New Wave Maths - Teachers Guides: Level G - Ages 11-12 by Teacher Superstore

RIC–1090 12.4/592

Teac he r

ew i ev Pr

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

w ww

. te

o c . che e r o t r s super

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

m . u

Robert Dayman 2003

RIC-1090 ISBN 978-1-86311-711-1 Copyright Notice No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying or recording, or by any information storage or retrieval system without written permission from the publisher.

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

• Curriculum Frameworks;

• Student Outcome Statements; and

• National Outcome Statements.

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

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

ew i ev Pr

Teac he r

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.

m . u

w ww

. te

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

o c . che e r o t r s super

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

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

New Wave Maths Book G – Teachers Guide • iii •

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

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

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

ew i ev Pr

Teac he r

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

m . u

w ww

. te

Grid Paper.........................................................................................................................................198 – 204 Number Charts and Cards......................................................................................................205 – 208 Place Value Charts.........................................................................................................................209 – 210 Fraction Chart and Number Line...................................................................................................... 211 Spinners – Blank............................................................................................................................................ 212 Calendar – Any year..................................................................................................................................... 213 Bingo Cards.......................................................................................................................................214 – 217 3-D Model Attribute Table..................................................................................................................... 218 Venn diagrams – Blank............................................................................................................................... 219 3-D Shapes...................................................................................................................................................... 220 Tangrams.............................................................................................................................................221 – 224 Nets.......................................................................................................................................................225 – 231 Paper Circles.................................................................................................................................................. 232 Curve Stitch and Line Pattern................................................................................................233 – 234 Graph and Table – Blank........................................................................................................................... 235

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

Expectations of Knowledge of Basic Facts.................................................................................... 238 Primary School Mathematics..................................................................................................239 – 240 Problem-solving Strategies..................................................................................................................... 241 Concrete to Mental................................................................................................................................... 242 Mathematical Learning Areas................................................................................................................ 243 Homework Policy........................................................................................................................................ 244

• iv • New Wave Maths Book G – Teachers Guide

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

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.

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

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.

ew i ev Pr

Teac he r

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;

4. acquire the mathematical knowledge, ways of thinking and confidence to use mathematics to: (a) conduct everyday affairs such as money exchanges, planning and organising events, and measuring; (b) make individual and collaborative decisions at the personal, civic and vocational levels; and

w ww

. te

m . u

(c) engage in the mathematical study needed for fur ther education and employment. 5. develop skills in presenting and interpreting mathematical arguments; 6. possess sufficient command of mathematical expressions, representations and technology to: (a) interpret information (for example, from a court case or media report) in which mathematics is used;

o c . che e r o t r s super

(b) continue to learn mathematics independently and collaboratively; and (c) communicate mathematically to a range of audiences.

7. appreciate:

(a) that mathematics is a dynamic field with its roots in many cultures; and (b) its relationship to social and technological changes.

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

New Wave Maths Book G – 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):

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

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

ew i ev Pr

Teac he r

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

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

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

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

w ww

. te

m . u

8. modelling positive attitudes towards mathematics.

o c . che e r o t r s super

• 2 • New Wave Maths Book G – Teachers Guide

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

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:

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

• 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.

ew i ev Pr

Teac he r

• 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

w ww

. te

m . u

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

o c . che e r o t r s super

• 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.

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

New Wave Maths Book G – 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, and with 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.

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

ew i ev Pr

Teac he r

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

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

•f orr evi ew pur posesonl y•

m . u

w ww

. te

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.

o c . che e r o t r s super

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

• 4 • New Wave Maths Book G – Teachers Guide

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

General Content Outline Goals and Guidelines After completing and understanding Year 6 well, students should then move onto Year 7. In this stage, students now begin to think abstractly rather than relying on visual perception or concrete experiences, although these aids will enhance the learning of new mathematical concepts. With the increased ability to think abstractly there is an improved capacity to think hypothetically and reason logically. Students value mathematics the more the learning experiences provided recognise their interests. The development of an ownership of their own problems and the solutions will occur if the problems attract and involve the students.

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

The teaching of processes is necessary to develop independent problem-solvers.Therefore, for children 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:

ew i ev Pr

Teac he r

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

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

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

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

w ww

. te

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

m . u

8. estimation of number and measurement activities;

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

Understanding, skills and knowledge relationships make up the content that builds up conceptual structures. In the New Wave Maths series the following areas of mathematical content are included: 1. Working Mathematically – develops mathematical thinking processes through conceptualising, investigating, applying and verifying and reasoning mathematically. 2. Space – describes and analyses the features of objects, environments and movements through location, shape, transformations and geometric reasoning. 3. Measurement – using direct and indirect measurement and estimation skills in length, area, mass, volume and capacity and time.

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

5. Number – using operations, number concepts and relationships in the number system to calculate, reason about number patterns and understand numbers and operations. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 5 •

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

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

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

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.

ew i ev Pr

Teac he r

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.

Problem-solving

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

w ww

. te

m . u

2. Prepare a plan to solve the problem – working from the known to the unknown, draw diagrams, tables, charts to assist. 3. Carry out the plan – using different strategies as appropriate.

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

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.

o c . che e r o t r s super

• 6 • New Wave Maths Book G – Teachers Guide

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

In accepting the individuality of each student, teachers should also accept that students will interpret teaching in different ways and need new content to be presented by easily understood, believable methods and for that content to be seen as more useful than knowledge already held. Knowledge that students already hold is important to later learning and should be used as the basis for subsequent teaching through learning activities which are relevant to the students’ environment. Encouragement of discussion within the class allows for reflection on experiences and understanding. Where students lack the skills required to complete a task satisfactorily, more effective alternative methods that nurture their understanding need to be used. 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):

r o e t s Bo r e p ok u S 1. observing and identifying;

3. making patterns and arrangements; 4. constructing models;

5. estimating and measuring; 6. recording and calculating; 7. inferring, predicting and hypothesising; and 8. discussing what they are doing.

ew i ev Pr

Teac he r

2. comparing, ordering and classifying;

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

4. drawing pictures, diagrams and graphs; 5. constructing models;

6. translating between relationships; 7. making lists and tables;

w ww

. te

9. interpreting results; and 10. communicating findings.

m . u

8. drawing conclusions;

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.

o c . che e r o t r s super

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

New Wave Maths Book G – 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.

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

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.

ew i ev Pr

Teac he r

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:

m . u

w ww

. te

o c . che e r o t r s super 3+2=

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

• 8 • New Wave Maths Book G – Teachers Guide

43 + 24

54 – 22

469 – 217

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

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

18 + 19 =

76 – 25 =

21 + 14 + 37 = 256 + 48 =

100 – 60 =

329 + 257 =

700 – 300 =

638 – 73 =

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

42 =

34 x 2 =

200 =

x 40

r o e t s Bo r e p ok u S 6 x 100 =

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

ew i ev Pr

Teac he r

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

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.

m . u

w ww

. te

5 76

o c . che e r o t r s super

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

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

40 720

3 60.24

50 267

New Wave Maths Book G – 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.

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

• 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

ew i ev Pr

Teac he r

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.

Students are now conservers of mass and area and are becoming conservers of volume. They still require concrete experiences to assist in mathematical learning and, while less reliant on concrete and pictorial experiences, they should continue to link concrete, pictorial, verbal and symbolic representations to enhance conceptualisation.

Chance and Data

w ww

. te

m . u

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. 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.

o c . che e r o t r s super

Students are able to locate data that has been published, interpret, analyse and draw conclusions from this data taking into account data collection techniques and chance processes involved. In the New Wave Maths series, students are directed to make sensible judgments about the quality of the data and then to make a decision and draw inferences from the data. Chance and data work should focus on collecting, representing and interpreting data. Data collection activities should lead to classification, organisation, summarising and displaying in a variety of ways. In the New Wave Maths series, students are introduced to activities that include an element of 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.

• 10 • New Wave Maths Book G – Teachers Guide

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

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.

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

ew i ev Pr

Teac he r

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. Angles and directions are related to compass directions. Estimation and measuring skills, using standard units, should be completed.

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

w ww

. te

m . u

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

o c . che e r o t r s super

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

New Wave Maths Book G – 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.

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

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:

ew i ev Pr

Teac he r

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.

m . u

w ww

. te

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.

o c . che e r o t r s super

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.

• 12 • New Wave Maths Book G – Teachers Guide

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

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

• Valid

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

• Educative

Assessment should make a positive contribution to students’ learning.

• Explicit

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

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

• Fair

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

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

ew i ev Pr

Teac he r

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.

m . u

w ww

. te

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.

o c . che e r o t r s super

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 G – 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.

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

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

ew i ev Pr

Teac he r

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.

w ww

. te

m . u

o c . che e r o t r s super

• 14 • New Wave Maths Book G – Teachers Guide

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

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

ew i ev Pr

Teac he r

Teachers Notes and Answers

w ww

. te

Contents

m . u

o c . che e r o t r s super

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

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

New Wave Maths Book G – 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.

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

Resources have been listed to aid organisation before the lesson.

Language terms relevant to the workbook page have been listed here. It is preferred these words be introduced before beginning the activity to ensure students have a clear understanding of the terminology used in the activities. A space for you to record notes relevant to the lesson has been provided. This space could be used for any purpose. Some suggestions: • 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. 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.

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

ew i ev Pr

Teac he r

Skills relevant to the main activity have been listed.

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

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. 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.

w ww

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.

. te

m . u

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

o c . che e r o t r s super

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 G – 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 235. Teachers may photocopy and use them with their class(es).

adhesive tape

newspapers (financial and other)

analog clocks and watches

number lines 1 – 100 or

attribute blocks

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

overhead projector

Base 10 MAB

paper – coloured or plain

boxes, tens etc.

– A4 and A3

building and shopping catalogues

paper circles • page 232

bundles of 10s and 1s

pattern blocks

calculator

pentomino shapes

calendar • page 213

place value charts • pages 209 – 210

clock stamp

protractor

coloured pencils

pyramids (various shapes and sizes)

coloured rods

ruler

compass (drawing)

Smarties™

containers – various shapes and sizes

spinners • page 212

counters

square tiles

dice

stopwatch

digital clocks and watches

straws

fraction cake

street directories, maps, atlases

fraction chart • page 211

string

fraction/decimal number line • page 209

tangrams • pages 221 – 224

fraction shapes

tape measure

glue

tessellating sets

heavy card – coloured or plain

tetrominoes

ew i ev Pr

Teac he r

1000 – 100 000 • page 208

w ww

. te

– A4 and A3

m . u

timetables (bus, train, TV etc.)

interlocking cubes

toothpicks or equivalent

isometric grid paper • page 202

transparencies

kitchen scale

trundle wheel

light card – coloured or plain

world map

o c . che e r o t r s super – A4 and A3

1-cm grid paper • page 199

magazines

2-cm cubes

marker cones

2-D shapes

marker pens

3-D shapes

measuring containers measuring stick metre rule mirror/mira modelling clay money (coins/notes) nets • pages 225 – 232 R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 17 •

Term One Week Unit Outcomes

Page

1 1 S4.1—Use grids on maps and plans to locate points.

N4.1a—Read, write, say, count and compare whole numbers and decimals.

M4.4a—Understand relationships involving the area of regions and use these for practical purposes.

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

C&D4.3, C&D4.4—Display frequency data and read and make sensible statements about the information provided in tables and bar graphs.

N4.3—Calculate with decimals, drawing mostly on mental strategies to add and subtract decimal numbers. 3

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

3 S4.1—Use direction on maps and plans in descriptions of locations.

4 5 6 7

N4.2—Understand the meaning, use and connections between the four operations on whole numbers, and use this understanding to choose appropriate operations to construct and complete equivalent statements.

M4.4a—Understand relationships involving the perimeter of regions and use these for practical purposes.

9 10

ew i ev Pr

4 N4.4—Recognise, describe and use patterns involving operations on whole numbers, and follow and describe rules for how successive terms in a sequence or paired quantities can be linked by a single operation.

Teac he r

C&D4.3, C&D4.4—Display measurement data using simple scores on axes, and summarise data with simple highest, lowest and middle scores; and means. Read and make sensible statements about information provided in tables and line graphs.

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

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, M4.4b—Understand relationships involving the area of regions based on squares and uses these for practical purposes. Understand and use scale factors involving small whole numbers for straightforward tasks. 6

6 WM4.2, WM4.3, S4.3—Work mathematically, take risks and organise data to investigate tessellating shapes. 16

N4.3—Calculate with whole numbers, drawing mostly on mental strategies for multiplications and divisions related to basic facts.

7 S4.1—Use distance and direction on maps and plans.

w ww

m . u

C&D4.2, C&D4.3, C&D4.4—Collaborate with peers to plan what data to collect and how to tabulate them to answer particular questions. Display frequency data using simple scales on axes. Read and make sensible statements about the information provided in tables and bar graphs.

N4.3—Calculate with fractions, drawing mostly on mental strategies and readily visualised fractions.

M4.2—Measure area by counting uniform units including part-units.

. te

8 N4.3—Calculate with whole numbers, drawing mostly on mental strategies for multiplications and divisions related to basic facts.

o c . che e r o t r s super

C&D4.3—Organise data to make a timetable.

N4.1a—Read, write, say and understand the meaning, order and relative magnitude of whole numbers and negative integers.

9 S4.2—Attend to the shape, size and placement of parts when matching 3-D models which can be seen using some basic conventions for drawing them.

N4.1b—Read, write, say and understand the meaning, order and relative magnitude of any percentages and the common equivalence between them.

M4.1—Take purpose and practicality into account when selecting units and instruments for measuring things.

10 N4.1b—Read, write, say and understand the meaning, order of any decimals and fractions and the common equivalence between them.

C&D4.2, C&D4.4—Collaborate with peers to classify data to answer particular questions. Read and make sensible statements about the information provided in diagrams.

N4.1b—Read, write, say and understand the meaning of fractions and shows equivalence between them.

• 18 • New Wave Maths Book G – Teachers Guide

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

Term Two Week Unit 1

Outcomes

Page

11 S4.3—Recognise and rotate and reflect figures and objects systematically to produce arrangements and patterns. 31

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

M4.2—Measure surface area by counting uniform units and volume using scales.

12 N4.1b—Read, write, say and understand the meaning, order and relative magnitude of any fractions.

C&D4.2, C&D4.3, M4.2—Measure, record and order length reading whole number scales.

N4.1b—Read, write, say and understand the meaning of fractions and shows equivalence between them.

13 S4.2—Visualise and make models of 3-D shapes and interprets and produces conventional mathematical drawings of them.

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

Teac he r

N4.2—Understand the meaning, use and connections between the four operations on whole numbers, and construct and complete equivalent statements. 4

ew i ev Pr

M4.2—Measure surface area by counting uniform units.

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

38 39 40

C&D4.3—Display frequency and summarises data with simple fractions, highest, lowest and middle scores; and means.

S4.3—Recognise rotations and patterns and translates to produce arrangements and patterns.

15 S4.4—Select, describe and compare figures and objects on the basis of spatial features, using conventional geometric criteria.

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

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

16 M4.2—Estimate, measure and record length.

C&D4.2, C&D4.3, C&D4.4—Record data in a table and use the information to find the mean, mode and median. 47 N4.1b—Subtract fractions. Complete fraction equivalents.

w ww

17 S4.4—Select, describe and compare figures and objects on the basis of spatial features, using conventional geometric criteria.

m . u

48 49

WM4.2, WM4.3, N4.1a, N4.1b, N4.2, N4.3—Research, select and explain, sporting equipment to be purchased for the class.

M4.2—Measure time reading a calendar.

. te

o c . che e r o t r s super

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

C&D4.2, C&D4.3—Record data in a table. Understand chance outcomes.

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

19 S4.4—Select, describe and compare figures and objects on the basis of spatial features, using conventional geometric criteria.

N4.1b—Read, write, say and understand the meaning of fractions and shows equivalence between them.

M4.4a—Understand relationships involving the perimeter of polygons, and uses these for practical purposes.

20 N4.1a, N4.4—Read, write, say and understand the meaning and relative magnitude of whole numbers.

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

58 59

N4.3—Read, write, say and understand the meaning and common equivalencies between decimals and percentages. 60

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

New Wave Maths Book G – Teachers Guide • 19 •

Term Three Week Unit Outcomes

Page

1 21

M4.4b—Use a grid to reduce a figure using unit fraction scales.

N4.1b—Read, write, say and understand the meaning, order and relative magnitude of any fractions, decimals and percentages, and knows the more common equivalencies between them.

M4.4a—Understand relationships involving the perimeter of polygons, and uses these for practical purposes. 63

2 22

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

C&D4.3—Display frequency and summarises data with simple fractions, highest, lowest and middle scores; and means.

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

3 23

S4.3—Identify the transformation(s) used to produce a spatial sequence.

N4.1a—Read, write, say and understand the meaning, order and relative magnitude of decimal numbers.

M4.4a—Understand relationships involving the perimeter of polygons and the area of regions based on squares. 69

4 24

N4.1a—Read, write, say, count and compare decimals.

C&D4.3, C&D4.4, M4.2—Collect and record data on a scattergraph. Answer questions directly related to the data collected.

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

5 25

M4.4b—Enlarge a diagram and draw it on a grid according to scale.

N4.3—Calculate with whole numbers and decimals, drawing mostly on mental strategies for whole numbers. 74

M4.2—Measure area by counting uniform units including where part-units are required, and measure length, reading whole number scales. 75

6 26

WM4.2, WM4.3, WM4.4, C&D4.2, C&D4.3, C&D4.4—Collect, display and summarise data.

C&D4.2, C&D4.3, C&D4.4, N4.1a—Collect data, record in a pie graph and answer questions relating to the data collected.

N4.1b—Add fractions. Order fractions.

7 27

S4.4—Select, describe and compare figures and objects on the basis of spatial features, using conventional geometric criteria.

m . u

w ww

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

ew i ev Pr

Teac he r

N4.1a, N4.1b—Use diagrams and/or manipulatives to solve word problems involving decimals, fractions and whole numbers.

. te

M4.2—Read and record times using 12- and 24-hour clocks.

8 28

N4.1b—Read, write, say and understand the meaning of fractions and, for readily visualised fractions, shows equivalence between them.

C&D4.3, C&D4.4, N4.4—Record data in tables, use the data to find patterns.

N4.4—Recognise, describe and use patterns involving whole numbers, decimals and fractions.

9 29

S4.3—Recognise rotations, reflections and translations in arrangements and patterns and translates, rotates and reflects figures systematically to produce arrangements and patterns.

C&D4.2—Decide how to collect, modify and record data to answer a specific question.

N4.3—Divide whole and decimal numbers. Estimate first.

10 30

M4.4a—Use chosen methods to solve a problem relating to area.

C&D4.2, C&D4.3, C&D4.4—Decide what information needs to be gathered, how it will be gathered and record the data in a Venn diagram.

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

o c . che e r o t r s super

• 20 • New Wave Maths Book G – Teachers Guide

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

Term Four Week Unit

Outcomes

1 31

S4.1—Draw a map or plan which shows a sense of scale.

N4.3—Multiply money and measurements by a one-digit number.

M4.2—Use a straightforward timetable.

2 32

N4.1a—Read, write, say and understand the meaning, order and relative magnitude of any fractions and percentages and know the more common equivalencies between them.

C&D4.1—Interpret and make numerical statements of probability based on equally likely outcomes and using fractions and percentages.

N4.3—Add and subtract fractions.

3 33

S4.4—Construct a diagram using conventional geometric criteria.

N4.3—Partition double-digit numbers to simplify problems.

M4.4a—Understand relationships involving the perimeter of polygons.

Teac he r

Page

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

N4.1a—Rewrite a decimal as a fraction and a fraction as a decimal.

C&D4.4, C&D4.1—Read and record data accurately and use it to answer questions.

101

N4.2—Understand the meaning, use and connections between the four operations on whole and decimal numbers, and uses this understanding to choose appropriate operations and constructs and completes equivalent statements.

102

5 35

WM4.2, WM4.3, S4.3—Work mathematically to create a symmetrical design.

103

N4.1b—Subtract and add fractions.

M4.2—Read a calendar and record answers about dates on the calendar.

ew i ev Pr

4 34

100

104 105 106

C&D4.3—Find the mean, mode and median of a set of data.

107

N4.1b—Calculate with fractions, drawing mostly on mental strategies for whole numbers and readily visualised fractions.

108

7 37

S4.1—Use coordinates in descriptions of locations.

109

N4.3—Use a calculator to plan and calculate a sequence of totals.

110

M4.2—Read a timetable and convert times to 24-hour.

8 38

N4.1b—Read, write, say and understand the meaning of fractions and, for readily visualised fractions, estimate their relative size and position on a number line.

w ww

. te

m . u

N4.1b—Calculate with fractions, drawing mostly on mental strategies for whole numbers and readily visualised fractions.

6 36

o c . che e r o t r s super

111 112

C&D4.3, C&D4.4—Represent data in an arrow diagram.

N4.1a, M4.2—Read and complete a table to include all relevant information. Order information in chronological order.

114

9 39

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

115

N4.1a—Read, write, say, count with and compare whole numbers, decimals and fractions.

116

M4.2—Distinguish time from elapsed time.

117

10 40

N4.4—Recognises, describe and uses patterns involving operations on whole numbers, and follows rules.

118

C&D4.1—Interpret and make numerical statements of probability based on lists of equally likely outcomes. 119

N4.4—Understand and apply directly length, area and volume relationships for shapes based on rectangles and rectangular prisms. 120

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

113

New Wave Maths Book G – Teachers Guide • 21 •

Unit 1–1

Student page 1

Outcomes N3.3, N4.3, S4.1

Skills • following directions • drawing • reasoning • locating and plotting points

Memory Masters (N3.3)

Indicators

Resources

Language

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 (including angle multiples of 45º) and common map grids.

• calculator • overhead projector • transparency

• add • grid • locate • reference points • order • traversable • quadrant • centre point • horizontal and vertical axes

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 (S4.1) Warm up

• To assist in developing the concept of grid reference points ask students to use their atlas to locate the following places and give their coordinates.These may be found in the index or by using the grid on the map page(s) where the place is located. (a) Perth 31º 57’ 115º 51’ (b) Canberra 35º 17’ 149º 13’ (c) Darwin 12º 14’ 130º 58’ (d) Brisbane 27º 28’ 153º 01’ (e) Adelaide 34º 56’ 138º 36’ (f) Melbourne 37º 49’ 144º 58’ (g) Sydney 33º 53’ 151º 13’ (h) Hobart 42º 50’ 147º 15’ • Discuss with students the use of grid reference points for locating points or places.

ew i ev Pr

Teac he r

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

What to do

w ww

• Draw a quadrant on the blackboard or whiteboard or project on a screen using an overhead. • Discuss the centre point, horizontal axis and vertical axis. Explain that points to the right of the centre point (origin) on the horizontal axis are positive as are those above the centre point on the vertical axis. Points to the left and below the centre point on the horizontal and vertical axes respectively are negative.This allows for ready identification of the quadrant that the point is found; for example, (+, +) is top right quadrant; (–, –) is the bottom left quadrant; (+, –) is the bottom right quadrant and (–, +) is the top left quadrant. • Points are always read on the horizontal axis then the vertical axis and are written in the same order; for example, point (5, –3) is five places to the right of the centre point on the horizontal axis and three places below the horizontal axis. • Ask students to locate point 1 on their grid. Check accuracy. • Repeat for points 2 and 3 then encourage students to complete the ten points. Join the dots in order and check traversability.

. te

m . u

o c . che e r o t r s super

Challenge • Make word codes using a grid. • Play ‘battleships’ game.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 2 – 3. • 22 • New Wave Maths Book G – Teachers Guide

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

Unit 1–2

Student page 2

Outcomes

Indicators

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

The student is able to: • say decimals. • order decimals according to instruction.

Skills • multiplying • ordering

Resources

Language

• calculator • Base 10 MAB

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

• add • digits • decimal numbers • total price • unit price • order • weight • lightest, heaviest • lowest, highest • equivalent fractions

Notes

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

ew i ev Pr

Teac he r

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

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

Main Activity (N4.1a) Warm up

• Distribute Base 10 MAB to small groups of students and allow a few minutes of free play.

What to do

Challenge

. te

m . u

w ww

• When buying items from a butcher the item has a label that shows the weight of the meat, the cost of the meat per kilogram and the total price of the meat in the package. Ask the students ‘Given the weight and unit price, how can you work out the total price?’. (Multiplying the two.) • Use your calculator to find the total price of the four meat packages as shown from the labels displayed by the Good Meats Butcher. • Arrange the meat in order from lightest to heaviest; from lowest total price to highest total price; and from lowest unit price to highest unit price.

• Use the numbers from 1 – 9 inclusive to make three equivalent fractions. Each digit may be used once only in making the three equivalent sets. • Share answers as discovered by the students but don’t provide the correct answer if students have not discovered it. Students may be referred back to the activity at a later date.

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 54 – 55. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 23 •

Unit 1–3

Student page 3

Outcomes

Indicators

N4.3, M4.4a

The student is able to: • given a rectangle with whole number length sides, explain why multiplying the length by height gives the area.

Skills • multiplying • problem-solving

Memory Masters (N4.3)

Resources • calculator • toothpicks (or equivalent)

• area • perimeter • square metres • total

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)

ew i ev Pr

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

• The focus for this unit is allowing students to explore and discover mental strategies to solve problems. • Students demonstrate facts they know which are related to the fact on the workbook page. They need to show how each calculation is related to each other; e.g. If I know 10 x 5 is 50, then I also know 9 x 5, 11 x 5, 5 x 5, 10 x 50, 10 x 0.5 etc.

Teac he r

Language

w ww

• Ask students how they would find the area of a rectangle given its length and width. • Given the rate of application of fertiliser to a lawn and knowing the area of lawn to which it is applied, how would you work out the amount of fertiliser required? (Multiply the two.) • If you know the cost of fertilising a lawn for every 100 m2, how would you work out the total cost? (Find the number of 100 m2 and multiply by the cost.) • Work out the answers to Exercise 3 using a calculator. • Exercise 4 requires the area of grass to be divided into a number of smaller rectangular areas. Ask students how this may be done. (There are several options: all should be accepted.) The simplest is to make a 15 m x 25 m rectangle on the right – find its area, and take the 5 m x 20 m missing piece from the area of the 30 m x 55 m large rectangle; then add the two areas. • Set students work to find the total area, using their preferred method.

. te

Challenge

m . u

• Discuss the concept of area as opposed to perimeter.

o c . che e r o t r s super

• Distribute the toothpicks (or equivalent) to all students or small groups. • Students are to make three squares using the 20 toothpicks. • Show final solutions.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 134 – 135. • 24 • New Wave Maths Book G – Teachers Guide

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

Unit 1—Answers

Student pages 1 – 3 Unit 1–1

1. (a) 6 (b) 5 (c) 4 (d) 8 (e) 8 (f) 11 (g) 12 (h) 14 (i) 2 (j) 14 19 616 (c) 20 991 (d) 20 692 2. (a) 23 224 (b) (e) 18 387 (f) 17 872 3. (a)

Unit 1–2 1. (a) 8 (b) 8 (c) 3 (d) 16 (e) 3 (f) 2 (g) 7 (h) 16 (i) 6 (j) 2 2. (a) 116 789 (b) 131 501 (c) 115 150 (d) 100 000 (e) 64 669 (f) 139 678 3. steak $4.33 pork $5.21 fish $7.42 lamb $8.58 (a) Fish, Steak, Pork, Lamb (b) Steak, Pork, Fish, Lamb (c) Pork, Lamb, Steak, Fish Challenge

r o e t s Bo r e p ok u S /18

/54

ew i ev Pr

Teac he r

(b) Yes Challenge Teacher check

. te

• Locate place in an atlas using latitude and longitude.

Consolidation 1–2 • Use grocery catalogues for various prices of meat, fish, poultry and vegetables. Ask students to work out costs for given weights.

Consolidation 1–3

m . u

w ww

1. 5 x 10 = 50, 50 ÷ 10 = 5 (other answers are possible.) (c) 18 200 2. (a) 3716 (b) 14 127 (d) 7793 (e) 11 052 (f) 10 100 3. (a) 26.25 kg (b) $301.88 4. (a) 1925 m2 Challenge

• Measure a grassed area in the school grounds and find the amount of fertiliser required and cost involved.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 25 •

Unit 2–1

Student page 4

Outcomes

Indicators

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

The student is able to: • say decimals.

Skills • recording • measuring • rounding

Memory Masters (N3.3)

Resources • calculator • place value charts (see pages 209 – 210) • stopwatch

Language • subtract • place value, tens, units • tenths, hundredths, thousandths, ten thousandths

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) Warm Up

• Distribute place value charts. • Ask students to name the place value positions to the right and left of the decimal point. Ask what the relationship is between the adjacent places. • Ask students to show or respond orally to the place value column of the following numbers: 0.2, 4, 0.003, 80, 0.7, 0.09 • Ask students to place the following digits in the correct place value columns, either on the blackboard/whiteboard or in the workbooks—24.863

ew i ev Pr

Teac he r

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

w ww

• Students are to complete Exercise 3. Work through the placement of the first couple of numbers to ensure understanding is there. • Students to work in groups of five or six and time each other to the nearest one-hundredth of a second to walk around the outside of a netball court. • Record each time. • Relate this form of timing to athletes and swimmers

Challenge

. te

m . u

What to do

o c . che e r o t r s super

• Briefly revise square numbers. • Remind students of the limits between which the square numbers lie: 40 and 140. • Ask them to record their working and findings for sharing later.

• 26 • New Wave Maths Book G – Teachers Guide

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

Unit 2–2

Student page 5

Outcomes

Indicators

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

The student is able to: • suggest what data to collect to help estimate numbers or quantities. • make quickly-produced ‘working’ graphs in order to explore data. • read the information provided on axes on bar and line graphs, including where all calibrations on the scale may not be labelled.

