1-cm grid paper (see page 199)

Language • subtract • larger • redraw • shape • grid • double • dimensions • tetromino shapes • square

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Notes

Teac he r

Memory Masters (N4.1a)

• The focus for this unit is determining which number in a pair is the greater.

• 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 (S4.3) Warm up

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• Explain to students that they will be redrawing the shape on the page with its dimensions doubled. • Ask students to identify a suitable starting point. The actual point does not matter except that students need to remember that the point will be twice as far from the edge and/or top as it is in the original diagram. • Once the starting point has been identified, students should draw the lines that emanate from this point. Remind students to count the grid units in the original diagram then work out how long the redraw line will be. For example, if the original line is 3 grid units long the redraw will be 2 x 3 or 6 grid units long. • Suggest to students that they continue to draw the double scale shape working from the end of each new line drawn until the shape is completed. • Answer the questions about the original and new shapes.

Challenge

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• Review the idea of scale.

What to do

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Number (N4.3)

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• Students require four small squares or 1-cm grid paper. • This activity may be worked in small groups. • Students will need to make four shapes similar to the one on page 73 of the workbook, cutting these from paper or card. Use these shapes to make a square. • Draw all attempts and write an explanation of your findings. • Students share their findings with the class.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 144 – 145. • 118 • New Wave Maths Book F – Teachers Guide

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

Unit 25–2

Student page 74

Outcomes

Indicators

N4.1a, N4.3

The student is able to: • use chosen method to multiply two-digits by a single digit.

Skills

Resources

Language

• calculator • 10 green counters • 16 blue counters

• round • subtract • regrouping • mentally

• working mentally

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

Notes

Teac he r

Memory Masters (N4.1a)

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

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Number (N4.3)

• 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 (N4.3) Warm up

© R. I . C.Publ i cat i ons What to• dof orr evi ew pur posesonl y• • Remind students that working addition, subtraction, multiplication and division problems mentally is the aim for each of these. • The best way to do this is to practise frequently.

Challenge

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• Page 74 of the workbook provides the opportunity to practise mental multiplication with regrouping. • Ask students to work Exercise 3(a) in their minds and write the answer down. • Ask students to explain how they worked the answer out. Ask who else used the same method. Ask for students who used a different method. Ask for the explanation and then who else used the same method. Continue until all methods have been discussed. • Repeat this for 4(a) and 5(a). • If students are confident, set them to work. Provide assistance as a group for students who are struggling. You may be required to work with some of this group for much of the time.

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• Students are to be encouraged to work this problem out and record their thought processes as they go. • For those who need them, provide counters or similar as a substitute for socks, to allow them to work out the problem in a practical manner. Students must record their actions.

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New Wave Maths Book F – Teachers Guide • 119 •

Unit 25–3

Student page 75

Outcomes

Indicators

The student is able to: • decide whether a shape is sufficiently close to a rectangle that adding adjacent sides and doubling will be a ‘good enough’ estimate for perimeter. • use cubes to copy other structures, attending to what can not be seen but must be there.

N3.3, N4.3, M4.4a, S4.2

Skills • measuring • calculating

Memory Masters (N3.3)

Resources • calculator • 2-cm cubes • toothpicks or similar

• subtract • square metre • procedure • cubes • models • squares

Number (N4.3)

• 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 (M4.4a, S4.2) Warm up

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

• The focus for this unit is addition of a whole number less than 100 to a multiple of 10 less than 100.

Teac he r

Language

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• Ask students how they will find the cost of carpeting the floor given the price is $110 per metre. Carpet comes in rolls 3.66 m wide, so carpet is priced by linear metre. • Students are to calculate the cost of the carpet then explain how they worked the cost out. • Exercise 4 requires the distribution of 2-cm cubes to small groups. • Explain to students they are to determine the total number of cubes required to make the models from the directions given. Ask students whether they need to make the models to find out how many cubes they require. If not, ask them how they would find the number. • Encourage students to make some or all of the models to check their working.

Challenge

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© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Revise calculation of area given length and width of a shape. Remind students that they may use a calculator to assist with calculations.

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• Students are to collect 10 toothpicks, or similar, to be used to make three squares. • All attempts should be drawn and include an explanation of what was done, for sharing with the class or teacher.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 136 – 137. • 120 • New Wave Maths Book F – Teachers Guide

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Unit 25—Answers

Student pages 72 – 75 Unit 25–2

Unit 25–1 1. (a) 7.1 (b) 97 486 (c) 39.4 (e) 90 000 (f) 8.802 (g) 74 (i) 3.481 (j) 675 2. (a) 0.43 (b) 0.36 (c) 0.28 (e) 0.19 (f) 0.25 3.

Teac he r

(d) 0.28

1. (a) 50 (b) 630 (c) 90 (d) 730 (e) 420 (f) 20 (g) 80 (h) 780 (i) 310 (j) 40 5.69 (c) 2.47 (d) 0.56 (e) 0.79 2. (a) 5.49 (b) (f) 2.64 (k) 72 4. (a) 282 3. (a) 98 (b) 48 (l) 72 (b) 252 (c) 95 288 (m)76 (c) (d) 60 (n) 84 (d) 267 (e) 54 486 (o) 51 (e) (f) 85 (p) 96 (f) 525 (g) 91 (q) 92 (g) 195 (h) 90 (r) 94 (h) 544 (i) 72 (s) 75 (i) 234 (j) 96 (t) 78 (j) 406 168 (i) 408 5. (a) 186 (e) (b) 249 (f) 368 (j) 126 (c) 568 (g) 88 (k) 216 (d) 427 (h) 128 (l) 729 Challenge 3

r o e t s Bo r e p ok u S (d) ≈ 40 (e) ≈ 80 (f) double

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4. (a) ≈ 46.5 squares (b) ≈ 186 squares (c) 4 times Challenge Yes – Possible answer:

(d) 0.07 (h) 0.27

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• Students draw their own diagram on the 1-cm grid on page 199. Swap with another student to enlarge.

Consolidation 25–2

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1. (a) 107 (b) 86 (c) 114 (d) 57 (e) 125 (f) 172 (g) 93 (h) 139 91 (i) 108 (j) 2. (a) 5.172 (b) 5.321 (c) 2.333 (d) 2.241 (e) 1.202 (f) 7.325 3. (a) $566.50 (b) Answers will vary; e.g. $110 x 5.15 = $566.50 4. (a) 125 (b) 1000 (c) 64 (d) 8 (e) 343 Challenge

• Develop more examples to provide students with further practice.

Consolidation 25–3

• Measure the classroom floor and work out the cost of recarpeting it.

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New Wave Maths Book F – Teachers Guide • 121 •

Unit 26–1

Student page 76

Outcomes

Indicators The student is able to: • use materials and diagrams to represent fractional amounts where the ‘whole’ may be an object, quantity or collection.

N3.3, N4.3, N4.1b

Skills • fractions

Memory Masters (N3.3)

Resources • calculator • cuisenaire rods • 1-cm grid paper (see page 199)

Teac he r

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

Main Activity (N4.1b)

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• Distribute 1-cm grid paper to the students. • Ask students to draw an outline around eight squares arranged in a horizontal line. • Draw another outline around eight squares on a horizontal line on the second line below the first square. • Colour five squares in the first line and three squares in the second line. • Explain to students that the first line shows 5/8 and the second line shows 3/8. Ask students what the difference is between the two shaded amounts in the diagram. ( 2/8) • Repeat the process with a line of six squares with four shaded and a second line of six squares with one shaded. • Repeat with three lines of eight outlines and 21 squares shaded to represent 25/8 and a line of eight below with one square shaded.

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• Ask students to work through the exercises on the page. It may be best to work with the class as a whole for each of the examples.

Challenge

Notes

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Number (N4.3)

What to do

• divide • shaded diagram • difference between • fractions • whole number • added • multiplied

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

• The focus for this unit is subtraction of a multiple of 10 less than 100 from a whole number less than 100.

Warm Up

Language

• Students will need to search their knowledge of addition and multiplication facts to find a whole number that fits the requirements as outlined. • Students are to record all their working for later discussions with the class or teacher.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 64 – 65. • 122 • New Wave Maths Book F – Teachers Guide

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

Unit 26–2

Student page 77

Outcomes

Indicators

The student is able to: • represent data in diagrams and tables which may include Venn diagrams and two-way tables. • select appropriate operations for whole number situations. • plan a short sequence of calculations needed for familiar situations.

N3.3, N4.3, C&D4.3, N4.2

Skills • graphing • calculating • measuring

Resources

Language

• calculator • pencil • ruler • protractor • large circle

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

• divide • information • construct • circle • pie graph • percentage • total area • degrees

Notes

Memory Masters (N3.3)

Teac he r Number (N4.3)

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• The focus for this unit is multiplication of a multiple of 10 less than 100 by a whole number 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&D4.3, N4.2) Warm up

What to do

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• Hold up a circle cut from card or stiff paper and explain to students that this may be used to show a graph commonly known as a pie graph. • If half or 50% of the data were to be shown on a pie graph it would be drawn as a line bisecting the circle – draw a line across the circle to show this. • Explain that this represents half of the total degrees in the circle – half of 360º or 180º. • Explain to students that this may be drawn using a protractor. Mark in a radius anywhere on the circle. Use a protractor to show 180º or half of the circle. The result is the same as previously shown. • Work through other segments and draw them at one-fifth or 72º, two-thirds or 240º.

• Work through the information provided to find the percentage of the total for each area and then convert to degrees. • Explain to students that to find the percentage they firstly need to find the total area by adding all of the areas given. Do this as a whole class using calculators. • The next step is to divide the individual areas by the total area and multiply by 100 to find the percentage. Record the percentage in the space provided. • The final step is to work out the number of degrees. This is done by entering 360, then x, then percentage total (e.g. 20), then the % key, to give the total required. • When all degree totals have been found these may then be transferred to the pie graph as explained earlier. • The whole of this activity may be completed with the teacher directing all the way or by initial guided instruction, then allowing the students to work through by themselves.

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Challenge • Enter the data into a spreadsheet and use the chart function to produce a pie graph. Compare it to the one you have drawn.

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New Wave Maths Book F – Teachers Guide • 123 •

Unit 26–3

Student page 78

Outcomes

Indicators The student is able to: • use materials and diagrams to represent fractional amounts where the ‘whole’ may be an object, quantity or collection.

N4.3, N4.1b

Skills • adding fractions

Teac he r

Memory Masters (N4.3)

Resources • calculator

Language • divide • fraction • mentally • add • diagrams

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

• The focus for this unit is division of a multiple of 100 by a whole number 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 (N4.1b) Warm up

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• Using the diagrams in Exercise 3 add the fractions shown.Work with the whole class asking what the total is – one-fourth, two-fourths and three-fourths. • Work through the next three questions with the whole class. • Direct students to Exercise 4 and ask for the answer to (a), then (b) and (c). Set students to complete up to (g). • In Exercise 4, from (h) onwards the fractions total one.This is added to the whole number to provide the answer. Work through the first three questions with the class as a whole. Ask what 3/4 and 1/4 equal. Add this to one to give the total. The class can then complete the rest of the exercises. • Work through the diagrams in Exercise 5 assisting students in converting the fractions to mixed numbers to provide the total. • Work with students for the first two or three questions in Exercise 6 to ensure understanding before setting them to complete the rest up to (h). • Exercise 6 (h), onward puts into practice the questions using diagrams in Exercise 5. Work with students for the first 2 or 3 questions before setting them to complete the rest.

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• Explain to the class that the exercise is to add fractions with like denominators. Remind them that when fractions have like denominators all that is required is to add the top numbers and write them on the same denominator. When the total of the top number is greater than the bottom number the fraction is changed to a mixed numeral by dividing the bottom number into the top number.

What to do

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Number (N4.3)

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Challenge • Students are to choose their own method to work this problem recording all their workings for sharing with the class or teacher.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 64 – 65. • 124 • New Wave Maths Book F – Teachers Guide

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Unit 26—Answers

Student pages 76– 78 Unit 26–2

1. (a) 24 (b) 49 (c) 41 (d) 64 (e) 13 (f) 42 (g) 37 (h) 53 (i) 26 (j) 19 2. (a) $1.32 (b) $1.12 (c) $3.14 (d) $3.21 (e) $1.22 (f) $4.12 (d) 2 3. (a) 3/5 5 (b) /10 (e) 2/3 (c) 1 4. (a) 11/4 (h) 3 (b) 1/3 (i) 21/3 (c) 1 (j) 2/3 (d) 3/4 (k) 1/2 2 (e) 1 /5 (l) 31/5 (f) 5/6 (m)1/5 (g) 1/3 Challenge 2

1. (a) 320 (b) 120 (c) 120 (d) 450 (e) 90 (f) 350 (g) 40 (h) 200 (i) 300 (j) 270 2. (a) $2.01 (b) $1.01 (c) $1.03 (d) $3.04 (e) $2.01 (f) $2.01 3.

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

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Teac he r

Unit 26–1

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• Provide students with further opportunities to subtract fractions.

Consolidation 26–2 • Develop a research question and collect and record data to produce a pie graph.

Consolidation 26–3

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1. (a) 200 (b) 100 (c) 200 (d) 300 (e) 100 (f) 400 (g) 200 (h) 300 (i) 100 (j) 100 2. (a) $0.81 (b) $0.81 (c) $0.91 (d) $0.91 (e) $0.61 (f) $0.91 3. (a) 3/4 5. (a) 21/4 (b) 2/6 or 1/3 (b) 21/6 (c) 8/8 or 1 (c) 3 (d) 4/5 4. (a) 2/5 (h) 2 2 (b) /7 (i) 2 (c) 2/3 (j) 3 2 (d) /9 (k) 4 3 (e) /5 (l) 2 (f) 6/7 (m)3 7 (g) /9 (n) 4 4 6. (a) 3 /7 (h) 21/4 (b) 51/2 (i) 21/8 21/5 (c) 47/10 (j) (d) 64/5 (k) 51/3 (e) 25/8 (l) 32/7 Challenge (f) 42/3 (m)42/5 6912 L/per day (g) 45/6 (n) 62/3

• Proved students with further opportunities to add fractions.

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New Wave Maths Book F – Teachers Guide • 125 •

Unit 27–1

Student page 79

Outcomes

Indicators

Resources

Language

N4.2, N4.3, S4.2, C&D4.3

The student is able to: • locate shapes in the school environment. • represent data in diagrams and tables which may include arrow diagrams, Venn diagrams and twoway tables.

• calculator • local environment • 3-D shapes as listed in workbook

• estimation • number sentences • brackets • shapes • cylinder • sphere • triangle • rectangle • square • hexagon • cube • rectangular prism • triangular prism • square pyramid • square number

Skills • observing • recording • comparing

• The focus for this unit is completion of number sentences using the distributive property of multiplication and division by working with brackets.

Number (N4.3)

• 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 (S4.2, C&D4.3) Warm up

Notes

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Teac he r

Memory Masters (N4.2)

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

• Mathematics is important to our society. One of the simplest places to see mathematics in use is in our environment. Shapes are very important for a number of reasons, not the least of which is their use for structural support and, less importantly, aesthetics. • Students may need to see models of the 3-D shapes to remind them of what they look like.

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• Explain to the class that today they will be working in small groups and moving around the school buildings and playground to find shapes that are listed in the workbook on page 79. • As each shape is identified, students are to write where it is found and why they think it was used in that particular location. • To allow for the recording of the same shape in different locations, suggest that one student in the group records the shape as it is located and the remainder of the students record other findings. At the end of the time, those students with blank spaces may complete them from the other students in their group. • Share findings across the class at the conclusion of the lesson.

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• Students will need to list square numbers to assist them with this activity. • Students then test addition of one square number to another to see if they are able to make another square number. • All attempts are to be recorded as is the thinking process for sharing with the class or teacher.

• 126 • New Wave Maths Book F – Teachers Guide

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Unit 27–2

Student page 80

Outcomes

Indicators

N4.1a, M4.1, N4.3, M4.2

Skills

The student is able to: • collaborate to make sensible estimates of quantities.

• measuring • comparing

Resources

Language

• calculator • 3 cylinders of different diameters and heights • 1 rectangular or square container • five squares or 1-cm grid paper (see page 199)

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

• millilitres • litres • convert • multiply • containers • cylinder • fractions • arrange • squares • whole sides • arrangements • perimeter • units

Notes

Teac he r

Memory Masters (N4.1a, M4.1)

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

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Number (N4.3)

• 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 (M4.3) Warm up

• Arrange the class into small groups. • Each group is to be given three cylindrical containers of different diameters and heights and one square or rectangular container. If possible, two of the cylindrical containers should be the same height and one cylinder and the square/rectangular containers should be the same height.