Skills • recording • collecting data • graphing • analysing

Resources

Language • number patterns • subtract • bar graph • total number • mean • total difference

• calculator • enrolment information

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

Notes

Memory Masters (N3.4) Number (N4.3)

ew i ev Pr

Teac he r

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

• Students 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 students that statistics are used to help determine distribution of resources, for planning future development, supporting particular ideas/stances, or presenting current status. • Interpretation of statistics can be done to provide a variance of opinions.Today’s exercise is a sample of collecting and showing total numbers of boys, girls and students in five classes.

. te

Challenge

m . u

w ww

• Provide the information required to the class to enter in the table provided. • Remind students how to construct a bar graph. The graph outline is provided. • Once the graphs are complete, answer the questions below the graph. • Students may need to be reminded how to find the mean of a set of numbers. Add all required numbers together (in this case five totals of boys and then five totals of girls); each total is divided by the number of individual items added together (in this case divide the boys and girls totals by five). This provides the mean. • The difference between boys and girls in each class, and on the averages, may favour one set at all times. The difference may vary in each case. Reasons for differences will be problematical, students may suggest fiction such as – local all boys/girls school, friendships, unaccountable reasons (left to chance) or other explanations they think are reasonable.

o c . che e r o t r s super

• The mean of three numbers is 20. What might the numbers be? • Note:This challenge will really test the student’s understanding of the mean. Many possible answers; e.g. 60, 0, 0 or 20, 20, 20 etc.

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

New Wave Maths Book G – Teachers Guide • 27 •

Unit 2–3

Student page 6

Outcomes

Indicators

N4.1a, N4.3

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

Skills • subtracting

Resources • calculator • financial business section of a newspaper • toothpicks or equivalent

Language • convert • kilograms • grams • subtract • exchange • rates • difference • between • triangles

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

Memory Masters (N4.1a)

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 (N4.3) Warm up

ew i ev Pr

• The focus for this unit is the conversion of grams to kilograms and kilograms to grams. • 1000 g = 1 kg so for questions 1 to 5 x by 1000 and for questions 6 to 10 ÷ by 1000.

w ww

What to do

. te

• The task for this exercise is to find the difference between the buy and sell exchange rates. This may be completed mentally, as written sums or by using a calculator. • Which countries have the widest difference between buying and selling rates? Find reasons why this might be – unstable politically, low economic development etc.

Challenge

m . u

• All countries have their own money and each country’s rate of exchange may vary considerably, as does the comparative monetary value, from place to place and day to day. • A collection of the exchange rate guides from a local newspaper over time, focusing on one currency and plotting its exchange rate for presentation to the class, may be a useful extra activity. • Some countries use the same name for their currency as other countries, but the value of the money differs. Students may be directed to see how many countries use the same name for their currency; for example, dollar. • Three countries listed in the workbook no longer have their own currency. Why not? (Germany, France and Italy are now part of the European monetary conglomerate that uses the Euro.)

o c . che e r o t r s super

• Issue students with toothpicks. • Arrange the toothpicks to make five triangles. • Keep recorded drawings of all attempts. • Share solutions.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 80 – 81. • 28 • New Wave Maths Book G – Teachers Guide

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

Unit 2—Answers

Student pages 4 – 6 Unit 2–1

4. Teacher check Challenge 81

Unit 2–2 1. (a) 125 (b) 25 (c) 15 (d) 23 (e) 24 (f) 20 (g) 192 (h) 75 (i) 39 (j) 75 (b) 491 (c) 676 (d) 566 2. (a) 785 (e) 684 (f) 366 3. Teacher check Challenge Possible answers are: 60, 0, 0 or 20, 20, 20.

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

ew i ev Pr

Teac he r

1. (a) 25 (b) 9 (c) 7 (d) 30 (e) 63 (f) 5 (g) 6 (h) 21 (i) 5 (j) 16 412 (c) 512 (d) 712 2. (a) 222 (b) (e) 121 (f) 712 3.

. te

• Students throw a ball, measure the distance and round to the nearest metre.

Consolidation 2–2 • Select classes of a different age group; how do the results compare?

Consolidation 2–3

m . u

w ww

1. (a) 0.046 kg (b) 0.827 kg (c) 0.006 kg (d) 2.473 kg (e) 0.084 kg (f) 2174 g (g) 843 g (h) 1006 g (i) 70 g (j) 9g 2. (a) 727 (b) 809 (c) 527 (d) 614 (e) 439 (f) 818 3. US 0.0040 Japan 1.52 UK 0.0046 Malaysia 0.0291 Canada 0.0142 NZ 0.0071 China 0.0901 PNG 0.0189 Fiji 0.0281 Singapore 0.0190 Mexico 0.0595 S/Africa 0.0463 Poland 0.0174 Switzerland 0.0156 Hong Kong 0.0888 Thailand 0.41 India 0.467 Tonga 2.01 Indonesia 39 Vanuatu 0.0806 Denmark 16 W/Samoa 0.0086 Europe 0.0168 Botswana 2.27 Brunei 0.0392 Solomon Is. 0.3961 Saudi Arabia 0.0641 Sweden 0.1432 Challenge

• Plan a trip to an overseas destination on the chart. Nominate an amount of spending money. Convert to the currency of the country to find the actual amount to spend.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 29 •

Unit 3–1

Student page 7

Outcomes N4.1a, N4.3, S4.1

Skills • locating and plotting points • logical reasoning • discussing • rationalising

Indicators

Resources

• 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 (including angle multiples of 45º) and common map grids.

• calculator • world guide (map)

Language • convert • cents, dollars • multiply • locate • position • longitude, latitude

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

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 (S4.1) Warm up

• Lead-up activity may involve following an around the world yacht race, inviting a noted sailor as a guest speaker or comparing the exploratory feats of past sailing explorers; for example, Captain Cook, Ferdinand Magellan and others. • Note: 2002 was the Year of Matthew Flinders. • Use a flat map of the world to investigate Antarctica and its surrounding ocean. At this stage it may be worthwhile plotting the positions of the yachts on the flat map to show relative positions and gain an idea of conditions likely to be faced. • Develop discussion to include an understanding of longitude and latitude and how these lines are related to time. Include east and west of the Greenwich Meridian, 0º.

ew i ev Pr

Teac he r

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

w ww

What to do

• Use the positions and means of location to plot the yachts’ positions on the map in the workbook. • Students may be set to answer the questions through their own research or use this as a whole-class activity for general discussion.

Challenge

. te

m . u

o c . che e r o t r s super

• Students should present a reasoned argument for their answer to the problem. • Suggest the use of a globe or world map to assist. • All working out is to be recorded.

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

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

Unit 3–2

Student page 8

Outcomes

Indicators

N4.3, N4.2

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

Skills

Resources

Language • multiply • factor pairs • factor tree • triangles • multiplies • prime factors

• calculator • toothpicks (or equivalent)

• multiplying • problem-solving

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

Notes

Memory Masters (N4.3)

Teac he r

ew i ev Pr

• The focus for this unit is allowing students to explore and discover mental strategies to solve problems. • Students are required to list as many calculations as they can which will make the original problem easier to solve; 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.

Main Activity (N4.2)

Warm up

m . u

• What are factors? (Numbers which when multiplied give the multiple.) • What are multiples? (Answers to multiplication sums or results of multiplying factors.) • What are factor pairs? (Two numbers which always give the same result when multiplied; for example, 2 x 4.) • What are the factors of 8? (1, 2, 4 and 8) What are the factor pairs of 8? (1 x 8), (2 x 4)

w ww

What to do

• Work systematically through the factor pairs for 10, 12 and 16. Ensure that students realise that there may be several pairs. A simple exercise to assist in finding pairs is to start with 1, move to 2, then 3 etc. until all pairs are found. Once the new number to be used has already been found as a factor then all pairs should have been noted; for example, factor pairs for 16—1 x 16, 2 x 8, 3 (no), 4 x 4, 5 (no), 6 (no), 7 (no), 8 already used. Factor pairs, 1 x 16, 2 x 8, 4 x 4. Complete the exercises. • Factor trees are a method of finding the prime factors of a number. • Factor pairs are written below the original number and each component of the pair is further broken down until only prime numbers are left. For example, 12 6 2 3 2 This is known as a factor tree because of the ‘branches’. • Complete the exercises.

. te

o c . che e r o t r s super

Challenge • Students are issued with toothpicks. • Toothpicks are arranged so that three triangles are shown using the seven sticks. • Show all attempts by drawing each arrangement. • Show findings.

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

New Wave Maths Book G – Teachers Guide • 31 •

Unit 3–3

Student page 9

Outcomes

Indicators

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

The student is able to: • given a rectangle with whole number length sides, explain why multiplying the length by the height gives the area.

Skills • multiplying • problem-solving

Memory Masters (N3.3)

Resources • calculator • 2-cm cubes

Language • multiply • perimeter • average speed • length • scale • surfaces

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) Warm up

• Why do we need firebreaks? (To slow the spread of fires.) • Where would you prepare a firebreak on a block of land? (Inside boundary fence line.) • Farmers prepare firebreaks around their properties. They use machinery and need to be aware of costs to ensure that they budget for preparation of the firebreak.

ew i ev Pr

Teac he r

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

w ww

• Can you assist the owner of the property shown in the workbook to find the cost of preparing the firebreak to protect the property? • Students should discuss as a class, or in groups, how they will calculate the perimeter, then the length of time to complete the firebreak, before finding the cost of ploughing the firebreak. Remember, the firebreak will be inside the perimeter of the property, so these measurements will need to be taken into consideration. • Students should use their preferred method to find answers to calculations. Notes should be kept of how each activity was undertaken.

Challenge

. te

m . u

o c . che e r o t r s super

• Students are to be encouraged to find their own means of finding the total surfaces showing of all cubes in the arrangement. Covered surfaces do not count. Surfaces on the bottom of the arrangement do count. • Distribute 2-cm cubes to those who wish to use them. • Students record all working and findings for sharing.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 134 – 135. • 32 • New Wave Maths Book G – Teachers Guide

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

Unit 3—Answers

Student pages 7 – 9 Unit 3–1

Unit 3–2 1. 36 x 2 x 3 24 x 9 3 x 8 x 9 2 x 6 x 3 x 6 6 x 6 x 6 36 x 6 27 x 8 3x4x2x9 6 x 4 x 9 8 x 3 x 9 72 x 3 and so on. 2. (a) 1350 (b) 1350 (c) 3920 (d) 4760 (e) 3540 (f) 1680 4. 3. (b) 1 x 10, 2 x 5 (c) 1 x 12, 2 x 6, 3 x 4 (d) 1 x 16, 2 x 8, 4 x 4 (e) 1 x 18, 2 x 9, 3 x 6 (f) 1 x 20, 2 x 10, 4 x 5 (g) 1 x 21, 3 x 7 (h) 1 x 24, 2 x 12, 3 x 8, 4 x 6 (i) 1 x 28, 2 x 14, 4 x 7 (j) 1 x 30, 2 x 15, 3 x 10, 5 x 6 (k) 1 x 32, 2 x 16, 4 x 8 (l) 1 x 35, 5 x 7 (m) 1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6 (n) 1 x 40, 2 x 20, 4 x 10, 5 x 8 (o) 1 x 42, 2 x 21, 3 x 14, 6 x 7 (p) 1 x 44, 2 x 22, 4 x 11 (q) 1 x 48, 2 x 24, 3 x 16, 4 x 12, 6 x 8 (r) 1 x 50, 2 x 25, 5 x 10 (s) 1 x 54, 2 x 27, 3 x 18, 6 x 9 (t) 1 x 56, 2 x 28, 4 x 14, 7 x 8

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

ew i ev Pr

Teac he r

1. (a) 215c (b) 827c (c) 379c (d) 146c (e) 482c (f) $3.24 (g) $8.16 (h) $1.74 (i) $9.28 (j) $6.57 2. (a) 946 (b) 713 (c) 726 (d) 384 (e) 492 (f) 273 3. Teacher check 4. To minimise the distance travelled. 5. To sail with the prevailing winds. Challenge the Equator

Challenge

. te

• Continue the yacht race by asking students to give new coordinates for each yacht each day over the week. Who will be the winner?

Consolidation 3–2

m . u

w ww

1. (a) 5 (b) 8 (c) 6 (d) 3 (e) 12 (f) 12 (g) 8 (h) 11 (i) 7 (j) 7 2. (a) 2852 (b) 5100 (c) 3478 (d) 4558 (e) 4958 (f) 5074 3. (a) 17.385 km (b) 3.477 hrs ≈ 3 hrs 28 mins (c) 104.31 L (d) $93.04 (e) Time: ≈ 6hrs 56 mins Cost: $186.08 Challenge 36 cm2

• Students select numbers which are relevant to them and make factor trees.

Consolidation 3–3

o c . che e r o t r s super

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

• Work out the area of the farm. • Work out the cost of diesel used (at today’s prices).

New Wave Maths Book G – Teachers Guide • 33 •

Unit 4–1

Student page 10

Outcomes

Indicators

N3.3, N4.3, N4.4

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 operation.

Skills • problem-solving • discussing

Memory Masters (N3.3)

Resources

Language • divide • pattern • rule • doubling • multiply • add • symbols

• calculator

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 (N4.4) Warm Up

• Mathematics is largely about logic and patterns. It is often defined as the mathematics of pattern. Exploring number patterns helps develop our understanding of mathematics.There are many different types of number patterns. For example; doubling of numbers, 2, 4, 8, 16 etc. and adding a constant 2, 4, 6, 8, 10 etc. • This is an important first step in the development of algebra. • Ask students to provide number patterns of their own. Listen to several.

ew i ev Pr

Teac he r

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

w ww

• Turn to the number pattern in the workbook.The numbers 1 to 9 are listed down the left-hand side. In the next column the numbers 7 and 13 are shown. Ask the students how they think these numbers were chosen to be placed where they are. (They are placed there by adding the two numbers joined to them.) What number would be placed in the top box of the second column? (3) In the second box? (5) In the first box of the third column? (8 (3 + 5)) • Complete the pattern. Write in your own words an explanation of the rule for the pattern. • Exercise 4 uses patterns to multiply large numbers. By breaking 23 into 1 + 2 + 4 +16 (numbers gained from doubling – 1, 2, 4, 8, 16 etc.) then doubling 296, answers are obtained without having to multiply. Add multiples of 1, 2 and 16 to give the multiple of 296 x 23. • Discuss how 17, 26, 47 and 35 may be broken up using doubling. For example; 1, 2, 4, 8. 17 = 1 + 8 + 8 ; 35 = 1 + 2 + 16 + 16 • Find the answers as asked.

. te

Challenge

m . u

What to do

o c . che e r o t r s super

• 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 5 using one or all of the operations; e.g. 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 supply answers that students can not find themselves.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 90 – 91. • 34 • New Wave Maths Book G – Teachers Guide

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

Unit 4–2

Student page 11

Outcomes

Indicators

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

The student is able to: • suggest what data to collect to help estimate numbers or quantities. • make quickly-produced ‘working’ graphs in order to explore data. • interpret and report on information provided in line graphs, informally describing trends in the data.

Skills • graphing • analysing • recording • discussing

Resources

Language

• calculator

• divide • information • line graph • mean • range • multiples

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

Notes

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

ew i ev Pr

Teac he r

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

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

Main Activity (C&D4.2, C&D4.3, C&D4.4) Warm up

• Use the blackboard/whiteboard or an overhead transparency to remind or show students how to construct a line graph. Discuss vertical and horizontal axes and the information they show. Remind students that points are plotted at the intersection of horizontal and vertical lines drawn respectively from points on the vertical and horizontal axes. Once points have been plotted they are joined in consecutive order, using a ruler. • Note: Only continuous or measured data should be shown on a line graph, such as temperature, otherwise the points can not be joined and information can not be read between the points.

What to do

w ww

• Assist students to plot the temperatures for January and February on their own graphs.The demonstration graph may be used for this. • Ask students to plot the remainder of the points and join the points to make a line graph. • Discuss a range of scores (spread from highest to lowest) and a mean score (the average of the average temperatures). • Direct students to answer the questions.

. te

Challenge

m . u

Technology Opportunity! ☛ In most real-world settings people use a spreadsheet to draw a graph. The data are entered into the spreadsheet and the charting function used to produce the graph. Unfortunately, many people choose the wrong type of graph to match the data. Students could use a spreadsheet such as Microsoft ® Excel to graph temperature.

o c . che e r o t r s super

• Students are to record each step as they attempt to reach 1000. • Those who realise that 1 x 1000 or 2 x 500 make 1000 in one step should be directed to find as many ways as they can to make 1000 in one step. • Show findings.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 158 – 159. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 35 •

Unit 4–3

Student page 12

Outcomes

Indicators

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

Resources

• round • nearest place • chart • whole number • symbols

• calculator

• rounding

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.

Main Activity (N4.1a)

What to do

w ww

• Round the numbers in the two exercises following the rules.

. te

m . u

• Revise rounding of whole numbers. What are the rules/guidelines for rounding? (When a number ends in 1–4 we round down. When a number ends in 6–9 we round up. When a number ends in 5 we round either up or down.) • If we are rounding decimal numbers will the same rules apply? ( Yes) Why? (The place value system works on factors of ten. Decimal places are part of the same system, therefore the same rules apply. Basically, ignore the decimal point and apply the same strategies with the decimal point remaining in its original position.)

Challenge

ew i ev Pr

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

• 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 = 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

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 56 – 57. • 36 • New Wave Maths Book G – Teachers Guide

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

Unit 4—Answers

Student pages 10 – 12 Unit 4–1

Unit 4–2 1. (a) 48 (b) 4 (c) 2 (d) 24 (e) 7 (f) 9 (g) 25 (h) 4 (i) 9 (j) 12 2. (a) 71 (b) 71 (c) 61 (d) 51 (e) 51 (f) 72 3.

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

4. (a) 8092 (b) 12 376 (c) 22 372 (d) 16 660 Challenge Answers will vary. Some examples are: 7/7 + 7/7 + 7/7 = 3 7 + 7+ 7 ÷ 7 = 3

(a) 18 º – 33º (b) 24.6 ºC (c) December, January, February (d) June, July, August (e) Answers will vary (f) Answers will vary Challenge 2 x 500

ew i ev Pr

Teac he r

1. (a) 4 (b) 8 (c) 24 (d) 4 (e) 54 (f) 5 (g) 10 (h) 40 (i) 27 (j) 2 2. (a) 221 (b) 211 (c) 432 (d) 332 (e) 332 (f) 243 3.

w ww

2738

81 624 9257

30 781

52 395 784

6500

42 791

(b)

Number

2.3794 0.8358 0.2314 8.7627

. te

Nearest whole number

• Find patterns in everyday events and numbers—street numbers are even on one side, odd on the other etc.

Consolidation 4–2 • Record temperature over one week, record results in a table, graph the results and work out the range, mean and mode.

Consolidation 4–3

m . u

1. Teacher check 66.6 (c) 46.8 (d) 46.6 (e) 54.1 2. (a) 97.8 (b) (f) 55.6 3. (a) Number Nearest 10 Nearest 100 Nearest 1000

• Use various measures; weight, capacity, time, length etc. to round to the nearest whole number.

o c . che e r o t r s super

Nearest 10th

Nearest 100th

Nearest 1000th

5.6075 0.0947 8.1950 4.8175

Challenge Answers will vary. Some examples are: 6 x 6 + 6/6 6 x (6 + 6/6 ) (6 + 66 + 6) x (6 + 6 + 6) + (6/6 + 6/6 ) + 6

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

New Wave Maths Book G – Teachers Guide • 37 •

Unit 5–1

Student page 13

Outcomes

Indicators

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

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

Skills • drawing • problem-solving • checking • constructing

Resources • calculator • card • scissors • sticky tape or glue • 3-D models (polyhedron geoshapes are handy) • display of nets of 3-D models (see pages 225 – 231)

Language • circle • greatest • pair • subtract • diagram • net • construct • cube • digit • prime number

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

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 (S4.2) Warm up

• Distribute and allow students to handle and discuss attributes of 3-D models in groups. • Display samples of nets for a number of the 3-D models. Ask students to match nets and models. • Possibly cut up cardboard boxes and packets to discover the various nets.

ew i ev Pr

Teac he r

• The focus for this unit is determining order relationships of decimal numbers.

w ww

• The net in the workbook is of a cube. There are 11 configurations that the net may be drawn in to make a cube without a side missing or doubling up on another side. • Ask students to draw as many different arrangements as they can. Remind them that flips and rotations are to be considered the same design. • When all design possibilities have been exhausted students are to select two of their designs and transfer each to stiff paper or card to cut and construct a cube to check their design.

Challenge

. te

m . u

o c . che e r o t r s super

• Students need to investigate prime numbers to find a fit for the description provided. • Ask students to set their own limits prior to examining possible prime numbers. • Note: A prime number is a number with only two factors, 1 and itself. • All exploratory findings are to be recorded and an explanation describing the solution written up for sharing.

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

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

Unit 5–2

Student page 14

Outcomes

Indicators

N4.1a, N4.3

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

Skills

Resources

Language

• calculator • coloured pencils

• round • nearest hundred • subtract • separate • paths

• following patterns • adding • subtracting

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

Notes

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

ew i ev Pr

Teac he r

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

• Students 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

• Write 742, 893 and 265 on the board. Ask how many hundreds, tens and ones in each number.

What to do

. te

Challenge

m . u

w ww

• Numbers are made up of components.These components may be separated to show the total value of each place within the number; for example, 786 is made up of 700 + 80 + 6. Renaming numbers is important: 786 also means 78 tens and 6 ones or 786 ones or 7 hundreds and 86 ones. • By breaking numbers into their components it is easier to add and subtract the given numbers. In the following example the hundreds are added together, the tens are added together and the ones are added together before combining the hundreds, tens and ones to provide an overall total. 682 + 594 (600 + 500) + (80 + 90) + (2 + 4) = 1276 • Note: working from the left, traditional algorithms work from the right. • When subtracting the same principle applies except each component is subtracted from its like component before the subsets are added together to give the final difference. • Work through several examples from the workbook with students to ensure understanding is there. Include some subtraction examples when working with the class. • Direct students to complete the exercises.

o c . che e r o t r s super

• Suggest to students that they use a different coloured pencil for each path they are drawing for ease of viewing later. • All paths attempted need to be shown. • Write an explanation of how paths may be found from your experimentation.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 82 – 83. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 39 •

Unit 5–3

Student page 15

Outcomes N4.3, M4.4a, M4.4b

Skills • drawing • following instructions • estimating • planning • problem-solving

Memory Masters (N4.3)

Indicators

Resources

The student is able to: • investigate rectangles drawn on a square grid and show that a short cut for finding the area is to multiply the number in each row by the number of rows. • understand that for the final figure to ‘look the same shape’ as the original, all lengths have to be scaled by the same amount.

• calculator • ruler • pencil • trundle wheel • tape measure (30 – 50 m) • marker cones • coloured pencils • paper (A3) • playing field measurements

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, M4.4b)

ew i ev Pr

Teac he r

• subtract • grid • squares • units • dimensions • doubled • hectare • area

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

Warm up

Language

• What can you tell me about a square? Discuss features of a square. Introduce other regular shapes if necessary.

w ww

• Students are to follow the instructions in their workbook and draw the four squares as requested. Suggest that a different coloured pencil used for each square may help. • Double the dimensions of each of the original squares and draw the new squares. Use the same colour for the new squares as for the originals. Try not to overlap any squares. • What happens to the area when the dimensions are doubled? (Area squared) Is this so in each case? Note: Area is the measure of covering a space. • Exercise 4 may be treated as a whole-class or a small-group activity. Under supervision, ask the students to mark out one hectare (or as the grounds permit). Discuss the size of the measure ‘hectare’ and where its main use is. • In small groups ask students to plan for a lightning carnival to be played at their school using the available ground area. Sports to be played are given. Space may not allow for all grounds to be marked. • Ask students to give reasons for their modifications to or deletion of some of the playing fields. Ask whether playing area is large enough for the game to be played successfully. Are the dimensions you used large enough? If not, what will happen? • Display plans.

. te

m . u

What to do

o c . che e r o t r s super

Challenge • Students are to list all the different ways they can think to find the area of a rectangle 7 m x 4 m. Diagrams and or written explanations are to be used. • Share the methods of finding the area and the reasoning used. OR • A rectangle has an area of 48 m2. What might its dimensions be?

• 40 • New Wave Maths Book G – Teachers Guide

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

Unit 5—Answers

Student pages 13 – 14 Unit 5–1

(b) 11 altogether Challenge 43, 61

Unit 5–2 1. (a) 72 900 (b) 9200 (c) 61 800 (d) 49 700 (e) 800 (f) 6500 (g) 400 (h) 59 200 (i) 86 500 (j) 4800 2. (a) 87 (b) 88 (c) 39 (d) 69 (e) 49 (f) 48 3. (a) 1141 4. (a) 643 (b) 1224 (b) 524 (c) 973 (c) 110 (d) 1363 (d) 261 (e) 992 (f) 1406 (g) 1276 (h) 851 (i) 1339 (j) 1479 Challenge Teacher check Possible answer: 7 m x 4 m = 28 m2

r o e t s Bo r e p ok u S (c) Teacher check

ew i ev Pr

Teac he r

1. (a) 6.274 (b) 37 209 (c) 42.82 (d) 2.8071 (e) 692 874 (f) 0.0021 (g) 8.407 (h) 15.1 (i) 15.1 (j) 201 2. (a) 397 (b) 395 (c) 298 (d) 598 (e) 498 (f) 396 3. (a) Possible answers:

w ww

. te

(b) Teacher check (c) Area is 4 times larger. 4. (a) Teacher check (b) Teacher check Challenge

• Select one of the nets. Copy onto card and make a cube.

Consolidation 5–2 • Practice further examples where students are required to use the technique of partitioning to solve the problem.

Consolidation 5–3

m . u

1. 6 x 11 = 66, 66 ÷ 11 = 6 (other answers are possible) 428 (c) 269 (d) 128 2. (a) 156 (b) (e) 458 (f) 527 3. (a)

• Try tripling the dimensions of the square. • Attempt the same activity with rectangles. Are the results the same?

o c . che e r o t r s super One possible solution.

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

New Wave Maths Book G – Teachers Guide • 41 •

Unit 6–1

Student page 16

Outcomes N4.2, N4.3, N3.2

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

Indicators

Resources

The student is able to: • ask organising questions to get him or her started. • continue questions in a brainstorming situation to help deal with a practical task. • comment on conjectures in light of results of testing them and revise if needed.

• cardboard • scissors • pencils • building catalogues, such as for driveways, bricks, pavers etc. (optional)

Language • tessellating • patterns • tiling • repeating • interlocking • area • gaps

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

Notes

What to do

• This activity can be completed by students working independently or collaboratively in pairs. As students will need to discuss their opinions and ideas, enough time should be allowed so they do not feel rushed and for ideas to evolve. • 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 students. – What does tessellate mean? (Shapes which cover an area with no gaps between them.) – What kind of shapes tessellate? (squares, rectangles etc.) – Do circles tessellate? – Do triangles tessellate? – How do you know if a tile tessellates? • For this activity, a shape tessellates or tiles if it can cover an area without any gaps.This does not include adding any smaller shapes to fit into the gaps (such as the tiles on a bathroom floor with larger shapes combined with smaller shapes to cover the area). • Allow students time to practise drawing a number of each of the shapes to see which tessellate. Students can draw the shape, trace it onto cardboard and cut it out. They will then be able to trace around this shape several times to create their pattern. • Shapes tessellate depending on the angle of their edges. If the angle is a multiple of 360, the shape will tessellate. For example, Square Equilateral triangle 4 x 90º = 360 6 x 60º = 360

ew i ev Pr

Teac he r

Main Activity (N4.2, N4.3, N3.2)

w ww

. te

m . u

o c . che e r o t r s super

The pentagon and octagon will not tessellate. • Students may wish to present their findings as a poster on coloured card. • Allow each pair to discuss and evaluate their ability to problem-solve and their 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.

• 42 • New Wave Maths Book G – Teachers Guide

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

Unit 6–2

Student page 17

Outcomes

Indicators

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

The student is able to: • suggest what data to collect to help estimate numbers or quantities. • represent data in diagrams and tables which may include arrows, Venn diagrams and two-way tables. • read the information provided on the axes of bar and line graphs, including where all calibrations on the scale may not be labelled.

Skills • recording • graphing • analysing • ordering • collecting data • working in a group • discussing

Resources

Language

• calculator • pencil • ruler • coloured pencils

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

• multiply • table • survey • compare • differences • tally • total • arrange • order

Notes

Memory Masters (N4.3)

Teac he r Number (N4.3)

ew i ev Pr

• Students 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

What to do

. te

m . u

w ww

• Discuss the construction and use of surveys with the class. • Ask students what they need to conduct a survey. (Topic, sample group to survey, tools for collection and recording data, purpose for survey.) • Exercise programs undertaken by people is a common survey subject run at different times. A sample survey can be undertaken at school by finding out how many students walk to school. • The results may be used to identify a Physical Education program or to negotiate better access from home to school to encourage more students to walk.

• A survey collection table is provided in the workbook to undertake a survey of students who walk to school. Suggest the class works in small groups and completes the survey. • Data may be collected prior to the lesson to minimise disruption to other classes. • Discuss the type of graph most suited to represent the information collected. • Collate data and present as a bar graph on the axes provided. • Answer the questions using the information collected.

Challenge

o c . che e r o t r s super

• A local tennis club wants to run a competition. If ten players sign up and each player must play each other once, how many games will need to be scheduled? • What if 20 players sign up?