What to do

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Challenge

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• Mark a height of 1/3 on the side of the containers with a marker pen. • Fill each container with water to the 1/3 mark. • Pour the water back into a large container then refill one of the containers to the 1/3 marker. Pour this amount into each of the containers in turn. • Write what you noticed about: – the size of the 1/3 of each container; and – the size of the 1/3 of water when poured from one container to another. • Talk to students about what a fraction represents. It is the means of showing part of a whole. If a group of students, such as a class, is broken into parts these parts are written as part of the whole class. • Allow students the opportunity to collect various containers.They are required to compare 1 /4 this time but will need to draw what they learned when comparing 1/3.

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• Each student is to have five squares or a sheet of 1-cm grid paper. • Follow the directions given to make arrangements of the five squares with a perimeter of 10 units. • Draw all shapes and record your findings for sharing with the class or teacher.

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New Wave Maths Book F – Teachers Guide • 127 •

Unit 27–3

Student page 81

Outcomes N4.1a, N4.3, M4.4

Skills

Indicators

Resources

The student is able to: • measure the perimeter of polygons.

• calculator • string • netball court, oval

• calculating

Language • multiply • perimeter • calculate • length • procedure

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

Memory Masters (N4.1a) Number (N4.3)

• 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 (M4.4) Warm up

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Teac he r

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

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• The first activity asks for the perimeter of the block of land to be found. From the discussions on finding perimeter, how would you find the perimeter of this block of land? Encourage discussion then focus on the second part of the activity. • How will you find the cost of fencing the block of land? Encourage student input. • For Exercise 4, the first task is to find the perimeter of the block. From the perimeter we know how many metres it is around the block. We can now multiply this answer by $4.25 to find the cost of fencing. • Tell students they may use their preferred method to find the answers. • Set the students to complete both activities.

Challenge

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© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Ask students how they would find the perimeter of the cover page of their workbook. • Ask students how they would find the perimeter of their desk top, the classroom, the school building. Encourage discussion on the different possible means offered.

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• Students may use whatever method they wish to determine the length of string required to find the perimeter of the oval or netball court. • Remind students to think creatively and not be focused on a narrow set of options as can occur by thinking only in terms of what the question asks. • Record all your thinking and actions.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 134 – 135. • 128 • New Wave Maths Book F – Teachers Guide

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Unit 27—Answers

Student pages 79 – 81

Unit 27–1

1. (a) 4270 m (b) 965 mL (c) 8400 mL (d) 2 mL (e) 73 mL (f) 0.069 L (g) 0.824 L (h) 0.003 L (i) 7.284 L (j) 0.081 L 2. (a) $107.80 (b) $279.22 (c) $521.95 (d) $491.79 (e) $593.04 (f) $236.06 3. (a) Teacher check (b) Answers will vary; e.g. 1/3 height varies according to the size and shape of the container 4. Teacher check Challenge a possible solution

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

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Teac he r

1. (a) _ < 27 < _ (f) _ < 31 < _ (b) _ < 22 < _ (g) _ < 25 < _ (c) _ < 19 < _ (h) _ < 28 < _ (i) _ < 39 < _ (d) _ < 17 < _ (e) _ < 30 < _ (j) _ < 13 < _ Teacher check also required. 2. (a) $262.20 (b) $478.80 (c) $393.30 (d) $460.80 (e) $562.40 (f) $309.60 3. Teacher check Challenge Yes – 9 + 16 = 25

Unit 27–2

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• Use the various shapes on the page to design a building.

Consolidation 27–2 • Repeat the activity to compare 5/6.

Consolidation 27–3

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1. (a) $27.46 (b) $40.00 (c) $0.29 (d) $8.64 (e) $30.04 (f) 2400c (g) 879c (h) 15c (i) 3c (j) 85c 2. (a) $348.48 (b) $282.24 (c) $498.12 (d) $439.68 (e) $475.38 (f) $423.51 3. 118.07 m, $3187.90 4. $590.75 5. 48 m, $1728 6. $2842 Challenge Teacher check

• Develop further questions of this type using areas within the school grounds.

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New Wave Maths Book F – Teachers Guide • 129 •

Unit 28–1

Student page 82

Outcomes

Indicators The student is able to: • use equivalences that are readily visualised to compare and order fractions.

N4.3, N4.1b

Skills

Resources • calculator • fraction grid (see page 209)

• fractions

Memory Masters (N4.3)

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.

Notes

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Number (N4.3)

• Ask students if they are able to say which fraction of a pair is bigger than another if the bottom number (denominator) is the same but the top number (numerator) is different. • Use a fraction grid (large one for the whole class or individual ones) so that students are able to show smaller and larger fractions with the same denominator; for example, 1/4 and 3 /4; 2/6 and 5/6; 7/10 and 3/10; 3/5 and 2/5.

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What to do

• Ask students to identify the larger fraction directly or use the fraction grid to write the bigger of the pair in the space provided. • When Exercise 3 is completed direct students to Exercise 4. Here mixed numbers are used and students are to identify the bigger of the pair.Where the whole numbers are different, identification is straight forward as the fraction is irrelevant. However, where the whole numbers are the same students will need to determine the bigger by saying which fraction part of the number is bigger. • Before providing this information to the students, ask them if they are able to determine the bigger of the pair and to provide the information to the class. • Students complete the activities.

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Main Activity (N4.1b)

Challenge

• divide • bigger • fractions • mixed numbers • symbols

r o e t s Bo 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.

Warm Up

Language

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• Students will need to experiment with adding subtracting, multiplying and dividing to find ways of making other numbers using a single number and the four operations. (Brackets may also be used.) • As a hint, ask students how they might make the number 1 from the number 9 using one or all of the operations; for example, 9 ÷ 9 = 1. • Leave students to experiment either by themselves or in small groups. • Share answers with the class and encourage the finding of different combinations. Praise efforts but don’t give answers that students can not find. For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 62 – 63. • 130 • New Wave Maths Book F – Teachers Guide

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Unit 28–2

Student page 83

Outcomes

Indicators The student is able to: • use cubes to copy other structures made from cubes. • represent data in diagrams and tables which may include arrow diagrams, Venn diagrams and two-way tables. • read the information provided on axes of bar and line graphs.

N3.3, N4.3, S4.2, C&D4.3, C&D4.4

Skills • modelling • scaling • recording data • measuring

Resources

Language

• calculator • 2-cm cubes • ruler • pencil

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• divide • scaling activity • model • rectangular prism • double • triple • quadruple • calculate • volume • base area • base perimeter

Memory Masters (N3.3)

Notes

Number (N4.3)

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• The focus for this unit is addition of a multiple of 10 less than 100 to a whole number less than 100.

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

Main Activity (S4.2, C&D4.3, C&D4.4) Warm up

• Distribute 2-cm cubes to small groups. • Allow a few minutes of directed play. Ask students to use the cubes to build models.

What to do

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• Explain the activity for the day outlining building of scale models. Students start with a very simple model of one cube. • Ask students to use their base model and ask them to write the volume (number of 2-cm cubes used) and the surface area (number of faces of individual cubes on the outside of the model) in the spaces provided on the Carroll diagram. • Direct students to build another model whose dimensions are double the original and record volume and surface area of this model. • Repeat this process for models three, four, five and six times the size of the original. • When all recordings have been made, ask students to find the ratio of surface area to volume. This is found by dividing the surface area by the volume of the model. Record this answer in the ratio column (use a calculator if required). • Once all calculations have been made, students are then to record the ratio against the scale of the model. For example, using the scale model, which is 1, on the horizontal axis move perpendicularly up this line to reach the number on the vertical axis showing the ratio. If using a base model of 1, this is 6. The graph point has been placed on the graph already. • Repeat this process for each of the other scale models then join each dot to the next using a ruler. • Finally, explain the relationship between the surface area and the volume as shown on the graph for each scale model.

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Challenge • Write a story/word question to match one of the division problems in Exercise 2.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 142 – 143. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 131 •

Unit 28–3

Student page 84

Outcomes

Indicators The student is able to: • use materials and diagrams to represent fractional amounts where the ‘whole’ may be an object, quantity or collection.

N3.3, N4.3, N4.1b

Skills • write fractions • represent fractions in diagrams

• calculator • toothpicks or equivalent • four sheets of paper

Number (N4.3)

• 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 (N4.1b)

What to do

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• Work with the whole class on Exercise 3(a) to show that 32/4 of the diagram is shaded; this is equivalent to how many fourths? Write this on the page. • Direct students to continue this for the rest of Exercise 3. • Exercise 4 is a reverse process where the fraction is given and students are required to shade the equivalent fraction parts on the diagrams provided. Work with students as a whole.

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• Hold up a sheet of paper. Fold it in half and ask students to tell you what they see. Tear the paper in half along the fold line. Ask students to tell you about the two halves of paper. Fold one of the halves in half and tear along the fold line. Hold the quarters and the half up and ask students to tell you about the paper now. If students have not been able to, then direct discussion to equivalence. Note that 2/4 are equivalent to 1/2. • Repeat this process with three pieces of paper, each torn/cut in half.Try to get from students an explanation that the six halves are equivalent to three wholes.

Challenge

• divide • diagrams • fraction • equivalent • mixed numeral

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• The focus for this unit is subtraction of a whole number less than 100 from a multiple of 10 less than 100.

Warm up

Language

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Memory Masters (N3.3)

Resources

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• Students may need materials to work this problem out or may be able to do it by drawing the equivalent of toothpicks. • Remind students to record and write down their reasoning for future sharing.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 62 – 63. • 132 • New Wave Maths Book F – Teachers Guide

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

Unit 28—Answers

Student pages 82 – 84 Unit 28–2

1. Teacher check 2. (a) $6.01 (b) $5.01 (c) $9.01 (d) $8.01 (e) $8.01 (f) $6.01 2 3. (a) 3/4 (e) /3 (i) 7/10 5 5 2 (b) /8 (f) /6 (j) /9 5 3 5 (c) /6 (g) /4 (k) /8 7 4 (d) /10 (h) /5 (l) 7/9 1 1 4. (a) 2 /3 (d) 6 /10 (g) 41/3 (b) 45/8 (e) 22/3 (h) 27/8 (c) 31/3 (f) 51/3 (i) 53/4 Challenge Answers will vary; (9 + 9) ÷ 9 = 2

1. (a) 76 (b) 95 (c) 142 (d) 123 (e) 127 (f) 125 (g) 97 (h) 119 (i) 122 (j) 143 2. (a) $1.03 (b) $1.05 (c) $3.06 (d) $2.04 (e) $1.08 (f) $1.06 3.

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Unit 28–1

5. The ratio of Surface Area (SA) to Volume (V) decreases because the SA increases rapidly as the number of cubes increases.

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• Provide students with further opportunities to order fractions.

Consolidation 28–2 • Repeat the activity with a different model.

Consolidation 28–3

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1. (a) 55 (b) 13 (c) 18 (d) 47 (e) 4 (f) 15 (g) 12 (h) 29 (i) 7 (j) 29 2. (a) $8.08 (b) $9.09 (c) $9.09 (d) $7.07 (e) $6.06 (f) $6.06 3. (a) 14/4 (b) 7/3 (c) 19/8 (d) 18/5 (e) 9/4 (f) 17/10 (g) 9/2 4. (a) (b)

• Provide students with fractions and ask them to draw diagrams to represent the fractions given.

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25/6

Challenge I = III – II

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New Wave Maths Book F – Teachers Guide • 133 •

Unit 29–1

Student page 85

Outcomes N4.2, N4.3, S4.3

Skills • working geometrically

Resources

The student is able to: • sort different arrangements of a fixed number of squares into groups that can be superimposed and re-sort into those which do or do not require reflection.

• calculator • 1-cm grid paper (see page 199) • 2-cm cubes • 5 squares • polydron or geoshape squares

• subtract • investigate • construction • pentominoes • adjacent • regular shapes

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

Number (N4.3)

• 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 (S4.3) Warm up

Language

Notes

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Memory Masters (N4.2)

Indicators

• Explain to students, or ask them to describe, what a pentomino is. • A pentomino is a shape made by joining five squares so that only whole sides adjoin to whole sides. Corner to corner only or half-side only contacts are not permissible. • Ask students to describe some pentomino shapes. Blackboard/whiteboard these for the class to see. After displaying two or three, ask the class to discover as many different pentomino shapes as they can. Rotations, reflections or translations are deemed to be the same.

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• Encourage students to use concrete materials such as 2-cm cubes or square shapes to assist their discoveries. Using 1-cm grid paper may also assist. • Polydron or geoshape squares can also be used. • As each different shape is found, draw it on the grid provided. • Share the discoveries with the whole class at the end of the lesson.

Challenge

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What to do

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• Which pentominoes may be folded to make open top boxes?

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 42 – 43. • 134 • New Wave Maths Book F – Teachers Guide

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

Unit 29–2

Student page 86

Outcomes

Indicators

N4.3

The student is able to: • add and subtract money and measures with equal numbers of decimal places.

Skills

Resources

Language

• calculator • pencil

• subtract • square • above • circle • triangle • inside

• problem solving • calculating

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

Notes

Teac he r

Memory Masters (N4.3)

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

Number (N4.3)

• 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 (N4.3) Warm up

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Challenge

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• Read through each word problem in turn with the whole class and ensure students have a clear understanding of what is being asked. • Set students to work, completing all problems using their preferred method.

• Students follow the directions given and share their finished drawing with their teacher.

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For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 72 – 73. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 135 •

Unit 29–3

Student page 87

Outcomes

Indicators

WM4.2, C&D4.2, C&D4.3

Skills

• reasoning • estimating • problem solving • recording data • working mentally • speaking and listening • taking risks • collaborative learning and working

Resources

Language • data • plan • classify • organise • questions • brainstorm • estimate • round • category

• calculator • dice

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

What to do

• Note: Ensure the students understand that the focus of this task is to consider and calculate the probability of rolling two sixes. The actual game in the picture is irrelevant. Students display how they calculate the answer (1/36) using techniques such as a tree diagram, a graph, writing all possible combinations or their own method. • This activity is designed for students working collaboratively in groups. As students will need to discuss their opinions and ideas, allow enough time so that they do not feel rushed and for ideas to evolve. • The activity is designed to be open-ended and investigative. Students may request resources such as dice or the Internet. • When completing investigative tasks, some students may be more successful in mixed-ability groups rather than same-ability groups. • Some groups will be able to work independently while others may need guidance. The stimulus questions below may prompt such groups. – What types of games involve using dice? – What are the chances of rolling a six on a die? – How would this affect the outcome of a game? – How could you record your findings? – Do you need to physically roll dice to find the answer? Why or why not? (Some groups may need to roll dice to visualise an answer.) • Groups may wish to collate their findings and present them as a poster with diagrams, graphs and calculations. • Allow each group to discuss and evaluate its ability to problem solve and success as a group. A ‘group’ or ‘self ’ assessment form could be completed.This information will be helpful for creating groups for future investigative tasks.

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Main Activity (WM4.2, C&D4.2, C&D4.3)

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• 136 • New Wave Maths Book F – Teachers Guide

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Unit 29—Answers

Student pages 85 – 87 Unit 29–2

1. (a) 240 (b) 420 (c) 180 (d) 280 (e) 540 (f) 540 (g) 560 (h) 400 (i) 240 (j) 630 2. (a) 0.075 (b) 0.059 (c) 0.048 (d) 0.077 (e) 0.034 (f) 0.078 3. Answers will vary

1. (a) 600 (b) 800 (c) 800 (d) 700 (e) 700 (f) 900 (g) 600 (h) 800 (i) 300 (j) 500 2. (a) 4.809 (b) 3.809 (c) 3.174 (d) 2.757 (e) 1.697 (f) 2.359 3. Answers will vary 4. $75 5. $2100 6. 20 weeks @ $2/week Challenge

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Unit 29–1

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• Use 1-cm cubes to make the pentominoes drawn on the grid.

Consolidation 29–2 /6 x 1/6 = 1/36

1 2 3 4 5 6 = /6

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• Ask students to develop their own word problems. Swap with a partner.

Consolidation 29–3

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1. 1/36 Recording techniques will vary. For example, Roll 1 1 2 3 4 5 6 = 1/6

• Brainstorm games which use dice. Discuss the chance involved.

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or 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6 1 = /36

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New Wave Maths Book F – Teachers Guide • 137 •

Unit 30–1

Student page 88

Outcomes

Indicators The student is able to: • add and subtract money and measures with equal numbers of decimal places. • plan sequences of calculations using a calculator memory facility.

N4.1a, N4.3

Skills • problem solving • calculating

Resources

Language • round • nearest tenth • divide • credit • debit • total • income • expenditure • symbols

• calculator

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Notes

Memory Masters (N4.1a) Number (N4.3)

• 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 (N4.3) Warm Up

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• Study the bank statement provided on the workbook page. • Discuss the terms ‘credit’, ‘debit’ and ‘total’. • Look at the income and expenditure. • Read through the questions to ensure students have a clear understanding of what they are being asked to do. • Set the students to work to complete the problems using their preferred method.

Challenge

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• Students will need to experiment with adding, subtracting, multiplying and dividing to find ways of making other numbers using a single number and the four operations. (Brackets may also be used.) • As a hint, ask students how they might make the number 1 from the number 7 using one or all of the operations; for example, 7 ÷ 7 = 1. • Leave students to experiment either by themselves or in small groups. • Share answers with the class and encourage the finding of different combinations. Praise efforts but don’t give answers that students can not find.

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• Discuss with students that adults earn a wage and from that wage are responsible for their living expenses.

What to do

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Teac he r

• The focus for this unit is rounding of decimals to the nearest tenth.

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For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 80 – 81. • 138 • New Wave Maths Book F – Teachers Guide

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

Unit 30–2

Student page 89

Outcomes

Indicators The student is able to: • represent data in diagrams and tables which may include arrow diagrams, Venn diagrams and twoway tables.

N4.2, N4.3, C&D4.3

Skills • logical order • recording

Resources

Language

• calculator • coloured pencils • ruler

• estimation • number sentences • divide • arrow diagram • shape • continuous line • intersection

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Notes

Memory Masters (N4.2)

Teac he r

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• The focus for this unit is completion of number sentences using the distributive property of multiplication and division by working with brackets.

Number (N4.3)

• 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&D4.3) Warm up

What to do

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• Explain to the class what an arrow diagram is. This may be shown on the blackboard/ whiteboard as follows. ‘I will show you the activities you will be doing today using an arrow diagram to show the sequence. Home room activities reading mathematics writing science art.’ • Ask students to display on the blackboard/whiteboard, or dictate for the teacher to record, the sequence they follow in their lunchtime routine. This may be repeated until students understand.

• Work through with the class explaining that they are to show, using arrows ruled onto the map, four different guided tours that a Western Australian tourist might make if visiting three different cities on a tour once leaving Perth. Suggest to students that using a different colour for each tour will help in the display. • The tours are then to be recorded at the bottom of the page in the space provided using an arrow diagram for each tour. • Ask several students to show and share their tours.

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• Remind students that they are to start and finish at point S. • Suggest to students that it may be useful to number each intersection as it is passed through to remind them that it has been passed through and also the order in which it was passed through for future reference. • Keep records of all attempts and explain why you chose the final one.

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

New Wave Maths Book F – Teachers Guide • 139 •

Unit 30–3

Student page 90

Outcomes

Indicators The student is able to: diagrams • use materials and to represent fractional amounts where the ‘whole’ may be an object, quantity or collection.

N3.3, N4.3, N4.1b

Skills • dividing • shading

• calculator • strip of paper

Language • divide • shade

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• The focus for this unit is addition and subtraction of a whole number less than 100 to or from a whole number less than 100.

Number (N4.3)

• 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 (N4.1b) Warm up

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Memory Masters (N3.3)

Resources

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• What to do • Hold up a strip of paper. Fold in half, then half again to show fourths. • Shade three sections to show 3/4. • Ask students how many are unshaded. (1/4)

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Challenge

• Students are to place the numbers 1 – 8 inclusively in the circles on the diagram so that each of the four line totals 12. • Show all attempts. • Keep notes of your thinking and working out.

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• Read through the questions on the page. • Set students to work to complete the exercise using their preferred method.

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For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 62 – 63. • 140 • New Wave Maths Book F – Teachers Guide

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

Unit 30—Answers

Student pages 88 – 90 Unit 30–2

1. (a) 6.8 (b) 5.8 (c) 2.7 (d) 9.0 (e) 24.3 (f) 17.5 (g) 6.2 (h) 2.1 (i) 8.7 (j) 26.2 2. (a) $3.22 (b) $2.21 (c) $3.34 (d) $3.21 (e) $4.23 (f) $1.21 3. (a) $514.19 (b) Yes (c) $425.11 x 26 pays/year = $11 052.86 Challenge Answers will vary; e.g. (7 + 7 + 7) ÷ 7 = 3

1. (a) _ < 18 < _ (f) _ < 20 < _ (b) _ < 13 < _ (g) _ < 36 < _ (c) _ < 18 < _ (h) _ < 38 < _ (i) _ < 21 < _ (d) _ < 18 < _ (e) _ < 28 < _ (j) _ < 31 < _ 2. (a) $3.02 (b) $1.18 (c) $1.26 (d) $1.15 (e) $1.17 (f) $1.13 3. Teacher check 4. Teacher check Challenge

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Unit 30–1

(b)

(c)

(d)

(e)

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• Use real-life forms and ask students to answer questions.

Consolidation 30–2 • Students develop a question which requires the use of an arrow diagram.

Consolidation 30–3

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1. (a) 22 (b) 34 (c) 13 (d) 14 (e) 22 (f) 97 (g) 113 (h) 80 (i) 60 (j) 141 2. (a) $1.04 (b) $1.03 (c) $1.06 (d) $1.06 (e) $1.07 (f) $1.09 2 3. (a) /5

• Provide students with further opportunities to represent fractions in diagrammatic form.

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/10

Challenge

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New Wave Maths Book F – Teachers Guide • 141 •

Unit 31–1

Student page 91

Outcomes

Indicators

N3.3, N4.3, S4.1, M4.2

The student is able to: • use the informal idea that a particular map is ‘to scale’ in interpreting it. • compare and order length, capacity and mass measurements provided in common standard units.

Skills • measuring

• multiply • order • path

• calculator • ruler

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

Number (N4.3)

• 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 (S4.1, M4.2) Warm up

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• Alice Springs is the central city that you will be working from to find the order of proximity of other Australian cities. • How will you be sure that you have your cities in the correct order? Use a ruler. Remind students to measure from the 0 (zero) point not the end of the ruler. • Measure the direct paths as shown on the map of Australia to find the distance from Alice Springs to each other city shown.Write these in order from closest to furthest in the space provided. • Using the paths shown, find out how many different ways you can travel to Sydney from Perth passing through two other cities on the way. Write each different journey as you find it. • Share different travel paths with the class by asking students to name the path they chose.

Challenge

Notes

• Ask students the following questions: Which Australian city is located nearest the centre of Australia? Which Australian city do you think is closest to this city? Which Australian city do you think is furthest from this city?

What to do

Language

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Teac he r

Memory Masters (N3.3)

Resources

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• Write a word question that would lead to one of the calculations found in Exercise 2.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 98 – 99. • 142 • New Wave Maths Book F – Teachers Guide

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

Unit 31–2

Student page 92

Outcomes

Indicators

N3.3, N4.3, N4.1a

The student is able to: • round decimal numbers accordingly.

Skills

Resources

Language

• calculator • place value chart (see pages 207 and 208)

• reading • rounding

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• multiply • units • tenth • hundredth • metre • degree • litre • maximum

• rounding • nearest • thousand • kilometre • second • tonne • total

Notes

Teac he r

Memory Masters (N3.3)

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• The focus for this unit is subtraction of multiples of 10 less than 100 from multiples of 10 less than 200.

Number (N4.3)

• 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 (N4.1a) Warm up

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What to do

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• Discuss rounding in context. For example, if you were painting and you needed 6 litres of paint, you would round up to a 10 litre can of paint rather than down to a 4 litre can. • Tell the class that today’s activity will be rounding of decimal numbers. • Ask the class to provide rules for rounding. First number to the right of the rounding place is the focus number. If this number is 1 – 4 then round down, if the number is 5 – 9 then round up. • Ask students whether the decimal point affects the rules of rounding. (Answer is no.)

• Use these examples with the class as refreshers – round to the nearest tenth 6.47, 3.483; round to the nearest hundredth 5.271, 2.694; round to the nearest one 63.24, 74.8. • Read through the activities with the students then set the class to work the activities.

Challenge

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• Set students to work out the maximum number of vehicles possible. Record working and final result.

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New Wave Maths Book F – Teachers Guide • 143 •

Unit 31–3

Student page 93

Outcomes

Indicators The student is able to: • order regions by using a unit including instances where partunits need to be combined.

N3.3, N4.3, M4.2

Skills • measuring

Language • multiply • shapes • area • perimeter • pentominoes • divide • straight lines • sum • sector

• calculator

r o e t s Bo Notes 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 less than 1000.

Number (N4.3)

• 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 (M4.2) Warm up

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Memory Masters (N3.3)

Resources

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• Discuss the 1-cm grid and explain to the students they will need to count each square to find the size of each state and Territory. • Any half or quarter squares should be added together to make one whole square. • Record the number of squares in each state and Territory. • Allow students the opportunity to work as a whole group to count the squares of one state or Territory. Record the number of squares. • Students can then work independently or in pairs to count the squares and record the total for each state and Territory. • Students can then write the states and Territory in order of size from biggest to smallest.

Challenge

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• Students are to look at combinations of numbers so that they are able to divide the clock face into sectors so the sum of each sector is the same. • All attempts are to be recorded as well as the reasoning behind the attempts. • Share results with the class or teacher.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 138 – 139. • 144 • New Wave Maths Book F – Teachers Guide

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Unit 31—Answers

Student pages 91 – 93

Unit 31–1

1. (a) 120 (b) 150 (c) 50 (d) 90 (e) 90 (f) 80 (g) 30 (h) 90 (i) 70 (j) 120 2241.58 (c) 2865.28 2. (a) 836.05 (b) (d) 1702.22 (e) 3734.40 (f) 4902.30 2. (a) 39 ºC (g) 2.7 km 1.2 m (b) 4.2 m (h) (c) $160.00 (i) 12.0 secs (d) $28 000 (j) 1.75 L (e) 5.7 tonnes (f) 3.36 kg Challenge 1 car + 28 motorbikes 29 vehicles

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1. (a) 530 (b) 150 (c) 140 (d) 1100 (e) 1200 130 (f) 150 (g) 1300 (h) 1000 (i) 120 (j) 2. (a) $2545.23 (b) $4834.70 (c) $6223.68 (d) $3995.88 (e) $1915.65 (f) $4052.91 3. (a) Darwin, Adelaide, Melbourne, Canberra, Brisbane, Sydney, Perth and Hobart (b) Sydney – Alice Springs – Darwin – Perth Sydney – Brisbane – Alice Springs – Perth Sydney – Canberra – Alice Springs – Perth Sydney – Canberra – Melbourne – Perth Challenge Teacher check

Unit 31–2

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• Find out the actual distances between the cities using the scale in an atlas.

Consolidation 31–2

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1. (a) 600 (b) 1000 (c) 1500 (d) 1600 (e) 2400 (f) 3500 (g) 1200 (h) 3600 (i) 2000 (j) 600 2. (a) 42.14 (b) 17.48 (c) 54.76 (d) 30.24 (e) 20.06 (f) 46.72 3. (b) Answers are approximate. • WA ≈ 241/2 • Qld ≈ 16 • NT ≈ 13 • SA ≈ 10 • NSW ≈ 81/2 • Vic. ≈ 3 • Tas. ≈ 3/4 • ACT ≈ 1/4 Challenge

• Engage students in various activities; e.g. running, jumping and so on. Time and measure results and ask students to round accordingly.

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Consolidation 31–3

• Use the 1-cm grid on page 199. Ask students to draw their own design. Swap with a partner and repeat the activity.

New Wave Maths Book F – Teachers Guide • 145 •

Unit 32–1

Student page 94

Outcomes

Indicators The student is able to: • rewrite a decimal as a fraction. • use judgments of length to estimate the position of fractions on a number line.

N4.3, N4.1a, N4.1b

Skills • rewriting fractions • converting fractions to decimals • recording

Teac he r

Memory Masters (N4.3)

Resources • calculator

Language • divide • number line • fractions • decimal values • equivalent • tenths • maximum

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

Notes

• The focus for this unit is division of a multiple of 1000 by a whole number 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 (N4.1a, N4.1b) Warm Up

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Number (N4.3)

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• Ask students to look at the number line. Work with the class as a whole to complete the decimal and fraction equivalents. • When converting fractions to decimals it is often easier to change the fraction to an equivalent fraction with a denominator of 10, 100 or 1000 if possible. Use this information to work through Exercise 3 with the class as a whole. • For Exercise 4, ask the class what the decimal equivalent of 1/4 is. How did they obtain the answer for this? For those who still have difficulties converting fractions to decimals, suggest they use their calculator and divide the bottom number into the top number. The answer is the decimal they are seeking; e.g. 1/4 = 1 ÷ 4 = 0.25. • Work with the class as a whole to complete questions (b) to (f). Remind students that fractions with denominators of 10, 100 or 1000 provide direct conversion to decimals with the numerator filling the equivalent decimal place tenth, hundredth or thousandth. For example, 1/100 = 0.01, 94/100 = 0.94 • Students continue to complete the rest of the questions.

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Challenge

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• Students are to record all their workings and their conclusion for sharing with the class or teacher.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 58 – 59. • 146 • New Wave Maths Book F – Teachers Guide

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

Unit 32–2

Student page 95

Outcomes

Indicators

N4.4, N4.3, C&D4.3, C&D4.4

The student is able to: • represent data in diagrams and tables. • interpret and report on information provided in tables and bar graphs where data are grouped in simple intervals which can be regarded as categories.

Skills • counting • recording • interpreting • reasoning

Resources

Language

• calculator • newspaper • reading book • magazine • library book

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

• divide • count • record • length • tally • total • greatest • probability • increasing order

Notes

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Memory Masters (N4.4)

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

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Number (N4.3)

• 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&D4.3, C&D4.4) Warm up

© R. I . C.Publ i cat i ons What to• dof orr evi ew pur posesonl y• • Discuss how to record data in a table using a tally. • Four downward strokes are drawn with one across, tying them together and forming a group of five. Ask students why they think groups of five are made (easier to count).

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Challenge

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• This activity may be best done in small groups. • For each of the reading materials, students are to open to any page. On each page, count down to the end of the one hundredth word and put a light pencil mark that can easily be removed. • Instruct students that for each publication they are to record, in tally form on the chart provided, out of the first hundred, the number of words that have 1 – 10 and more than 10 letters. • When all publications have been counted, write the total of the number of words with each number of letters. • Using the information recorded, answer the questions on the page. • Share the results from each group with the class as a whole to see if the same results are shown in each recording for individual groups.

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• Which letter occurs most in English? Design a method for collecting data to find out and then try it.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 162 – 163. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 147 •

Unit 32–3

Student page 96

Outcomes

Indicators The student is able to: • estimate and check calculations.

N3.3, M4.1, N4.3

Resources • calculator

• divide • division • sums • calculator • accurate • round • nearest • multiply • symbols

Skills • estimating • dividing • rounding • checking • multiplying

Language • estimate

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

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

Number (N4.3)

• 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 (N4.3) Warm up

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Memory Masters (N3.3, M4.1)

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• Discuss with the class how to estimate when dividing. Round the number being divided to the nearest 10. Use knowledge of basic facts to give an approximation of the answer. For example, 294 ÷ 4 – round to 290 ÷ 4; 4 x 7 = 28 therefore approximation 70. This figure is the estimate. Use this process for making estimates. • Use this process to make estimates in Exercise 3. • Check estimates with a calculator to see how accurate they were. • Exercise 4 requires students to round the numbers to the nearest 10 then multiply. Answers can be checked by calculator. • Work through each activity with the class as a whole; for example, 4(a): 43 rounding 40 multiply 1200 x 27 30 Repeat for the rest of the exercise. • Use the same methods from Exercises 3 and 4 to complete 5.

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Challenge

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© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Discuss with the class the importance of estimating. • Estimating helps to guide us when considering the correctness of our answer.

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• Students will need to experiment with adding, subtracting, multiplying and dividing to find ways of making other numbers using a single number and the four operations. (Brackets may also be used.) • As a hint, ask students how they might make the number 1 from the number 8 using one or all of the operations; for example, 8 ÷ 8 = 1. • Leave students to experiment either by themselves or in small groups. • Share answers with the class and encourage the finding of different combinations. Praise efforts but don’t give answers that students can not find.

• 148 • New Wave Maths Book F – Teachers Guide

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

Unit 32—Answers

Student pages 94 – 96

Unit 32–1

Unit 32–2

1. (a) 2000 (b) 4000 (c) 1000 (d) 1000 (e) 3000 (f) 2000 (g) 2000 (h) 1000 (i) 1000 (j) 2000 2. (a) $4.00 (b) $5.00 (c) $5.00 (d) $6.00 (e) $5.00 (f) $8.00 3.

1. (a) 5.0 (b) 10.0 (c) 1.9 (d) 10.5 (e) 25.5 7.0 (f) 5.5 (g) 3.5 (h) 2.0 (i) 4.5 (j) 2. (a) $30.21 (b) $20.12 (c) $20.12 (d) $30.23 (e) $10.21 (f) $10.23 3. Answers will vary 4. Teacher check Challenge E is the most commonly used letter in the English language.