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

New Wave Maths Book G – Teachers Guide • 43 •

Unit 6–3

Student page 18

Outcomes

Indicators

N3.3, N4.3

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

Skills

Resources

• multiply • calculator • estimate • actual • value

• calculator

• using a calculator • estimating

Memory Masters (N3.3)

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 (N4.3) Warm up

• Start with a quick mental quiz. Answers to basic facts of multiplication or division. Throw in some multiples of 10 or 100 to test skills. • Mental calculations are important to our everyday lifestyle and these skills need to be developed.

ew i ev Pr

Teac he r

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

w ww

• The activities in the workbook focus on mental division requiring an estimate of the answer first, followed by an accurate mental calculation. Work with the class on several examples from both Exercise 1 and 2 to assist in focusing attention. • Once students have calculated answers mentally these may be checked with a calculator. Allow students who are unable to handle the mental calculations to use a calculator after they have given an estimate.

Challenge

. te

• Students are to record all their attempts at finding the value of the statement given. • Explain the reasoning behind the final solution.

m . u

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 76 – 77. • 44 • New Wave Maths Book G – Teachers Guide

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

Unit 6—Answers

Student pages 16 – 18 Unit 6–1

Unit 6–2 1. 14 x 7 x 3; 2 x 7 x 7 x 3; 2 x 7 x 21 and so on. 2. (a) 30 000 (b) 67 200 (c) 46 000 (d) 43 800 (e) 51 300 (f) 73 800 3. Teacher check Challenge 10 players = 45 20 players = 190

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

ew i ev Pr

Teac he r

1. Equilateral triangles, rhombus and hexagon will tessellate due to the angle of their edges being a factor of 360º. See teachers notes on page 42.

. te

• Measure the angles of the edges of a number of regular shapes. Consider the relationship between the angles of the shapes that do tessellate with the degrees of a circle (360º).

Consolidation 6–2

m . u

w ww

1. (a) 49 (b) 12 (c) 9 (d) 36 (e) 9 (f) 8 (g) 4 (h) 14 (i) 8 (j) 6 2. (a) 33 600 (b) 37 500 (c) 51 200 (d) 27 300 (e) 42 300 (f) 38 000 3. Teacher check estimations (a) 17.35 (k) 22.4 (b) 43.45 (l) 28.83 (c) 21.6 (m) 16.23 (d) 38.35 (n) 32.13 (e) 27.4 (o) 25.2 (f) 21.025 (p) 9.32 (g) 11.575 (q) 16.86 (h) 19.725 (r) 11.04 (i) 22.9 (s) 19.48 (j) 15.625 (t) 14.58 3. (a) 2.9 (b) 1.2 Challenge (c) 1.6 –50 1 – 2 = –1 (d) 2.5 3 – 4 = –1 (e) 1.14 5 – 6 = –1 (50 x) (f) 0.55 add 50 lots of –1 = –50

• Complete the activity to find out the number of students who ride to school, catch the bus or get a lift.

Consolidation 6–3

o c . che e r o t r s super

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

• Students will require many opportunities to practise estimation skills. Students can estimate the cost of a number of items or length of time of a number of activities etc.

New Wave Maths Book G – Teachers Guide • 45 •

Unit 7–1

Student page 19

Outcomes N4.2, N4.3, S4.1

Skills

Indicators

Resources

The student is able to: • place or describe key features on a map or path with sufficient care that others can use them.

• calculator • pencil • ruler • street directories or maps of old and new suburbs

• planning • drawing • analysing

Memory Masters (N4.2)

Language • divide • grid system • bird’s-eye view

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 (S4.1) Warm up

• Take students in an imaginary flight over their town or suburb. From their aerial view advantage, ask students what they might see. Students should be encouraged to sketch a rough outline of the main features they would see on the blackboard/whiteboard with added assistance from classmates. • Ask students what features of their town or suburb they particularly like. List these. Ask them what they would change or like to see included. List these. • Note: The bird’s-eye view worksample in the EDWA book could provide a useful sample.

ew i ev Pr

Teac he r

• The focus for this unit is reordering of multiples (associate property) to simplify calculations.

w ww

• Set students the task of planning their own suburb using the page from the workbook or their own starting point. Ask them to ensure their location has all the necessar y infrastructure, is safe and environmentally friendly for them. Details of essential infrastructure are listed in the workbook. • Finished plans may be displayed and/or sent to the local town council for their perusal.

Challenge

. te

Technology Opportunity! ☛ Encourage students to play computer simulation games, such as SimCity™.

m . u

What to do

o c . che e r o t r s super

• Most simple calculators only have an eight-digit display. Explain how you would calculate the answer to 10 000 015 x 347. • There are several possible ways to calculate this result. A knowledge of place value will assist in multiplying 10 000 000 by 347 and 15 x 347 may be calculated in many different ways.

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

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

Unit 7–2

Student page 20

Outcomes

Indicators

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

The student is able to: • mentally find unit fractions of numbers which are multiples of the fraction.

Skills • displaying fractions • writing fractions

Resources

Language

• calculator • coloured pencils • pattern blocks • fraction shapes

• divide • diagrams • fractions • subtract • inclusive

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

Notes

Memory Masters (M4.1a, N4.1a)

Teac he r

ew i ev Pr

• The focus for this unit is conversion of millilitres to litres and litres to millilitres (1000 mL = 1 L).

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

• Distribute pattern blocks to small groups of students. After initial free play ask them to make pairs of fractions; e.g. 22/3 and 11/3. Compare the two fractions and find the difference between them. Ask students how they might do this. (Lie fractions side by side or one on top of the other.) • Repeat for 3 and 13/4; 25/6 and 3/6; 31/4 and 13/4.

What to do

Challenge

. te

m . u

w ww

• The same process can be achieved by using diagrams. Appropriate diagrams have already been drawn in the workbook. • Work with the whole class and ask them to shade (colour) 23/4 of the 4 shapes in 3(a). How much is left unshaded? (11/4) • Repeat this for each activity.

o c . che e r o t r s super

• Students have 11 circles to place the numbers 1 to 11 inclusively into so that each line gives a total of 14. • All combinations and attempts are to be shown. • An explanation as to how the final solution was reached is to be written up.

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

New Wave Maths Book G – Teachers Guide • 47 •

Unit 7–3

Student page 21

Outcomes

Indicators

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

The student is able to: • find the area of a triangle using the measures of a rectangle.

Skills • finding area • cutting • drawing • problem-solving • explaining

Resources

Language • convert • dollars, cents • divide • area • triangles, rectangles • network • continuous • adjacent

• calculator

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) Warm up

• Draw some triangles and enclose them in a rectangle.

ew i ev Pr

Teac he r

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

• Use paper folding or cutting to show that the area of a triangle is half the area of the associated rectangle.

w ww

• One method of finding the area of a triangle is to place it in a rectangle so that the base of the triangle forms one side of the rectangle and the vertical height of the triangle is the same as the adjacent side of the rectangle. • There are three examples in the workbook showing triangles drawn inside rectangles in this manner. • Use these diagrams to find the area of the rectangle and then the area of the triangle. • Ask students to explain the relationship between the triangle and the rectangle in each case. • By drawing one of the examples onto paper and cutting around the triangle and the rectangle the pieces may be placed over each other to show that both the triangle and the leftover pieces of the rectangle are the same area.

. te

Challenge

m . u

What to do

o c . che e r o t r s super

• Students are to experiment with different starting points to see if they can trace over the network with one continuous line, finishing at the starting point. • Intersection points are numbered to assist in explaining the path followed. • Share solutions with the class.

• 48 • New Wave Maths Book G – Teachers Guide

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

Unit 7—Answers

Student pages 19 – 21 Unit 7–1

1. (a) 60 (b) 70 (c) 120 (d) 140 (e) 270 (f) 900 (g) 1200 (h) 1400 (i) 3000 (j) 1500 665.57 2. (a) 456.375 (b) 432.16• (c) 877.4 (d) • (e) 842.6 (f) 559.2 3. Teacher check Challenge Answers will vary; students should display an understanding of place value.

Unit 7–2 1. (a) 0.043 L (b) 0.005 L (c) 0.294 L (d) 0.003 L (e) 2.714 L (f) 3 mL (g) 802 mL (h) 5000 mL (i) 10 mL (j) 240 mL 2. (a) 71 (b) 71 (c) 61 (d) 51 (e) 51 (f) 72 3. (a) 11/4

(b) 5/6

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

(d) 18/10

(e) 6/8

ew i ev Pr

Teac he r

Challenge

. te

• Make a 3-D model of the map.

Consolidation 7–2 • Devise further examples for students to complete.

Consolidation 7–3

m . u

w ww

1. (a) $0.24 (b) $3.56 (c) $0.08 (d) $72.42 (e) $8.97 (f) 6742c (g) 2410c (h) 683c (i) 47c (j) 299c 2. (a) 7.4 (b) 5.3 (c) 6.6 (d) 5.8 (e) 6.5 (f) 6.5 3. (a) rectangle = 60 cm2 triangle = 30 cm2 (b) rectangle = 55 cm2 triangle = 27.5 cm2 (c) rectangle = 64 cm2 triangle = 32 cm2 Challenge 6, 5, 4, 11, 10, 4, 3, 2, 10, 9, 2, 1, 8, 9, 12, 8, 7, 6, 12, 11, 6

• Discuss any findings the students may have; e.g. area of triangle = 1/2 area of rectangle.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 49 •

Unit 8–1

Student page 22

Outcomes

Indicators

N4.2, N4.3

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

Skills • using a calculator • estimating

Memory Masters (N4.2)

Resources • calculator • 2-cm cubes

Language • subtract • estimate • mentally • remainders • calculator • actual • surfaces

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

• When finding the answer to numerical sums that are beyond our mental scope we generally make an estimate of the answer. If we need an accurate answer we use another form of calculation.

ew i ev Pr

Teac he r

• The focus for this unit is the use of brackets to determine order of subtraction and addition.

w ww

• Practice in this is provided in the workbook with division of large numbers, division of decimals and multiplication of large numbers. • Remind students that estimates are just that, a rough guide to the approximation of the actual answer. To aid in estimation rounding of the numbers to be used helps simplify the problem; e.g. 3(a) 847 ÷ 314 may be viewed as 800 ÷ 300. The approximate answer is 3 or between 2 and 3. Using a calculator, the actual answer is 2.617. • Work with students through several examples from 3, 4 and 5 before letting them complete the activity themselves.

Challenge

. te

m . u

o c . che e r o t r s super

• Distribute 2-cm cubes to small groups of students if they are requested. • Students are to use the cubes to assist them in determining the number of exposed surfaces of the 2-cm cubes in the drawn arrangement. • Surfaces under the arrangement are classed as exposed.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 76– 77. • 50 • New Wave Maths Book G – Teachers Guide

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

Unit 8–2

Student page 23

Outcomes

Indicators

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

The student is able to: • represent data in diagrams and tables which may include arrows, Venn diagrams and two-way tables.

Skills

Resources

Language • subtract • data • timetable • period

• calculator • TV guides

• planning

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

Notes

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

ew i ev Pr

Teac he r

• The focus for this unit is the addition of four addends each less than 10.

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

Main Activity (C&D4.3) Warm up

• Timetables are an important part of your lives. We have a timetable in our classroom to assist in ensuring that the activities that are required to be covered are covered in a given time. Without planning, organisation becomes chaotic. • It takes time and care in planning for a day’s activities. One of the hardest timetable tasks of all is fitting in the specialist program within a school week so that all students have a fair share of the time available. • The deputy principal, principal or a specialist teacher may be asked to speak to the class about timetabling.

m . u

What to do

w ww

• A task is set out in the workbook to test the timetabling skills of the students. This task may be completed as a group activity. • Planning must ensure that all the times required are used and a spread of activities is given so that classes don’t have the same activity on the same day and if possible have at least a day’s break between. • Giving the class the task of planning the schools own DOTT (specialist) timetable may be a useful follow-up activity.

Challenge

. te

o c . che e r o t r s super

• Use the TV guide to plan a single day of programs. Choose the TV shows that you like from any channel and create the ‘perfect’ day’s viewing.

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

New Wave Maths Book G – Teachers Guide • 51 •

Unit 8–3

Student page 24

Outcomes

Indicators

N4.3, N4.1a

The student is able to: • understand the multiplicative nature of the relationship between places for whole numbers; i.e. as they move from right to left, each place is 10 times the one before.

Skills • multiplying

Memory Masters (N4.3)

Resources • calculator

• subtract • power • total value • number sentence • expanded • notation

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)

w ww

What to do

• Exercise 3 asks for the number 10 raised to different powers to be written in expanded form, then as a total value; for example, 102 = 10 x 10 = 100. • Work through the other four parts of Exercise 1 with the class as a whole. • Exercise 4 is an extension of Exercise 3. The whole number is written then multiplied by a single multiple of 10 with the number of zeros transferred unchanged. The result gives the whole number x 10 to the given power. • Work through the exercise with the class as a whole.

. te

m . u

• Very large numbers take a lot of room to write. Mathematicians have devised a means of writing these numbers in simple forms based on multiples of 10. Each multiple of 10 is given a power of 1. 10 is 101; 100 is 10 x 10 or 102; 1000 is 10 x 10 x 10 or 103. Note: 100 = 1. • A simple means of determining what the power of 10 is to count the zeros at the end of the number. Two zeros is 10 to the power of 2 written as 102. Five zeros is 10 to the power of 5 and is written as 105.The power indicator tells how many times 10 is multiplied by itself. • Students could look up very large distances; e.g. How far is it to the nearest star?

Challenge

ew i ev Pr

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

• 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 = 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

o c . che e r o t r s super

• Students are to draw or write each attempt to solve the problem so that the solution may be shared and compared with that of others.

• 52 • New Wave Maths Book G – Teachers Guide

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

Unit 8—Answers

Student pages 22 – 24 Unit 8–1

Unit 8–2 1. (a) 18 (b) 22 (c) 15 (d) 25 (e) 22 (f) 20 (g) 22 (h) 22 (i) 20 (j) 17 2. (a) 327 (b) 317 (c) 348 (d) 537 (e) 448 (f) 335 3. One possible solution: Specialist Timetable Monday

Tuesday

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

Wednesday

Thursday

1.00

PE1, A3, M2 PE3, A1, M4 PE5, A7, M6 PE7, A5, M8

1.30

PE2,

Friday

#AA1

#AA2, ºC 7 & 8

2.10

PE7, A2, M6 PE6, A4, M5 PE4, A6, M3 PE2, A8, M4

#AA3, *PE 5 – 8

2.40

PE8

M1 PE4,

PE6,

PE8,

2.00

Recess Time

M1 PE5

3.10

M2 PE3

PE1

#AA4

ew i ev Pr

Teac he r

1. (a) 7 (b) 8 (c) 12 (d) 5 (e) 5 (f) 6 (g) 2 (h) 9 (i) 6 (j) 9 2. (a) 466 (b) 643 (c) 562 (d) 271 (e) 726 (f) 576 3. (a) 2.70 4. (a) 0.089 m (b) (b) 1.63 0.059 m (c) (c) 3.92 0.07 m (d) 2.05 (d) 0.064 m (e) 2.03 (e) 0.074 m (f) (f) 0.33 0.028 m (g) (g) 0.32 0.006 m (h) 0.47 (h) 0.020 m (i) (i) 0.28 0.009 m (j) (j) 0.27 0.010 m 5. (a) 30 456 Challenge (b) 17 745 18 surfaces (c) 61 758 (d) 48 357 (e) 521 136 (f) 188 643

Key: M = Music *PE = Joint phys ed class PE = Phys Ed #AA = Art appreciation A = Art ºC = Choir session Challenge Teacher check

. te

• Students will require many opportunities to practise estimation skills. Students can estimate the cost of a number of items in a shopping list or distances etc.

Consolidation 8–2

m . u

w ww

1. Teacher check 2. (a) 396 (b) 396 (c) 197 (d) 594 (e) 694 (f) 599 3. (a) 10 x 10 = 100 (b) = 1000 (c) = 10 000 (d) = 100 000 (e) = 1 000 000 4. (a) 6 x 10 000 = 6 x 104 (b) 2 x 10 000 = 2 x 104 (c) 8 x 100 = 8 x 102 (d) 3 x 100 000 = 3 x 105 (e) 7 x 1 000 000 = 7 x 106 (f) 42 x 1 000 000 000 = 4.2 x 109 (g) 15 x 100 000 000 = 1.5 x 108 (h) 9 x 10 000 000 = 9 x 107 (i) 6 x 10 = 6 x 101 (j) 5 x 100 000 000 = 5 x 108 (k) 8 x 1000 000 = 8 x 106 (l) 9 x 10 000 = 9 x 104 Challenge 2 children go across the river 1 child returns in the canoe 1 adult goes across the river 1 child returns in the canoe —repeat this process eight times to get everyone across the river

• Students develop their own weekly timetable, allowing for school, homework, home duties, activities, relaxation, sleep etc.

Consolidation 8–3

o c . che e r o t r s super

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

• Students can find examples of large numbers in newspapers and reference material and write them in expanded form.

New Wave Maths Book G – Teachers Guide • 53 •

Unit 9–1

Student page 25

Outcomes

Indicators

N3.3, N4.3, S4.2

The student is able to: • attend to shape, structure and scale in making recognisable models of things.

Skills • manipulating • checking • reasoning • explaining

Memory Masters (N3.3)

Resources

Language

• calculator • modelling clay • card • glue • tape • scissors • pencil • models of a tetrahedron and an octahedron

• multiply • regular polyhedrons, tetrahedron, hexahedron, octahedron, dodecahedron, icosahedron • congruent models • corresponding • angles, vertex, area, face

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 (S4.2) Warm up

• Ask students to describe what a regular shape is. (A regular 2-D shape has all sides and internal angles of the same size , e.g. a square or an equilateral triangle.) How would you expect a regular 3-D shape to be described? (Lengths of sides, size of angles and area of faces must all be the same.) • For a 2-D shape to be congruent (same size and shape) the length of corresponding sides and the size of corresponding angles must be equal. How would you describe a congruent 3-D shape? (The length of corresponding sides are equal, size of corresponding angles are equal and area of corresponding faces is equal.) • Show students examples of the models they are expected to make. Dice come in various shapes and sizes and could be used for this purpose.

ew i ev Pr

Teac he r

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

w ww

What to do

• Distribute models of an octahedron and a tetrahedron along with modelling clay and card so that students can make their own model of a tetrahedron and an octahedron. • After making the initial models make a congruent model of each. • Check models for congruence. • Students draw a net of a cube. Ensure students can confirm their net will make a cube. Provide net from Teachers Guide if required. • Encourage students to discuss in their group or with the class the congruence of their nets.

. te

Challenge

m . u

o c . che e r o t r s super

• Use the grid provided to place the crosses as directed. • Keep a record of all attempts and provide a description of what you attempted and reasons for acceptance or rejection of your findings.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 28 – 29. • 54 • New Wave Maths Book G – Teachers Guide

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

Unit 9–2

Student page 26

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 • shading • counting • recording • brainstorming

Resources

Language • multiply • shade • chart • parts per hundred • per cent • percentage

• calculator • coloured pencils • toothpicks (or equivalent)

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

Notes

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

ew i ev Pr

Teac he r

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

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

Main Activity (N4.1b) Warm up

• The recording of scores, parts of a whole and many other mathematical representations may be shown as a decimal, a fraction or a percentage. • A percentage is a representation against the whole or complete total. The whole is represented by 100% or 100 parts of a hundred. • Percentages are parts of 100; 9 parts out of 10 is the same as 90 out of 100 and is shown as 90%. • Students could also use the flat from a Base 10 MAB set to act as a base board. Lay longs and units onto the board to represent the percentage.

What to do

m . u

w ww

• Grids of 10 x 10 are useful in representing percentages. • The activity in the workbook allows students to show the given percentages by colouring in the given number of squares. The shaded squares represent the parts of a hundred of the percentage. • Work with students to colour the first two or three grids then set them to finish by themselves. • The fourth exercise requires students to think of all the places that percentages are used in everyday life.

. te

Challenge

o c . che e r o t r s super

• Students are required to use their knowledge of Roman numerals to complete the statement. • Allow students to experiment for themselves without providing the above hint. • Students are to record all attempts together with an explanation of what they are trying to do. • Share findings.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 68 – 69. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 55 •

Unit 9–3

Student page 27

Outcomes N4.1a, N4.3, M4.1

Skills • measuring • estimating • recording • explaining • describing

Indicators

Resources

Language

The student is able to: • select a unit of a suitable size to enable the required comparison to be made.

• calculator • objects as listed in the workbook • ruler, tape measure, metre rule • trundle wheel (ensure it is set so it clicks after 1 m) • squares • grid paper

• multiply • place value • measure • chart • width, height, length, depth, perimeter, diameter, circumference • unit of measure • estimate • actual

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 (M4.1) Warm up

• Different measures of length, height, width and circumference require different units of measure and different measuring tools. • Ask students to choose an object and then describe the units of length measure they would use to make the measurements and what measuring device they would use. Share information in small groups. Choose one example from each group to share with the class.

ew i ev Pr

Teac he r

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

w ww

• Working in small groups choose the unit of measure to be used to make the measurements of the objects given in the workbook. Write the estimate of the lengths then make the actual measurements. • Describe how the object was measured. • Share two or three examples from each group once measuring is complete.

Challenge

. te

m . u

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 94 – 95. • 56 • New Wave Maths Book G – Teachers Guide

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

Unit 9—Answers

Student pages 25 – 27 Unit 9–1

Unit 9–2 1. (a) 3 (b) 3 (c) 9 (d) 3 (e) 2 (f) 6 (g) 24 (h) 63 (i) 64 (j) 1 2. (a) 61 608 (b) 14 245 (c) 32 224 (d) 36 984 (e) 40 890 (f) 46 464 3.

r o e t s Bo r e p ok u S (a) 50%

(b) 25%

(d) 45%

(e) 59%

(f) 69%

(g) 91%

(h) 7%

4. interest, sales, exam results, and so on. Challenge

ew i ev Pr

Teac he r

1. (a) 6 (b) 72 (c) 4 (d) 6 (e) 40 (f) 5 (g) 36 (h) 35 (i) 2 (j) 28 2. (a) 57 800 (b) 23 220 (c) 67 940 (d) 21 660 (e) 48 380 (f) 54 020 3. Teacher check 4. Teacher check Challenge One possible answer:

. te

(5 + 5) – (5 – 5/5 )

• Students can use the modelling clay to make the other 3-D shapes mentioned in the workbook.

Consolidation 9–2 • Record results obtained in class as a percentage.

Consolidation 9–3

m . u

w ww

1. (a) 1/100 (b) 10 000 1 /1000 (c) 1/10 000 (d) 1 1 (e) /10 000 (f) /10 (g) 1 000 000 (h) 1/ 100 (i) 1 (j) 10 2. (a) 20 276 (b) 53 701 (c) 34 190 (d) 69 204 (e) 52 292 (f) 33 858 3. Teacher check Challenge Answers will vary. Some examples are: 5/5 + 5 5 5x 5 + 5/5

• As a class, brainstorm other items for students to add to the table.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 57 •

Unit 10–1

Student page 28

Outcomes N4.1a, N4.3, N4.1b

Skills • ordering • discussing • working independently

Indicators

Resources

The student is able to: • order fractions where the denominator changes and explain the order in objects, diagrams or words.

• calculator • place value charts (see pages 209 – 210) • fraction chart

Language • round • nearest thousand • divide • symbols • order relationship

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

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 (N4.1b) Warm Up

• Revise place value using place value chart. • Discuss and explore fraction decimal equivalents. Conversion of decimals to fractions by writing as tenths, hundredths etc. • Revise fraction values using fraction chart. • Revise use of symbols < (less than) and > (greater than).

ew i ev Pr

Teac he r

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

w ww

• Work through several examples with the class as a whole. Open discussion as to how each number’s size compared to its partner is determined. • Direct students to work through remainder of activities by themselves. Provide assistance as required. • These activities provide the perfect opportunity for teachers to pick up any misconceptions on the students’ part where decimals are concerned. Some students may think that 3.607 is a larger number than 3.61 because it has more digits in the number and that 1/3 is bigger than 1/2 because 3 is larger than 2. • As a final activity, students write the numbers in the correct place on the number line provided.

. te

Challenge

m . u

What to do

o c . che e r o t r s super

• Students may require squares or grid paper to assist in completing this activity. • Record all experiments and findings. • Write a brief report or conclusion for sharing with the class.

• 58 • New Wave Maths Book G – Teachers Guide

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

Unit 10–2

Student page 29

Outcomes

Indicators

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

The student is able to: • suggest what data to collect to help estimate numbers or quantities. • describe information from diagrams which may include arrow diagrams, tree diagrams, Venn diagrams or Carroll diagrams.

Skills • organising • interpreting • problem-solving

Resources

Language • reorder • fractions • divide • diagram • point • network • continuous

• calculator • coloured pencils

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

Notes

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

ew i ev Pr

Teac he r

• The focus for this unit is the commutative property 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&D4.4) Warm up

• Games of tennis, badminton, and similar court games require a draw that is shown as a tree diagram to provide an organised sequence of games. • The draw allows organisers to arrange games so that the top seeds will not meet each other until the final games of the draw.

Challenge

. te

m . u

w ww

• Using the tree diagram provided in the workbook, arrange a tournament draw for 16 players. • As part of the draw, organise the seeds so that the top four players will not meet until the final games of the tournament if they win all their games. • Answer the questions at the foot of the diagram.

• Choose a starting point so that the network can be traced over using a continuous line. • To allow easier viewing of attempts use a different colour for each attempt. • Write a brief explanation of your solution.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 59 •

Unit 10–3

Student page 30

Outcomes

Indicators

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 • writing fractions

Memory Masters (N4.3)

Resources • calculator • coloured pencil • apple • knife

Language • diagrams • equivalent fractions • numbers • inclusive • total

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 (N4.1b) Warm up

• Hold up an apple to the class and ask ‘What will I have when I cut the apple in half?’ (2 halves, which is still the same as a whole apple.) • Cut one half in halves again. Hold the two halves of the half up and ask what you have now. (2 fourths) Explain to the class that 2 fourths are equivalent to one half and that 2 halves are equivalent to one whole. • It may help to make up a rod mat of 1/8s, 1/4s, 1/2s, 1/3s, 1/6s and 1/9s and use these as reference points for equivalent fractions. • This analogy can be applied to fractions. Draw a square on the blackboard/whiteboard. Mark a line to cut the square in half. Mark another line to cut the square into fourths. Shade 2 fourths and write it as 2/4 on the board. Ask the class what this is equivalent to. (one half) • Note: Fractions are really about relationships while equivalent fractions are really about different names for the same thing.

ew i ev Pr

Teac he r

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

w ww

What to do

. te

o c . che e r o t r s super

• The workbook activities use this principle to find equivalent fractions. Work with the class as a whole having a student talk his/her way through the explanation of each equivalent fraction, much as has been done above. Class colours the equivalent fraction pieces.

Challenge

m . u

• Students will need to experiment with combinations to find the answers. • All attempts are to be recorded and explanations of findings given. • Share results with the class.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 62 – 63. • 60 • New Wave Maths Book G – Teachers Guide

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

Unit 10—Answers

Student pages 28 – 30

Unit 10–1

Unit 10–2 1. (a) 7 x 8 = 8 x 7 = 56 (f) 4 x 6 = 6 x 4 = 24 (b) 9 x 5 = 5 x 9 = 45 (g) 7 x 9 = 9 x 7 = 63 (c) 6 x 4 = 4 x 6 = 24 (h) 6 x 7 = 7 x 6 = 42 (d) 3 x 7 = 7 x 3 = 21 (i) 4 x 8 = 8 x 4 = 32 (e) 8 x 5 = 5 x 8 = 40 (j) 9 x 6 = 6 x 9 = 54 2. (a) 56 (b) 62 (c) 45 (d) 46 (e) 42 (f) 46 3.

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

Challenge 20 units 10 units —

4. (a) 15 (b) Put best at top, 2nd best at bottom, 3rd best at 2nd top place, 4th best at 2nd bottom place etc. (c) Allow opponent to go straight into the next round. Challenge One possible solution: 2, 3, 4, 2, 3, 1, 4, 1, 2

ew i ev Pr

Teac he r

1. (a) 877 000 (b) 26 000 (c) 62 000 (d) 44 000 (e) 5000 (f) 83 000 (g) 10 000 (h) 897 000 (i) 7000 (j) 31 000 2. (a) 82 (b) 77 (c) 93 (d) 74 (e) 86 (f) 76 3. (a) < (f) = (b) < (g) < (c) < (h) < (d) < (i) = (e) > (j) < 4. (a) > (f) < (b) > (g) = (c) < (h) = (d) = (i) > (e) > (j) > 3 5. /4 > 0.6 > 50% > 28/100 > 0.1 = 10% < 11/100 < 7/10

(b)

. te

• Students work in pairs. Each person writes a number in secret. Show each other the numbers. Decide which is greater.

Consolidation 10–2 • Plan a tennis tournament within the class. How will the configuration look for the number of students?

Consolidation 10–3

m . u

w ww

1. (a ) 6 x 2 = 2 x 6 = 12 (b) 9 x 5 = 5 x 9 = 45 7 x 5 = 5 x 7 = 35 (c) 8 x 7 = 7 x 8 = 56 (d) (e) 6 x 8 = 8 x 6 = 48 (f) 6 x 9 = 9 x 6 = 54 (g) 9 x 8 = 8 x 9 = 72 (h) 4 x 9 = 9 x 4 = 36 (i) 7 x 9 = 9 x 7 = 63 (j) 4 x 7 = 7 x 4 = 28 2. (a) 147 (b) 34 (c) 119 (d) 156 (e) 138 (f) 148 3. (a)

• Provide fraction squares to demonstrate equivalent fractions further.

o c . che e r o t r s super Challenge Yes. Brett: $4.00 Rochelle: $5.00 (or $20 and $25 or $40 and $50) etc.