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6 4. (a) 0.2 (b) 8/10 0.8 (c) /10 0.6 (d) 4/10 0.4 5. (a) 0.25 (g) 0.5 (m)0.75 (s) 0.5 (b) 0.875 (h) 0.2 (n) 0.5 (t) 0.1 (o) 0.19 (u) 0.04 (c) 0.01 (i) 0.05 (d) 0.5 (j) 0.75 (p) 0.8 (v) 0.46 (e) 0.94 (k) 0.13 (q) 0.37 (w) 0.71 (f) 0.03 (l) 0.09 (r) 0.9 (x) 0.3 Challenge 1x2= 2 10 x 3 = 30 11 = 32 wheels

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• Provide students with further opportunities to convert fractions to decimals.

Consolidation 32–2

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1. (a) 0.06 m (b) 52.84 m (c) 0.28 m (d) 6.52 m (e) 4.96 m (f) 800 cm (g) 725 cm (h) 99 cm (i) 4300 cm (j) 8 cm 2. (a) $3.36 (b) $3.59 (c) $3.55 (d) $3.85 (e) $6.37 (f) $4.78 3. (a) 62r4 ≈ (62.57) (c) 67r2 ≈ (67.25) (b) 49r1 ≈ (49.17) (d) 84r7 ≈ (84.78) 4. (a) 1161 40 (d) 3243 50 x 30 x 70 1200 3500 (b) 2376 70 (e) 2392 50 x 30 x 50 2100 2500 (c) 2173 50 (f) 3599 60 x 40 x 60 2000 3600 5. (a) 57r5 ≈ 57.83 (b) 1008 ≈ 1200 Challenge Answers will vary; e.g. (8 – 8/8 ) – 8/8 = 6

• Discuss students’ responses to Exercise 4(d).

Consolidation 32–3

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

• Provide students with further estimating activities.

New Wave Maths Book F – Teachers Guide • 149 •

Unit 33–1

Student page 97

Outcomes

Indicators

The student is able to: • identify the transformation(s) used to produce a spatial sequence and continue the sequence. • provided with a single copy of a shape that will tile, produce a tiling pattern by systematically translating, rotating or reflecting the shape.

N3.1, N4.3, S4.3

Skills • tessellating • drawing

Teac he r

Memory Masters (N3.1)

Resources • calculator • paper • coloured pencils • pencil • tape • ruler • light card • Escher drawings (if available)

Language • subtract • Escher drawings • shapes • trace • transformations • template • design • corner • adjacent • side • opposite

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

Notes

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

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

Main Activity (S4.3) Warm up

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Number (N4.3)

• Escher drawings are based on tessellations and transformations of shapes. • They are based on starting with a shape that tessellates. This shape is changed by taking from one side and adding to the other. • Discuss what it means to tessellate. • Show pictures of Escher drawings if available.

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• Simple Escher drawings can be completed by students if the following example is followed. • Use a rectangle – large piece of card or demonstrate on an overhead or blackboard/ whiteboard. – cut a simple shape from one end (make the cut start from the top corner and finish at the bottom corner). – slide the cut out shape to the opposite end of the rectangle and tape it in place. – use the new shape as a template to trace tessellating shapes onto a larger sheet of paper (between A4 and A3 size). – decorate and colour the design. • The original shape must tessellate itself. For first attempts, the square or rectangle is the best option. • When first attempts are finished, suggest to the students they try another more complex design that does not necessarily start at or finish at a corner. Remind students that when the cut section is reattached it must be the same distance from a corner as when it was cut or the shape will not tessellate. • Try using more than one cutout in the design. • Use the finished designs for colourful decorations in the class.

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What to do

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Challenge • Do all triangles tessellate? For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 38 – 39. • 150 • New Wave Maths Book F – Teachers Guide

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

Unit 33–2

Student page 98

Outcomes

Indicators

N3.3, N4.3

The student is able to: • estimate and check calculations.

Resources

Language • subtract • estimate • sums • calculator • symbols

• calculator

Skills • estimating • calculating

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

Notes

Teac he r

Memory Masters (N3.3)

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• The focus for this unit is addition of a multiple of 10 greater than 100 to a whole number less than 100.

Number (N4.3)

• 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 (N4.3) Warm up

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• What to do • Discuss with the class the importance of estimating. Estimating helps to guide us when considering the correctness of our answer. • Brainstorm situations when we might estimate.

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Challenge

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• Working with Exercise 3(a), ask students how they would estimate the answer. If there is more than one suggestion, ask students to outline the alternative. Using Exercise 3(a), ask students to show how their chosen method works. • Students may then be set to work to complete the questions on the page. • Alternatively, focus students’ attention on rounding to find an estimate.This process is done mentally as follows. Round each number to the highest place value possible; for example, 460 x 26 ≈ 500 x 30 = 1500. • Check each answer using a calculator. • Students may work independently through the questions or work under teacher direction as a whole class with students giving the instructions and the whole class responding as they work through each activity.

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• Students will need to experiment with adding, subtracting, multiplying and dividing to find ways of making other numbers using a single number and the four operations. (Brackets may also be used.) • As a hint, ask students how they might make the number 1 from the number 7 using one or all of the operations; for example, 7 ÷ 7 = 1. • Leave students to experiment either by themselves or in small groups. • Share answers with the class and encourage the finding of different combinations. Praise efforts but don’t give answers that students can not find.

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

New Wave Maths Book F – Teachers Guide • 151 •

Unit 33–3

Student page 99

Outcomes

Indicators

N3.3, N4.3, M4.4a, C&D4.3

Skills • measuring • recording • counting • using scale

Memory Masters (N3.3)

The student is able to: • calculate perimeter and area of various polygons within a design. • represent data in diagrams and tables.

Resources

• subtract • squares • rectangles • diagram • width • length • perimeter • area • table

• calculator • ruler

Number (N4.3)

• 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 (M4.4a, C&D4.3) Warm up

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• The focus for this unit is subtraction of a whole number less than 100 from a multiple of 10 greater than 100.

Teac he r

Language

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• Focus student attention on the scale diagram on the page. Ask students what they see. • Read the instructions to them, explaining that there are many rectangles and squares drawn in the diagram, some obvious, some hidden. The task is to find as many of these squares and rectangles as you can and then record their length, width, perimeter and area on the table provided. • If students are unable to determine the length of lines from the information provided, remind them that the diagram is drawn to scale. Ask how long and how wide the diagram is from the given measurements. Ask students to measure the length and width with their ruler. The drawn length and width is 4 cm. Therefore it is drawn with a scale of what? (1 cm = 4 cm) This scale may be used if students wish to measure lengths and widths rather than use the measures given. • The table may be used to record the shape of the rectangle or square that is being counted by outlining/shading the rectangle or square in the left-hand column. This will assist in ensuring that the same shapes are not counted twice and as a reminder of which shapes have been counted. For example a 5 x 5 cm square may be shown by shading/outlining the top left square on the appropriate diagram on the left. • Good luck and happy searching!

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Challenge • How many squares are on a chessboard? (The answer is not 64!)

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 134 – 135. • 152 • New Wave Maths Book F – Teachers Guide

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Unit 33—Answers

Student pages 97 – 99

Unit 33–1

1. (a) 167 (b) 196 (c) 179 (d) 192 (e) 176 (f) 193 (g) 197 (h) 169 (i) 186 (j) 191 2. (a) $32.08 (b) $64.27 (c) $23.17 (d) $25.18 (e) $33.35 (f) $43.47 2. (a) 11 960 (f) 23 940 (k) 44 280 (p) 28 080 (b) 32 224 (g) 33 538 (l) 29 711 (q) 45 184 (c) 25 002 (h) 17 802 (m) 40 552 (r) 16 284 (d) 35 972 (i) 74 994 (n) 30 966 (s) 73 866 (e) 30 723 (j) 53 088 (o) 21 812 (t) 53 568 Challenge Answers will vary; e.g. (7 x 7) + 7 + 7 – 7/7 = 6

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Teac he r

1. (a) $0.42 (b) $0.76 (c) $0.16 (d) $0.94 (e) $0.32 (f) 73c (g) 69c (h) 14c (i) 86c (j) 41c 2. (a) $4.22 (b) $4.58 (c) $2.55 (d) $2.07 (e) $1.15 (f) $3.63 3. Answers will vary

Unit 33–2

Consolidation 33–2 • Ask students to develop a word problem to match one of the problems on the page.

Consolidation 33–3

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Width

Length

Shape

Perimeter

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Area

256

Perimeter

Length

Area

Shape

Perimeter

Width

Length

Shape

Width

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1. (a) 142 (b) 183 (c) 158 (d) 104 (e) 117 (f) 135 (g) 124 (h) 128 (i) 117 (j) 112 2. (a) $49.99 (b) $29.99 (c) $19.96 (d) $19.98 (e) $39.98 (f) $29.99 3.

• Discuss the process used by the students to complete the task.

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4. 1 cm = 4 cm 5. P = (L + W) x 2 A = L x W Challenge 204 squares = 64 – 1 x 1 squares, 49 – 2 x 2 squares, 36 – 3 x 3 squares, 25 – 4 x 4 squares, 16 – 5 x 5 squares, 9 – 6 x 6 squares, 4 – 7 x 7 squares, 1 – 8 x 8 square. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book F – Teachers Guide • 153 •

Unit 34–1

Student page 100

Outcomes

Indicators The student is able to: • estimate and check calculations.

N3.3, N4.3

Resources

• divide • estimate • division • sums • calculator • accurate • round • nearest • multiply

• calculator

Skills • estimating • dividing • rounding • checking • multiplying

Memory Masters (N3.3)

Language

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

Notes

Number (N4.3)

• 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 (N4.3) Warm Up

• Discuss with the class the important of estimating. • Estimating helps to guide us when considering the correctness of our answer. • Note: Techniques change according to the operation used.

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Teac he r

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

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• Discuss with the class how to estimate when dividing. Round the number being divided to the nearest 10. Use knowledge of basic facts to give an approximation of the answer. For example, 275 ÷ 4 – round to 280 ÷ 4; therefore, approximation 70. This figure is the estimate. Use this process for making estimates. • Check estimates with a calculator to see how accurate the estimate was. • Exercise 4 requires students to round the numbers to the nearest 10 then multiply. Answers can be checked with a calculator. • Work through each activity with the class as a whole; for example 4(a): 84 rounding 80 multiply 4000 x 48 50 Repeat for the rest of the exercise.

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Challenge

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• Students will need to experiment with adding, subtracting, multiplying and dividing to find ways of making other numbers using a single number and the four operations. (Brackets may also be used.) • As a hint, ask students how they might make the number 1 from the number 4 using one or all of the operations; for example, 4 ÷ 4 = 1. • Leave students to experiment either by themselves or in small groups. • Share answers with the class and encourage the finding of different combinations. Praise efforts but don’t give answers that students can not find.

• 154 • New Wave Maths Book F – Teachers Guide

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

Unit 34–2

Student page 101

Outcomes

Indicators The student is able to: • represent data in tables and diagrams which may include arrow diagrams, Venn diagrams and two-way tables. • complete construction of a pie graph.

N4.4, N4.3, C&D4.3

Skills • counting • recording • graphing

Resources

Language

• calculator • protractor • ruler • coloured pencils

• divide • circle or pie graph • information • percentage • approximate • area • hectares

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

Memory Masters (N4.4)

Notes

Number (N4.3)

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Teac he r

• The focus for this unit is division of a multiple of 10 less than 100 by a multiple of 10 less than 100.

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

Main Activity (C&D4.3) Warm up

• Hold up a circle cut from card or stiff paper and explain to students this may be used to show a graph commonly known as a pie graph. • If half or 50% of the data were to be shown on a pie graph it would be drawn as a line bisecting the circle – draw a line across the circle to show this. • Explain this represents half of the total degrees in the circle – half of 360º or 180º. • Explain to students this may be drawn using a protractor. Mark in a radius anywhere on the circle. Use a protractor to show 180º or half of the circle. The result is the same as previously shown. • Work through other segments and draw them at one-fifth or 72º, two-thirds or 240º.

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• Work through the information provided to find the percentage of the total for each area and then convert to degrees. • Explain to students that to find the percentage they firstly need to find the total area by adding all of the areas given. Do this as a whole class using a calculator. The next step is to divide the individual areas by the total area and multiply by 100 to find the percentage. Record the percentage in the space provided. The final step is to work out the number of degrees. This is done by entering 360, then x, then percentage total (e.g. 20), then the % key, to give the total required. • When all degree totals have been found they may be transferred to the pie graph as explained earlier. • The whole of this activity may be completed with the teacher directing all the way or providing guided instruction and allowing the students to work through by themselves. • Collect information from the class by a show of hands to indicate those families that have one child, two children, three children, four children and more than four children to complete the table and pie graph.

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Challenge • Use a spreadsheet and charting option to draw a pie graph showing the area of A Reserve Parklands in Australia. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 155 •

Unit 34–3

Student page 102

Outcomes

Indicators The student is able to: • round to simplify calculations.

N4.3

Resources

• divide • calculate • round • nearest ten • approximation • symbols

• calculator

Skills • estimating • rounding • multiplying

Memory Masters (N4.3)

Teac he r

Number (N4.3)

• 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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• Estimating answers is used frequently in everyday activities. A good example is when shopping where the shopper may have a limited amount of money to spend so it is necessary to keep a rough running total in his/her head. • Today’s exercise investigates two different means of estimating answers. One, rounding off, follows standard practices of rounding to the nearest 10. The second rounds all numbers down to the nearest 10. • By calculating the correct answer for each sum, then rounding off and rounding down, you will be able to say which of the methods of rounding gives the better approximation or estimate.

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Main Activity (N4.3)

What to do

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• The focus for this unit is to encourage students to find ways to simplify number problems and to support their approach; e.g.: If you had one wish and could change one digit in the following question which one would you change? Explain why. 14 x 6 I would change the 6 to a 5 because it is easier to multiply by fives.

Warm up

Language

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• When doing the correct calculations, use your calculator. • Working with the class as a whole, calculate the correct answer, then round off (asking students for the answer) and round down.Work out the answer to the two sets of rounded numbers (again use a calculator if unable to complete the working in your head). • Repeat for (b) and (c) then set the class to work by themselves.

Challenge • Students will need to experiment with adding, subtracting, multiplying and dividing to find ways of making other numbers using a single number and the four operations. (Brackets may also be used.) • As a hint, ask students how they might make the number 1 from the number 6 using one or all of the operations; for example, 6 ÷ 6 = 1. • Leave students to experiment either by themselves or in small groups. • Share answers with the class and encourage the finding of different combinations. Praise efforts but don’t give answers that students can not find. • 156 • New Wave Maths Book F – Teachers Guide

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Unit 34—Answers

Student pages 100 – 102 Unit 34–2

1. (a) 4200 (b) 4500 (c) 6400 (d) 6300 (e) 1800 (f) 2800 (g) 1600 (h) 5400 (i) 4900 (j) 4800 2. (a) 0.2 (b) 0.3 (c) 0.2 (d) 0.3 (e) 0.2 (f) 0.4 3. (a) 68r3 = 68.75 (b) 72r6 ≈ 72.67 (c) 54r6 ≈ 54.86 (d) 77 = 77 (e) 46r6 = 46.75 (f) 73r5 ≈ 73.71 4. (a) 4032 80 (d) 4002 70 x 50 x 60 4000 4200 (b) 2262 60 (e) 1836 40 x 40 x 50 2400 2000 (c) 1968 80 (f) 4307 70 x 20 x 60 1600 4200 Challenge Answers will vary; e.g. 4 + 4 + 4/4 = 9

1. (a) 4 (b) 1 (c) 2 (d) 1 (e) 1 (f) 2 (g) 3 (h) 2 (i) 3 (j) 1 2. (a) 3.2 (b) 1.2 (c) 2.1 (d) 4.2 (e) 2.2 (f) 3.1 Approximate area in 3. State or Territory Percentage (%) millions of hectares New South Wales

1.4

8.75

Victoria

0.2

1.25

Queensland

1.0

6.25

Western Australia

5.2

32.5

South Australia

3.6

22.5

Tasmania

0.4

2.5

Northern Territory

4.2

26.25

16.0

100%

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Vic.

NSW

Qld

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Unit 34–1

Tas.

4. Teacher check

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• Discuss the methods used by the students to check their work. How successful were they?

Consolidation 34–2

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1. Teacher check 2. (a) 2.4 (b) 1.2 (c) 1.6 (d) 1.9 (e) 1.2 (f) 1.4 3. (a) 1161 1200 800 (g) 2173 2000 2000 (b) 2262 2400 1500 (h) 1968 1600 1600 (c) 2376 2100 2100 (i) 4002 4200 3000 (d) 1161 1200 800 (j) 2444 2500 2000 (e) 3588 4000 2800 (k) 2688 2400 2400 (f) 2184 1800 1800 (l) 5244 5600 4200 4. Answers will vary Challenge Answers will vary; e.g. 6 – 6/6 = 5

• Students develop a question and collect data which will result in a recording on a pie graph.

Consolidation 34–3

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• Discuss students’ responses to Exercise 4.

New Wave Maths Book F – Teachers Guide • 157 •

Unit 35–1

Student page 103

Outcomes

Indicators

WM4.2, C&D4.2, N4.1, N4.2, N4.3

Skills • reasoning • estimating • problem solving • recording data • working mentally • speaking and listening • taking risks • collaborative learning and working

Resources • calculator • supermarket catalogues (optional)

• This activity is designed for students working collaboratively in groups. As students will need to discuss their opinions and ideas, allow enough time so they do not feel rushed and for ideas to evolve. Open-ended tasks such as these are a good opportunity for students to ‘take a risk’ with maths. • The activity is designed to be open-ended and investigative. Students may request resources such as supermarket catalogues. • When completing open-ended tasks, some students may be more successful in mixedability groups rather than same-ability groups. • Some groups will be able to work independently while others may need guidance. The stimulus questions below may prompt such groups. – What items would be required for a sausage sizzle? – What items would need to be purchased? Cost involved? – How would people know there was a sausage sizzle fundraiser? – How much money needs to be raised to fund or partially-fund a school camp? – How would you work out the cost per sausage to make a profit? – Does the profit margin increase as items are purchased in bulk? – How would you calculate all the costs involved and the profit made? • Groups may wish to collate their findings and present them as a poster with diagrams and information or as a series of graphs and calculations. • 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.

Notes

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Teac he r

• data • plan • classify • organise • questions • brainstorm • estimate • round • category

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Main Activity (WM4.2, C&D4.2, N4.1, N4.2, N4.3) What to do

Language

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• 158 • New Wave Maths Book F – Teachers Guide

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Unit 35–2

Student page 104

Outcomes

Indicators

N3.3, N4.3, N4.1b

The student is able to: • subtract fractions. • estimate and check calculations.

Skills • estimating • subtracting • calculating • recording

Resources

Language

• calculator

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• round • nearest hundredth • multiply • subtraction • mentally • sum • estimation • calculator • symbols

Memory Masters (N3.3)

Notes

Number (N4.3)

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• The focus for this unit is rounding of decimals to the nearest hundredth.

• 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 (N4.1b, N4.3)

Warm up

• Discuss the components of a fraction with the class. vinculum numerator 3

denominator

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Challenge

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• In all the examples, the fractions have like denominators. From activities already completed you know that when subtracting fractions with like denominators you need to do what with the numerators? (Take them away.) • What do you do with the denominator? (You write it down.) Why? (Because it is the naming part of the fraction.) The fraction name being the same allows you to simply take the numerators away from each other. This is the same as taking oranges from oranges. However, you can not take apples from oranges even though they are fruit because they are different. Similarly, you can not take away decimals with different denominators. • Work with the class as a whole to do Exercise 3 (a), (i) and (q) before setting the class to complete the exercise. • Exercise 4 requires students to estimate then check their answers with a calculator. • Remind students that estimation is best done by rounding to the place value at the left of the number, in each case here to the nearest 10, then make an estimation. • Work through the first two questions with the class as a whole before asking what the numbers are when rounded to the nearest ten and what the answer is to multiplying the two estimates before checking with a calculator. Complete the set.

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• Students will need to experiment with adding, subtracting, multiplying and dividing to find ways of making other numbers using a single number and the four operations. (Brackets may also be used.) • As a hint, ask students how they might make the number 1 from the number 8 using one or all of the operations; for example, 8 ÷ 8 = 1. • Leave students to experiment either by themselves or in small groups. • Share answers with the class and encourage the finding of different combinations. Praise efforts but don’t give answers that students can not find. For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 64 – 65. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 159 •

Unit 35–3

Student page 105

Outcomes

Indicators The student is able to: • use straightforward timetables and programs with both 12- and 24hour times.

N3.3, N4.2, M4.2

Skills

Resources • calculator

Language • add • timetable

• reading a timetable

Memory Masters (N3.3)

Teac he r

• The focus for this unit is addition of a multiple of 10 greater than 100 to a multiple of 10 less than 100.

Number (N4.2)

• 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 (M4.2) Warm up

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• Shown on this page is a partial timetable for the Derksen to Perth line. • What details are shown on the timetable? (Key, stopping pattern, codes displayed on trains, passenger responsibilities, stations and departure times for Monday to Friday) • Where else would you find a timetable useful?

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• Work with the class as a whole to find the answers to the questions. Read each question in turn and allow time for students to search for the answer. Select a student to provide the answer and give his/her explanation for choosing the answer.

Challenge

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• The timetable shown is for Monday to Friday. Explain what you would change on a timetable for the weekend.

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What to do

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For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 120 – 121. • 160 • New Wave Maths Book F – Teachers Guide

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Unit 35—Answers

Student pages 103 – 105

Unit 35–1

1. (a) 2.88 (b) 6.15 (c) 3.22 (d) 8.61 (e) 7.29 (f) 46.81 (g) 23.26 (h) 0.86 (i) 0.93 (j) 0.48 2. (a) 1020.24 (b) 1210.64 (c) 2936.64 (d) 1478.25 (e) 5012.94 (f) 2322.46 4 2 3. (a) /10 or /5 (i) 24/6 or 22/3 (q) 1 (b) 1/3 (j) 22/10 or 21/5 (r) 4 (c) 2/7 (k) 11/3 (s) 6 2 (d) /9 (l) 44/8 or 41/2 (t) 2 (e) 4/8 or 1/2 (m)23/5 (u) 5 2 1 2 (f) /6 or /3 (n) 5 /7 (v) 1 (g) 1/5 (o) 31/4 (w) 3 (p) 16/9 or 12/3 (x) 4 (h) 1/4 4. (a) 1200, 1176.12 (d) 1200, 1473.868 (b) 2800 3051.30 (e) 2000, 2007.046 (c) 5400, 5010.88 (f) 3500, 3612.752 Challenge Answers will vary; e.g. 8 x (8 – 8/8 ) = 56

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1. Answers will vary. Possible responses may include: – how much money to raise – best time/place to hold a sausage sizzle – cost of sausages – what condiments to have available – inclusion of cheese and onions – the best place to purchase food – hire or source a barbecue and barbecue equipment – purchase gas – and so on 2. Possible responses may include: – cost per sausage to make a profit – cost of equipment, food, hiring etc. – include estimated cost of people’s time – percentage ‘mark-up’ to make a profit etc.

Unit 35–2

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• Students can develop their own fundraising idea and provide all calculations to show their idea could be successful.

Consolidation 35–2

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1. (a) 160 (b) 260 (c) 170 (d) 240 (e) 190 (f) 200 (g) 190 (h) 200 (i) 210 (j) 270 24 998 (c) 4830 (d) 1002 2. (a) 1167 (b) (e) 18 891 (f) 9469 3. (a) 7.13, 7.21. 7.29, 7.37. 7.45, 7.53, 8.01, 8.09. 8.17 (b) 26 minutes (c) 24 minutes (d) (i) 5.13 a.m. (ii) 5.39 p.m. (e) 17 minutes (f) 7.56 a.m. (g) 7.20 a.m. (h) 7.38 a.m.

• Provide students with further opportunities to subtract fractions.

Consolidation 35–3

• Complete a similar activity using a bus timetable.

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New Wave Maths Book F – Teachers Guide • 161 •

Unit 36–1

Student page 106

Outcomes

Indicators The student is able to: • subtract fractions. • estimate and check calculations.

N3.3, N4.3, N4.1b

Skills • subtracting • estimating • multiplying • calculating

Memory Masters (N3.3)

Resources • calculator

• divide • fractions • mentally • multiply • round • estimate • accurate

Teac he r

Number (N4.3)

• 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 (N4.1b, N4.3)

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• Exercises 3(o) to 3(z) require regrouping. If the class is not ready for this, leave these exercises out. For those students who are ready, remind them that regrouping in fractions is the same process as regrouping with whole numbers. • Use this example to assist: 41/3 – 2/3. What whole numbers are to be subtracted from 4? (none) You have 2/3 and you wish to take it away from 1/3, can you? (No) You will have to trade 1 whole from the 4, leaving 3 and 1 whole. The 1 whole is then traded into thirds giving 3/3. You now have 2/3 to take from 4/3. Can this be done? (Yes) It leaves 2/3.Your answer now reads 32/3. In written form you may record the sum something like this. 41/3 – 2/3 Regroup 4 to make 3 + 1 1 2 3 + 1 /3 – /3 Trade 1 for 3 thirds 4 2 3 + /3 – /3 = 3 + 2/3 = 32/3 • Exercise 4 requires rounding of the two numbers before multiplying to find an approximate answer.

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• Remind students about subtraction of fractions with like denominators.

What to do

Notes

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• The focus for this unit is subtraction of a multiple of 10 less than 100 from a multiple of 10 greater than 100.

Warm Up

Language

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• Students record all attempts and reasoning. If need be, show in diagrammatic form the answer to the problem.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 64 – 65. • 162 • New Wave Maths Book F – Teachers Guide

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

Unit 36–2

Student page 107

Outcomes

Indicators

The student is able to: • represent data in diagrams and tables which may include arrow diagrams, Venn diagrams and two-way tables. • interpret and report on information provided in tables and bar graphs where data are grouped into simple intervals which can be regarded as categories.

N3.3, N4.3, C&D4.3, C&D4.4

Skills • gathering data • recording data • analysing

Resources

Language

• calculator • class mates • pencil • ruler

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• divide • tally • information • total • inference • shape • equal parts • Carroll diagram

Notes

Memory Masters (N3.3)

Teac he r Number (N4.3)

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

• 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&D4.3, C&D4.4) Warm up

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• What to do • Discuss how to record data in a table using a tally. • Four downward strokes are drawn with one across, tying them together and forming a group of five. Ask students why they think groups of five are made (easier to count).

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• This activity may be completed in one of two ways. Individual students may move quietly around the class collecting the information for the tally, or the teacher may direct students to stand as their preferred sandwich filling is called out. In the latter case there is little need for a tally; rather, the total may be written down immediately. • When all the information has been collected, hold a general discussion about the information. How many students chose each filling? Are the most liked your favourite? Do boys and girls like the same fillings? (Not collected but a rough idea may be available from remembering the collection reactions.) • Ask students to use the information gathered to make an inference about the sandwich fillings. An inference is something that may be decided from information known without actual proof. It may be possible to prove the inference with more data. • Share the inferences.

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• Students are asked to keep a record of all their attempts at finding the answer to this problem. Keep brief notes of the thinking and reasoning behind each attempt. • Share efforts with the class or teacher.

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

New Wave Maths Book F – Teachers Guide • 163 •

Unit 36–3

Student page 108

Outcomes

Indicators The student is able to: • subtract fractions. • select appropriate operations to deal with situations involving whole large numbers where intuition about the size of the solution may not help in choosing the operation.

N4.3, N4.1b, N4.2

Skills • subtracting • converting fractions • problem solving

Language • divide • subtraction • fractions • regroup • trade • total value • difference between • equally • divided

• calculator

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• The focus for this unit is division of a multiple of 10 less than 1000 by a multiple of 10 less than 100.

Number (N4.3)

• 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 (N4.1b, N4.2) Warm up

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Teac he r

Memory Masters (N4.3)

Resources

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• Work through 3(a), as follows.Take 1 from 3 leaves 2 (write 2 in the answer space).Take 3/6 from 5/6 leaves 2/6 (write 2/6 to the right of the 2 in the answer space to give your complete answer). • Exercises 4(a) to (j) require a different approach. Ask students if they can suggest how they would find the answer. Work through the first three with the class as a whole as follows. As there are no whole numbers or fractions to subtract directly, we need to regroup the whole number into 5 + 1. The 1 is then traded to give 2/2; 4 + 2/2 that may then have 1/2 taken away; giving 4 + 2/2 – 1/2; which leaves 4 + 1/2 or 4 1/2. • Repeat until satisfied the class or individuals can continue by themselves. • Exercise 4 requires another step to be added to the process. Working with Exercise 4(k), take the whole numbers from each other (7 – 3) leaving 4. We now have 4 – 1/4. This may now be completed as for the previous set. • When satisfied that individuals are able, allow them to complete the questions, otherwise continue to work through the questions with them. • Exercise 5 is a simple matter of finding the total cost of the four book orders and the difference between the Years 6 and 7 book orders.

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© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Refresh subtraction of fractions for 3(a) by noting that the sums are in two parts—take whole numbers from whole numbers and fractions from fractions to give the answer.

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Challenge • Students may require blocks or squares or grid paper to assist them with this problem. • A little creative thinking will provide a solution. Again, all attempts are to be recorded in diagrammatic or written form.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 64 – 65. • 164 • New Wave Maths Book F – Teachers Guide

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

Unit 36—Answers

Student pages 106 – 108

Unit 36–1

1. (a) 4000 (b) 2800 (c) 2000 (d) 5400 (e) 1600 (f) 4200 (g) 1800 (h) 900 (i) 7200 (j) 3000 2. (a) 0.04 (b) 0.03 (c) 0.01 (d) 0.03 (e) 0.02 (f) 0.02 3. Teacher check 4. Teacher check 5. Teacher check Challenge

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1. (a) 110 (b) 60 (c) 130 (d) 50 (e) 130 (f) 100 (g) 90 (h) 80 (i) 140 (j) 110 2. (a) 0.8 (b) 0.9 (c) 0.9 (d) 0.8 (e) 0.9 (f) 0.9 1 (s) 32/9 3. (a) 2/7 (j) (b) 4/9 (k) 3 (t) 14/10 or 12/5 (c) 1/3 (l) 5 (u) 24/8 or 21/2 3 (d) /5 (m)2 (v) 12/4 or 11/2 (e) 3/7 (n) 1 (w) 32/6 or 31/3 (f) 2/5 (o) 22/4 or 21/2 (x) 26/8 or 23/4 (g) 4/9 (p) 11/3 (y) 44/7 1 (h) 1 (q) 1 /5 (z) 36/9 or 32/3 (i) 2 (r) 21/7 4. (a) 6400, 6396 (c) 2700, 2816 (b) 1800, 1888 (d) 4500, 4048 Challenge 9 minutes

Unit 36–2

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• Provide students with further opportunities to subtract fractions.

Consolidation 36–2 • Develop a survey question and collect data in tally form.

Consolidation 36–3

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1. (a) 8 (b) 7 (c) 5 (d) 3 (e) 7 (f) 6 (g) 8 (h) 8 (i) 9 (j) 4 0.06 (c) 0.09 (d) 0.09 (e) 0.08 2. (a) 0.06 (b) (f) 0.08 3. (a) 22/6 or 21/3 4. (a) 41/2 (k) 33/4 (b) 33/10 (b) 32/3 (l) 12/3 2 3 (c) 1 /7 (c) 2 /4 (m)24/5 (d) 22/8 or 21/4 (d) 55/6 (n) 27/8 (e) 11/9 (e) 67/8 (o) 29/10 (f) 21/5 (f) 47/10 (p) 25/8 2 1 3 (g) 2 /4 or 2 /2 (g) 3 /8 (q) 23/10 (h) 42/5 (h) 71/3 (r) 11/5 (i) 22/8 or 21/4 (i) 21/4 (s) 11/3 2 1 1 (j) 2 /6 or 2 /3 (j) 5 /6 (t) 21/4 5. (a) $279.25 (b) 55.60 Challenge

• Develop further word problems for students to complete.

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Same area (60 squares ÷ 3 = 20 squares each; 60 squares ÷ 4 = 15 squares each, some of which consist of halves.

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

New Wave Maths Book F – Teachers Guide • 165 •

Unit 37–1

Student page 109

Outcomes

Indicators

N4.2, N4.3, S4.1

The student is able to: • use grid references to locate items on a grid.

Skills

Resources

• subtract • locate • grid • reference points

• calculator • atlas

• locating • writing coordinates

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• The focus for this unit is the distributive property of multiplication, addition and subtraction using brackets.

Number (N4.3)

• 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 (S4.1) Warm up

Notes

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Teac he r

Memory Masters (N4.2)

Language

• Ask students to take out their atlas and open it to a map of Australia. What can they tell about the map? Take all answers, but focus on the lines of latitude and longitude. Explain, if students can’t, that these lines are used to locate places and features on maps. Give some examples such as Perth, Sydney, Brisbane etc.