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

New Wave Maths Book G – Teachers Guide • 61 •

Unit 11–1

Student page 31

Outcomes

Indicators

N4.3, S4.3

The student is able to: • identify the transformation(s) used to produce a spatial sequence and continue the sequence.

Skills • following instructions • finding patterns

Memory Masters (N4.3)

Resources • calculator • compass • pencil • ruler • 2-cm cubes • coloured pencils

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)

Notes

ew i ev Pr

Teac he r

• add • points • circumference • circle • arrange • equal

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

Warm up

Language

• Explain to the class that patterns may be found in many places.The following activity should lead to a pattern being found.

w ww

• The exercise is best completed using coloured pencils so that the number of lines drawn on the diagrams is easier to count. • Work with the whole class to start the activity. Choose a colour and, using a ruler, join the top dot in circle A to the dot on its left; then join the top dot to the second dot on its left; then top dot to the third dot and so on. • Ask students how many lines were drawn. (5) • Start at the next dot, moving clockwise around the circle and join all dots around the circle to this one. Use a different colour. How many lines drawn this time? (4 because the top dot is already joined.) • Continue until all the dots are joined to each other. • How many lines drawn? • Complete circle B and answer questions. • Draw a third circle, using a compass, and mark 18 points evenly around its circumference. Join all the dots to each other. How many lines? • Is there a pattern developing? Describe what you have found.

. te

m . u

What to do

o c . che e r o t r s super

Challenge • Distribute 2-cm cubes to small groups of students. • Students arrange cubes as directed keeping drawings of their arrangements. • Write an account of the working out and final construction.

• 62 • New Wave Maths Book G – Teachers Guide

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

Unit 11–2

Student page 32

Outcomes

Indicators

N3.3, N4.3

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

Skills

Resources

Language

• calculator

• add • constant • percentage • subtraction

• subtracting decimals

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

Notes

Memory Masters (N3.3)

Teac he r

ew i ev Pr

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

Number (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

w ww

What to do

m . u

• Display the following sums on the blackboard/whiteboard. 47 – 29 82 – 29 65 – 36 To simplify subtraction, a constant (the same value) may be added to each number. In the first instance, by adding 1 to both numbers the sum becomes 48 – 30. By adding 3 to 82 – 47 the sum reads 85 – 50, and by adding 4 to 65 and 36 it now reads 69 – 40. Subtraction is now simpler. • When using decimals the same applies. In all cases look, to making the number being subtracted equal to the next whole number by adding the required constant.

• Work through the example in the workbook and then the first few questions in Exercise 3 with the class. Ask students what the constant should be and what the total of the numbers to be subtracted will now be. Explain that this helps mental calculations. • Students complete the exercise. • Students are to use the exercise just completed to assist in showing why the addition of a constant to each number in a subtraction sum does not alter the answer. • All thinking and working is to be shown.

. te

Challenge

o c . che e r o t r s super

• Discuss the origin of ‘per cent’ as it related to the Roman tax system, which is now our tax system.

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

New Wave Maths Book G – Teachers Guide • 63 •

Unit 11–3

Student page 33

Outcomes

Indicators

N3.3, N4.3, M4.2

The student is able to: • find or make different things with one measurement the same but the other different.

Skills • weighing • recording • analysing

Memory Masters (N3.3)

Resources • calculator • Base 10 MAB large cubes • kitchen scales • balance scales

• add • prism • cube • double • base area • height • mass • triangle

ew i ev Pr

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

• The focus for this unit is basic facts of multiplication and division with addition or subtraction of a whole number less than 10. Note rule of order: Rule of Order This is an artificial construct to ensure calculations are handled in the same way. 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: • brackets first • then indices (squares, cubes etc.) • multiplication or division (left to right) • addition or subtraction (left to right)

Teac he r

Language

• Students choose and use a repertoire of mental, paper and calculator strategies, meeting needed levels of accuracy and judging the reasonableness of results. (M4.2)

w ww

• Distribute Base 10 MAB large cubes to small groups of students. Provide kitchen scales for each group (if available). • Explain to the class that they will be building prisms and finding the surface area and volume of these prisms. Revise prisms – rectangular blocks.

What to do

. te

o c . che e r o t r s super

• In each case the group is asked to make a prism, find the surface area and volume of the prism then construct another prism with double the base area of the original prism. Then find the surface area and volume of the new prism. Record each one in the space provided on the table in the workbook. • Ask students what happens to the surface area and volume. Is this so in all cases? Why?

Challenge

m . u

Warm up

• Students will require the use of balance scales rather than a weigh scale for this activity.

• 64 • New Wave Maths Book G – Teachers Guide

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

Unit 11—Answers

Student pages 31 – 33

Unit 11–1

Unit 11–2 1. (a) 37 (b) 53 (c) 77 (d) 41 (e) 30 (f) 19 (g) 64 (h) 46 (i) 86 (j) 25 2. (a) 254 887 (b) 71 240 (c) 281 645 (d) 255 285 (e) 54 682 (f) 142 911 3. (a) $46 – $19.60 (40c) = ($46.40 – $20) = $ 26.40 (b) $52 – $29.70 (30c) = ($52.30 – $30) = $ 22.30 (c) $81 – $59.40 (60c) = ($81.60 – $60) = $ 21.60 (d) $64 – $37.20 ($2.80) = ($66.80 – $40) = $ 26.80 (e) $25 – $16.80 (20c) = ($25.20 – $17) = $ 8.20 (f) $34 – $18.10 ($1.90) = ($35.90 – $20) = $ 15.90 (g) $76 – $58.50 ($1.50) = ($77.50 – $60) = $ 17.50 (h) $55 – $27.30 ($2.70) = ($57.70 – $30) = $ 27.70 (i) $67 – $38.40 ($1.60) = ($68.60 – $40) = $ 28.60 (j) $93 – $45.90 (10c) = ($93.10 – $46) = $ 47.10 4. If both are made bigger by the same amount then the difference will not change. Challenge The final price will be 1% less than the original price.

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

ew i ev Pr

Teac he r

1. 2 x 15 = 30, 6 x 5 = 30, 3 x 10 = 30 and so on. 2. (a) 174 781 (b) 139 347 (c) 157 629 (d) 75 084 (e) 91 113 (f) 108 946 3. (a) A = 15, B = 66 (b) 153

Challenge

w ww

. te

• Complete the activity with other shapes; e.g. pentagon, hexagon octagon etc.

Consolidation 11–2 • Provide more examples for further practice as required.

Consolidation 11–3

m . u

1. (a) 14 (b) 10 (c) 1 (d) 9 (e) 31 (f) 14 (g) 25 (h) 17 (i) 4 (j) 11 2. (a) 2178.8 (b) 277.69 (c) 247.64 (d) 17.256 (e) 2801.7 (f) 185.58 3.

• Complete tasks which provide students with opportunities to further explore and understand the concept of surface area.

o c . che e r o t r s super

Challenge 1. One weighing by chance. 2. If first weighing equal; coin not being weighed is counterfeit. If unbalanced, follow the diagram.

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

New Wave Maths Book G – Teachers Guide • 65 •

Unit 12–1

Student page 34

Outcomes

Indicators

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

The student is able to: • understand that fractions are relative to particular wholes.

Skills

Resources

• subtract • round • mixed • numbers • nearest whole number • approximate answer • decimal • percentage • diagrams • inclusive • totals

• calculator

• rounding • approximating • converting decimals

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 (N4.1b) Warm Up

Notes

ew i ev Pr

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

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

Teac he r

Language

w ww

What to do

• Apply this to mixed numbers – see example in workbook. • Work through several examples with the class. Set class to complete the activity. • Exercise 4 requires the conversion of decimals to percentages. Remind students that decimals can be written as fractions, tenths over ten, and hundredths over a hundred; e.g. 0.1 = 1/10 and 0.27 = 27/100. Percentages are parts per hundred – 0.27 is 27 parts of a hundred or 27%. • Work through several with the class before allowing them to complete the task. • Exercise 5 requires students to shade the required fraction of each box then find the total shaded parts. This may be converted to a mixed number.

. te

Challenge

m . u

• Revise rounding rules. 0 – x/4 round down x/5 – x/9 round up • When rounding fractions the same basics of the rules apply – numerators up to half the denominator are rounded down, numerators greater than half the denominator are rounded up. Consider the idea of under and over estimates and the effect these may have on the outcome. A number line may be helpful. • Try these 1/7, 2/10, 7/8, 5/9, 3/6, 2/4.

o c . che e r o t r s super

• Students are to record all attempts and provide notes on how they attempted to reach the solution to this problem. • Share answers.

• 66 • New Wave Maths Book G – Teachers Guide

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

Unit 12–2

Student page 35

Outcomes

Indicators

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

The student is able to: • suggest what data to collect to help estimate numbers or quantities. • represent data in diagrams and tables which may include arrow diagrams, Venn diagrams and twoway tables. • read scales to the nearest graduation, including instances in which the graduations are not labelled.

Skills • measuring • ordering

Resources

Language

• calculator • measuring stick • tape measure

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

• number patterns • subtract • measure • height • ascending order • mass • digits • divided • remainder

Notes

Memory Masters (N3.4, N4.4) Number (N4.3)

ew i ev Pr

Teac he r

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

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

Main Activity (C&D4.2, C&D4.3, M4.2) Warm up

• Ask all students to stand and arrange themselves in a line so that they are in ascending (smallest to tallest) order of height.

What to do

Challenge

. te

m . u

w ww

• Return to seats. Students may be arranged in groups if there is enough measuring equipment so that their height and arm span can be recorded. Give each student a number (use class roll/attendance register Technology Opportunity! to enable future reference to be made for those who ☛ If data were entered into a forget their number). When height and arm span is spreadsheet, the sort function measured it is recorded on the table next to their could be used for ascending allocated number. Students can collect all measures order. from within their group. • When all measures have been made, one person from each group can call height and arm span measures for the numbers in the group for the rest of the class to record. • After all recordings have been made, students arrange height and arm span in ascending order.

o c . che e r o t r s super

• Students are to use their mathematical knowledge, trial and error or other means to find the answer to this problem, there is more than one solution. • Students are to show all working and keep notes explaining how they attempted to reach the solution.

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

New Wave Maths Book G – Teachers Guide • 67 •

Unit 12–3

Student page 36

Outcomes

Indicators

N4.1a, 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. • state fractional equivalents in words and symbols.

Skills • following directions • writing fractions • renaming fractions

Resources • calculator • coloured rods • grid paper (see page 199) • coloured pencils

Language • convert • subtract • metres • centimetres • fraction • numbers • thirds, fourths, halves • equal • denominator • mixed numerals • improper fraction • intersection

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 (N4.1b) Warm up

• Distribute coloured rods to groups of students and allow free play for a few minutes. • Distribute grid paper to groups.

w ww

• Ask students to draw four rectangles of three squares close to each other on the grid paper. Each rectangle represents a pie; each square one-third of the pie. How many thirds are there altogether? (12, this is shown as 12/3 on blackboard/whiteboard) Such a number is called an improper fraction – the top number (numerator) is larger than the bottom number (denominator). • Draw three rectangles of four squares in length. How many fourths are there? (12, written as 12/4) • Coloured rods may be used by using one rod to make items or by choosing the three and four rods in the above examples. • Work with students using both aids for 3(c). • As a class discuss how 3(d) to (f) would be written. • Ask students to attempt Exercise 4 using the examples from Exercise 3. • Using your knowledge try Exercise 3. Assist those with difficulties by drawing 31/3 on grid paper. The three whole numbers to be made of three squares to correspond with 3(a). • Exercise 6 can be shown as a shortcut – divide the denominator into the numerator and answer as a whole number and a fraction. Students may be shown a drawing of 7 halves as 3 whole units and 1 half unit.

. te

Challenge

m . u

What to do

ew i ev Pr

Teac he r

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

o c . che e r o t r s super

• Students should use coloured pencils to assist in solving this problem. • Show all attempts. • Write a brief report on how you reached the solution to share with the class.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 64 – 65. • 68 • New Wave Maths Book G – Teachers Guide

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

Unit 12—Answers

Student pages 34 – 36

Unit 12–1

Unit 12–2 1. (a) 480 (b) 85 (c) 10 000 (d) 4.5 (e) 6.0 (f) 150 (g) 91 (h) 3.0 (i) 2.0 (j) 10.5 2. (a) 476 (b) 259 (c) 339 (d) 137 (e) 465 (f) 278 3. Teacher check Challenge Possible answers: 11, 20, 29, 38

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

ew i ev Pr

Teac he r

1. (a) 6 (b) 20 (c) 16 (d) 28 (e) 43 (f) 15 (g) 1 (h) 22 (i) 15 (j) 3 2. (a) 306 (b) 307 (c) 408 (d) 207 (e) 608 (f) 506 4. (a) 30% 3. (a) 5 x 3 = 15 (b) 60% (b) 2 x 4 = 8 (c) 97% (c) 2 x 1 = 2 (d) 42% (d) 4 x 4 = 16 (e) 5% (e) 6 x 2 = 12 (f) 40% (f) 4 x 5 = 20 (g) 35% (g) 8 x 5 = 40 (h) 67% (h) 4 x 8 = 32 (i) 78% (i) 5 x 4 = 20 (j) 4% (j) 2 x 6 = 12 1 (k) 362% 5. 1 /4 (l) 117% 3 2 Challenge One possible solution:

. te

• Discuss situations where students may need to use the techniques practised in the workbook.

Consolidation 12–2 • Students measure stride length, length of foot etc. and order from longest to shortest.

Consolidation 12–3

m . u

w ww

1. (a) 428.73 m (b) 63.51 m (c) 0.86 m (d) 2.74 m (e) 0.04 m (f) 4 cm (g) 8729 cm (h) 46 380 cm (i) 72 cm (j) 814 cm 2. (a) 3953 (b) 2574 (c) 3846 (d) 6746 (e) 3843 (f) 2554 3. (a) 12/3 (b) 12/4 (c) 16/2 (d) 15/3 (e) 32/4 (f) 12/2 10 18 21 16 4. (a) /2 (b) /6 (c) /3 (d) /4 18 (e) /3 13 22 21 5. (a) 10/3 (b) /5 (c) /5 (d) /4 19 21 23 9 (e) /5 (f) /8 (g) /6 (h) /2 (i) 8/3 6. (a) 31/2 (b) 23/4 (c) 51/3 (d) 41/3 3 1 3 1 (e) 4 /5 (f) 11 /5 (g) 2 /6 or 2 /2 3 1 (h) 3 /5 (i) 4 /4 (j) 63/4 1 5 (k) 11 /3 (l) 7 /6 Challenge Possible answer.

• Provide students with further opportunities to practise writing fractions, converting mixed numerals and improper fractions.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 69 •

Unit 13–1

Student page 37

Outcomes

Indicators

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

The student is able to: • select and cut suitable lengths to make a skeleton of a provided 3-D shape.

Skills • manipulating • drawing • constructing • following and formulating directions • analysing

Resources • calculator • straws or toothpicks • modelling clay or frozen peas • 3-D model of tetrahedron

Language • convert • dollars • cents • multiply • construct • tetrahedron • directions

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

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 (S4.2) Warm up

• Distribute straws and modelling clay or toothpicks and frozen peas to students. • Allow time for free play.

w ww

• Display a solid model of a tetrahedron for all students to see. Discuss the features of the tetrahedron – number of sides (edges), faces and vertices. Relative sizes of each of the edges, vertices and faces. • Ask students to write a report of how they would construct a tetrahedron using straws and modelling clay. • In the report, materials are to be listed as well as a step-by-step list of directions used to construct the tetrahedron. A diagram of the finished model is to be included. • Students are to follow their directions to see if they are able to make a tetrahedron. • Display models.

Challenge

. te

m . u

What to do

ew i ev Pr

Teac he r

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

o c . che e r o t r s super

• Use the Internet to discover what a geodesic dome is and how it is constructed. • Make your own geodesic dome.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 22 – 23. • 70 • New Wave Maths Book G – Teachers Guide

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

Unit 13–2

Student page 38

Outcomes

Indicators

N3.3, N4.3, N4.2

Resources

The student is able to: • use properties of operations to complete and justify number sentences without completing the calculations. • generate missing numbers which obey a constraint.

Skills • calculating • recording • checking

Language • multiply • number sentences • true statements • greater than > • less than < • squares

• calculator

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

Notes

Memory Masters (N3.3)

Teac he r Number (N4.3)

ew i ev Pr

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

• Students 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

Rule of Order

☛ This is an artificial construct to ensure calculations are handled in the same way.

Challenge

. te

m . u

w ww

• Brackets first • Write the following number sentence on the blackboard/whiteboard, • Then indices (squares, cubes etc.) (8 + 7) x 4 = (6 + _ ) x 4 • Then multiplication or division Remind students that the contents of the brackets (left to right) must be worked out first; in this case 8 + 7 and then multiply by 4.Where brackets are found within brackets • Then addition or subtraction (left to right) the inner brackets are worked out first. To balance a number sentence, both sides must be equal. 15 x 4 = 60. What is to be added to 6 to make 15? (9) • Work through several examples before allowing them to complete the exercises. • Exercise 4 is a simple substitution of > or < in the box to ensure the statements are true. The number sentence figures and operations stay the same.

o c . che e r o t r s super

• Students are to find the greatest number of squares they are able to in the shape given. • All recordings and an explanation of the process used are to be kept for sharing later. • This could be extended further to find out how many squares there are on a chessboard.

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

New Wave Maths Book G – Teachers Guide • 71 •

Unit 13–3

Student page 39

Outcomes

Indicators

N3.3, N4.3, M4.2

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

Skills • manipulating • drawing • comparing • discussing • sharing

Memory Masters (N3.3)

Resources • calculator • 2-cm cubes • squares or tetrominoes • grid paper (see page 199)

• multiply • cube • square • shape • surface area • tetromino

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

ew i ev Pr

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

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

Teac he r

Language

w ww

• Explain to the class that they will be making shapes using the five cubes. All five cubes must be used in each shape. When the shape has been made, a drawing of the shape is required in the workbook (keep drawings to a size where about ten may be drawn on the page). • The surface area of each shape is to be found. (Count the surface squares or faces of the cubes that are exposed – include those underneath.) Write the surface area next to the drawn shape. • Compare shapes found by others in your group. • Discuss the different shapes and compare their surface areas. Discuss findings about surface area and give explanations as to why this is so. Develop the idea that the more surfaces joined together the less the surface area. • Each group to share their findings with the class.

Challenge

. te

m . u

© R. I . C.Publ i cat i ons orr evi ew pur posesonl y• What to do •f • Distribute 2-cm cubes to students. Class may be organised in groups – provide five 2-cm cubes for each student. • Allow time for investigation.

o c . che e r o t r s super

• Use tetromino shapes or make four tetromino shapes using grid paper. • Manipulate the shapes to see if a square can be made. • Keep records of attempts and explanations of how each solution was reached. • Display and/or share results.

• 72 • New Wave Maths Book G – Teachers Guide

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

Unit 13—Answers

Student pages 37 – 39

Unit 13–1

Unit 13–2 1. (a) 68 (b) 83 (c) 44 (d) 43 (e) 64 (f) 35 (g) 33 (h) 87 (i) 18 (j) 47 2. (a) $182.10 (b) $204.00 (c) $242.10 (d) $723.20 (e) $425.40 (f) $123.20 3. (a) 1 (k) 2 (b) 3 (l) 3 (c) 2 (m) 2 (d) 70 (n) 6 (e) 2 (o) 9 (f) 5 (p) 5 (g) 3 (q) 4 (h) 2 (u) 6 (i) 4 (s) 5 (j) 76 (t) 4 3. (a) > (b) < (c) > (d) > (e) > Challenge (f) < 30

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

ew i ev Pr

Teac he r

1. (a) $0.03 (b) $0.42 (c) $200.05 (d) $10.04 (e) $0.09 (f) 93c (g) 4729c (h) 30 042c (i) 64 705c (j) 3c 2. (a) 189 681 (b) 290 844 (c) 568 308 (d) 248 755 (e) 382 305 (f) 233 712 3. Teacher check Challenge Teacher check

. te

• Students use knowledge gained to make other models.

Consolidation 13–2 • Ask students to write their own number sentences.

Consolidation 13–3

m . u

w ww

1. (a) 14 (b) 10 (c) 31 (d) 9 (e) 1 (f) 14 (g) 25 (h) 17 (i) 4 (j) 11 2. (a) $300.00 (b) $340.00 (c) $246.00 (d) $399.00 (e) $432.00 (f) $228.00 3. Teacher check 4. Teacher check 5. Teacher check Challenge Yes

• What happens if six or seven cubes are used? How many more shapes can be used?

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 73 •

Unit 14–1

Student page 40

Outcomes

Indicators

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 • completing data • subtracting • ordering

Memory Masters (N4.3)

Resources

• divide • table • order • shortest, longest • date • decade

• calculator

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)

Notes

ew i ev Pr

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

Warm Up

Language

w ww

What to do

• In the workbook is a list of all Australia’s Prime Ministers. Who served for the longest period? (Robert Menzies ) Who was the shortest serving Prime Minister? (Francis Forde) • Complete the table by writing in the decade in which the Prime Minister took office; e.g. 1920 – 1930 or 1920s. Then write in the decade the Prime Minister finished in office. • Finally calculate the length of service as Prime Minister in years and months. • When writing the order of service for Prime Ministers, only include names once. For those who served more than one term you may choose to write longest or shortest term or combine the length of all terms in office and provide a total term.

. te

Challenge

m . u

• Who was Australia’s first Prime Minister? (Edmund Barton) • What is a decade? (10 years) • Up until 2002, Australia has had thirty changes of Prime Minister. Some of these served more than one term with a break between terms. • Who is the current Prime Minister? (Check on the Australian Government website http://www.aph.gov.au/library/parl/hist/primmins.htm)

o c . che e r o t r s super

• Some dates are palindromic; e.g. 20/02/2002. A palindrome is a word or number that can be read the same forwards as backwards; e.g. mum, dad, noon, 323, 747. • Try to find some more.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 54 – 55. • 74 • New Wave Maths Book G – Teachers Guide

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

Unit 14–2

Student page 41

Outcomes

Indicators

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

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

Skills • calculating • recording

Resources

Language

• calculator

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

• divide • range • mode • mean • table • tally • total • number • symbol

Memory Masters (N3.3)

• calculate • median

Notes

Number (N4.3)

ew i ev Pr

Teac he r

• The focus for this unit is the addition or subtraction of a whole number less than 10 to or from a basic fact of multiplication or 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) Warm up

• Discuss with the 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:

. te

Mode: The most frequently occurring score and there may be more than one mode;

o c . che e r o t r s super

e.g. The data are bimodal, having 8 and 9 occurring the most frequently.

What to do

m . u

w ww

Range: The spread of scores from the lowest to the highest, or a spread in which a category may be placed. Median: The middle score when scores arranged in ascending or descending order; e.g. 1, 3, 6, 8 (median), 8, 9, 9 Note: If there were only six scores then the median would be found — 1, 3, 6, 8, 8, 9 — 6 + 8 — 14 ÷ 2 = 7 2 Mean: The total of a set of scores divided by the number of scores; e.g. 1 + 3 + 6 + 8 + 8 + 9 + 9 7

• The two exercises in the workbook ask for the range, median, mean, and mode to be found for a set of given data and for the information for the second set to be collected and then calculated. (Students will need to convert to decimals first.) • Share the calculations from the class age collected data to ensure understanding is there.

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 5 using one or all of the operations; e.g. 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 supply answers that students can not find themselves. For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 160 – 161. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 75 •

Unit 14–3

Student page 42

Outcomes

Indicators The student is able to: • record and observe any patterns or relationship between symmetry and shapes.

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

Skills • completing tables • analysing • deducing • interpreting • drawing

Resources

Language • Roman numeral • Hindu-Arabic numeral • divide • fraction • relationship • shapes • regular polyhedron • lines of symmetry • sides • square, equilateral triangle, pentagon, hexagon, octagon, dodecagon

• calculator

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

Memory Masters (N3.1a)

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 (S4.3) Warm up

ew i ev Pr

• The focus for this unit is the conversion of Hindu-Arabic numerals to Roman numerals and Roman numerals to Hindu-Arabic numerals.

• What are lines of symmetry? In a square, how many lines of symmetry are there? Show me on the blackboard/whiteboard. • Discuss the lines of symmetry shown. Are there others? (No) There are other forms of symmetry – rotational. • How many sides in a square?

w ww

• In an equilateral triangle there are how many sides? How many lines of symmetry? Show these on the blackboard/whiteboard. • Enter the information collected into your workbook. Now complete the exercise for the regular pentagon and regular hexagon. • What is the relationship between the number of lines of symmetry and the number of sides of regular Technology Opportunity! shapes? ☛ Use a standard package with • Test your findings in a regular octagon and a regular drawing tools to draw regular dodecagon. Use the computer to draw these shapes and flip or rotate them. accurately if possible.

. te

Challenge

m . u

What to do

o c . che e r o t r s super

• Students show all their workings and explanations in solving this problem. • Show findings.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 34 – 35. • 76 • New Wave Maths Book G – Teachers Guide

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

Unit 14—Answers

Student pages 40 – 42

Unit 14–1 1. 4 x 4 x 11, 8 x 22, 2 x 8 x 11 and so on. 2. (a) 106 (b) 205 (c) 305 (d) 206 (e) 307 (f) 304 Decade took Date finished Decade finished Length of time as Prime Date took PM 3. office office office Minister in yrs/mths office 11 17 5 6

Barton

01/1901

Deakin

09/1903

Watson

04/1904

Reid

08/1904

Deakin

07/1905

18 Fisher

11/1908

Deakin

06/1909

Fisher

04/1910

Cook

06/1913

Fisher

09/1914 10/1915

Menzies

12/1949

02/1923 10/1929 01/1932 04/1939 04/1939

Teac he r

08/1941

9 Holt 3 McEwen 13 Gorton 8 McMahon 12 Whitlam 22 Fraser 24 Hawke 15 Keating 23 Howard

Challenge

10/1941

07/1945 07/1945 01/1966 12/1967

01/1968 03/1971 12/1972

11/1975 03/1983

12/1991 03/1996

09/1903 04/1904 08/1904 07/1905 11/1908 06/1909 04/1910

1 1 1 1 1 1 2 2 2 2 3 3 4 4 4 5 5 5 5 5 7 7 7 8 8 8 9 10 10

2 years 8 months 7 months 4 months 11 months 3 years 4 months 7 months 10 months 3 years 2 months 1 years 3 months 1 years 1 months 7 years 4 months 6 years 8 months 2 years 3 months 7 years 3 months 19 days 2 years 4 months 2 months 3 years 9 months 7 days 4 years 5 months 16 years 1 months 1 years 11 months 1 month 3 years 2 months 1 year 9 months 2 years 11 months 7 years 4 months 8 years 9 months 4 years 4 months

r o e t s Bo r e p ok u S 06/1913 09/1914 10/1915 02/1923 10/1929 01/1932 04/1939 04/1939 08/1941 10/1941 07/1945 07/1945 12/1949 01/1966 12/1967 01/1968 03/1971 12/1972 11/1975 03/1983 12/1991 03/1996

ew i ev Pr

21 Hughes 19 Bruce 10 Scullin 20 Lyons 2 Page 25 Menzies 4 Fadden 14 Curtin 1 Forde 16 Chifley

1 1 1 1 1 1 1 2 2 2 2 3 3 4 4 4 5 5 5 5 5 7 7 7 8 8 8 9 10 10

Unit 14–2 1. (a) 1 (b) 11 (c) 1 (d) 16 (e) 16 (f) 13 (g) 19 (h) 55 (i) 59 (j) 9 2. (a) 248 (b) 130 (c) 196 (d) 152 (e) 116 (f) 148 3. (a) 6 (b) 28.5 (c) 26 (d) 28.6 4. Teacher check Challenge Answers will vary. Possible solutions are: (9 + 9/9 ) x { (99+ 9) + (99+ 9) + 9/9} – 9/9 = 49 9 x ( 9 – 9/9 ) = 72 (9 + 9 + 9 + 9) – 9/9 = 35

Teacher check

. te

• Complete a similar activity using Australia’s governor-generals.

Consolidation 14–2 • Repeat the activity with actual class enrolment in your school.

Consolidation 14–3

m . u

w ww

1. (a) XC (b) IC (c) C (d) CX (e) CC (f) 596 (g) 2250 (h) 79 (i) 222 (j) 550 2. (a) 26.13 (b) 18.86 (c) 24.38 (d) 14.12 (e) 19.65 (f) 29.1 3.

• Discuss any knowledge gained. Share ideas and results.

o c . che e r o t r s super

4. Octagon: 8 Dodecagon: 12 Teacher check Challenge Yes.There is an infinite number of points on the number line and therefore there is a point between any two given points. You just need to make the divisions smaller; i.e. denominator larger.

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

New Wave Maths Book G – Teachers Guide • 77 •

Unit 15–1

Student page 43

Outcomes

Indicators

N4.1a, N4.3. S4.4

The student is able to: • visualise and select figures and objects which meet geometric criteria.

Skills • measuring angles • using a protractor

Resources

Language • greater than >, less than <, equal = • order • subtract • protractor • measure • angles • superimpose • congruent • symbols

• calculator • protractor • thin paper • scissors

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

Notes

Memory Masters (N4.1a)

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 (S4.4)

w ww

• Revise congruent shapes (corresponding angles, faces and sides are all equal. ) What will be the criteria for congruent angles. (Size of the angle to be equal.) • Ask students how they can check congruency of angles. (Superimpose or measure rotation with a protractor.) • Show students how to use a protractor correctly. Demonstrate on the blackboard/whiteboard or use an overhead projector. Ensure 0º – 180º base line is over the base arm of the angle. Ensure the 0º – 90º and 0º – 180º line intersection is over the point of the angle arms’ intersection. Read the angle from the 0º – 180º anticlockwise to the upper arm to find the internal angle size. Some angles may be upside down. Show how the page may be reoriented to assist in reading the angle size. • If you also teach students to recognise right angles, acute angles, obtuse angles, these can act as benchmarks for estimating to avoid reading 150° for 30°.

What to do

. te

m . u

Warm up

ew i ev Pr

• The focus for this unit is determination of equality and inequality of pairs of numbers. • Note: This is the ideal opportunity to clear up any misconceptions the students may have with numbers.

o c . che e r o t r s super

• Measure the size of the angles on the page and write the measurement in the space provided. Assist students who still require help. • Trace over the three angles in the box on the page and superimpose each on the angles above to find congruent angles. Circle congruent angles as they are found. Mark them as A, B or C to identify which angle they are congruent with.

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 5 using one or all of the operations; e.g. 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 supply answers that students can not find themselves. For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 124 – 125. • 78 • New Wave Maths Book G – 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: • estimate sums and products by rounding to single-digit multiples of 10 and give upper and lower bounds.

Skills • estimating • adding • subtracting

Resources

Language

• calculator • Base 10 MAB • coloured pencils • place value charts (see pages 209 – 210)

• round • nearest tenth • subtract • estimate • path • vertically, horizontally • cells

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

Notes

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

ew i ev Pr

Teac he r

• The focus for this unit is rounding of decimal numbers to the nearest tenth.

• Students 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

• Distribute Base 10 MAB to groups of students. Ask them to make the following numbers: 2438; 56; 259. • Find the total of the three lots of wood. • Repeat for 29, 6, 1578. • A place value mat may be handy for this activity. • Use your calculator to check your answer. • What did you notice about adding each of the sets of numbers? (The were not all of the same place value – unlike extension of place values for the numbers. When adding you add places of like value only to other places of like value.)

w ww

What to do

m . u

• Complete Exercise 3 in the workbook. Estimate answers first. Use Base 10 MAB if required. Check answers with your calculator. • Exercise 4 is a subtraction exercise with an unlike extension of decimal places. Demonstrate using Base 10 MAB using questions (a) and (b). • Encourage students to complete the exercise using the Base 10 MAB, estimate first and check using student’s preferred method.

. te

Challenge

o c . che e r o t r s super

• Students experiment using different colours to find a path that moves horizontally and vertically through all the non-coloured squares. • Try to find a path that does not pass through any square more than once. • If unable to do this, find a path that doubles up on the least number of squares possible. • Share results.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 80 – 81. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 79 •

Unit 15–3

Student page 45

Outcomes

Indicators

N4.3, M4.2

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

Skills • reading time • reading Roman numerals • writing time

Memory Masters (N4.3)

Resources • calculator • analog clock • digital watch • coloured pencils

Teac he r

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

Main Activity (M4.2)

ew i ev Pr

Number (N4.3)

w ww

. te

m . u

• Ask students if they can tell the time using both an analog clock (one with hands) and a digital clock. • Show the analog clock and move the hands to a number of different settings, asking students to give the time. • Ask students how they know whether it is morning or afternoon! (Use of a.m. or p.m. when writing times or speaking about the time.) • Definitions: a.m. – short for ante meridiem, Latin words meaning ‘before noon’; p.m. – short for post meridiem, Latin words meaning ‘after noon’. • Do timetables and the armed forces use this sort of time? (No, they use 24-hour time, not 12-hour time.) • Explain that morning times are written and said in similar terms but p.m. or afternoon time is different. After noon in 24-hour time, the p.m. (12-hour time) is added to the 12 (noon) to give 24-hour time; e.g. 3 p.m. plus 12 noon is 12 + 3 = 15. This is written in hours as 1500 hours, 3.20 p.m. is written as 1520 hours.

What to do

• subtract • analog, digital • time • network • continuous line • 12-hour time, 24-hour time • clock face

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

Warm up

Language

o c . che e r o t r s super

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

Challenge • Encourage students to use coloured pencils to show the paths they attempted in trying to trace the network. • Keep notes of attempts. Share results.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 114 – 117. • 80 • New Wave Maths Book G – Teachers Guide

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

Unit 15—Answers

Student pages 44 – 45

Unit 15–1

Unit 15–2 1. (a) 147.9 (b) 15.9 (c) 6.7 (d) 0.9 (e) 1.3 (f) 61.4 (g) 2.6 (h) 9.5 (i) 17.2 (j) 8.9 2. (a) 3358 (b) 2879 (c) 4867 (d) 2375 (e) 1777 (f) 4364 3. (a) 16 637 4. (a) 0.861 (b) 17 298 (b) 0.045 (c) 14 897 (c) 4.96 (d) 13 716 (d) 3.97 (e) 5832 (e) 19.14 (f) 841 (f) 15.56 (g) 872 (g) 54.11 (h) 633 (h) 37.68 (i) 57.04 Challenge

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

ew i ev Pr

Teac he r

1. (a) = (b) > (c) < (d) < (e) > (f) = (g) < (h) = (i) < (j) < 2. (a) 5772 (b) 7962 (c) 6782 (d) 4442 (e) 6625 (f) 7372 3. (a) 35º (b) 90º (A) (c) 20º (B) (d) 130º (C) (e) 130º (C) (f) 130º (C) (g) 20º (B) (h) 90º (A) (i) 70º (j) 50º (k) 90º (A) (l) 20º (B) Challenge Answers will vary. Possible solutions are: 2 + 2 + 2 + 2 + 2/2 2 x 2 x 2 + 2/2 22 + 22 + 2/2

Note: Impossible to find a path that passes through each square once only.

• Draw angles found around the classroom and measure using a protractor.

(e)

8 7

. te XII

(c)

XII

11 55 a.m. (i)

XII

o c . che e r o t r s super (j)

(h)

III

(f)

• Practice reading time from various clocks through the day.

0548

8 55 a.m.

2111

(k)

1 30 p.m.

XII

III

4 42 a.m.

III

0025

III

Consolidation 15–3

12 35 p.m.

III

3 00 p.m.

(b)

• Provide students with opportunities to estimate sums and check their estimates.

XII

Consolidation 15–2

m . u

w ww

1. 3 x 4 x 8, 3 x 4 x 2 x 4, 3 x 2 x 2 x 2 x 2 x 2, (other answers are possible.) 2. (a) $7.22 (b) $4.43 (c) $2.43 (d) $2.31 (e) $2.33 (f) $3.41 (d) (g) 3. (a)

1534

1026 (l) XII

Challenge Possible solution:

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

III

2300

New Wave Maths Book G – Teachers Guide • 81 •

Unit 16–1

Student page 46

Outcomes

Indicators

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

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

Skills • estimating • measuring • rounding • recording • reasoning • logical thinking

Memory Masters (N3.3)

Resources • calculator • ruler • tape measure • scales • metre rule • trundle wheel • coloured pencils

• multiply • estimate • round • nearest • measurement • length, span, width, mass, height • distance • kilometre • path • intersection

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

Notes

ew i ev Pr

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

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

Teac he r

Language

w ww

• Students will need access to a range of measuring instruments to measure various body parts and other items. • The activity may be completed in groups. • Prior to measuring, estimate each measure. • After all measurements have been made, round the actual measure to the nearest unit. Revise the rules for rounding. • Use the measure of number of steps to cover 50 m as a guide to measuring one kilometre around the school oval. How can this be done? (Number of steps to cover 50 m multiplied by 20.) • Check the final measure taken by stepping with a trundle wheel.

Challenge

. te

m . u

© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Ask students to estimate how tall the door, ceiling, top window etc. are. Explain that they will be required to estimate measurements in today’s activity.

o c . che e r o t r s super

• Use coloured pencils to show each attempt at tracing the path around the shape shown. Keep notes to explain how this is achieved. • Share final results.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 100 – 101. • 82 • New Wave Maths Book G – Teachers Guide

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

Unit 16–2

Student page 47

Outcomes

Indicators

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

Resources

The student is able to: • suggest what data to collect to help estimate numbers or quantities. • find the mean where there is sufficient data to make summarising sensible. • put data in order and describe the highest, lowest and middle scores. • use a mean to get an estimate of a number. • interpret and report on information provided in line graphs, informally describing trends in the data.

Skills • analysing • recording

Language

• calculator

• multiply • maximum temperature • order • range • median, mean, mode • triangles • shape

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

Notes

Memory Masters (N3.3)

Teac he r

ew i ev Pr

• The focus for this unit is the multiplication of a whole number less than ten by a multiple of 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 (C&D4.2, C&D4.3, C&D4.4) Warm up

w ww

• Ask students to collect temperatures for seven consecutive days or have a week’s supply of weather reports from a local newspaper.

What to do

. te

m . u

• Revise statistical terminology – range, median, mean and mode:

Technology Opportunity!

o c . che e r o t r s super ☛ Students could use the Internet to find the temperatures over the week.

• The recording of the temperatures shows a range of temperatures recorded over the week. • Students are to arrange the temperatures in ascending order. • Students record the range of temperatures then find and record the median, mean and mode. • Exercise 4 shows the annual average daily minimum temperatures for London and Paris. Use the information to find the median and mean recording for both cities. • Explain the variance of mean and median between the two cities.

Challenge • Students are to find as many triangles as they can in the shape provided. • Show all recordings of actions in searching for and counting triangles. • Share results. For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 160 – 161. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 83 •

Unit 16–3

Student page 48

Outcomes

Indicators

N4.3, N4.1b

The student is able to: • subtract fractions. • understand that fractions are relative to particular wholes.

Skills • subtracting fractions • converting fractions

Memory Masters (N4.3)

Resources • calculator • coloured rods • 10 x 10 grid transparency

Language • multiply • fraction • decimal • percentage • subtraction • triangle, circle, square

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 (N4.1b) Warm up

• Revise subtraction of fractions and mixed numerals. Use coloured rods to make models to assist students if required. • Remind students that like fractions can be taken from like denominated fractions.

ew i ev Pr

Teac he r

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

w ww

• Work through examples 3(a), (b), (j), (k), (u) and (v) following these steps. – subtract whole numbers – subtract fractions – add whole number and fractions. For example: 65/7 – 5/7 = (6 – 0) + (5/7 – 5/7) = 6 + 0/7 = 6 or 42/3 – 1/3 = (4 – 0) + (2/3 – 1/3) regroup = 3 + (1/3) = 31/3 • Work with students needing assistance, leave examples on the board for others to follow. • Work with class as a whole to make fraction, decimal, percentage conversions. Use a 10 x 10 grid to show each form, fraction, decimal and percentage. Remind students to work fractions to hundredths or as decimals. Decimal convert to percentages as parts per hundred; e.g. 0.1 is 10 parts per hundred.

. te

m . u

o c . che e r o t r s super

Challenge • Follow the instructions given, keeping a record of each attempt and notes describing what was attempted and the results obtained.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 64 – 65. • 84 • New Wave Maths Book G – Teachers Guide

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

Unit 16—Answers

Student pages 46 – 48

Unit 16–1

Unit 16–2 1. (a) 450 (b) 60 (c) 300 (d) 630 (e) 560 (f) 240 (g) 60 (h) 3500 (i) 320 (j) 210 2. (a) $408.60 (b) $349.74 (c) $553.38 (d) $283.92 (e) $441.65 (f) $414.18 3. Teacher check 4. (a) London median: 13.5ºC London mean: 13.8ºC Paris median: 16ºC Paris mean: 15.6ºC (b) Yes. Both cities had a ‘normal’ distribution; therefore median and mean were close. Challenge 20

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

ew i ev Pr

Teac he r

1. (a) 65 (b) 69 (c) 85 (d) 64 (e) 84 (f) 17 (g) 65 (h) 81 (i) 42 (j) 50 2. (a) $374.40 (b) $540.20 (c) $562.80 (d) $442.40 (e) $360.40 (f) $262.20 3. Teacher check 4. Teacher check Challenge

. te

m . u

w ww

• Measure other body parts and complete a table similar to the one 1. (a) 70 (b) 70 (c) 60 (d) 90 in the workbook. 80 (g) 30 (h) 80 (e) 90 (f) (i) 70 (j) 80 Consolidation 16–2 2. (a) $666.12 (b) $385.65 (c) $368.79 (d) $509.08 • Use the Internet to find average temperatures for their favourite (e) $652.62 (f) $375.82 cities over the year. Complete the information as per the workbook. 3. (a) 6 (j) 4 (s) 24/7 3 1 2 1 3 1 (b) /6 = /2 (k) /4 = /2 (t) 1 /9 = 1 /3 Consolidation 16–3 (c) 4 (l) 2 (u) 41/3 • Discuss fraction, decimal and percentage equivalents. (d) 5 (m) 7 (v) 24/9 1 3 2 1 (e) 2 /5 (n) /8 (w)1 /8 = 1 /4 (f) 2 (o) 2 (x) 32/4 = 31/2 3 (g) 8 (p) 2 /8 (y) 22/6 = 21/3 (h) 2/9 (q) 12/5 (z) 11/7 3 (i) 4 (r) 2 /7 4. Challenge Possible solution:

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 85 •

Unit 17–1

Student page 49

Outcomes N4.2, N4.3, S4.4

Skills

Indicators

Resources

The student is able to: • make figures and objects which meet criteria related to sides, faces, angles, edges etc.

• calculator • ruler • quadrilaterals (2-D shapes with four sides) • mirror

• drawing • using a ruler

Memory Masters (N4.2)

• reorder • divide • symmetry • quadrilaterals • numbers • inclusive • totals

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 (S4.4) Warm up

Notes

ew i ev Pr

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

• The focus for this unit is reordering of factors (commutative property) and finding the answer to the problem.

Teac he r

Language

w ww

• From these drawings and using your own examples, draw five separate quadrilaterals in your workbook so that one has no lines of symmetry, one has one line of symmetry, one has two lines of symmetry, one has three lines of symmetry; and one has four lines of symmetry. Use a mirror or mira to assist in finding lines of symmetry. The class may work well in small groups. • Share drawings with the class.

Challenge

. te

• Keep in mind that both lines must have a total of 25; place the numbers 1 – 9 in the places provided. • Keep a record and notes of all attempts. • Share solution with the class.

m . u

© R. I . C.Publ i cat i ons orr evi ew pur posesonl y• What to do •f • Ask students to draw a number of quadrilaterals on the blackboard or whiteboard. • Remind students that quadrilaterals do not have to be regular. Think beyond the square and oblong.

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 34 – 35. • 86 • New Wave Maths Book G – Teachers Guide

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

Unit 17–2

Student page 50

Outcomes

Indicators

WM4.2, WM4.3, N4.1a, N4.1b, N4.2, N4.3

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

The student is able to: • ask organising questions to get him/her started. • contribute questions in a brainstorming situation. • sort the information in a problem in a useable form.

Resources

Language

• calculator • pencil and paper

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

• data • plan • classify • organise • questions • brainstorm • estimate • round • category • percentage • discount • purchase • money • dollars, cents • total • cost

Notes

What to do

ew i ev Pr

Teac he r

Main Activity (WM4.2, WM4.3, N4.1a, N4.1b, N4.2, N4.3)

• This activity is designed for students working independently or collaboratively in groups. As students will need to discuss their opinions and ideas, allow enough time so they don’t feel rushed and for ideas to evolve. Investigative tasks such as these are a good opportunity for students to ‘take risks’ with maths. • When completing investigative tasks, some students may be more successful in mixed-ability groups rather than in 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 can you find out what kind of sports the students in the class enjoy? – What equipment does the school have already? – How many of each piece of equipment is needed? – How do you calculate 10% off a price? – How do you know if you have spent the money well? • 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 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.

w ww

. te

m . u

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 87 •

Unit 17–3

Student page 51

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

Skills • reading a calendar • counting • calculating

Indicators

Resources

The student is able to: • use straightforward timetables and program with both 12- and 24hour times.

• calculator • school holiday dates

Language • convert • dollars, cents • divide • calendar • events • calculate

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) Warm up

• Hold a general discussion about calendars, why we have them, months of the year, days in each month, weeks in a year and special days (Anzac Day, Christmas, birthday etc.).

• Using the calendar in the workbook, circle the days listed. Provide students with the dates of the school holidays. • Ask students to write their birth date in the space provided then calculate the time in weeks and days between their birthday and the event. Use the calendar to count on in weeks and then single days to make calculation easier. For longer periods, count the months then weeks and days if this is more practical.

w ww

Challenge

• The prefix ‘oct’ means eight and yet October is the 10th month of the year. Use the Internet to find out why.

. te

m . u

What to do

ew i ev Pr

Teac he r

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

o c . che e r o t r s super

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

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

Unit 17—Answers

Student pages 49 – 51

Unit 17–1

Unit 17–2 1. Answers will vary but not exceed a total of $450. 2. The student should be able to justify his/her choices of sporting equipment so they reflect the interests of all class members.

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

ew i ev Pr

Teac he r

1. (a) 312 = 3 x 104 (f) 1510 = 302 x 5 (b) 1220 = 4 x 305 (g) 1218 = 3 x 406 (c) 1836 = 3 x 104 (h) 1624 = 203 x 8 (d) 3521 = 7 x 503 (i) 1216 = 304 x 4 (e) 828 = 207 x 4 (j) 1812 = 3 x 604 2. (a) 43.48 (b) 18.89 (c) 28.86 (d) 23.63 (e) 14.49 (f) 22.82 3.

Challenge One possible solution:

. te

• Complete other activities where students search for symmetry in logos, nature, the classroom, structures etc.

Consolidation 17–2 • Students use a catalogue of their choice (within reason) to complete a similar task with a specified budget.

Consolidation 17–3

m . u

w ww

1. (a) $3.02 (b) $80.04 (c) $0.06 (d) $0.19 (e) $0.05 (f) 16c (g) 251c (h) 7c (i) 83c (j) 7421c $1.09 (c) $1.07 (d) $1.05 2. (a) $1.05 (b) (e) $1.08 (f) $1.08 3. (a) Teacher check (b) 25/12 (c) 26/1 (d) 25/4 (e) Teacher check (f) Teacher check (g) Teacher check 4. Teacher check Challenge Original Roman calendar had only 10 months until January and February were added around 700BC. October then became the 10th month.

• Provide daily opportunities for students to use and read a calendar.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 89 •

Unit 18–1

Student page 52

Outcomes

Indicators

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 • recording

Memory Masters (N4.3)

Resources • calculator

• subtract • patterns • consecutive • palindrome • addition • multiplication • combination • symbols

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)

Notes

ew i ev Pr

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

• The ‘Today’s number is …’ activity asks students to list all they know about a particular number; Today’s number is 12 … 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

• Mathematics is really about pattern.There are many interesting patterns and combinations to be found in numbers. Using dates provides us with many such situations. For example; consecutive numbers can be found in dates – 1/2/34, palindromic numbers can also occur in dates – 1/10/11 (read backwards the digits are in exactly the same order), addition sums may be found as number sentences 21/10/11 (21 = 10 + 11), multiplication number sentences can also be found in dates 7/6/42 (7 x 6 = 42).

w ww

• Find as many similar dates as you can that are consecutive numbers, palindromic numbers, show addition or multiplication number sentences. Write these in your workbook. • Palindromic numbers may occur more frequently than many realise. For days of the month 01 to 28 there are up to ten palindromic dates each year; e.g. 13/1/31 where the year is the reverse of the day. Find the ten for 2002 written as 02. • For Exercise 4, students are to use the four digits to make as many 4-digit numbers as possible. There are 24 combinations. Students record the combinations they make.

Challenge

. te

m . u

What to do

o c . che e r o t r s super

• 90 • New Wave Maths Book G – Teachers Guide

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

Unit 18–2

Student page 53

Outcomes

Indicators

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

The student is able to: • realise that different classifications may tell different things and suggest an alternative classification to answer new questions. • represent data in diagrams and tables which may include arrow diagrams, Venn diagrams and twoway tables.

Skills • reasoning • recording

Resources

Language

• calculator • 2-D shapes • mirror or mira

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

• lines of symmetry • shapes • chance • segment • subtract • space • sides • equilateral triangle, square, pentagon, hexagon, octagon, rectangle, rhombus, isosceles triangle

Notes

Memory Masters (N3.3)

Teac he r

ew i ev Pr

• The focus for this unit is addition and subtraction of a whole number less than 10 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 (C&D4.2, C&D4.3) Warm up

© R. I . C.Publ i cat i ons orr evi ew pur posesonl y• What to• dof • Distribute 2-D shapes to groups of students. Each group is also to be provided with a mirror or mira. • Allow time for exploration of miras. Encourage students to discuss what they see.

. te

Challenge

m . u

w ww

• Use the mirror or mira to assist in finding lines of symmetry for each of the shapes. • Use the table in the workbook to record lines of symmetry and the number of sides of each shape. • If it helps, draw the shapes on the page in the workbook and draw in the lines of symmetry. • Exercise 4 looks at chance. Using a class of students the chance of certain events happening is proposed. Students need to understand that chance is guided by the number of students involved. In this case there are 28 students in total: 16 boys and girls—16/28 boys against 12 /28 girls gives boys a greater chance of notice. • However, within each group a boy has less chance of notice than a girl – 1/16 against 1/12. • Ask students to find the answers to the questions individually, or treat as a whole-class activity.

o c . che e r o t r s super

• Students are to place one number in each of the segments shown in the Venn diagram (named after John Venn). When all numbers in each circle are totalled, the total for each circle must be different. • Show all working and keep notes explaining your thinking.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 34 – 35.

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

New Wave Maths Book G – Teachers Guide • 91 •

Unit 18–3

Student page 54

Outcomes

Indicators

N3.3, 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

Resources

• subtract • digit • numbers • inclusive • circle • total

• calculator • bar codes

• brainstorming • recording • analysing

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 (N4.1a) Warm up

ew i ev Pr

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

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

Teac he r

Language

w ww

• If a company chose to use the letters of the alphabet—a single letter only—and two digits, how many employees would the company be able to identify? Would they use the digits 00 as an identity? (Not likely.) • What would you do if you owned the company and grew beyond the identity code? (Add another letter or digit.) Which would allow the greater room for expansion? (Added digit.) • Answer the questions in the workbook. • Codes are used to identify many things including all resources at the Australian National Library, which are given an ISBN. • Bar codes are used to identify items for sale. Collect a variety of bar codes – make sure you know what they came from. Use this information to suggest what the bar codes in your workbook represent.

Challenge

. te

m . u

© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Refresh the use of identity numbers – car registration, bank accounts etc. • A simple identity tag used is to give people a number and/or a letter from the alphabet.

o c . che e r o t r s super

• Students are to place the numbers 1 – 7 in each segment of the drawing shown. • The total of the digits in each circle must be the same. • Arrange the digits, keeping records of your workings and findings.

• 92 • New Wave Maths Book G – Teachers Guide

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

Unit 18—Answers

Student pages 52 – 54

Unit 18–1 1. Teacher check $3.37 (c) $2.47 (d) $4.23 2. (a) $3.16 (b) (e) $2.19 (f) $2.15 3. Teacher check. Some possible answers:

(3 x 3) – (3/3 + 3/3 )

4. (a) boy there are more (b) girls fewer to be chosen from

Challenge

ew i ev Pr

Teac he r

1. (a) 86 (b) 34 (c) 49 (d) 32 (e) 93 (f) 57 (g) 81 (h) 80 (i) 44 (j) 55 2. (a) $1.99 (b) $4.96 (c) $4.98 (d) $3.99 (e) $2.94 (f) $5.99 3.

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

4. 9762, 9726, 9672, 9627, 9267, 9276, 7962, 7926, 7692, 7629, 7296, 7269, 6972, 6927, 6792, 6729, 6297, 6279, 2976, 2967, 2796, 2769, 2697, 2679 Challenge Answers will vary. Possible answers are: 3 + 3 + 3/3 (3 x 3) – (3 3+ 3)

Unit 18–2

. te

• Discuss and share results.

Consolidation 18–2 • Complete other activities where students search for symmetry in logos, nature, the classroom, structures etc.

Consolidation 18–3

m . u

w ww

1. (a) 20 (b) 80 (c) 75 (d) 35 (e) 32 (f) 72 (g) 76 (h) 46 (i) 36 (j) 86 2. (a) $3.67 (b) $2.58 (c) $3.75 (d) $5.47 (e) $1.27 (f) $2.67 3. (a) 2600 (b) 26 000 (c) 00 – 99 = 100 number combinations x 26 letters 000 – 999 = 1000 number combinations x 26 4. Teacher check. Code represents information about the product’s name, size, price etc. and is part of a system for organising and ordering. Challenge One possible solution:

• Search for any patterns in ISBNs.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 93 •

Unit 19–1

Student page 55

Outcomes

Indicators

N4.3, S4.4

The student is able to: • provide a description of a diagram or shape so that a peer could reproduce or recognise it.

Skills • using a ruler • using a protractor • following directions • working independently

Memory Masters (N4.3)

Resources • calculator • protractor • pencil • ruler

• multiply • fractions • mixed numerals • part of the whole • diagram • set • subset

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 (S4.4) Warm up

w ww

• Students may be encouraged to complete this activity by themselves as an exercise in following directions, or work through the exercise, directed by the teacher, as a whole-class activity. Teacher direction may be required for students who are having difficulties. • Note: Students should be directed to complete the activity using a pencil for ease of correction. • Parallel lines (have to be the same length), perpendicular lines (are joined) as well as horizontal lines will all need to be discussed to ensure understanding of terms. • Remind students that when constructing drawings pencils need to be sharp and measurements accurate. Always make initial drawings light. Rubbing out should not be necessary.

. te

m . u

• Review use of a protractor.

Challenge

Notes

ew i ev Pr

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

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

What to do

Language

o c . che e r o t r s super

• To tessellate means to tile. Do all types of triangles tessellate? (Yes, all tessellate.) • Note: There are seven different types of triangles. OR • Create your own simple straight line design and then write a set of instructions so someone else could draw your design.

• 94 • New Wave Maths Book G – Teachers Guide

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

Unit 19–2

Student page 56

Outcomes

Indicators

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 • displaying fractions

Resources

Language

• calculator • coloured rods • fraction cake

• divide • multiply • fraction • part, whole • share

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

Notes

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

ew i ev Pr

Teac he r

• The focus for this unit is the division of a decimal 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

• Distribute coloured rods to groups of students. Ask students to show 3/4, 7/8, 1/2, 2/3 with the rods. Check displays, discuss why arrangements were made as they are shown. If the

net is shown as

/4, direct students to show their models in the same way.

For 21/4, represents the model to be used. 1 3 • Ask students to display 2 /4, 1 /4, 31/2 with the rods. Discuss displays.

m . u

What to do

w ww

• Ask students to shade the fractions and mixed numerals shown in their workbook. • Check students’ work as they proceed. • Use the diagrams in Exercise 4 to show the distribution of the items among the numbers given. Whole-class discussion of the activity may assist.

Challenge

. te

o c . che e r o t r s super

• Students are to provide a detailed analysis of how they reached their answer. • Share findings with the class.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 62 – 63. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 95 •

Unit 19–3

Student page 57

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 at hand.

N4.1a, N4.3, M4.4a

Skills • multiplying • problem-solving

Resources

Language • ascending order • multiply • dimensions • length • perimeter • square

• calculator • string

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 (M4.4a) Warm up

• Discuss with the class how the perimeter of a rectangle may be found given you know the length of the sides. Encourage a range of answers before coming back to focusing on the addition of the two different length sides and doubling the total. • Ask students how they would find the cost of fencing an area of land if they knew its perimeter and the cost of the fence per metre.

ew i ev Pr

Teac he r

• The focus for this unit is ordering of decimal numbers.

w ww

Challenge

• Students experiment with various fencing solutions. Some students may prefer to use manipulatives, such as string, to help them solve the problem.

. te

m . u

• Direct students to the activities in their workbook. Suggest that they use calculators to find the answers to the problem.

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 136 – 137. • 96 • New Wave Maths Book G – Teachers Guide

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

Unit 19—Answers

Student pages 55 – 57

Unit 19–1 1. (a) 2.4 (b) 1.6 (c) 2.7 (d) 4.8 (e) 3.5 (f) 3.6 (g) 2.1 (h) 6.3 (i) 5.6 (j) 5.4 2. (a) $6392.82 (b) $1766.73 (c) $3073.32 (d) $3790.88 (e) $6414.74 (f) $3213.76 3. Teacher check

Unit 19–2 1. (a) 0.4 (b) 0.7 (c) 0.9 (d) 0.9 (e) 0.9 (f) 0.4 (g) 0.6 (h) 0.5 (i) 0.4 (j) 0.6 2. (a) 447.12 (b) 282.80 (c) 518.80 (d) 158.60 (e) 331.66 (f) 437.78 3. (a) 2/3 (e) 21/2 5 (b) /8

3 (f) 2 /4

r o e t s Bo r e p ok u S 7 (c) /10

(g) 32/3

5 (d) /6

Challenge There are seven different types of triangles which all tessellate.

(b)

ew i ev Pr

Teac he r

4. (a)

Challenge Yes. 1 000 000 ÷ 60 ÷ 24 ÷ 7 ÷ 52 = 1.9 years

. te

• Ask students to write a set of instructions to draw a diagram. Give it to a partner to draw the diagram from the instructions.

Consolidation 19–2 • Use real-life examples of sharing items among groups.

Consolidation 19–3

m . u

w ww

1. 6.027, 6.049, 6.094, 6.409, 6.49, 6.493, 6.613, 6.827, 6.904, 6.94 2909.4 (c) 6595.2 (d) 4489.1 2. (a) 5198.8 (b) (e) 3815.5 (f) 5023.2 3. (a) 370 m (b) $9 250 4. (a) $14 280 Challenge The chicken run will have a perimeter of 25 m. A rectangle 6 m by 6.5 m gives an area of 39 m2.