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• The map of Namyad Island has a centre point grid reference that may be used to locate places and features on the map. • Note the use of four quadrants. • A centre point reference breaks the map into four quadrants: a positive, positive quadrant; a positive, negative quadrant; a negative, negative quadrant; and a negative, positive quadrant. When reading reference points, you read the horizontal axis reference first, then the vertical reference. • With this in mind, point to Lake Colbert. The dot in the middle identifies the location point of the lake. Reading along the horizontal axis it is at grid reference ... ? (4) Reading along the vertical axis it is at grid reference ... ? (–1) The reference point is written as (4, –1). Record this in your workbook beside Lake Colbert. • Continue to find and write the reference points for the rest of the locations given.

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• Use a road map to locate your house. Give the map number and grid reference.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 6 – 7. • 166 • New Wave Maths Book F – Teachers Guide

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

Unit 37–2

Student page 110

Outcomes

Indicators

N4.1a, M4.1, N4.3, N4.1b

Skills • adding • recording

The student is able to: • add fractions. • count forwards and backwards from any whole number.

Resources

Language

• calculator • twenty-seven 2-cm cubes per student

• minutes • hours • subtract • thousands • between • addition • cubes

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Notes

Teac he r

Memory Masters (N4.1a, M4.1)

• The focus for this unit is conversion of hours to minutes and minutes to hours.

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Number (N4.3)

• 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 (N4.1b, N4.1a) Warm up

© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Remind students that adding, like subtracting, with like denominators requires only the addition of the numerators and then the placement of the total over the denominator.

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• When adding fractions and mixed numbers, add whole numbers together and fractions together. • Using 3(a) as an example. There is only one whole number, 2. 5/6 and 4/6 may be added to give 9/6. In this case the numerator is larger than the denominator, making an improper fraction. An improper fraction must be changed to a mixed numeral. How is this done? It is changed by dividing the denominator into the numerator. In this case 6 into 9 leaves 13/6. • Repeat this until students are able to work by themselves. • Exercise 4 requires the addition of two whole numbers and from (c) on changing of improper fractions. Work through examples as above until satisfied students can work by themselves. • Exercise 5 requires students to name the nearest thousand below and above the number given; e.g. 3456, nearest below is ... ? Nearest above is ... ? 3456 lies between 3000 and 4000. Set students to work to complete the activity.

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• Provide students with twenty-seven 2-cm cubes to assist them with this activity. • Record all results and display them in a logical manner for sharing.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 64 – 65. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 167 •

Unit 37–3

Student page 111

Outcomes

Indicators The student is able to: • show commonsense in the choice of units for familiar practical measurement tasks.

N4.1a, N4.3, M4.1

Skills

Resources • calculator • coloured pencils

• estimating • recording • discussing

Language • subtract • units of measure • measurements • estimate • actual measurement • least • shape • adjacent

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Memory Masters (N4.1a) Number (N4.3)

• 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 (M4.1) Warm up

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Teac he r

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

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• Tell students that in this activity they will be deciding on the best standard measuring units to use to measure the items given in the three tables. • After determining the best units for measuring, the students are then to make an educated estimate of the actual length, area, volume/capacity, time, mass, distance, age as required. • Select students to share both units chosen and estimates with the whole class, to confirm the best unit of measure and a good estimate of the required measure.

Challenge

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© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Hold a general discussion with students on measurement and how we measure different items.This may be by direct comparison, using an arbitrary unit of our own choosing, using standard units, or using general terms such as bigger, lighter, longer etc.

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• Students are to work out the least number of colours to colour the shape. No two adjacent sections are to have the same colour. • Share results with the class or teacher.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 94 – 95. • 168 • New Wave Maths Book F – Teachers Guide

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

Unit 37—Answers

Student pages 109 – 111

Unit 37–1

1. (a) 180 mins (b) 300 mins (c) 90 mins (d) 150 mins (e) 195 mins (f) 10 hours (g) 5 hours, 20 mins (h) 3 hours, 10 mins (i) 4 hours (j) 1 hour, 40 mins 2. (a) 225 (b) 126 (c) 3181 (d) 3508 (e) 6164 (f) 4353 (i) 4/5 5. (a) 3000, 4000 3. (a) 31/2 1 6 (b) 2 / (j) / (b) 1000, 2000 3 7 4 4 (c) 3 /7 (k) /5 (c) 6000, 7000 3 1 (d) 4 /5 (l) /2 (d) 8000, 9000 1 8 (e) 6 / (m) / (e) 4000, 5000 2 9 1 2 (f) 3 /2 (n) /3 (f) 5000, 6000 1 3 (g) 4 /8 (o) /4 (g) 7000, 8000 1 9 (h) 2 /5 (p) /10 (h) 2000, 3000 4. (a) 54/5 (h) 61/4 (i) 5000, 6000 7 1 (b) 5 /10 (i) 4 /2 (j) 9000, 10 000 2 1 (c) 5 /5 (j) 6 /2 3 (d) 8 / (k) 62/5 5 2 (e) 5 /7 (l) 52/3 Challenge 4 1 (f) 6 /9 (m)6 /9 (a) 8 (b)12 (c)6 (d) 1 1 5 (g) 7 /2 (n) 5 /7

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Teac he r

1. (a) 78 >_ (b) 60 >_ (c) 57 >_ (d) 75 >_ (e) 60 >_ (f) 76 >_ (g) 66 >_ (h) 103 >_ (i) 129 >_ (j) 93 >_ 2. (a) 125 (b) 391 (c) 273 (d) 436 (e) 261 (f) 252 3. (a) (4, –1) (e) (–2, –2) (i) (–7, –5) (b) (4, 3) (f) (–1, –7) (j) (3, –6) (c) (0, 2) (g) (–7, 6) (d) (–4, 5) (h) (–3, –3)

Unit 37–2

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• Use an atlas and longitude and latitude to locate towns and cities.

Consolidation 37–2

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1. (a) $29.46 (b) $59.83 (c) $90.00 (d) $0.63 (e) $5.64 (f) 8700c (g) 20 603c (h) 4917c 1c (i) 61c (j) 2. (a) 4158 (b) 6448 (c) 2569 (d) 2996 (e) 0.3 (f) 7.2 3. Note: Estimates – Teacher check.

• Provide students with further opportunities to add fractions.

Consolidation 37–3

• Students could measure a selection of items to check estimation.

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Challenge 4

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New Wave Maths Book F – Teachers Guide • 169 •

Unit 38–1

Student page 112

Outcomes

Indicators The student is able to: • use the symbols =, < and > to state comparisons. • order fractions.

N4.3, N4.1a, N4.1b

Skills • comparing fractions • ordering

Memory Masters (N4.3)

Resources • calculator • fraction grid (see page 209 of workbook) • 2-cm cubes

Number (N4.3)

• 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 (N4.1a, N4.1b)

Notes

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• divide • symbols • > greater than • < less than • order • fractions • surfaces • construction

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• The ‘Today’s number is ... ’ activity asks 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

• Ask students to turn to the fraction grid on page 34 of their workbook. • Discuss the components of the fraction grid, comparing the different fractions.

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• Ask students what they can tell you about fractions with like denominators and different numerators.The numerator indicates the size of the fraction compared to others with the same denominator. • Where fractions have unlike denominators it is more difficult to compare them. How might you compare fractions with unlike denominators? Accept students’ responses. • Focus on finding fraction equivalents for those who understand; for example 1/5 = 2/10 if comparing fifths and tenths. • Using the fraction grid allows for direct comparison of fraction sizes and is the best method for comparison, particularly for students who have difficulties. • Work with students who have problems and allow the rest to work through the activities by themselves.

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Challenge

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What to do

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• If required, distribute 2-cm cubes to students for this activity. • Students are required to record the strategies they used to find their answer.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 62 – 63. • 170 • New Wave Maths Book F – Teachers Guide

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Unit 38–2

Student page 113

Outcomes

Indicators

Resources

The student is able to: • represent data in diagrams and tables and describe the information collected from diagrams such as arrow diagrams, tree diagrams, Venn diagrams or Carroll diagrams.

N4.4, N4.3, C&D4.2, C&D4.3, C&D4.4

Skills • reading • problem solving • analysing

Language

• calculator • coloured pencils

• descending order • divide • arrow diagram • combinations • shape • continuous

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Memory Masters (N4.4)

Notes

Number (N4.3)

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• The focus for this unit is ordering whole and decimal numbers.

• 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&D4.3, C&D4.4) Warm up

• Display on blackboard/whiteboard or by overhead projector, a very simple family tree with grandparents, mother, father and two children.

Grandmother A + Grandfather A

Grandmother B + Grandfather B

+ Father

Mother

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Child 1 Child 2 • Explain to the class that this is an arrow diagram showing the descendants of family A and family B over three generations. There are other ways to show this as an arrow diagram. • An alternative arrow diagram may be used to show the possible routes between towns A and D.

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• Direct students to read the directions for completion of the arrow diagram. • Suggest to the students that it may be easier to see the combinations that they choose by using a different coloured pencil for each combination. • All combinations must include one jumper, one shirt and one pair of trousers. • Share completed combinations with the class.

Challenge • Trace over the shape, starting at any point and not lifting the pencil off the page or redrawing any line. • Give a description of how you completed or attempted to complete the task. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 171 •

Unit 38–3

Student page 114

Outcomes

Indicators The student is able to: • interpret and report on information provided in tables and bar graphs where data are grouped into simple intervals which can be regarded as categories. • order students according to criteria.

N3.3, N4.3, M4.2, M4.3, C&D4.4

Skills • reading a timetable • discussing • recording • comprehending • working cooperatively • ordering • graphing

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Memory Masters (N3.3)

Resources • calculator • pencil

Language • divide • time line • decade • arrange • ascending order • youngest • oldest • trace • network • continuous

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• The focus for this unit is addition of two multiples of 100 each less than 1000.

• 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&D4.4, M4.2, M4.3) Warm up

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• Read each question in turn in Exercise 3 to the class and ask them, collectively, to find the answers from page 40. • As the information is located it is to be recorded on page 114. • Check with the students to see what responses they have recorded prior to reading the next question. • Exercise 4 requires groups of eight to be formed in the class. • When the groups are settled, instruct the students within each group to record the names of all the students in their group, and their birthdates, on the table provided, in ascending order—the youngest through to the oldest. • It may be beneficial to record the student names on the time line provided next to the month of birth. Information may be more easily transferred to the table. Remember to check correct order where more than one student has the same birth month or for students born a year earlier or later. • Students crosscheck within their group.

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Challenge

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• Ask students what a decade is.

What to do

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Number (N4.3)

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• Students are to provide written details of how they proceeded in answering this problem. • It may assist to keep other drawings of the network to show attempts.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 122 – 123. • 172 • New Wave Maths Book F – Teachers Guide

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Unit 38—Answers

Student pages 112 – 114

Unit 38–1

1. (a) 83 (b) 47 (c) 42.01 (d) 24.51 (e) 8.4 (f) 8.16 (g) 8.06 (h) 5.064 (i) 4.2159 (j) 2.93 2. (a) 36.4 (b) 58.9 (c) 82.3 (d) 94.7 (e) 72.4 (f) 45.3 3. (a) 10 (a) 40 Challenge Possible answer. Note: Start at E.

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1. Teacher check 3.37 (c) 1.62 (d) 5.47 (e) 2.56 2. (a) 2.77 (b) (f) 3.48 (d) < (g) > (j) > 3. (a) > (b) > (e) < (h) > (k) < (c) < (f) < (i) < (l) < 4. (a) <, < (d) <, > (g) <, < >, > (b) >, > (e) <, > (h) (c) >, > (f) >, > (i) <, > 5. (a) >, > (d) <, < (b) >, < (e) >, < (c) >, < (f) <, < Challenge 34

Unit 38–2

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• Provide students further opportunities to order fractions.

Consolidation 38–2 • Students develop another question which will result in the use of an arrow diagram.

Consolidation 38–3

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1. (a) 400 (b) 900 (c) 600 (d) 700 (e) 900 (f) 1600 (g) 1300 (h) 1500 (i) 1200 (j) 1200 (b) 6 (c) 7 (d) 4 (e) 4 2. (a) 7 (f) 6 3. (a) 1610 – 1620 (b) 1600 – 1610, 1610 – 1620 (c) 80 years (d) Jansz, Hartog and Houtman 4. Teacher check Challenge No

• Devise another feature which could be used to order students.

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New Wave Maths Book F – Teachers Guide • 173 •

Unit 39–1

Student page 115

Outcomes

Indicators

N3.3, N4.3, S4.1

The student is able to: • use the informal idea that a particular map is ‘to scale’ in interpreting it. • draw maps and plans which show a sense of scale.

Skills • measuring • using scale • inferring

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Memory Masters (N3.3)

Resources • calculator • red pencil • blue pencil • ruler

Language • multiply • network • scale

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Notes

• The focus for this unit is subtraction of a multiple of 100 from a multiple of 100.

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

Main Activity (S4.1) Warm up

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Number (N4.3)

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• Read through the questions with the class to ensure that each question is understood. • Ask students how they will find the length of the journey. • Talk to students about scale. After having measured the distance of the journey, how will they determine the actual length of the route? 3(c) tells us that one centimetre is equivalent to one kilometre. Students should be able to state that the measured length in centimetres gives the converted actual length in kilometres. • Ask students if anyone knows how the scale might be represented in written form. Note: This question may be treated as a Challenge and left for students to determine at the end of the lesson themselves. In recording in the format that is used in maps, the resultant scale would be 1:100 000. Simplistically it may be written as 1 cm: 1 km.

Challenge

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© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Explain to the students that the diagram shows a network of roads joining eight schools. • Discuss various situations that may use a network map such as this.

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• Write a word question that would result in a calculation like Exercise 2(e).

• 174 • New Wave Maths Book F – Teachers Guide

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Unit 39–2

Student page 116

Outcomes

Indicators

N4.1a, M4.1, N4.3, N4.1a

The student is able to: • order decimal numbers.

Resources

Language

• calculator • weather maps

Skills • reading a weather map • interpreting data

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• multiply • hectopascals • temperature • highest, lowest • air pressure • digits • counting order • number sentence • equals

Notes

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Memory Masters (N4.1a, M4.1)

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

Number (N4.3)

• 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 (N4.1a) Warm up

What to do

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• Either ask students to collect copies of a weather map from newspapers or bring in copies for students to use. • Ask students to find the maximum temperature for Perth. • Ask students what the highest maximum of any capital city in Australia was. Which city recorded this? • What is the temperature measured in? (degrees Celsius – ºC) • Direct students to look at the lines drawn in circular or long waves on the weather map. Ask if anyone knows what these lines represent. (Air pressure.) What is air pressure measured in? (Millibars or hectopascals—both provide the same readings.) • What is the highest air pressure reading on the weather map that they are viewing? What is the lowest? A general background on high and low air pressures may be given.

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• Here you are given a weather map. From this map, order given temperatures from highest to lowest and record the highest air pressure shown.

Challenge

• Remind students that a number sentence is a statement that equals a given number, in this case 4. • Digits 1 to 4 are to be arranged in their correct counting order to make a number sentence that equals 4. • Students are to record all their attempts for sharing with the class and/or teacher.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 58 – 59. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 175 •

Unit 39–3

Student page 117

Outcomes

Indicators The student is able to: • explain why money and measures use decimal notation.

N4.3, N4.1a

Skills • converting

Memory Masters (N4.3)

Resources • calculator • 2-cm cubes • place value charts (see pages 207 and 208)

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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 (N4.1a)

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Number (N4.3)

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• Have on display a place value chart or ensure that students have access to one. • – How many centimetres in a metre? (100) – How many millilitres in a litre? (1000) – How many metres in a kilometre? (1000) – how many grams in a kilogram? (1000) • Ask students to show you and the class how they would write 500 grams as kilograms. Use blackboard/whiteboard. Repeat for 30 cm, 600 mL, 425 mL, 18 cm, 126 g, 7 g and more if required. • If students are unsure, encourage them to use the place value chart to assist. For example, 7 g is 7/1000 of a kilogram; therefore 7 is in the thousandths place column and written as 0.007. Note the use of 0 before the decimal point to highlight the fact that there is a decimal point.

What to do

• multiply • decimal • metres • centimetres • kilometres • litres • millilitres • kilograms • grams • surfaces • shape

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• The focus for this unit is division of a multiple of 100 less than 1000 by a multiple of 10 less than 100.

Warm up

Language

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• Work through the first two or three questions of each exercise as a whole class before setting students to work by themselves.