• Measure an area within the school grounds and complete the activity again.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 97 •

Unit 20–1

Student page 58

Outcomes

Indicators

N4.1a, N4.3, N4.1a, N4.4

The student is able to: • understand the multiplicative nature of the relationship between places for whole numbers, i.e. as the numbers move from right to left. • follow the rule based on multiplication, division or simple fractions to generate a sequence.

Skills • making models • multiplying • using a calculator

Resources • calculator • 1-cm cubes

Language • round • nearest hundredth • divide • diagrams • powers • expanded form • total value • symbols

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

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, N4.4) Warm Up

• Mathematicians have found simple means of writing large numbers. One means is to express a number as a power or index.The power or index indicates the number of times a number is multiplied by itself. For example 23 is the same a 2 x 2 x 2. • Ask students to show how they would write 3 x 3 x 3 x 3 x 3. (35) How would they write 4? (41, four only occurs once therefore is raised to the power of one; 40 = 1)

ew i ev Pr

Teac he r

• The focus for this unit is rounding of decimal numbers to the nearest hundredth.

Challenge

. te

m . u

w ww

• Distribute 1-cm cubes to small groups. • Show 2, 2 x 3, 2 x 2 x 2; 3, 3 x 3, 3 x 3 x 3; 4, 4 x 4, 4 x 4 x 4; and 5, 5 x 5, 5 x 5 x 5 using the 1-cm cubes. • Can you see where the terms square and cube numbers come from? • Complete activities in the workbook.

o c . che e r o t r s super

• 98 • New Wave Maths Book G – 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: • suggest what data to collect to help estimate numbers or quantities. • represent data in diagrams and tables which may include arrow diagrams, Venn diagrams and twoway tables. • describe information from diagrams which may include arrow diagrams, tree diagrams, Venn diagrams or Carroll diagrams.

Skills • collecting data • recording data • analysing data • logical reasoning

Resources

Language

• calculator • students in class • toothpicks or equivalent

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

• greater than >, less than <, equal • number sentences • divide • round • nearest cent • Venn diagram • squares

Notes

Memory Masters (N4.2)

Teac he r Number (N4.3)

ew i ev Pr

• The focus for this unit is inequality of number statements using brackets to determine order of operations.

• Students 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 students what a Venn diagram is. Draw a two overlapping circles Venn diagram on the blackboard/whiteboard. 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 overlay segment.

m . u

What to do

w ww

• If we record those students who have brothers only in one segment, sisters only in another segment, those who have both brothers and sisters would be recorded in the overlapping segment. Those who are only children would be recorded outside the two circles. • You are to complete your own Venn diagram in your workbook by finding information from the class on the musical instruments they play. Find the information and record it on the Venn diagram. • Use the information collected to answer questions about the Venn diagram.

Challenge

. te

o c . che e r o t r s super

• Provide students with toothpicks or the equivalent. • Students use the toothpicks to make three squares. • Keep recordings of all attempts and make notes of the process followed. • Share results.

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

New Wave Maths Book G – Teachers Guide • 99 •

Unit 20–3

Student page 60

Outcomes

Indicators

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

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

Skills • converting decimals • finding percentages

Resources • calculator • toothpicks or equivalent

Language • greater than >, less than <, equal • order relationship • divide • round • nearest cent • decimals • percentage • two places • squares

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

Memory Masters (N4.1a)

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 (N4.3) Warm up

ew i ev Pr

• The focus for this unit is the comparison of decimals, fractions and decimals and fractions to determine equality or inequality of statements.

w ww

What to do

• Work with students on the conversion of several examples from their workbook before allowing them to continue.

If required

. te

m . u

• Explain the use of percentage (out of 100) as a way of describing ratio of part to whole. Percentages may be used to show comparisons of different things; e.g. 1/2 and 0.4 as 50% and 40% respectively. • Decimals may be converted to percentages. If decimals are written to the hundredth place value conversion to percentage is easy as conversion is a simple as writing the numbers without the decimal place; e.g. 0.25 is 25%; 0.48 is 48%; 0.8 is 0.80 or 80%. This may be done by multiplying the decimal, on the calculator, by 100. • Where whole numbers are written as part of the decimal number the same still applies; e.g. 2.81 is 281%, 5.45 is 545%. (Note: In these cases you have more than one whole.)

o c . che e r o t r s super

• Finding percentages of totals is best performed on the calculator by entering the number x percentage value and percentage key; e.g. 1000 x 25 % = answer. (Note:The percentage key may operate differently on different calculators. Check this before beginning the lesson.) • If working mentally, simplify percentage and find the answer; e.g. 50% is half. Find a half of the value. Complete Exercise 5.

Challenge • Distribute toothpicks to students. • Students make the shape as shown in the workbook. • Show all attempts at making two squares by removing eight toothpicks.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 68 – 69. • 100 • New Wave Maths Book G – Teachers Guide

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

Unit 20—Answers

Student pages 58 – 60

Unit 20–1

1. (a) > (b) < (c) < (d) > (e) = (f) < (g) < (h) < (i) < (j) = 2. (a) $3.34 (b) $3.76 (c) $3.88 (d) $5.53 (e) $6.58 (f) $3.95 3. Teacher check 4. Teacher check Challenge

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

ew i ev Pr

Teac he r

1. (a) 8.27 (b) 2.94 (c) 1.82 (d) 9.21 (e) 3.57 (f) 6.28 (g) 1.00 (h) 2.08 (i) 5.01 (j) 6.15 $1.23 (c) $1.48 (d) $1.28 2. (a) $1.53 (b) (e) $1.32 (f) $1.48 3. Teacher check 4. (a) 32 (b) 625 (c) 6561 (d) 4096 (e) 3125 (f) 81 (g) 512 (h) 1 (i) 216 (j) 49 Challenge Answers will vary. Possible solutions are: (4 x 4) + (4 + 4) + (4 – 4/4) = 27 (4 + 4/4) x (4 + 4 + 4/4) = 45 4 x (4/4 + 4/4 x 4) + 4/4 = 33

Unit 20–2

. te

• Continue making models 6, 7, 8 etc. to study the patterns.

Consolidation 20–2 • Complete Venn diagrams to show information decided by the class.

Consolidation 20–3

m . u

w ww

1. (a) > (b) = (c) < (d) > (e) < (f) > (g) < (h) > (i) > (j) = 2. (a) $0.12 (b) $0.09 (c) $0.09 (d) $0.09 (e) $0.08 (f) $0.09 3. (a) 70% (l) 15% 4. (a) 13% (l) 63% (b) 20% (m) 3% (b) 59% (m) 5% (c) 50% (n) 9% (c) 72% (n) 9% (d) 90% (o) 1% (d) 32% (o) 2% (e) 10% (p) 637% (e) 85% (p) 29% (f) 80% (q) 925% (f) 19% (q) 50% (g) 25% (r) 847% (g) 56% (r) 6% (h) 75% (s) 329% (h) 81% (s) 2% (i) 45% (t) 207% (i) 94% (t) 22% (j) 84% (u) 533% (j) 13% (k) 69% (v) 487% (k) 88% 5. (a) $1000 Challenge (b) $800 (c) $150 (d) $200

• Find set percentages of amounts to give more practice.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 101 •

Unit 21–1

Student page 61

Outcomes

Indicators

N3.3, N4.3, M4.4b

The student is able to: • use a grid to enlarge or reduce a figure in a specified way.

Skills • using a ruler • using a protractor • following directions • working independently

Memory Masters (N3.3)

Resources

• subtract • reduce • scale • space • shape

• calculator • pencil

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 (M4.4b) Warm up

w ww

• The task is to draw the shape shown one-third the size it is now. That is, to use a scale so that one represents three of the original units. • First task is to find the overall dimensions of the current shape (18 units high and 18 units wide). The reduced shape will be how many units long and how many units wide? (6 x 6) • Draw a box around a 6 x 6 grid in the top left of the grid on the page. • Select a relatively simple starting point. Mark this line in. • Check students’ work. • Remind students that three squares of the original fit one of the reduced drawing. Suggest a marked 3 x 3 grid on the original may assist in determining what will fit in each square of the reduced drawing. • Set the class to work, checking and assisting as required.

. te

m . u

• Ask students about scale models. What does the scale mean?

Challenge

Notes

ew i ev Pr

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

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

What to do

Language

o c . che e r o t r s super

• What are the fewest number of colours needed to colour the drawing so that no two adjacent areas share a common colour?

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 144 – 145. • 102 • New Wave Maths Book G – Teachers Guide

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

Unit 21–2

Student page 62

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 • writing fractions • converting fractions

Resources

Language

• calculator • pencil

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

• subtract • fraction • decimal • percentage • square, circle, triangle • overlapping • enclosed • denominator • numerator

Notes

Memory Masters (N3.3)

Teac he r

ew i ev Pr

• 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 (N4.1b) Warm up

• Fractions, decimals and percentages may all be used to represent the same value. • Each may be used in different instances to show the representation in its strongest form. Because of this it is necessary to know how to convert from one to the other forms so that you are able to make comparisons in the simplest, or best, form for understanding. • Present the following test results and ask students to brainstorm how they would compare 16 18 /20 /25 78% 27/40. them.

What to do

. te

Challenge

m . u

w ww

• In the workbook there are a series of diagrams that have been partially shaded. Spaces have been provided to write the fraction, decimal and percentage equivalents. • Fractions may be readily found by writing the shaded part as part of the whole. • Fractions may be converted to decimals by dividing the numerator by the denominator of each fraction (use a calculator). • Percentages may be found by multiplying the decimal by 100 or converting each decimal to its hundredths equivalent; e.g. 0.5 is 0.50 and writing the hundredths equivalent as a percentage is 50%; another example is 0.28 which is 28%. • Complete the task.

o c . che e r o t r s super

• Follow the instructions to complete the drawing. • Show all attempts and keep notes to explain what you did.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 68 – 69. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 103 •

Unit 21–3

Student page 63

Outcomes

Indicators

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

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 at hand.

Skills • multiplying • following instructions

Memory Masters (N4.3)

Resources • calculator • string

• subtract • diagram • dimensions • length • perimeter • approximately • numbers • inclusive

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

ew i ev Pr

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

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

Teac he r

Language

• Refresh students’ memories on the calculation of the perimeter of an oblong rectangle. Lead a general discussion on different methods but focus on adding adjacent sides and doubling. • Discuss the calculation of costs incurred to fence or brick a perimeter. Remind students to read all instructions carefully to ensure all items are included.

w ww

• Check the two examples in the workbook. Ensure students understand the requirements of both activities. Focus on cost of fencing in the first activity, in particular the number of strands of barbed wire. • In the second activity note the spaces left in the wall for access to the sunken garden. • Stress the significance of finding the total number of bricks required before calculating the cost of bricks and the cost of laying the bricks. • Students should use a calculator for calculations and the space provided to explain how the answer was obtained.

Challenge

. te

m . u

What to do

o c . che e r o t r s super

• Students experiment with various fencing solutions. Some students may prefer to use manipulatives, such as string, to help them solve the problem.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 134 – 137. • 104 • New Wave Maths Book G – Teachers Guide

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

Unit 21—Answers

Student pages 61 – 63

Unit 21–1 1. (a) 87 (b) 53 (c) 68 (d) 89 (e) 55 (f) 77 (g) 77 (h) 45 (i) 93 (j) 25 $3.26 (c) $3.43 (d) $6.24 2. (a) $1.43 (b) (e) $5.18 (f) $1.63 3. Teacher check Challenge 3

Unit 21–2 1. (a) 79 (b) 38 (c) 58 (d) 91 (e) 63 (f) 49 (g) 88 (h) 35 (i) 51 (j) 29 2. (a) $62.38 (b) $46.67 (c) $23.67 (d) $25.46 (e) $54.68 (f) $56.85 3. 1/ 3/ 50% 30% 2 0.5 10 0.3 1/

0.25 25%

0.75 75%

0.2

20%

0.8

80%

Teac he r

40%

0.8

80%

0.6

60%

100%

ew i ev Pr

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

0.4

0.6

0.67 67%

60%

Challenge

. te

• Students draw their own design on 1-cm grid paper (see page 199) and give to a friend to reduce by a set scale.

Consolidation 21–2 • Refer to students’ own results to convert from a score to a percentage.

Consolidation 21–3

m . u

w ww

1. (a) 5600 (b) 5400 (c) 5400 (d) 2400 (e) 4200 (f) 4500 (g) 8100 (h) 3200 (i) 1400 (j) 4800 2. (a) $22.07 (b) $26.49 (c) $16.67 (d) $31.78 (e) $17.68 (f) $32.65 3. (a) 230 m (b) $4140 4. Teacher check working 31 m 1705 bricks $622.33 cost of bricks $358.05 cost of laying $980.38 total cost Challenge 36 m2

• Students work out the cost to fence a particular area within the school grounds.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 105 •

Unit 22–1

Student page 64

Outcomes

Indicators

N4.3, N4.3, N4.4

The student is able to: • identify and use patterns linking pairs of letters in a table.

Skills

Resources

• multiply • code • message • number • inclusive

• calculator

• developing codes • using codes

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.

Main Activity (N4.4)

Notes

ew i ev Pr

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

Warm Up

Language

• Discuss codes with the class, perhaps use the Morse code as an example of sending messages to distant locations. A copy of the Morse code for students may assist. Braille is another form of communication, using raised dots in various arrangements to make numbers and letters of the alphabet.

CODES Letter Morse Braille

Letter Morse Braille

w ww

a .- h .... o --- v ...- b -... i .. p .--. w .-- c -.-. j .--- q --.- x -..- d -.. k -.- r .-. y -.-- e . l .-.. s ... z --.. f ..-. m -- t - g --. n -. u ..-

. te

What to do

Braille period comma

? !

m . u

Letter Morse Braille

o c . che e r o t r s super

• A simple code can be developed using the alphabet by substitution of numbers for letters in the alphabet; e.g. A–1, B–2 and so on. • The code in the workbook is a little more complex. Using the code and the letters it stands for, a message may be written. Write a message using the code and pass to a friend to decode. • Develop your own code and write a message. See if your friend can break the code without assistance.

Challenge • Students are to show all attempts when solving the problem. Notes of their procedures also need to be made. • Share findings with the class. • 106 • New Wave Maths Book G – Teachers Guide

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

Unit 22–2

Student page 65

Outcomes

Indicators

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

Skills • calculating • recording

Resources

Language

• calculator • coloured pencils

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

• number patterns • multiply • data • range • median • mean • mode • paths

Notes

Memory Masters (N3.4) Number (N4.3)

ew i ev Pr

Teac he r

• The focus for this unit is to complete basic number patterns involving 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) Warm up

• Revise statistical terms – range, median, mean and mode. • Revise statistical terminology – range, median, mean and mode:

w ww

e.g. The data are bimodal, having 8 and 9 occurring the most frequently.

What to do

. te

m . u

o c . che e r o t r s super

• The table drawn in the workbook shows the average monthly rainfall for both Alice Springs and Darwin. Use the information to find the range, median, mean and mode recordings for each centre. • Compare the mean and median scores of Alice Springs and again for Darwin. Were both recordings for both places close to each other? • Explain why these recordings are different for each centre. Use students’ findings as the basis for open class discussion covering range of rainfall, total amounts of rainfall and variance between months of high and low rainfall.

Challenge • Students should be encouraged to use coloured pencils to show the paths they choose to pass through days of the month following the given instructions. • Keep notes to explain your thinking and the process you used.

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

New Wave Maths Book G – Teachers Guide • 107 •

Unit 22–3

Student page 66

Outcomes

Indicators

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

The student is able to: • plan sequences of calculations using a calculator.

Skills • adding • subtracting • multiplying • dividing • problem-solving

Resources • calculator • place value chart (pages 209 –210)

Language • convert • kilograms, grams • number chains • added, subtracted, divide, multiply

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

Memory Masters (M4.1, 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

• Ask the students to take out their calculators and a place value chart or look at a large class place value chart. • Ask students to enter 24 into their calculator and then suggest what will happen if x and 10 are then entered into the calculator. Repeat the process with 16 x 100, 47 x 10 and 83 x 100. • Using the place value chart, show where each number will be on the chart; e.g. 24, two tens and four ones. The 24 x 10 becomes 2 hundreds, 4 tens and 0 ones. • Ask students to enter 3742 into their calculator and then add 1000. What happens? (The number in the thousands place changes.) • The game ‘Wipeout’ could be played using the calculator. Ask students to press in a number—e.g. 4562—ask students which number will be wiped out if we subtract 60? (The 6 of course!)

ew i ev Pr

Teac he r

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

w ww

What to do

. te

• Open workbooks and ask students to attempt the activities shown using their calculator to assist if necessary. Fill in the blank spaces.

Challenge

m . u

o c . che e r o t r s super

• Using the information in the third activity above students are to provide a sound mathematical shortcut to multiplying and dividing by 1000. • Note: Avoid rules that suggest the decimal point moves—this leads to misconceptions later. • Record all thinking processes and workings for sharing with the class.

• 108 • New Wave Maths Book G – Teachers Guide

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

Unit 22—Answers

Student pages 61 – 66

Unit 22–1

1. (a) 2.5 (b) 7.4 (c) 10.5 (d) 4.5 (e) 5.2 (f) 6.0 (g) 7.0 (h) 3.9 (i) 4.5 (j) 4.7 2. (a) 1047.2 (b) 2546.7 (c) 4568.4 (d) 2502.5 (e) 2580.3 (f) 3977.5 Alice Springs Darwin 3. (a) (i) 17.8 mm 383.5 mm (ii) 8.9 mm 53.35 mm (iii) 15.2 mm 0.0 mm (iv) 9.1 mm 114.1 mm (b) no (c) The range for Darwin was very large and the median did not take into account the very high rainfall of Dec – Mar, which the median did. 3. Teacher check Challenge Three possible solutions:

r o e t s Bo r e p ok u S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

ew i ev Pr

Teac he r

1. 2 x 7 x 12, 14 x 3 x 4, 2 x 7 x 3 x 4, 14 x 2 x 6, 2 x 7 x 3 x 2 x 2, 2 x 7 x 2 x 6, 14 x 3 x 2 x 2 and so on. 603.36 (c) 549.26 (d) 226.54 2. (a) 295.63 (b) (e) 470.40 (f) 245.76 3. Teacher check 4. Teacher check Challenge One possible solution: inner ring = 22 outer ring = 44

Unit 22–2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

(b)

(c)

. te

• Ask students to devise their own number codes.

Consolidation 22–2 • Brainstorm and test various data that could be used to find range, mean and median.

Consolidation 22–3

m . u

w ww

1. (a) 0.04 kg (b) 0.008 kg (c) 0.792 kg (d) 3.654 kg (e) 0.024 kg (f) 9270 g (g) 5 g (h) 902 g (i) 81 g (j) 4 6003 g 649.92 (c) 612.54 (d) 239.12 2. (a) 498.12 (b) (e) 657.54 (f) 408.24 3. (a)

• Students could develop their own calculation chains following principles similar to those in the workbook.

o c . che e r o t r s super

Challenge Multiply: Numbers move 3 spaces to the right. Divide: Numbers move 3 spaces to the left.

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

New Wave Maths Book G – Teachers Guide • 109 •

Unit 23–1

Student page 67

Outcomes

Indicators

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

The student is able to: • identify the transformation(s) used to produce a spatial sequence and continue the sequence.

Skills • drawing • analysing

Resources • calculator • blackboard or whiteboard

Language • convert • dollars, cents • divide • round • nearest cent • topology • equivalent • distorted

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

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 (S4.3) Warm up

• Topology is the study of topography or the relative relationship of points and features; in this case the relationship between points. Such points may be drawn on a page with curved connecting lines or straight connecting lines.Two drawings are topologically equivalent if they have the same number of points in the same order and the same number of connections. • Draw a couple of simple topology diagrams on the blackboard/whiteboard. For example;

ew i ev Pr

Teac he r

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

w ww

. te

What to do

m . u

o c . che e r o t r s super

• Look at the examples in the workbook. • Redraw the shapes shown in a topologically distorted manner. Generally, if curved lines have been used use straight lines and curved lines if straight lines have been used.

Challenge • Ask students to trace over designs without lifting their pencil. For example;

• 110 • New Wave Maths Book G – Teachers Guide

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

Unit 23–2

Student page 68

Outcomes

Indicators

N4.3, N4.1a

The student is able to: • say decimals. • read scales, including instances where each calibration may not be labelled.

Skills • plotting number lines • ordering

Resources

Language • divide • round • nearest cent • number line • order relationship • smallest to largest (ascending order)

• calculator

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

Notes

Memory Masters (N4.3)

Teac he r

ew i ev Pr

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

• Draw a number line on the blackboard/whiteboard or display on the overhead. Mark the numbers 0 to 1.0 and 0.1 intervals on the line. Ask students to place the following numbers on the number line – 0.45, 0.65, 0.13, 0.72 and 0.98.

What to do

Challenge

. te

m . u

w ww

• Use the number lines in the workbook to place the numbers given at their correct points along the respective number lines. • Once the numbers have been placed in order, ask students to write them in ascending order in the box under the end of the number line. • Check regularly to ensure understanding.

o c . che e r o t r s super

• Using the information given, students should record all their thinking and solutions for sharing later.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 58 – 59. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 111 •

Unit 23–3

Student page 69

Outcomes

Indicators

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

The student is able to: • investigate the area of triangles drawn on a square grid.

Skills

Resources • calculator • ruler

• divide • round • nearest cent • area • triangular shape

• finding area • describing processes

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

ew i ev Pr

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

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

Teac he r

Language

w ww

What to do

. te

m . u

• Ask students to explain how they would find the area of a rectangle if they were given the measurements of adjacent sides. • Previously, students found the area of triangles drawn inside rectangles. Revise the method used to determine the area of triangles. Demonstrate by using a sheet of card (rectangular) with a triangle drawn inside. Cut the triangle out and manipulate the cut out pieces to show that the cut-offs are the same size as the triangle itself. This shows that the triangle is half the area of a rectangle with the same base and perpendicular height. It also shows that the perpendicular height of the triangle is the same length as the side of the rectangle; e.g.

o c . che e r o t r s super

• Ask students how they might find the area of the triangles shown in their workbook. (Grid drawn and squares counted, drawing rectangles around, or finding the area using the perpendicular height and base and halving.) Share students’ workings.

Challenge

• Students explore numbers to find the answer required. • Keep a record of all workings to show later.

• 112 • New Wave Maths Book G – Teachers Guide

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

Unit 23—Answers

Student pages 67 – 69

Unit 23–1

1. 2 x 7 x 3 = 42 6 x 7 = 42 42 ÷ 14 = 3 42 ÷ 3 = 14 6 x 7 = 42 42 ÷ 6 = 7 (other answers are possible.) 2. (a) $0.39 (b) $0.24 (c) $0.47 (d) $0.28 (e) $0.25 (f) $0.20 3.

r o e t s Bo r e p ok u S Challenge 6

ew i ev Pr

Teac he r

1. (a) $0.02 (b) $0.46 (c) $8.29 (d) $0.09 (e) $0.77 (f) 8c (g) 74c (h) 56c (i) 7200c (j) 5140c 2. (a) 0.47 r35 (b) 0.67 r15 (c) 0.29 r31 (d) 0.48 r28 (e) 0.43 r35 (f) 0.57 r21 3. Teacher check Challenge Teacher check

Unit 23–2

. te

• Use topology to study changes to 3-D shapes. (Modelling clay is great for this.)

Consolidation 23–2 • Use decimal scores from students’ results to order on a number line.

Consolidation 23–3

m . u

w ww

1. (a) 53 (b) 13 (c) 33 (d) 22 (e) 24 (f) 51 (g) 11 (h) 15 (i) 51 (j) 21 2. (a) 4.96 (b) 4.27 (c) 3.53 (d) 3.00 (e) 4.95 (f) 4.80 3.

• Encourage students to attempt to work out the area of triangles without the grid.

o c . che e r o t r s super

Challenge 18

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

New Wave Maths Book G – Teachers Guide • 113 •

Unit 24–1

Student page 70

Outcomes

Indicators

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

The student is able to: • say and write decimals as fractions and fractions as decimals.

Skills

Resources • calculator • fraction/decimal number line

• writing decimals • writing fractions

Memory Masters (N4.3)

Language • add • fraction • decimal • number • symbols • numerator, denominator

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) Warm Up

• Ask students to write the following decimals as fractions – 0.1, 0.6, 0.48, 0.73; and the following fractions as decimals – 1/2, 3/4, 81/100, 7/50. • Remind students that converting decimals to the hundredths place makes for easy conversion to a fraction.The decimal number needs to be placed over 100.When converting fractions to decimals, divide the numerator into the denominator. • If whole numbers form part of the conversion these are not affected and remain as the given whole number.

ew i ev Pr

Teac he r

• The focus for this unit is the multiplication of two multiples of 10 each less than 100.

w ww

• Use the fraction/decimal number line to help if required. • Change the fractions to decimals and decimals to fractions as asked in the workbook. Work through three or four of each with the whole class before setting them to work and assisting those having difficulty. • Encourage students to use diagrams or concrete materials to show their decision.

Challenge

. te

m . u

What to do

o c . che e r o t r s super

• 114 • New Wave Maths Book G – Teachers Guide

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

Unit 24–2

Student page 71

Outcomes

Indicators

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

The student is able to: • display data in a scattergraph. • describe information recorded in a scattergraph. • read scales to the nearest graduation, including instances in which the graduations are not labelled.

Skills • measuring • collecting data • plotting points • analysing data

Resources

Language

• calculator • pencil • tape • height measuring stick

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

• divide • add • measure • scattergraph • axis • plot • position • concentration

Notes

Memory Masters (N4.3)

Teac he r Number (N4.3)

ew i ev Pr

• The focus for this unit is division of a multiple of 10 less than 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 (C&D4.3, C&D4.4, M4.2) Warm up

• Ask, ‘How would you measure your height and arm span?’

What to do

. te

Challenge

m . u

w ww

• Organise the class into groups so that height and arm span may be measured and recorded. Individual student names are not required. • When all students have been measured, one person from each group takes turns to read a student’s height and arm span. The rest of the class plots the point of intersection of the mass and arm span on their graph. The teacher may wish to do the same on a master graph drawn on the blackboard/whiteboard or overhead. When the point is marked the next pair of measurements is read out. This continues until all points from the group are plotted, then the next group continues. • When all points have been plotted there will be a scattergraph recorded on the page. • Scattergraphs are used as a means of looking for trends in data. You can not always read cause and effect into the graph. • The scattergraph may be used to answer the questions on the page. Answering the questions can be done individually, as a group or as a whole-class discussion with students recording answers while the discussion takes place.

o c . che e r o t r s super

• Ask students to devise a word problem that would result in Question 2(e).

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

New Wave Maths Book G – Teachers Guide • 115 •

Unit 24–3

Student page 72

Outcomes

Indicators

N4.3, N4.1a

The student is able to: • understand the multiplicative nature of the relationship between places for whole numbers; i.e. as they move from right to left, each place is 10 times the one before.

Skills • rounding

Memory Masters (N4.3)

Resources • calculator • number line 1 – 100 or 1000 to 100 000 or both (page 208)

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)

ew i ev Pr

Teac he r

• add • round • nearest place value

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

• 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 = 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

• Use a number line, 1 – 100 or 1000 –100 000, to explain rounding of numbers to the class. Numbers that end in 1 – 4 are rounded down to the nearest place. Numbers that end in 5 – 9 are rounded up to the nearest place. • Numbers greater than 100 or 1000 are rounded to the nearest hundred or thousand, unless they are multiples of 10 or 100 respectively. Those greater than 60 or 600 are rounded up and those less than 50 or 500 are rounded down.

w ww

• Use the crowd figure of 78 000 to show how, by rounding the three weeks’ attendance figures to the nearest thousand, the final approximation is reached. • Work through the first two examples of each set with the class to assist understanding, then let the class work. • Provide assistance using a number line for those who still have difficulties. • Use rounding skills to complete the final activity.

Challenge

. te

m . u

What to do

o c . che e r o t r s super

• Students are to provide their explanation for rounding to the nearest tenth.The explanations should be the guide that they would use in all instances.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 56 – 57. • 116 • New Wave Maths Book G – Teachers Guide

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

Unit 24—Answers

Student pages 70 – 72 Unit 24–2

1. (a) 5600 (b) 1800 (c) 8100 (d) 1400 (e) 1800 (f) 4000 (g) 1000 (h) 1600 (i) 1800 (j) 1500 2. (a) 122.925 (b) 162.110 (c) 119.216 (d) 138.281 (e) 169.024 (f) 76.397 3. (a) 3.5 (b) 5.25 (c) 10.1 (d) 7.8 (e) 2.75 (f) 8.6 (g) 12.9 (h) 6.50 (i) 11.16 (j) 4.35 4. (a) 65/10 = 61/2 5. Either—both are half. 25 1 (b) 3 /100 = 3 /4 Challenge (c) 108/10 = 104/5 Possible answers: (d) 46/10 = 43/5 6 + 6/6 + 6/6 75 3 (e) 2 /100 = 2 /4 (66x 6) + (6 6+ 6) (f) 129/10 6/6 + 6/6 + 6/6 + 6/6 + 4 1 6 (g) 8 /100 = 8 /25 /6 + 6/6 + 6/6 + 6/6 (h) 755/100 = 711/20 (i) 142/100 = 141/50 (j) 537/100 (k) 424/100 = 46/25

1. (a) 70 (b) 70 (c) 60 (d) 90 (e) 90 (f) 80 (g) 30 (h) 80 (i) 70 (j) 80 2. (a) 193.473 (b) 22.7107 (c) 23.8649 (d) 1680.12 (e) 186.985 (f) 16.4088 3. (a) Teacher check (b) Teacher check (c) Teacher check (d) Most people in the class are around this height with around this arm span. (e) As height increases, arm span increases (positive correlation). (f) Yes. As people grow taller their arms grow longer proportionately. (g) Teacher check Challenge Teacher check

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

ew i ev Pr

Teac he r

Unit 24–1

w ww

(b)

. te

• Give students a number of fractions to be written in words.

Consolidation 24–2 • Students select other data that could be collected and represented in a scattergraph.

Consolidation 24–3

m . u

1. Teacher check 2. (a) 241.133 (b) 16.1258 (c) 1353.51 (d) 1663.46 (e) 121.543 (f) 968.13 3. (a)

• Ask students to devise their own word problems.

o c . che e r o t r s super

4. (a) Each week was rounded to the nearest 1000. 23 000 + 34 000 + 21 000 = 78 000 (b) 103 000 (c) 25 000 Challenge Teacher check

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

New Wave Maths Book G – Teachers Guide • 117 •

Unit 25–1

Student page 73

Outcomes

Indicators

N4.1a, N4.3, M4.4b

The student is able to: • use a grid to enlarge or reduce a figure in a specified way.

Skills • enlarging • drawing • using a ruler

Resources • calculator • pencil • ruler • tangram pieces

Language • circle • smaller • subtract • grid • enlarge • dimensions • perimeter • area • parallelogram • tangram

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

Notes

Memory Masters (N4.1a)

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 (M4.4b) Warm up

ew i ev Pr

• The focus for this unit is the ordering of decimal numbers. • Discuss the misconceptions with students; e.g. a longer number means it is larger than a shorter number – 1.0235 is larger than 1.12.

w ww

• Ask students to identify a suitable starting point.The actual starting point does not matter, but it must allow three times the length and height of the original drawing to fit on the page. Guidance may be needed in suggesting the point at the left of the diagram or the flat top or right of diagram. In all cases, one square in from the edge and top then count the required squares to find the starting position. For example, front (left) point – four squares up from the bottom and one square in from the left. • Once the starting point has been established, remind students that all lines radiating from this point should be drawn and then to continue with the lines that connect to the drawn lines. • Students should be reminded that they need to count the squares of the original boxes then triple the length when drawing the large picture. • When the drawing is complete, answer the questions.

. te

Challenge

m . u

© R. I . C.Publ i cat i ons orr evi ew pur posesonl y• What to do •f • Explain to students that they will be enlarging the small drawing in their workbook so that its dimensions are three times the original.

o c . che e r o t r s super

• Use the seven tangram pieces to make a large parallelogram. • Keep drawings of all attempts for showing later.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 144 – 145. • 118 • New Wave Maths Book G – Teachers Guide

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

Unit 25–2

Student page 74

Outcomes N4.1a, N4.3, N4.3

Skills • rounding • using a calculator • estimating • dividing

Indicators

Resources

The student is able to: • enter divisions in a calculator and interpret the part after the decimal point in answers.

• calculator • newspapers • protractor • ruler • large circle on card

Language • multiply • table • pie graph • proportion • percentage • radius

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

Notes

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

ew i ev Pr

Teac he r

• The focus for this unit is rounding of decimal numbers to the nearest one.

• Students 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

• Remind students that the quickest form of finding answers to addition, subtraction, multiplication and division problems is to work them out mentally. Part of learning how to work things out mentally is to make estimates of answers. • When making estimates we usually round numbers for ease of working. • In this case, however, exact answers are simple to calculate.

What to do

. te

Challenge

m . u

w ww

• Work with students to find the estimates or approximations of the answers to several of the questions in Exercise 3. Ask students what they would round the number to in 3(a)—600—ask how many 60s in 600. (10) Approximate answer is 10. Calculator check is then made and the answer including remainder (written as a decimal) is written. • Repeat process. • Exercise 4 requires a similar process. Encourage students to ignore the decimal place when rounding. • Students should work sums mentally or use short division. Work through the first couple with them. Rounding to nearest hundred in 4(a) is 200. There are about 30 sixes in 200. Actual working then follows.

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 76 – 77. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 119 •

Unit 25–3

Student page 75

Outcomes

Indicators

N3.3, N4.3, M4.2

The student is able to: • count uniform units to find area and perimeter.

Skills • describing • problem-solving

Memory Masters (N3.3)

Resources • calculator • ruler • string • pencil

Language • subtract • area • perimeter • shapes • digit • square number

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) Warm up

• Ask students how they might find the perimeter and area of a variety of different shapes. Open discussion to encompass a variety of methods.

• Organise the class into groups and distribute string and any other materials students ask for to measure perimeter and area of the different shapes. • Students are to further discuss the method(s) they use to find both area and perimeter and to make a summary of these methods on or near the respective shape. • Actual perimeters and areas are to be noted inside or beside each shape. • Share final workings with reports from each group to the class as a whole.

w ww

Challenge

• Students are to record the working they used to find their answer. • Share findings with the class.

. te

m . u

What to do

ew i ev Pr

Teac he r

• The focus for this unit is addition of two whole numbers both less than 100.

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 138 – 139. • 120 • New Wave Maths Book G – Teachers Guide

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

Unit 25—Answers

Student pages 73 – 75

Unit 25–1

1. (a) 567 (b) 209 (c) 76 (d) 82 (e) 33 (f) 8 (g) 8 (h) 275 (i) 86 (j) 325 (b) 5.5 (c) 8.2 (d) 8.1 2. (a) 6.4 (e) 7.3 (f) 5.2 3. (a) 10 = 10.6 (k) 12 = 12.3 (b) 13 = 13.1 (l) 17 = 16.98 (c) 16 = 15.95 (m) 14 = 13.8 (d) 14 = 14.4 (n) 16 = 15.66 (e) 12 = 12.2 (o) 14 = 14.1 (f) 12 = 12.6 (p) 11 = 10.7 (g) 13 = 13.3 (q) 12 = 11.98 (h) 14 = 13.8 (r) 12 = 12.3 (i) 12 = 11.9 (s) 29 = 28.6 (j) 11 = 11.3 (t) 11 = 10.5 Est. 48 (c) Est. 70 (d) Est. 54 4. (a) Est. 36 (b) Ans. 36.4 Ans. 48.3 Ans. 69.4 Ans. 53.8 Challenge Possible solutions: (2 x 2 x 2 x 2) + (2 + 2/2) = 19 22 = 22 (2 x 2 x 2 x 2 x 2) – (2 + 2/2) = 29

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

ew i ev Pr

Teac he r

1. (a) 27.01 (b) 8.279 (c) 65 937 (d) 0.0416 (e) 52 999 (f) 2.214 (g) 627 (h) 4.287 (i) 1.596 (j) 132.87 0.6 (c) 0.1 (d) 0.5 2. (a) 0.3 (b) (e) 0.1 (f) 0.2 3. Teacher check Challenge

Unit 25–2

. te

• Select a drawing from a magazine. Draw a grid over it and reproduce it reduced or enlarged.

Consolidation 25–2 • Use real-life numbers to continue estimating activities.

Consolidation 25–3

m . u

w ww

1. (a) 84 (b) 66 (c) 74 (d) 85 (e) 87 (f) 54 (g) 91 (h) 86 (i) 81 (j) 73 2. (a) 1.6 (b) 3.2 (c) 6.2 (d) 2.1 (e) 5.2 (f) 3.3 3. Approximate answers: (a) A: 18.7 cm2 (d) A: 19.6 cm2 P: 16.2 cm P: 15.7 cm (b) A: 16 cm2 (e) A: 15.4 cm2 P: 16 cm P: 17.5 cm (c) A: 11 cm2 (f) A: 16.5 cm2 P: 12.2 cm P: 15.8 cm Area – count whole squares, then find part squares that look like they make a whole if put together. Perimeter – measure with a ruler or string. Challenge 49 ( 4 = 2 x 2; 9 = 3 x 3)

• Brainstorm other methods students could use to find the perimeter and area of the shapes on the page.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 121 •

Unit 26–1

Student page 76

Outcomes

Indicators

WM4.2, WM4.3, WM4.4, C&D4.2, C&D4.3, C&D4.4

The student is able to: • ask organising questions to get him/her started. • contribute questions in a brainstorming situation. • use examples to explain why he/ she made a conjecture. • draw on mathematical knowledge to check reasonableness of answers.

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

Resources • ruler • pencil

Language • data • graph • axis • intervals • questions

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

Notes

What to do

• This activity is designed for students working collaboratively in groups but students will record their personal responses in their workbooks. Allow enough time so that students can discuss their opinions and for ideas to evolve. Open-ended tasks such as these are a good opportunity for students to ‘take a risk’ with maths. • When completing open-ended, investigative tasks, some students may be more successful in mixed-ability groups rather than in same-ability groups. • Some groups will be able to work independently while others may need guidance. The stimulus questions below may prompt such groups. – At what times do you feel hungry during the day? – At what times do you not feel hungry during the day? – What could we label these items and the times in between? (not hungry, mildly hungry, hungry, extremely hungry etc.) – How hungry are you when you first wake up in the morning? – Do you think you feel hungry when you are asleep? – What hours are you awake? • Groups may wish to collate and summarise their findings and present them as a poster with a series of graphs, diagrams and information. • Allow each group to discuss and evaluate its ability to problem-solve and 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.

ew i ev Pr

Teac he r

Main Activity (WM4.2, WM4.3, WM4.4, C&D4.2, C&D4.3, C&D4.4)

w ww

. te

m . u

o c . che e r o t r s super

• 122 • New Wave Maths Book G – Teachers Guide

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

Unit 26–2

Student page 77

Outcomes

Indicators

N4.3, C&D4.2, C&D4.3, C&D4.4, N4.1a

The student is able to: • collect, record and represent data. • answer questions related to the data.

Skills • collecting data • recording data • graphing • analysing data

Resources

Language

• calculator • paper circles (see page 232)

• divide • round • nearest cent • number line • order relationship • smallest to largest (ascending order)

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

Notes

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

ew i ev Pr

Teac he r

• The focus for this unit is the division of a multiple of 100 by a multiple of 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&D4.2, C&D4.3, C&D4.4, N4.1a) Warm up

• Hold up a circle cut from card and explain that this may be used to show 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 is represented as 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 in the circle. Use a protractor to show 180º or half of the circle. The result is the same as previously.

What to do

m . u

w ww

• Before constructing the pie graph, information must be collected. Use a newspaper to make approximations of the number of pages allocated to the areas shown in the table. Work this amount out as a percentage of the total paper. • Calculate the number of degrees to be drawn on the graph showing this percentage; e.g. 15% is 15 x 360º ÷ 100 = 54º or as 360º x 15% = 54º. • Draw the segments on the graph once they have been calculated. • To follow on, choose a different day, or days, of the week and make comparisons of the relative proportions of the newspaper allocated to different areas of news and advertising. Is there a difference? Give reasons why. • If other newspapers are available make comparisons between different newspapers. Is there a difference? Give reasons why.

. te

Challenge

o c . che e r o t r s super

• Beginning with a circle, what size angles can you make by paper folding?

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 166 – 167. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 123 •

Unit 26–3

Student page 77

Outcomes

Indicators

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

The student is able to: • add fractions with like denominators.

Skills • adding fractions • converting fractions

Memory Masters (N4.3)

Resources • calculator • coloured rods • coloured pencils • grid paper • pencil • fraction chart (page 211)

Language • multiply • fraction • diagrams

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 (N4.1b) Warm up

• Ask students to explain what a fraction is. Brainstorm for answers.

w ww

• Use grid paper or coloured rods to show 1/5 and 3/5. Add the two together to make 4/5. • Continue to work with students to find answers to 3(b), (c) and (d) of the workbook. • When adding whole numbers and fractions, suggest that separating whole numbers and fractions and adding each separately helps to find the total; e.g. 32/5 + 21/5 = (3 + 2) + (2/5 + 1/5) = 5 + 3/5 = 53/5 • Use a fraction grid to aid students when ordering fractions in Exercise 4. • Complete the diagrams by shading the nominated fraction in each component of the diagram. The total of the shaded parts provides the answer. If the answer is an improper fraction, change this to a mixed numeral.

Challenge

. te

m . u

What to do

ew i ev Pr

Teac he r

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

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 64 – 65. • 124 • New Wave Maths Book G – Teachers Guide

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

Unit 26—Answers

Student pages 75 – 77

Unit 26–1

1. (a) 3000 (b) 28 000 (c) 12 000 (d) 56 000 (e) 45 000 (f) 14 000 (g) 45 000 (h) 18 000 (i) 16 000 (j) 63 000 2. (a) 125.950 (b) 261.182 (c) 145.656 (d) 383.076 (e) 606.886 (f) 523.12 3. Teacher check Challenge 1 /2 fold = 180º 1 /4 fold = 90 º etc.

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

ew i ev Pr

Teac he r

1. Answers will vary. Students should have labelled the ‘Y’ axis of the graph with terms such as very hungry, mildly hungry, full etc. 3. Summary should reflect the peaks and troughs in the graph.

Unit 26–2

. te

• Students draw a hunger graph for a weekend, considering different sleeping times and more regular access to food.

Consolidation 26–2 • Follow the same procedure with magazines—check advertising versus articles.

Consolidation 26–3

m . u

w ww

1. (a) 200 (b) 200 (c) 300 (d) 300 (e) 100 (f) 200 (g) 400 (h) 100 (i) 100 (j) 100 2. (a) 343.643 (b) 362.295 (c) 379.783 (d) 294.424 (e) 657.462 (f) 291.228 4. (a) < (j) 47/8 (s) 3/5 3. (a) 4/5 6 7 6 (b) = (b) /8 (k) 2 /8 (t) /8 (c) 4/6 (l) 16/7 (u) 76/8 (c) < (d) 3/4 (m) 34/6 (v) 45/6 (d) < 5 2 (e) /7 (n) 4 /3 (w) 98/9 (e) = (f) 53/5 (o) 28/10 (x) 86/7 (f) = 6 5 (g) 3 /7 (p) /6 (y) 64/5 (g) < (h) > (h) 58/10 (q) 7/7 = 1 (i) < (i) 45/6 (r) 6/9 5. (a) 11/3 Challenge (b) 17/8 Possible solutions: 7 x (7 + 7/7 + 7/7 + 7/7 + 7/7 + 7/7 ) = 84 (c) 21/4 (7 + 7 +7 7) x 7 = 63

• Encourage students to use diagrams to solve problems with fractions.

o c . che e r o t r s super

7 x 7 + (7/7 + 7 +7 7) = 52

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

New Wave Maths Book G – Teachers Guide • 125 •

Unit 27–1

Student page 79

Outcomes

Indicators

N4.2, N4.3, S4.4

The student is able to: • use mathematical equipment to measure angles.

Skills

Resources • calculator • protractor

• divide • diagrams • protractor • angle • diagonal • square, rectangle • measure • intersecting lines • parallel lines • patterns • rows, columns

• using a protractor • explaining discoveries

Memory Masters (N4.2)

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 (S4.4) Warm up

• Show students a blackboard protractor. Explain the correct use of a protractor. The point of intersection of the 0º – 180º line is placed over the vertex (point of intersection of the arms of the angle) of the angle to be measured.The 0º – 180º line is directly over the base line. The second arm of the angle is used to read the size of the angle. • Ask students to use the blackboard protractor to read the size of a series of angles drawn on the blackboard/whiteboard.

ew i ev Pr

Teac he r

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

What to do

w ww

• Ask the students to open their workbook and measure the size of the angles formed when the diagonal is drawn in a square or rectangle. Record all new angles. • Write about what they discovered. (Both angles formed at the diagonal are the same at each end but on opposite sides of the diagonal. In a square the angles are the same size.) • Ask students to draw more rectangles with greater variance in the size of sides to see if the relationship still holds.

Challenge

. te

m . u

o c . che e r o t r s super

• Students are to place the letters as directed. • Keep records of all attempts and notes to explain what you did to solve the problem.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 124 – 125. • 126 • New Wave Maths Book G – Teachers Guide

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

Unit 27–2

Student page 80

Outcomes

Indicators

N4.1a, M4.1, N4.3, N4.1b

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

Skills • problem-solving • calculating ratios

Resources

Language • convert • litres • diagrams • subsets • ratio • triangle • surrounds

• calculator

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

• millilitres • divide • sets • fraction • circles • square

Notes

Memory Masters (N4.1a, M4.1) Number (N4.3)

ew i ev Pr

Teac he r

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

• Students 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

• To introduce ratios try the following activities. Ask a group of five boys and five girls to come to the front of the class. Ask the groups to make a smaller group that has 2 boys to each girl. How many different groups can be made? (Two, one of 2:1 and one of 4:2.) Explain the make up as 2:1 or 4:2. This terminology—write on board for class to see—is called a ratio. • Ask the group to make a ratio of 1 boy to 4 girls. • What is the ratio of the main group: 5:8 or 1:1? • Explain to the class that sometimes when finding ratios it is necessary to split wholes into parts of wholes to receive an even distribution.

w ww

What to do

m . u

• Look at workbook Exercise 3(a).There are four pies to share among 3 children. Each child receives 1 pie and a third of a pie or 4:3 or 4/3 of a pie. • Work through each activity with the class as a whole. Ask students to provide verbal explanations of the answer. Answers are to be written when found.

Challenge

. te

o c . che e r o t r s super

• Students are to keep all records of attempts and find a solution for sharing.

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

New Wave Maths Book G – Teachers Guide • 127 •

Unit 27–3

Student page 81

Outcomes

Indicators

N4.1, N4.3, M4.2

The student is able to: • use straightforward timetables and programs with both 12- and 24hour times.

Skills • reading time • writing time

Memory Masters (N4.1)

Resources • calculator • analog clock

Language • convert • dollars, cents • divide • decimal place • 24-hour time, 12-hour time • digital, analog • square

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) Warm up

• Display a large analog clock and ask the class to give the time as the hands are changed. Provide several opportunities for the class. • Ask class what most people use to tell the time today. (A digital watch or clock.) • How is the time read? In most cases as a.m. or p.m. time. However, timetables are written in 24-hour time. Can they give a reason for this? (Possibly to avoid confusion with a.m. or p.m. reading of time.) • How is 24-hour time read differently from 12-hour time? (After noon, the hours continue from 12 to midnight—2400 hours or 0000 hours. ) Simply remember to add p.m. time to 12 to obtain 24-hour afternoon time. • How would you write or say 3 p.m. in 24-hour time? (1500 hours)

ew i ev Pr

Teac he r

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

w ww

m . u

What to do

• Work out the missing time from analog clocks or digital clocks in the workbook. • Check work as and after it is completed.

Challenge

. te

o c . che e r o t r s super

• Students use the solution from Question 3(d) to write a word problem. • Share results.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 114 – 117. • 128 • New Wave Maths Book G – Teachers Guide

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

Unit 27—Answers

Student pages 79 – 81

Unit 27–1 1. (a) 0 < (2 x 8) ÷ 4 – 3 < 2 (b) 4 < (6 x 7) ÷ 6 – 2 < 6 (c) 4 < (9 x 4) ÷ 6 – 1 < 6 (d) 6 < (5 x 4) ÷ 2 – 3 < 8 (e) 3 < (8 x 3) ÷ 4 – 2 < 5 (f) 12 < (56 ÷ 7) + 5 < 14 (g) 12 < (63 ÷ 9) + 6 < 14 (h) 10 < (49 ÷ 7) + 4 < 12 (i) 15 < (54 ÷ 6) + 7 < 17 (j) 13 < (72 ÷ 8) + 5 < 15 2. (a) 81.2 (b) 71.1 (c) 41.1 (d) 92.3 (e) 72.1 (f) 61.1 3. (a)

Unit 27–2 1. (a) 74 mL (b) 8020 mL (c) 40 mL (d) 2706 mL (e) 8 mL (f) 0.087 L (g) 0.006 L (h) 0.042 L (i) 3.27 L (j) 0.8 L (b) 66.8 (c) 62.6 (d) 74.2 2. (a) 47.6 (e) 45.9 (f) 52.7 3. (a) 3/4 (b) 1/4 (3/12) (c) 20 out of 30 or 2/3 (d) 5 (e) 40 boys, 60 girls (f) 80 (g) 6/10 (h) 60 goals, 48 shots Challenge

Teac he r

ew i ev Pr

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

(b) Add up to 90º. Diagonal opposites are congruent. (c) Teacher check Challenge Possible answer

a.m.

9.44

(b)

. te

(f) 2

3 4

8 7

a.m.

(h)

p.m.

5.56

• Convert analog times into digital for the purposes of recording a television program.

p.m.

(i) 4

Consolidation 27–3

2359

3 7

• Using situations which arise every day in the classroom, create more word problems involving ratio.

12.17

1450

a.m.

Consolidation 27–2

o c . che e r o t r s super 5

9.15

(c)

4 5

7.10

3 8

(e)

• Students can explore and study angles used in real-life. Discuss how angles can be used to make a structure stronger.

m . u

w ww

1. (a) $82.74 (b) $5.96 (c) $0.04 (d) $0.61 (e) $0.89 (f) 6c (g) 51c (h) 203c (i) 89 400c (j) 1827c 0.114 (c) 0.115 (d) 0.205 2. (a) 0.215 (b) (e) 0.105 (f) 0.116 3. (a) 12 12 12 (d) (g)

0225

1 2

10 9

3 4

8 7

1155

Challenge Teacher check

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

New Wave Maths Book G – Teachers Guide • 129 •

Unit 28–1

Student page 82

Outcomes

Indicators

N4.3, N4.1b

The student is able to: • order fractions with like denominators.

Skills • demonstrating equivalence • ordering

Memory Masters (N4.3)

Resources • calculator • coloured pencils • grid paper (see page 199) • pattern blocks

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)

Notes

ew i ev Pr

Teac he r

• subtract • greater than >, less than <, equal = • order • fractions • like denominators, unlike denominators • adjacent sections • numerator

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

• 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 = 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

• Distribute pattern blocks to small groups of students. After initial free play ask them to make pairs of fractions; e.g. 2/3 and 1/3. Compare the two fractions and find the difference between them. Ask students how they might do this. (Lie fractions side by side or one on top of the other.) • Repeat this for several different examples.

w ww

• Ask students to look at Exercise 3 in their workbook. Suggest that as the denominators are the same on all questions they may think of the denominator as cubes and work out their answers. All that is required is to focus on the numerators (top numbers). • Test scores may be shown as a fraction as the number correct out of the total score. • Ask whether the scores can be ordered using >, < or = by looking only at the top number. If so, ask students to order the scores. • Use the knowledge they have developed to arrange the scores in descending order. What happens if two scores are the same? (Write it twice.) • Ask students to use the diagram to show 5/8 and then 3/4. (Draw lines using different colours.) Which is the larger?

. te

Challenge

m . u

What to do

o c . che e r o t r s super

• Students experiment with colours to find the least they require to colour the drawing as directed. • Keep notes to explain your thinking. • Share findings.

• 130 • New Wave Maths Book G – Teachers Guide

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

Unit 28–2

Student page 83

Outcomes

Indicators

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

The student is able to: • represent data in diagrams and tables which include arrow diagrams, Venn diagrams and two-way tables. • interpret and report on information provided in bar graphs where data are grouped into simple intervals which can be regarded as categories. • identify, describe and continue patterns linking pairs of numbers on a coordinate grid or in a table by a single operation.

Skills • using a ruler • recording data • analysing data

Resources

Language

• calculator • newspapers • protractor • ruler • large circle on card

• multiply • table • pie graph • proportion • percentage • radius

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

Notes

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

ew i ev Pr

Teac he r

• The focus for this unit is addition of whole numbers both 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, C&D4.4, N4.4) Warm up

• Ask students to describe a diagonal. Ask whether there can be more than one diagonal starting and finishing at a corner. (Yes) • Once this is understood ask students how many diagonals are there in a square. (two)

Challenge

. te

m . u

w ww

• When finding diagonals in a shape it helps to draw all new diagonals from each corner in a different colour. When drawing the diagonals for the hexagon start at one corner and draw the three diagonals in red. Move clockwise around the hexagon then draw the three diagonals from the next corner in green, move to the next corner and there is already a diagonal drawn, so draw the remaining two in blue. At the next corner there is only one diagonal to draw, use yellow for this one. Counting is easy – 3 red + 3 green + 2 blue + 1 yellow for a total of 9 diagonals. • When all recordings have been made in the table ask students to find patterns from the information collected. • In Exercise 5 ask students to complete the table to show the diagonals drawn from each succeeding corner to see if they are able to find any patterns in this set of figures. Share results as class discussion.

o c . che e r o t r s super

• Students use the information they have discovered from the activities on the page to solve the problem. • Keep records of all workings and notes explaining how you reached your solution to share with the class.

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

New Wave Maths Book G – Teachers Guide • 131 •

Unit 28–3

Student page 84

Outcomes

Indicators

N3.3, N4.3, N4.4

The student is able to: • follow a rule based on multiplication, division or simple fractions to generate a sequence. • represent constant addition or multiplication sequences of decimal fraction with materials or diagrams.

Skills • analysing patterns • calculating • recording

Memory Masters (N3.3)

Resources • calculator

• subtract • patterns • decimals • fractions • constant addition

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

ew i ev Pr

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 100 from a whole number less than 100.

Teac he r

Language

w ww

What to do

• Questions 3 and 4 relate to finding the fractional pattern of 1/4 of 64 and 1/5 of 125. • Students input 64 ÷ 4 = , = , = to find an answer of 1. At this point ask the students to press the = button again. (0.25 or 1/4) Ask them what happened. Keep pressing the = key and record what is happening. (Progressively smaller fractions.) • Complete Question 4 independently. • Questions 5 and 6 relate to constant addition of 0.2 and 0.4. • Students complete the activities, recording on the number line as they go. • Students should note that there are half as many jumps for 0.4 as for 0.2. Discuss why.

. te

Challenge

m . u

• Write this sequence of numbers on the board: 81 27 9 3 1 • Ask students what is happening. (81 is being divided by three; i.e. we are finding 1/3 each time.) • Explain to students that the sequence creates a pattern. • Discuss the concept of constant addition. We add the same number each time. This technique is also seen in multiplication.

o c . che e r o t r s super

• Students are to show all attempts when finding the solution to the problem. • Notes should be kept to describe what has taken place. • Share results.

• 132 • New Wave Maths Book G – Teachers Guide

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

Unit 28—Answers

Student pages 82 – 84

Unit 28–1 1. Teacher check 4.5 (c) 3.6 (d) 1.7 2. (a) 2.7 (b) (e) 2.4 (f) 4.5 3. (a) > (b) < (c) > (d) > (e) > (f) < (g) < (h) < (i) < 4. (a) 17/20 < 19/20 < 20/20 = 20/20 > 15/20 > 11/20 =

/20 < 18/20

Unit 28–2 1. (a) 64 (b) 86 (c) 83 (d) 83 (e) 91 (f) 46 (g) 63 (h) 61 (i) 64 (j) 61 (b) 0.33 (c) 0.52 (d) 0.32 2. (a) 0.33 (e) 0.32 (f) 0.53 3.

Teac he r

5. Teacher check, 3/4 6. (a) Teacher check (b) 1/6 Challenge 4

Challenge 35

4. Number of diagonals equals number of sides minus three.

ew i ev Pr

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

(b) 11/20, 11/20, 15/20, 17/20, 18/20, 19/20, 20/20, 20/20,

Teacher check 6.

. te

Teacher check Challenge

• Write pairs of fractions. Swap with a partner and order the fractions. Check each other’s work.

Consolidation 28–2 • Continue with finding the diagonals on a 50c coin.

Consolidation 28–3

m . u

w ww

1. (a) 51 (b) 51 (c) 56 (d) 50 (e) 46 (f) 26 (g) 41 (h) 44 (i) 25 (j) 26 2. (a) 0.27 (b) 0.26 (c) 0.25 (d) 0.49 (e) 0.36 (f) 0.53 3. 64 16 4 1; decimals are formed 0.4, 0.16, 0.64 4. 125 25 5 1; decimals are formed 0.5, 0.25, 0.125 5.

• Students use numbers of their own choosing to follow the pattern of Exercises 3 and 4.

o c . che e r o t r s super

9 2

8 7

1 11

3 4

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

New Wave Maths Book G – Teachers Guide • 133 •

Unit 29–1

Student page 85

Outcomes N4.3, S4.3

Skills • tessellating • investigating • drawing • problem-solving

Memory Masters (N4.3)

Indicators

Resources

The student is able to: • sort different arrangements of a fixed number of squares (such as pentominoes) into groups that can be superimposed and re-sort into those which do or do not require a reflection.

• calculator • pentomino shapes • paper • pencils • coloured pencils

• divide • investigate • tessellating • properties • pentominoes • collectively, individually • patterns • rectangle

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

Notes

ew i ev Pr

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

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

Teac he r

Language

w ww

• Students will need a supply of pentomino shapes or to be instructed to investigate to find all the different configurations. (12) • Use the shapes to find which of the 12 will tessellate with itself. When drawing the shapes it is suggested that a different colour is used to show neighbouring pentominoes so that they are not lost on the page. • Drawings using about 10 of the same pentomino should provide the answers sought. • When each of the individual pentomino shapes has been tested, use the whole set to make a rectangle. • How many different rectangles can you make? Draw around each pentomino shape so that the final rectangle can still show the individual pentomino placements. • Display finished patterns and rectangles.

Challenge

. te

m . u

o c . che e r o t r s super

• Choose one pentomino; double the length and height. What happens to the area?

• 134 • New Wave Maths Book G – Teachers Guide

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

Unit 29–2

Student page 86

Outcomes

Indicators

N4.3, C&D4.2

The student is able to: • suggest what data to collect to help estimate numbers or quantities. • revise a survey question so it can be answered by yes/no or as a simple multiple choice. • construct and use his/her own categories to answer specific questions.

Skills • investigating • recording • collecting • modifying • reviewing

Resources

Language

• calculator • coloured pencils • Smarties™ (1 box per student)

• divide • common • favourite • data • record

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

Notes

Memory Masters (N4.3)

Teac he r

ew i ev Pr

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)

Warm up

• Explain to students that this activity is requiring them to develop their own strategy for collecting and collating data. They may find that they need to alter their questioning to refine their results.

w ww

• Distribute a packet of Smarties™ to each student. • Students complete the activity.