Challenge

• Distribute 2-cm cubes if required. • Remind students that it is the external faces of individual cubes that they are required to identify. • Students are to record their working strategies and their findings.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 60 – 61. • 176 • New Wave Maths Book F – Teachers Guide

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Unit 39—Answers

Student pages 115 – 117 Unit 39–2

1. (a) 400 (b) 400 (c) 300 (d) 100 (e) 100 (f) 300 (g) 700 (h) 900 (i) 900 (j) 700 2. (a) 285 (b) 2616 (c) 4560 (d) 5920 (e) 4002 (f) 4452 3. Start at NS

1. (a) 1500 (b) 400 (c) 2800 (d) 1500 (e) 1800 (f) 4500 (g) 800 (h) 3200 (i) 1200 (j) 1000 2. (a) 48 420 (b) 62 400 (c) 27 300 (d) 42 630 (e) 33 768 (f) 33 031 3. 37.2 ºC, 33.8 ºC, 33.1 ºC, 29.2 ºC, 27.6 ºC, 21.4 ºC, 18.9 ºC 17.6 ºC, 15.4 ºC 4. 1028.7 hPa Challenge 1+2–3+4=4

(a) Yes (b) 81 cm (c) 80.5 km (d) 34.2 km HS – NS – DS – ES – CS – JS – HSS – RS – HS (e) No (f) No

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1. (a) 30 (b) 10 (c) 20 (d) 10 (e) 40 (f) 10 (g) 10 (h) 30 (i) 20 (j) 20 28.8 2. (a) 34 748 (b) (c) $422.40 (d) $144.00 (e) $636.40 (f) $546.72 3. (a) 3.56 m 5. (a) 4.627 L (b) 9.21 m (b) 7.954 L (c) 1.92 m (c) 9.256 L (d) 6.35 m (d) 8.317 L (e) 4.9 m (e) 12.3 L (f) 59.007 L (f) 7.06 m (g) 6.08 m (g) 82.089 L 4. (a) 6.829 km 6. (a) 57.329 kg (b) 3.302 km (b) 6.072 kg (c) 9.556 km (c) 92.046 kg (d) 127.008 kg (d) 5.5 km (e) 97.039 km (e) 836.009 kg (f) 156.006 km (f) 6.049 kg (g) 729.284 km (g) 93.3 kg Challenge 24

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• Students design a similar diagram displaying all the places they may visit in one week.

Consolidation 39–2

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Unit 39–1

• Collect a new weather report from the newspaper and complete the activity again.

Consolidation 39–3

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

• Provide students with further opportunities to convert measures to decimal form.

New Wave Maths Book F – Teachers Guide • 177 •

Unit 40–1

Student page 118

Outcomes

Indicators The student is able to: • plan a sequences of calculations using a calculator memory facility.

N4.1a, N4.3

Skills

Resources

Language • round • nearest • tenth • divide • verify • divisible • digits

• calculator

• verify

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Memory Masters (N4.1a)

Notes

Number (N4.3)

• 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 (N4.3) Warm Up

• Explain to students that they are going to verify that a given rule is correct.The rule is that for a number to be divisible by 4 its last two digits must be divisible by 4. • Ask students what verify means. • Ask students how they might go about verifying the rule. Encourage discussion and debate on ideas put forward by individuals. • Ask how many times they think a test of their ideas needs to be carried out to be confident that the rule applies.

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• The focus for this unit is rounding of decimals to the nearest tenth.

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• Students may be set to work in small groups. • All working must be recorded and a written explanation or verification provided.

Challenge

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• Students are to apply their confirmation of the rule for divisibility by 4 to the numbers given. • Keep records of all working and findings.

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• 178 • New Wave Maths Book F – Teachers Guide

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Unit 40–2

Student page 119

Outcomes N4.2, N4.3, C&D4.3, C&D4.4

Skills • graphing • recording • measuring • analysing

Indicators

Resources

The student is able to: • represent data in diagrams and tables which may include arrow diagrams, Venn diagrams and twoway tables. • comment sensibly upon how well their questions are answered by data provided/collected and how it can be improved.

• calculator • classmates’ fingers • classmates’ feet • ruler

Language

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• estimation • number sentence • subtract • scatter graph • measure • length • plot • significant relationship

Memory Masters (N4.4)

Notes

Number (N4.3)

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Teac he r

• The focus for this unit is the distributive property of multiplication, addition and subtraction using brackets.

• 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&D4.2, C&D4.4) Warm up

• Explain to the students that scatter graphs are similar to line graphs in that they use their horizontal and vertical axes to plot data points. Scatter graphs show how much one variable is affected by another.The relationship between the two variables is called their correlation.

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• Use the class register to assist students to put the initials of each class member in order on the table provided as they are called out by you. • Ask students to measure in millimetres the length of their middle finger (the longest one) and record this under their own initials on the table. Then record the length of their foot in millimetres and record this on the table under their own initials. • Reading from the class register, ask each student in turn to call out the length of his/her finger and then the length of his/her foot. The rest of the students record the two measures under the correct initials. • When all measures have been recorded, turn the class’s attention to the scatter graph. • Explain that a scatter graph allows for each individual to be recorded as a separate point on a graph. No lines or bars are required, just a scatter of dots. • Point out that both axes are shown on broken axes; that is they don’t start at zero but at a nominated point so that the graph can be represented at a reasonable size for reading. Broken axes are shown by a jagged line. • Assist the students to plot the first four points before leaving them to plot the rest for themselves. Find the finger length of the first person on the vertical axis, then the foot length on the horizontal axis. Ensure the point is level with both readings then mark it on the page and neatly put the person’s initials beside the point. Continue. • When all points are plotted, look at the graph and determine what it shows you, if anything, about finger length and foot size.

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Challenge • Draw a scatter graph to show what you think the relationship between smoking and lung cancer might look like. For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 166 – 167. R.I.C. Publications® www.ricpublicaions.com.au

New Wave Maths Book F – Teachers Guide • 179 •

Unit 40–3

Student page 120

Outcomes

Indicators The student is able to: • estimate and check calculations.

N4.3

Resources

• subtract • estimate • calculator

• calculator

Skills • estimation • rounding • calculating • recording

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• The focus for this unit is subtraction or addition of a whole number less than 100 from or to a whole number less than 100.

Number (N4.3)

• 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 (N4.3) Warm up

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Memory Masters (N4.3)

Language

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• Remind students first then complete the sum. • In the exercises to be completed, they will be multiplying a decimal by a whole number. In each case, round the decimal to the nearest whole number and the whole number they are multiplying by to the nearest 10. • In 3(a) what are the two numbers you will use when multiplying? Yes, 3 and 50.Your estimate is ... ? 150. Check this number with your calculator. • Repeat with 3(b). • In 4(a) what are the two numbers you will use when multiplying? Yes, 8 and 60.Your estimate is ... ? $480. Check the number with your calculator. • Direct students to complete the activities, providing assistance as required.

Challenge

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• This is a logic problem. You will need to show all your working, reasoning and final answer to be able to explain to another person how you reached the answer.

• 180 • New Wave Maths Book F – Teachers Guide

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Unit 40—Answers

Student pages 118 – 120

Unit 40–1

1. (a) 82>_ (b) 81>_ (c) 41>_ (d) 94>_ (e) 80>_ (f) 41>_ (g) 38>_ (h) 129>_ (i) 117>_ (j) 140>_ 2. (a) 2.4 (b) 2.6 (c) 0.25 (d) 0.24 (e) 2.39 (f) 4.411 3. Teacher check Challenge Teacher check

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1. (a) 3.3 (b) 18.8 (c) 15.1 (d) 4.4 (e) 7.1 (f) 5.8 (g) 9.2 (h) 14.3 (i) 6.8 (j) 0.3 2. (a) 0.04 (b) 0.04 (c) 0.08 (d) 0.06 (e) 0.06 (f) 0.07 3. Teacher check Challenge Correct = 7916, 196, 3576 Incorrect = 2454, 326

Unit 40–2

Brett

Consolidation 40–2

Consolidation 40–3

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• Devise a set of data to collect which can be displayed in a scatter graph.

• Provide students with further estimating opportunities.

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Bob

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Coleen Rochelle

• Discuss students’ working methods.

Chicken

Pizza

Hamburger

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Fish and chips

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1. (a) 25 (b) 13 (c) 55 (d) 25 (e) 42 (f) 122 (g) 82 (h) 137 (i) 156 (j) 102 4.649 (c) $3.54 (d) $3.26 2. (a) 0.046 (b) (e) $43.07 (f) $19.96 3. (a) 150, 154.50 (e) $120, $141.00 (b) 150, 151.20 (f) $240, $228.00 (g) 360, 360.80 (c) 560, 562.40 (d) $200, $212.00 (h) $420, $434.00 4. (a) $480, $481.40 (e) 120, 132.01 (b) $480, $465 (f) 240, 229.14 (g) 480, 506.52 (c) $630, $680.80 (d) $300, $313.60 (h) 280, 254.16 Challenge

✔ ✔

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New Wave Maths Book F – Teachers Guide • 181 •

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• 182 • New Wave Maths Book F – Teachers Guide

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

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Additional Activities

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

New Wave Maths Book F – Teachers Guide • 183 •

Space Activities S4.1

Using a local street map, instruct students to find the shortest path to school. Once marked on the map, direct students to find three different routes. Find the length of each route.

S4.2

This activity may be completed using a variety of three-dimensional shapes or completed after constructing a variety of three-dimensional shapes using the nets provided in the blackline masters on pages 223 – 229 of New Wave Maths for WA Teachers Guide. 1. Use three-dimensional shapes to see if a relationship between edges, faces and vertices can be determined.

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Students may be able to discover that faces + vertices = edges.

Shape

Faces

Vertices

Edges

cube

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Results may be recorded on a table such as this.

2. Make models of three-dimensional shapes using modelling clay. Using fishing line, make cross-sectional cuts. Try to determine what the cross-section will look like before the cut is made.

Alternatively, have the children make a cut that will give a predetermined shape— for example, triangle, rectangle, square—from a given shape.

3. Use the three-dimensional shapes, or models, to show axes of symmetry and planes of symmetry. Three-dimensional axes of symmetry will display rotational symmetry, as shown below.

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A plane of symmetry will effectively slice the model into two identical halves.

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half-turn

rectangular prism

square pyramid

S4.2, C&D4.3

Using a wide variety of three-dimensional shapes, students classify shapes according to their attributes using either a Venn or Carroll diagram found on pages 216 – 217 of New Wave Maths for WA Teachers Guide. Use features of the shapes such as regular, irregular or straight edges, curved edges or flat faces, curved faces (plane, not plane) or by shapes of faces or bases.

• 184 • New Wave Maths Book F – Teachers Guide

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Space Activities cont. S4.2 1. Using different three-dimensional shapes, investigate the faces of the shapes to find out which faces are congruent (i.e. shape and size are identical in every way). 2. If all faces of the shape are congruent, the shape is a regular polyhedron. Have the students find examples of regular polyhedra; for example, cube, tetrahedron, octahedron, dodecahedron or an icosahedron. 3. Students may make their own polyhedra using squares or equilateral triangles that meet at a common point. Students should discover that if the sum of angles at the meeting point is greater than 360°, the shape cannot be made.

S4.3

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1. Do a sphere, pyramid and prism have a plane of symmetry? (YES)

Can there be more than one plane of symmetry? (YES)

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Make models using modelling clay, cut across where you think the plane of symmetry is and check to see if you are correct.

2. Use pattern blocks to make simple patterns with one, two, three and four lines of symmetry. If time, record your patterns and give to a partner for him/her to find the line(s) of symmetry. 3. Use a square piece of paper to show lines of symmetry by:

(a) folding down the centre and making three cut-outs of simple shapes on the fold; or

(b) folding on both diagonals and making one simple cut-out on each fold.

Open your paper out and see the symmetrical pattern. Repeat the activity with different-shaped paper.

S4.4

1. Students are to draw all the possible nets of a cube. Use the grid paper on page 199 of New Wave Maths for WA Teachers Guide.

S4.4

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2. Investigate features of the school grounds and buildings. 1. Use pipe-cleaners or rolls of modelling clay to make numbers or letters of the alphabet without breaking or adding to the original piece. For example, B

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2. Write your name on a deflated balloon. Blow up the balloon and discover what happens to your name. 3. On the grid paper provided on page 198 of New Wave Maths for WA Teachers Guide, draw a square with sides of four units in length; a rectangle with sides of three and five units; a triangle with a base of four units and a vertical height of four units and an octagon with sides of two units. Redraw these shapes on the distorted grids provided on pages 202 – 203 of New Wave Maths for WA Teachers Guide to see what changes and what remains the same in distortion.

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New Wave Maths Book F – Teachers Guide • 185 •

Measurement Activities M4.2 1. Cooking presents a good opportunity to carry out practical measuring activities; for example, making pizza, muffins or scones. 2. Measure the mass of objects using kitchen scales, or bathroom scales for heavier objects. Estimate mass first then measure and record. Objects may be selected by students. M4.3

1. Compare 2 L soft drink bottles, 2 L cordial bottles and 2 L ice-cream containers. 2. Compare 1 L milk cartons, 1 L standard measuring containers and 1 L soft drink bottles. 3. Compare 500 mL containers with each other.

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Collect a variety of containers to compare capacities.

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M4.3

Use a trundle wheel to mark a one-kilometre distance around the school oval. Estimate how long it will take you to walk the one kilometre. Record your estimate, then record the actual time taken. How accurate were you? Estimate how long it will take you to run one kilometre. Time yourself while you run one kilometre. Record your results. How accurate were you?

4. Discuss the actual capacity of each of the containers, making suggestions as to why they change. Estimate prior to measuring in all activities. Using straight-sided transparent containers, containers to catch the overflow and containers marked in standard units; measure the volume of a variety of different objects in any of the following ways.

1. Marking the level water is raised when placing an object in the water. Alternatively, pour off the water above the original water mark after immersing the object and measure the displacement in standard units. 2. Placing an object in an empty container, partly filling the container with water, removing the object and noting how much the water level drops.

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3. Filling a straight-sided container with water, placing an object in the water, collecting the overflow and measuring it in standard units. Use all three methods. 1. Students use balance scales (commercial or self-made) to compare and order the mass of a variety of self-chosen objects. Compare to see if larger objects are heavier.

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2. Students measure mass of given objects against readily available materials; for example, sealed container of rice, nails, 2-cm cubes or other materials. Record results.

• 186 • New Wave Maths Book F – Teachers Guide

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Measurement Activities cont. M4.3 1. Measure the mass of objects using a spring balance. Repeat the measure with the object suspended in water. Record the results each time. Measure a number of other objects the same way.

What conclusions can be drawn from the two measures of each object?

2. Improvise a spring balance by joining strong elastic bands together to make your own scales. Calibrate by weighing objects of known mass. M4.4a

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1. Build a variety of prisms using sixty 2-cm cubes. Record the number of cubes used for the base and the height of each prism. 2. Use 2-cm cubes to build a prism six units long, four units high and two units wide.

M4.4a

Halve the volume of the prism. What happens to the dimensions?

Compare the surface area and mass of the original prism and the second one.

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1. Use a piece of wool, cotton or string 24 cm long. Join ends to make a loop. Use a sheet of grid paper to find the smallest and largest area you can cover using the loop. Draw the outlines on the grid paper. Record the perimeter and area of each. 2. Take twelve 2-cm cubes and make a rectangular prism with a volume of 12 cubic units.

What is the surface area of the prism?

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Make other prisms with the same volume but different surface areas.

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New Wave Maths Book F – Teachers Guide • 187 •

Number Activities N4.2

Give directions for students to take out amounts of wood (Base 10 MAB) to add or subtract given decimal numbers with an unlike number of decimal places. Impress on students that the wood has the value they give it – longs can be ones, tens, hundreds, tenths or hundredths. It may help understanding to use a counter as a decimal point. Recording of results is not important; development of understanding of concepts is the focus. Samples of addition and subtraction: Add 2.34 and 1.48; 1.862 and 2.4; 3.2 and 1.78; 2.761 and 1.89; 3.001 and 0.65 From 6 take 0.4; from 7 take 0.61; from 4.2 take 1.16; from 2.87 take 1.395; from 0.79 take 0.297; from 6.1 take 2.56

N4.3

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2. Use calculators to generate large numbers. For example, show a number on the calculator that is between 847 639 and 857 639; 768 274 and 768 374; 900 000 and 900 003; 420 762 and 420 027; 549 387 and 594 387; 600 000 and 500 000; 999 999 and 888 888; 7 269 486 and 7 296 486.

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1. Provide activities where students estimate then calculate the quantity of grams of rice in a packet, number of flakes in a cornflake packet, grams of sand in a cup and the like. Use strategies to work answers out and verify the accuracy of the strategy.

N4.4

1. Use the triangular grid available on page 201 of New Wave Maths for WA Teachers Guide to make a pattern. Record the pattern you have made in numerical form if you are able. 2. Use the square grid available on page 199 of New Wave Maths for WA Teachers Guide to make patterns of your own choosing. If you are able, record the number pattern in numerical form.

3. Blackboard or copy these number patterns, then have students continue the patterns. ,

19, 27, 35, 5, 27, 49,

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1, 5, 2, 10, 3, 15, 4, 20,

2, 3, 4, 6, 6, 9, 8,

106, 212, 318, 94, 87, 80,

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2, 5, 11, 23,

Students make patterns of their own and share with their classmates. Perhaps make up a small booklet of student-made patterns.

• 188 • New Wave Maths Book F – Teachers Guide

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New Wave Maths Book F – Teachers Guide • 189 •

Student Outcomes Working Mathematically WM4.1 The student compares the ways in which familiar mathematics is done or used in own and other communities.

Chance and Data C&D4.1 The student places events in order from those least likely to those most likely to happen on the bases of numerical and other information about the events.

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WM4.2 The student asks questions to clarify the essential C&D4.2 The student collaborates with peers to plan mathematical features of a problem and uses problem-solving strategies which include those what data to collect and how to classify, based on identifying and organising key information. sequence and tabulate them to answer particular questions, and sees the need to vary methods to WM4.3 The student uses examples to support or refute answer different questions. mathematical conjectures and attempts to make simple modifications of conjectures on the basis of C&D4.3 The student displays frequency and examples. measurement datausing simple scales on axes and some grouping, and summarises data with WM4.4 The student checks, when prompted, that answers simple fractions; highest, lowest and middle are roughly as expected and that methods and scores; and means. answers make sense. C&D4.4 The student reads and makes sensible statements about the information provided in Space tables, diagrams, line and bar graphs, fractions S4.1 The student uses distance, direction and grids on and means, and comments on how well the data maps and plans and in descriptions of locations and answers their questions. paths.

Number The student attends to the shape, size and placement of parts when matching, making and N4.1a The student reads, writes, says, counts with and drawing things, including making nets of 3-D models compares whole numbers into the millions and which can be seen and handled and using some decimals (equal number of places). basic conventions for drawing them. N4.1b The student reads, writes, says and understands S4.3 The student recognises rotations, reflections and the meaning of fractions and, for readily visualised translations in arrangements and pattens and fractions, estimates their relative size and position translates, rotates and reflects figures and objects on a number line and shows equivalence systematically to produce arrangements and between them. patterns. N4.2 The student understands the meaning, use S4.4 The student selects, describes and compares figures and connections between the four operations and objects on the basis of spatial features, using on whole and decimal numbers, and uses this conventional geometric criteria. understanding to choose appropriate operations (whole multipliers and divisors) and construct Measurement and complete equivalent statements. S4.2

M4.2

M4.3

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The student selects appropriate attributes, N4.3 distinguishes perimeter from area and time from elapsed time, and chooses units of a sensible size for the descriptions and comparisons to be made.

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The student calculates with whole numbers, money and measures (one-digit multipliers and divisors), drawing mostly on mental strategies to add and subtract two-digit numbers and for multiplications and divisions related to basic facts.

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The student measures area by counting uniform units including where part-units are required, and measures length, mass, capacity, time and angle, reading whole number scales.

The student uses the known size of familiar things to help make and improve estimates, including centimetres, metres, kilograms, litres and minutes.

N4.4

The student recognises, describes and uses patterns involving operations on whole and fractional numbers, and follows and describes rules for how successive terms in a sequence or paired quantities can be linked by a single operation.

M4.4a The student understands relationships involving the perimeter of polygons, the area of regions based on squares and the volume of prisms based on cubes, and uses these for practical purposes. M4.4b The student understands and uses scale factors involving small whole numbers and unit fractions for straightforward tasks, including making figures and objects on grids and with cubes. • 190 • New Wave Maths Book F – Teachers Guide

Extracted from Mathematics Outcomes and Standards Framework – Student Outcome Statements, Education Department of Western Australia 1998.

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Working Mathematically—Record Sheet

Apply and Verify

Reason Mathematically

Mathematical Strategies

Comment

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Student Name

Contextualise Mathematics

Outcome Category

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New Wave Maths Book F – Teachers Guide • 191 •

Space—Record Sheet

Reason Geometrically

Represent Transformations

Comment

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Student Name

Represent Shape

Represent Location

Outcome Category

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• 192 • New Wave Maths Book F – Teachers Guide

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Measurement—Record Sheet

Indirect Measure

Estimate

Direct Measure

Comment

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Student Name

Understand Units

Outcome Category

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New Wave Maths Book F – Teachers Guide • 193 •

Chance and Data—Record Sheet

Interpret Data

Summarise and Represent Data

Comment

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Student Name

Collect and Organise Data

Understand Chance

Outcome Category

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• 194 • New Wave Maths Book F – Teachers Guide

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Number—Record Sheet

Reason about Number Patterns

Calculate

Understand Operations

Comment

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Student Name

Understand Numbers

Outcome Category

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New Wave Maths Book F – Teachers Guide • 195 •

Proforma This proforma has been provided for you to copy and use with your class. You can either: • select an activity and evaluate the whole class; or • select a small group of students and evaluate their work. The indicators are found on the relevant page in the New Wave Maths for Teachers Guide. 1. 2. 3. 4. 5. 6. 7. 8.

r o e t s Bo r e p ok u Mathematics Proforma S Date

Strand

Outcome(s) Demonstrated

Indicators

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Photocopy this page. Write the appropriate date, strand, outcome(s) and indicators. Photocopy enough for one per student being assessed. Inform the students they are being assessed on the activity they are about to complete. Students complete the activity in the workbook. Mark the work completed by the student. Attach the proforma to the appropriate workbook page. Record evaluation as required.

Needs Further Opportunity

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Outcome(s)

Demonstrated

Needs Further Opportunity

Classroom Teacher • 196 • New Wave Maths Book F – Teachers Guide

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Photocopiable Resources

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1-cm dot grid paper.

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• 198 • New Wave Maths Book F – Teachers Guide

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1-cm grid paper.

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New Wave Maths Book F – Teachers Guide • 199 •

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2-cm grid paper.

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• 200 • New Wave Maths Book F – Teachers Guide

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1-cm triangle grid paper.

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New Wave Maths Book F – Teachers Guide • 201 •

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Distorted grid.

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• 202 • New Wave Maths Book F – Teachers Guide

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Distorted grid.

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New Wave Maths Book F – Teachers Guide • 203 •

100 Chart

9 10

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

r o e t s B r e 52 53 54 55 56 57 58 o 59o60 p u S 63 64 65 66 67 68 69 k 62 70

41 42 43 44 45 46 47 48 49 50 51

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71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

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0 – 99 Chart

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10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

. t41 40 e

30 31 32 33 34 35 36 37 38 39 50

42 43 44 45 46 47 48 49 co

. c e r 51 52h 53r 54 55 56 s 57 58 59 e o t super

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 • 204 • New Wave Maths Book F – Teachers Guide

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Basic Facts

0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 0 2 4 6 8 10 12 14 0 3 6 9 12 15 18 21 0 4 8 12 16 20 24 28 0 5 10 15 20 25 30 35 0 6 12 18 24 30 36 42 0 7 14 21 28 35 42 49 0 8 16 24 32 40 48 56 0 9 18 27 36 45 54 63 0 10 20 30 40 50 60 70

8 0 8 16 24 32 40 48 56 64 72 80

X 0 1 2 3 4 5 6 7 8 9 10

8 0 8 16 24 32 40 48 56 64 72 80

10 0 10 20 30 40 50 60 70 80 90 100

9 0 9 18 27 36 45 54 63 72 81 90

10 0 10 20 30 40 50 60 70 80 90 100

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9 0 9 18 27 36 45 54 63 72 81 90

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X 0 1 2 3 4 5 6 7 8 9 10

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New Wave Maths Book F – Teachers Guide • 205 •

Number Cards

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one

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zero • 206 • New Wave Maths Book F – Teachers Guide

Photocopy onto coloured card. Cut out and laminate. You may wish to enlarge to A3.

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1000

100

•

/10

/100

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10 000

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New Wave Maths Book F – Teachers Guide • 207 •

Tens

Ones

•

Tenths

Hundredths

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Ten thousands Thousands Hundreds

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Ten thousands Thousands Hundreds

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• 208 • New Wave Maths Book F – Teachers Guide

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Fraction Chart 1/ 10 1/ 9 1/ 8 1/ 6 1/ 5

1/ 3

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1/ 4

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Number Line

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1/ 5

1/ 2

3/ 8

5/ 8

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0.44

1/ 3

0.3 3/ 10

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4/ 9

0.5

1/ 2

0.67

0.55 5/ 9

0.78

2/ 3

0.4

0.5

0.6

2/ 5

1 / 2

3/ 5

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0.7 7/ 10

0.8

4/ 5

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5/ 6

0.89 8/ 9

0.9 9/ 10

New Wave Maths Book F – Teachers Guide • 209 •

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

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Spinners

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• 210 • New Wave Maths Book F – Teachers Guide

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Calendar January

March

15 22 29

12 19 26

16 23 30

13 20 27

10 17 24 31

14 21 28

11 18 25

15 22

15 22 29

12 19 26

16 23

16 23 30

13 20 27

10 17 24

10 17 24 31

14 21 28

11 18 25

16 23

14 21 28

10 17 24

15 22 29

11 18 25

16 23 30

12 19 26

10 17 24 31

13 20 27

11 18 25

14 21 28

12 19 26

15 22 29

June 4

11 18 25

12 19 26

13 20 27

14 21 28

15 22 29

16 23 30

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20 © R. I . C.P6u13bl i c27at i ons 3 10 17 24 July August September • f o r r e v i e w p u r p o s e s o n l y• 30 2 9 16 23 6 13 20 27 3 10 17 31

10 17 24

11 18 25

5 6

14 21 28

11 18 25

15 22 29

12 19 26

16 23 30

13 20 27

10 17 24 31

14 21 28

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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

o c . e October c November December her r o st super 7

14 21 28

11 18 25

15 22 29

12 19 26

16 23 30

15 22 29

12 19 26

10 17 24

16 23 30

13 20 27

11 18 25

10 17 24 31

14 21 28

12 19 26

11 18 25

15 22 29

13 20 27

12 19 26

16 23 30

14 21 28

13 20 27

10 17 24

15 22 29

14 21 28

11 18 25

16 23 30

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New Wave Maths Book F – Teachers Guide • 211 •

27 100 14

16 100 28

100 90

100

14 . t18

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r o e t s Bo r e p ok u 36 2 80 60 21 12 S

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• 212 • New Wave Maths Book F – Teachers Guide

Note: These cards can be cut along the dark, solid lines to make bingo cards.

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100 12

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o c . che e r o t 6 r 2s 4 8 s 2 40 uper

28 100 16

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New Wave Maths Book F – Teachers Guide • 213 •

r o e t s Bo r e p o u 5 4 35 40 25 k9 S

100

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• 214 • New Wave Maths Book F – Teachers Guide

Note: These cards can be cut along the dark, solid lines to make bingo cards.

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r o e t s Bo r e p o u k 25 S 4 25 100 21 25

28 . t 32

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New Wave Maths Book F – Teachers Guide • 215 •

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Attributes

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3-D Model

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• 216 • New Wave Maths Book F – Teachers Guide

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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New Wave Maths Book F – Teachers Guide • 217 •

triangular prism r o e t s Bo r e p ok u S cylinder

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cube

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• 218 • New Wave Maths Book F – Teachers Guide

sphere

square pyramid

Note: Students list or draw examples of these shapes found in their environment.

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Circular Tangram

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Enlarge to A3. These circles are used together as the circular tangram.

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New Wave Maths Book F – Teachers Guide • 219 •

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• 220 • New Wave Maths Book F – Teachers Guide

Enlarge to A3. Use this tangram puzzle to make various shapes.

Pythagorean Puzzle

Enlarge to A3. Use this Pythagorean puzzle to make various shapes.

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Tangram

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Broken Heart Puzzle

Enlarge to A3. Use this broken heart puzzle to make various shapes.

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Enlarge to A3. Use this circle puzzle to make various shapes.

Circle Puzzle

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New Wave Maths Book F – Teachers Guide • 221 •

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The Magic Egg Puzzle

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• 222 • New Wave Maths Book F – Teachers Guide

Enlarge to A3. Use this magic egg puzzle to make various shapes.

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Teac he r

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Cut out these nets to make an enclosed cone. Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

Cone Net

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New Wave Maths Book F – Teachers Guide • 223 •

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• 224 • New Wave Maths Book F – Teachers Guide

Cut out these nets to make an enclosed cylinder. Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

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Cylinder Net

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Cube Net

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Square Prism Net

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Triangular Prism Net

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Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

Triangular Pyramid Net

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Paper Circles

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Curve Stitch and Line Pattern 10 9 8 7 6 5 4 3 2

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Curve Stitch and Line Pattern 10 9 8 7 6 5 4 3 2

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20 19 18 17 16 15 14 13

Units of Measure:

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10 9 8 7 6 5

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Tally

Total

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Parent Information

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New Wave Maths Book F – 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

Year 5

Recall basic addition, subtraction, multiplication and division facts.

Years 6 and 7

Automatic response is desirable.

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Consolidate basic sums to 18 and differences taking from 18. Extend recall of basic multiplication and division facts to facts of 6, 7, 8 and 9 times tables.

The following suggestions can be used at home to assist your child in becoming more proficient at gaining automatic recall of the basic number facts. The ideas are not exclusive; many alternatives may be used.

4 5

6 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 F – 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 .5 1 + 2 1 .0 8 2 8 .5 9

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 .8 0 – 2 .9 3 3 .8 7

783 – 217 566

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2 .5 6 4 1 0 .3 5 1 2 .4 + 5 .7 2 2 1 .0 3 4

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Year 7

1+1=2 3 3 3

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4– 2=2 5 5 5

Subtract the numerators 4 – 2 = 2 and write the denominator as it appears.

11 + 22 = 33 5

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 5 5

Consolidation of above.

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New Wave Maths Book F – 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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468

468 ÷ 3

Starting with the leftmost digit (4, really 400) share the four hundreds among the 3 people. How many hundreds does each person receive? (1) How many hundreds were shared altogether? (4) How many hundreds are left? (1)

76 x 8 = 608, which is written directly under the line. 76 x 50 = 3800. The two results are then added together to get a final result.

76 x 58 608 + 3800 4408

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.

1 468 –300 168

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This is recorded as:

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:

0 .5 8 x 69 5 .2 2 + 3 4 .8 0 4 0 .0 2

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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

This is recorded as:

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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)

Year 6

Year 7

15 468 –300 168 – 150 18

156 468 –300 168 –150 18 – 18 0

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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 F – Teachers Guide

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Parent Information Problem-solving Strategies To assist your child in solving mathematical problems, the following strategies may help.

1 Understand the problem. (a) Ask relevant questions to determine the operation, pattern, sequence, form of measurement or other mathematical means to begin to work out the problem. (b) Choose a plan or strategy to work out all or part of the problem. (c) Simplify the problem by breaking it into smaller parts and working out each small part. (d) Guess. (e) Work backwards.

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Use appropriate computation: addition, subtraction, multiplication, or division, to work out the problem.

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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New Wave Maths Book F – 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.

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Children who have great difficulties in completing algorithms are to be encouraged to use the calculator to find the solutions after first estimating the answer. Estimation skills are essential in showing the development of mathematical knowledge. Should you encounter any problems, please contact me. Kind regards

Classroom Teacher ✄

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Dear Parent(s)

There are a number of different means of completing the four algorithms.

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Concrete to Mental – Including the Calculator

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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 F – Teachers Guide

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Parent Information Mathematical Learning Areas Mathematics is comprised of a series of learning areas. These learning areas are outlined for teachers in the Student Outcome Statements document produced by the Education Department. There are seven learning areas, each of which is outlined briefly below.

Appreciating Mathematics Appreciate the role of mathemtics in their own and other communities.

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Working Mathematically

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Thinking about ideas, investigating, applying, verifying and reasoning mathematically. In brief, problem solving.

Knowledge of location (place), shape, transformations (changes), and reasoning geometrically (angles, constructions and other geometrical relationships).

Measurement

Understand units of measure, measure objects using measuring units, estimate measures and calculate measurements.

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Understand chance events. Collect and organise data and information. Summarise and represent data. Interpret data.

Number

Understand number and the relationships, order, count, place value. Understand addition, subtraction, multiplication and division and be able to calculate using these operations. Work out number patterns.

Pre-Algebra

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Understand symbols and graphs. Represent variation. Solve equations and inequalities.

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New Wave Maths Book F – 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'.

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Should you encounter any problems, please contact me. Kind regards

Classroom Teacher

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Your encouragement and positive support are crucial to the continued development of your child's mathematical skills.

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Homework Policy

As part of my Homework Policy I encourage students to regularly undertake given exercises in the reinforcement of mathematical concepts learnt at school. These activities will be within the expected competency level of the children; however, there may be times when, due to unforeseen circumstances, your child does encounter difficulties with the homework. Please take the time to assist with the processes involved, but please encourage your child to 'have a go'.

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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. Should you encounter any problems, please contact me. Kind regards

Classroom Teacher

• 242 • New Wave Maths Book F – Teachers Guide