Challenge

. te

m . u

What to do

• Collate all class data to average the results of colours found in packets and students’ favourite Smartie™ colour.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 135 •

Unit 29–3

Student page 87

Outcomes

Indicators

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

The student is able to: • multiply and divide measurements and amounts of money by a 1-digit number.

Skills • estimating • dividing • rounding

Resources

Language • place value • divide • round decimal places • division • estimate • actual • consecutive prime numbers • multiplied • equalled

• calculator

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

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

• Estimation is a skill that we use every day to give us an approximation when using numbers. To assist in estimating, it is usual to round numbers first. • Revise rounding with the class—0 – 4 round down, 5 – 9 round up.

ew i ev Pr

Teac he r

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

w ww

• The workbook has a number of examples.The first task is to estimate the answer for 3(a); round to 6345 to the nearest 1000—6000. Ask, ‘How many fours in six?’ (approximately 11/2). ‘Therefore there are approximately 1500 fours in 6345.’ • Work through several examples as a whole class before setting students to complete the page. • You may want some of the sums worked out in full. Answers can be checked by the students’ preferred method. • The fourth exercise may be treated in the same manner as the first.

Challenge

. te

m . u

o c . che e r o t r s super

• Students will need to investigate different numbers, keeping a record of their findings and activities for sharing later.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 76 – 77. • 136 • New Wave Maths Book G – Teachers Guide

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

Unit 29—Answers

Student pages 85 – 87

Unit 29–1

1. (7 + 10) x (9 + 10), (20 – 3) x (20 – 1) and so on. (b) 0.4• (c) 0.16• (d) 0.3• 2. (a) 0.6• • • (e) 0.3 (f) 0.83 3. Teacher check Challenge Teacher check

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

ew i ev Pr

Teac he r

1. (a) 1800 (b) 3500 (c) 3000 (d) 6400 (e) 1800 (f) 6300 (g) 2400 (h) 800 (i) 5400 (j) 2100 0.2 (c) 0.5 (d) 0.5 2. (a) 0.5 (b) (e) 0.4 (f) 0.6 3. Teacher check. Possible answers:

Unit 29–2

Challenge Teacher check. Area increases by 4.

. te

• Students develop other pentominoes that will tessellate. Create a pattern.

Consolidation 29–2 • Brainstorm with students other information they may be able to obtain using Smarties™.

Consolidation 29–3

m . u

w ww

1 1. (a) 1/100 (b) 100 000 (c) 1/10 000 (d) /10 (e) 1 000 000 (f) 100 000 (g) 1 (h) 1/100 (i) 100 (j) 1 • • 1.16 (c) 2.6• (d) 1.3• 2. (a) 2.3 (b) (e) 1.29 (f) 1.14 3. Teacher check estimates (a) 135 (i) 142 (b) 208 (j) 204 (c) 119 (k) 15.63 (d) 207 (l) 17.02 (e) 128 (m) 5.66 (f) 341 (n) 10.47 (g) 123 (0) 20.69 (h) 131 (p) 15.38 4. Teacher check estimates (a) 3.85 m (e) 2.8 m Challenge (b) 3.56 m (f) 0.62 m House numbers: (c) 6.07 m (g) 1.97 m 37 41 43 (d) 3.4 m (h) 0.9 m Telephone number: 65231

• Provide students with everyday opportunities to estimate.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 137 •

Unit 30–1

Student page 88

Outcomes

Indicators

N4.1a, N4.3, M4.4a

The student is able to: • given a rectangle with whole number length sides, explain why multiplying the length by height gives the area.

Skills • subtracting decimals

Resources • calculator • Base 10 MAB • counters

Language • subtract • round • nearest hundredth

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

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 (M4.4a) Warm Up

• Ask students to use their calculators to find the answers to: 300 x 250, 450 x 200, 150 x 250 and 200 x 350

w ww

• What is the first task to complete? Find the area of the shape. • Direct students to think of a way to find the area of the shape. Accept all suggestions. Ask which is best. (Divide the shape into two.) • How can you find the area? Multiply the length by the width of each shape, then add the two figures. The answer is in square metres. • How can you find the amount of fertiliser to use? (Multiply the area by 100 g.) • If the cost is per kilogram, what needs to be done? (Change the grams to kilograms.) How? (Divide by 1000.) • How can the cost be found? (Multiply the tonnes by $450.)

Challenge

. te

m . u

What to do

ew i ev Pr

Teac he r

• The focus for this unit is rounding of decimal numbers to the nearest hundredth.

o c . che e r o t r s super

• Students experiment with colours to find the least they require to colour the map as directed. • Students keep notes to explain their thinking. • Share their findings.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 134 – 137. • 138 • New Wave Maths Book G – Teachers Guide

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

Unit 30–2

Student page 89

Outcomes

Indicators The student is able to: • suggest what data to collect to help estimate numbers or quantities. • represent data in diagrams and tables which may include arrow diagrams, Venn diagrams and twoway tables. • describe information which may include arrow diagrams, tree diagrams, Venn diagrams or Carroll diagrams.

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

Skills • planning • collecting • collating • recording • analysing

Resources

Language

• calculator

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

• estimation • number sentences • subtract • Venn diagram • criteria • information • number • symbols

Notes

Memory Masters (N4.2) Number (N4.3)

ew i ev Pr

Teac he r

• The focus for this unit is completion of number statements using rule of order.

• Students 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

• Venn diagrams are used to represent collected data.They are a means of showing different data with some common points. • Use the example from page 59 Unit 20–2 of the workbook to show how the collection of information about students’ musical instruments is displayed. • Discuss with the class what information they might collect to show on the Venn diagram. Remind students they need four different criteria, one of which may be ‘no information available on’. This group is listed on the outside surrounding the overlapping circles.

What to do

Challenge

. te

m . u

w ww

• Once students have settled on the information they are to collect, set them to work collecting it. Each student will need a recording sheet to assist with the collection. Prior to collecting the information, students should complete the sections ‘What I need to find out’, and ‘How will I gather my information?’.These may be used as a whole-class discussion points. • Once information has been collected, students are to display the results on the Venn diagram. • Share representations in groups. The group may select one for sharing with the class.

o c . che e r o t r s super

• If a number is divided by 10, or 9 is subtracted from it, the result is the same. What is the number? • Record all attempts. • Share and discuss results.

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

New Wave Maths Book G – Teachers Guide • 139 •

Unit 30–3

Student page 90

Outcomes

Indicators

N4.1a, N4.3

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

Skills • writing numbers • problem-solving

Resources • calculator • coloured pencils

Language • symbols • code • greater than >, less than <, equal = • order relationship • subtract • palindromic number • single-digit • networks • continuous

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

Memory Masters (N4.1a)

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) Warm up

ew i ev Pr

• The focus for this unit is determining greater than, less than or equal to quality of whole numbers, decimal numbers and fractions.

• Write DAD, MUM, BOB, POP, NON, BIB on the blackboard/whiteboard.Then write 1991, 2002, 1881, 59395 on the board as well. • Ask students what is common with all items written on the board? (They read the same forwards as they do backwards.) Who knows the name for this occurrence? (Palindrome. Words, numbers and sentences may all be palindromes.) • There are ten numbers that are always palindromes—the ten single-digit numbers. (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

w ww

• The task is to find palindromic numbers and write them in their workbook as directed. • Work this exercise as a whole-class activity so that everyone shares the fun of some different maths. Alternatively, have the class work in groups and report to the whole class on their findings. • If working in groups, direct each section prior to the class moving on. • The code breaking activity is a fun one for students to test their skills. Clues have been provided.

Challenge

. te

m . u

What to do

o c . che e r o t r s super

• Suggest to students that they used coloured pencils to show the different attempts that they made in trying to solve the problems of tracing the networks. • Keep notes of attempts to ensure the final solution is not lost.

• 140 • New Wave Maths Book G – Teachers Guide

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

Unit 30—Answers

Student pages 88 – 90

Unit 30–1 1. (a) 13 < (64 ÷ 8) + 6 < 15 (b) 16 < (81 ÷ 9) + 8 < 18 (c) 13 < (42 ÷ 6) + 7 < 15 (d) 16 < (32 ÷ 4) + 9 < 18 (e) 14 < (45 ÷ 5) + 6 < 16 (f) 65 < (9 x 8) – 6 < 67 (g) 40 < (7 x 7) – 8 < 42 (h) 32 < (5 x 8) – 7 < 34 (i) 48 < (9 x 6) – 5 < 50 (j) 49 < (8 x 7) – 6 < 51 2. (a) 1.805 (b) 4.140 (c) 3.014 (e) 4.672 (f) 5.381 3. Teacher check Challenge 10 10 ÷ 10 = 1 10 – 9 = 1

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

(d) 3.998

ew i ev Pr

Teac he r

1. (a) 8.28 (b) 1.98 (c) 4.29 (d) 6.11 (e) 0.72 (f) 0.82 (g) 0.41 (h) 18.76 (i) 7.72 (j) 5.09 6.18 (c) 5.16 (d) 5.45 2. (a) 2.46 (b) (e) 4.18 (f) 2.36 3. Teacher check 4. 120 000 m2 + 137 500 m2 257 500 m2 x 100 g/m2 = 25 750 000 g = 25 750 kg = 25.75 tonne x $4.50 = $115.88 Challenge 3

Unit 30–2

. te

• Find the cost of fertilising the school’s grassed areas.

Consolidation 30–2 • Display the data gathered for the Venn diagram in another way.

Consolidation 30–3

m . u

w ww

1. (a) > (b) > (c) < (d) > (e) < (f) < (g) < (h) = (i) = (j) > 2. (a) 0.047 (b) 0.079 (c) 0.056 (d) 0.077 (e) 0.065 (f) 0.054 3. (a) Teacher check. Possible answers are: 313, 5005, 924429 etc. (b) 2112 (c) 1991 (d) Possible answers are: 1881, 1771, 1661 (e) Possible answers are: 12. 12. 2121, 18.6.81 etc. (f) Teacher check (g) Teacher check 4. (a) Possible answers are: 123 or 523 + 523 + 123 646 646 (b) 5184 Challenge + 923 B 6107

• Brainstorm and chart as many palindromic numbers as possible. Try finding palindromic sums; e.g. 4 + 1 – 1 = 4

o c . che e r o t r s super start 1

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

2 B

4 finish

New Wave Maths Book G – Teachers Guide • 141 •

Unit 31–1

Student page 91

Outcomes

Indicators

N4.3, S4.1

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

Skills • drawing • using a ruler • problem-solving • planning

Memory Masters (N4.3)

Resources • calculator • ruler • tape measure • trundle wheel • pencil • toothpicks (or equivalent) • large sheets of paper

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 (S4.1)

Notes

ew i ev Pr

Number (N4.3)

w ww

. te

• The task in the workbook is to draw a map showing accurately the position of all buildings and main features of the school grounds.To do this you need to (ask students to list needs): – measure grounds size and buildings – measure distance of buildings from boundaries, each other and other key features – measure key features such as bitumen areas, ovals, play equipment, large trees, garden plots and so on. • Students may be best served working in groups using their workbook as a rough sketch plan, marking in key features and measures then drawing the actual plan on a large sheet of paper. • Display and discuss plans.

m . u

• Ask students what a bird’s-eye view means. Outline an example of such a view as looking down into the classroom with the roof taken off. All items would be seen as flat (2-dimensional) and located in their present position. • Such a view may be transferred to paper as a map. How can it be made to fit on a smaller piece of paper? (Draw the map to scale.) • Ask students to explain what scale drawing means. Scale should allow a map/drawing to be large enough to allow key features to be identified. In the case of a classroom roughly 8 m x 8 m a suitable scale would be 1 m = 2 cm.

What to do

• divide • scale map • accurately • position

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

Warm up

Language

o c . che e r o t r s super

Challenge • Distribute toothpicks (or equivalent) and direct students to solve the problem given. • Students are to record all attempts and keep notes explaining their actions.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 14 – 15. • 142 • New Wave Maths Book G – Teachers Guide

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

Unit 31–2

Student page 92

Outcomes

Indicators

N4.3

The student is able to: • multiply and divide measurements and amounts of money by a 1-digit number.

Skills • estimating • multiplying • problem-solving

Resources

Language • divide, multiply • estimate • actual • number • symbols

• calculator

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

Notes

Memory Masters (N4.3)

Teac he r

ew i ev Pr

• The focus for this unit is subtraction of a multiple of 10 less than 100 from a multiple of 10 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 (N4.3) Warm up

w ww

What to do

m . u

• Discuss with the class the use of longer forms of finding answers to difficult algorithms and the way we tackle the solutions. Estimation and calculators provide the keys.We estimate so that we have a fair idea of where the answer might be. The actual calculation is completed by a calculator, very seldom by pencil and paper. • When estimating we usually round to the appropriate place value. For example, if we multiply $5.30 by 4 we would round to …? ($5 by 4. Estimated answer is $20) If multiplying 47.82 m x 5 we would round to 50 m x 5, estimated answer 250 m. • Work through several examples with the class before allowing students to complete the activity by themselves.

• Exercise 4 should be set up as a problem for the students to solve by themselves or in small groups. Share solutions with the class.

Challenge

. te

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 76 – 77. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 143 •

Unit 31–3

Student page 93

Outcomes N4.3, N4.3, M4.2

Skills • reading timetables • problem-solving

Memory Masters (N4.3)

Indicators

Resources

The student is able to: • use straightforward timetables and programs with both 12- and 24hour times.

• calculator • bus/train timetables

• multiply • partial • timetable • time • departure • arrive • approximate • hours • distance • number • symbols, • 24-hour time

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

ew i ev Pr

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 100 by a whole number less than 10.

Teac he r

Language

• Display copies of bus/train timetables. If available, distribute timetables to groups of students. • Ask students to find suitable departure times to arrive at a destination by a set time such as 0830; 1200 hours; 1900 hours. • Remind students that some timetables are written in 24-hour time. A brief explanation of 24-hour time may be required – all morning times are written with 0 before the single digit hours,—00 to 09 hours—and past noon times continue, counting in hours until midnight—13 to 23 hours.

w ww

• Students may continue to work in groups to answer the questions regarding the train timetables.

Challenge

. te

• Students find the solutions recording each attempt and keeping notes explaining their thinking.

m . u

What to do

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 120 – 121. • 144 • New Wave Maths Book G – Teachers Guide

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

Unit 31—Answers

Student pages 91 – 93

Unit 31–1

1. (a) 190 (b) 370 (c) 270 (d) 160 (e) 80 (f) 380 (g) 190 (h) 90 (i) 280 (j) 170 (b) 0.75 (c) 0.71 (d) 0.91 2. (a) 4.64 (e) 0.41 (f) 0.61 3. Teacher check estimates (a) $21.20 (k) 24.3 m (b) $24.30 (l) 183.7 m (c) $38.00 (m) 25.26 m (d) $20.20 (n) 499.74 m (e) $30.80 (o) 28.72 m (f) $56.35 (p) 379.44 m (g) $26.10 (q) 45.84 m (h) $15.25 (r) 250.32 m (i) $31.20 (s) 11.64 m (j) $16.40 (t) 109.96 m 4. Either—both the same 3/2 = 11/2 = 1.5 Challenge 8 Possible solutions are: /8 + 8/8 8+8 8 (8 + 8) ÷ 8

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

ew i ev Pr

Teac he r

1. 2 x 9 x 2, 4 x 3 x 3, 6 x 3 x 2, 2 x 3 x 3 x 2 and so on. 0.163 (c) 0.129 (d) 0.218 2. (a) 0.143 (b) (e) 0.153 (f) 0.253 3. Teacher check Challenge

Unit 31–2

. te

11 7

4 3

• Make 3-D models of the building to add to the map.

Consolidation 31–2 • Provide students with estimating opportunities in daily activities wherever possible.

Consolidation 31–3

m . u

w ww

1. (a) 96 (b) 86 (c) 88 (d) 48 (e) 66 (f) 99 (g) 84 (h) 93 (i) 88 (j) 69 282.862 2. (a) 579.304 (b) 387.276 (c) 80.215 (d) (e) 329.323 (f) 340.292 3. (a) 26 mins (b) 5 mins (c) 3.19 p.m. (d) Express trains don’t stop in between. (e) 7.47 a.m. Challenge

• Plan a journey which requires the use of bus and train timetables to reach a destination.

o c . che e r o t r s super

2 6

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

New Wave Maths Book G – Teachers Guide • 145 •

Unit 32–1

Student page 94

Outcomes

Indicators

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

The student is able to: • rewrite a decimal as a fraction and a fraction as a decimal.

Skills • converting fractions • writing fractions • converting decimals • writing decimals • ordering

Memory Masters (N4.3)

Resources • calculator • number line with decimal fraction equivalents (page 211)

• subtract • number line • fraction • decimal • equivalent • longer • number • symbols

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

Notes

ew i ev Pr

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

• The focus for this unit is division of a multiple of 10 less than 1000 by a multiple of 10 less than 100.

Teac he r

Language

w ww

• When converting fractions to decimals, it is often easier to change the fraction to decimals by changing the fraction to an equivalent fraction with a denominator of 10, 100 or 1000 if possible. Use this information to work through the first three examples of Exercises 3, 4 and 5 with the class as a whole. • Ask the class what the decimal equivalent of 1/2 is. Ask how they obtained the answer. For those who have difficulties converting fractions to decimals, suggest that they use their calculator and divide the numerator by the denominator; e.g. 1 ÷ 2 = 0.5. • In Exercise 5 students will need to convert the fraction to a decimal or the decimal to a fraction to determine the larger number of the pair. Suggest working with both numbers as decimals may be the best option. • Check students’ work carefully as they proceed.

Challenge

. te

m . u

© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Distribute or display a large number line showing key fractions and decimals. Ask students to complete the number line by filling in other missing examples; e.g. 0.45, 45/100, 0.69, 69/100, 0.13, 13/100, 0.65, 13/20.

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 62 – 63. • 146 • New Wave Maths Book G – Teachers Guide

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

Unit 32–2

Student page 95

Outcomes

Indicators

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

The student is able to: • design a probability device such as a die, spinner or bag of coloured beads to produce a specified order of probability.

Skills • graphing • recording • predicting • discussing

Resources

Language

• calculator • Base 10 MAB or counters • 2-cm cubes • box or tin • marker pen

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

• subtract • record • results • graph • how often • prediction • accurate • tally

Notes

Memory Masters (N4.4) Number (N4.3)

ew i ev Pr

Teac he r

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

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

Main Activity (C&D4.1) Warm up

• Brainstorm with students the language related to chance events. • Ask students to explain what they understand about chance. Draw the discussion together by providing a simple overview – chance events are not accurately predicted. Some are more likely to occur than others, but each event that might occur is governed by the fact of uncertainty. Use the example of a coin. It has two sides, when tossed it may come down as a head or a tail. Which will it be? There is no certainty. Every throw is a 50:50 chance.

What to do

. te

m . u

w ww

• To show chance in a practical way organise the class into groups. Provide each group with ten 2-cm cubes. Each cube has a different number from 1 to 10 marked on it. The 10 cubes are placed in a box, tin or a bag. In 50 draws, with replacement, how often would they expect each numbered cube to be taken out? Write their prediction in the space provided. • Without looking, take one cube and colour a square on the graph to show which number was drawn. • Repeat this 50 times. After each draw replace the cube. • Check predictions and explain the results. • If five extra cubes with the number 4 are added to the ten in the container what do you think will happen? Record your prediction in the space provided. • Check your prediction by repeating the 50 draws as before. Keep a tally and see if you are correct. • Discuss findings in the group and with the whole class.

o c . che e r o t r s super

Challenge • List as many chance words as you can; e.g. might, possibly, impossible. • Order them from the least likely to the most likely. OR • Newspaper search: Look for the use of chance words – try the sports pages, weather etc.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 150 – 151. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 147 •

Unit 32–3

Student page 96

Outcomes

Indicators

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

The student is able to: • add and subtract fractions with like denominators.

Skills

Resources • calculator • Base 10 MAB • stopwatch

• adding fractions • subtracting fractions

Language • convert • metres • centimetres • subtract • add • constant • digits • sum

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

Memory Masters (M4.1, 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

• Encourage students to think mentally and find the answer to 32/10 + 24/10. (3 + 2) + (2 + 4/10) = 5 6/10. • Likewise, the final answer to 67/10 – 23/10. (6 – 2) + (7 – 3/10) = 4 4/10.

ew i ev Pr

Teac he r

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

m . u

• Work with the whole class on 3(a – c) before allowing students to move on. • Exercise 3 works on the same principle except that the fractions are being subtracted. • Work with students on Exercise 4(a – c) before allowing them to proceed.

w ww

Challenge

• Students will need to count beats for 1 minute. • Multiply this to find the time for 1000 minutes. • Share their results.

. te

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 64 – 65. • 148 • New Wave Maths Book G – Teachers Guide

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

Unit 32—Answers

Student pages 94 – 96

Unit 32–1

1. (a) 25.6 (b) 2.3 (c) 3.0 (d) 7.0 (e) 12.5 (f) 0.6 (g) 2.0 (h) 2.7 (i) 5.3 (j) 3.2 2. (a) 27.28 (b) 47.24 (c) 28.23 (d) 17.42 (e) 28.11 (f) 39.55 3. (a) 5 (b) Teacher check (c) Teacher check (d) More 4s would be drawn out. (e) Teacher check Challenge Teacher check

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

9 – (9 +9 9 + 9 +9 9 + 9/9)

(9 +9 9) + (9 9+ 9)

9+9+9+9 9

ew i ev Pr

Teac he r

1. (a) 9 (b) 4 (c) 9 (d) 6 (e) 4 (f) 7 (g) 6 (h) 6 (i) 6 (j) 9 33.22 (c) 43.14 (d) 23.15 2. (a) 23.46 (b) (e) 41.18 (f) 41.63 3. (a) 0.5 (e) 0.2 (i) 0.11 (b) 0.1 (f) 0.04 (j) 0.14 (c) 0.01 (g) 0.75 (k) 0.32 (d) 0.25 (h) 0.7 (l) 0.45 45 56 /100 = 9/20 (g) /100 = 14/25 4. (a) 8/10 = 4/5 (d) 18 (b) 1/2 (e) /100 = 9/50 (h) 9/10 89 3 (c) /100 (f) /4 (i) 22/100 = 11/50 5. (a) 0.55 (e) 0.65 (i) 0.79 (b) 3/4 (f) 0.9 (j) 9/10 27 (c) 0.1 (g) 0.8 (k) /100 30 (d) 0.8 (h) 0.97 (l) /50 Challenge Possible solutions are:

Unit 32–2

. te

• Students work in pairs to make up fractions. Swap work and convert to decimals.

Consolidation 32–2 • Repeat the activity using different items in a tin.

Consolidation 32–3

m . u

w ww

1. (a) 0.04 m (b) 0.23 m (c) 4.82 m (d) 27.5 m (e) 82 m (f) 80 cm (g) 5 cm (h) 410 cm (i) 82 400 cm (j) 307 cm 55.2 (c) 66.3 (d) 84.1 2. (a) 40.4 (b) (e) 75.7 (f) 61.7 6 3. (a) 5 (f) (b) 87/8 (g) 73/4 (c) 64/8 (h) 13 (d) 8 (i) 6 (e) 3 (j) 10 1 4. (a) /4 (f) 22/8 2 (b) 2 /6 (g) 45/8 2 (c) 32/7 (h) /5 2 (d) 2 /10 (i) 14/8 (e) 11/5 (j) 21/6 Challenge Answers will vary.

• Provide students with further examples to complete.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 149 •

Unit 33–1

Student page 97

Outcomes

Indicators

N4.1a, N4.3, S4.4

The student is able to: • make figures and objects which meet criteria related to sides, faces, angles and edges.

Skills • following instructions • using a compass • using a ruler • explaining

Resources • calculator • compass • ruler • pencil • protractor

Language • convert • instruction • add • compass • dollars, cents • construct • base line • distance • point • arc • measure • lengths • equilateral triangle • angle • order

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

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 (S4.4) Warm up

• Revise the use of compass (compass point and pencil point to be level, turn from top of compass to obtain light line) and protractor (correct placement on angle to read accurately). • When undertaking construction drawings all line work should be as light as possible so that there is no need to erase working lines.The final construction (drawing) may be made darker for easy reading.

w ww

• Follow the instructions as set out in the workbook to draw an equilateral triangle. The teacher may direct the activity for the whole class using it as an English lesson in following directions. Alternatively, students may work independently. • Once the triangle has been constructed, the lengths of the sides and the size of angles are to be measured and findings recorded. • Discuss results. Ask – ‘Why do you think the triangle is called equilateral?’ (equal sides)

Challenge

. te

m . u

What to do

ew i ev Pr

Teac he r

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

o c . che e r o t r s super

• To solve this logic problem students need to read and isolate the clues, and use these to determine the order of the elephants. • Keep notes explaining the reason for decisions. • Write the elephants in order and keep all attempts at a solution.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 128 – 129. • 150 • New Wave Maths Book G – 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: • partition double-digit numbers in order to mentally multiply and divide by small single-digit numbers.

Skills

Resources

Language

• calculator • blocks • counters

• add • subtract • partition

• partitioning • multiplying • adding

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

Notes

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

ew i ev Pr

Teac he r

• The focus for this unit is the addition of two whole numbers 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 (N4.3) Warm up

• Ask the students to mentally work out the following: 8 x 3; 22 x 5; 17 x 4 • Ask them to explain how they reached their answer. • Focus on one technique, that of partitioning two-digit numbers in preparation for multiplying by a single-digit number. For example, 22 x 5 = (20 x 5) + 2 x 5) = 100 + 10 + 110 • Repeat the example, using 17 x 4.

What to do

Challenge

. te

m . u

w ww

• Direct students to start on Exercise 3 and to check working carefully. If necessary, work through several more examples with students as a whole or in small groups for those who need assistance. • Students complete the exercises with help as required.

o c . che e r o t r s super

• Students may wish to use blocks and counters to represent tables and people. • Hint: A table can be used to seat 4 people where necessary. • Share their results.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 82 – 83. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 151 •

Unit 33–3

Student page 99

Outcomes

Indicators

N4.3, M4.4a

The student is able to: • demonstrate the ability to calculate the perimeter of polygons.

Skills

Resources • calculator

• subtract • simplest • calculate • perimeter • shapes

• calculating perimeters

Memory Masters (N4.3)

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

ew i ev Pr

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 multiple of 10 less than 1000.

Teac he r

Language

© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Discuss how the perimeter of different shapes may be found. Discussion should include adding sides; adding adjacent sides of rectangles and doubling; knowing one side of a regular shape multiplying by the number of sides of the shape (e.g. 4 for square, 8 for octagon).

w ww

Challenge

• Trace around your closed hand (fingers closed). • Find the perimeter. • Trace around your open hand (fingers spread). • Find the perimeter. • Find the area of your hand. Does it change? What can you say about area and perimeter? • Note: Students should realise area and perimeter are not related.

. te

m . u

• Students are to use different methods to find the shapes given noting the simplest. • Once all perimeters have been found. Use a whole-class discussion to determine the simplest method for each shape.

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 136 – 137. • 152 • New Wave Maths Book G – Teachers Guide

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

Unit 33—Answers

Student pages 97 – 99

Unit 33–1

1. (a) 93 (b) 163 (c) 142 (d) 141 (e) 134 (f) 145 (g) 121 (h) 180 (i) 125 (j) 135 2. (a) 1241.87 (b) 145.581 (c) 29.564 (d) 616 (e) 618 (f) 375 3. (a) (40 x 6) + (7 x 6) = 240 + 42 = $282 (b) (30 x 5) + (8 x 5) = 150 + 40 = $190 (c) (50 x 4) + (2 x 4) = 200 + 8 = $208 (d) (20 x 7) + (9 x 7) = 140 + 63 = $203 (e) (60 x 3) + (4 x 3) = 180 + 12 = $192 (f) (90 x 8) + (8 x 8) = 720 + 64 = $784 4. (a) (70 x 4) + (5 x 4) = 280 + 20 = 300 m (b) (80 x 5) + (6 x 5) = 400 + 30 = 430 cm (c) (10 x 9) + (9 x 9) = 90 + 81 = 171 L (d) (30 x 8) + (7 x 8) = 240 + 56 = 296 km (e) (90 x 7) + (8 x 7) = 630 + 56 = 686 mm (f) (80 x 9) + (5 x 9) = 720 + 45 = 765 mL Challenge 9

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

Each side is 8 cm. Each angle is 60º. Challenge Dumbo, Rumbo, Mumbo, Gumbo, Jumbo

ew i ev Pr

Teac he r

1. (a) $0.07 (b) $62.47 (c) $873.94 (d) $0.41 (e) $8.24 (f) 29c (g) 34 716c 826 900c (h) (i) 7600c (j) 95c 2. (a) 8034 (b) 25 138 (c) 1 668 875 100 019 (e) 141 327 (f) 231.37 (d) 3.

Unit 33–2

. te

• Use a compass and protractor to construct other types of triangles.

Consolidation 33–2 • Provide students with further examples where partitioning can be used.

Consolidation 33–3

m . u

w ww

1. (a) 234 (b) 621 (c) 267 (d) 133 (e) 225 (f) 188 (g) 209 (h) 452 (i) 346 (j) 284 2. (a) 16.64 (b) 35.08 (c) 47.43 (d) 67.17 (e) 83.29 (f) 55.71 3. (a) 16 cm (c) 13.5 cm (b) 21 cm (c) 15 cm (c) 9 cm (c) 14 cm Challenge Teacher check

• Use the simplest methods found to calculate the perimeter of objects in the classroom.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 153 •

Unit 34–1

Student page 100

Outcomes

Indicators

N4.3, N4.1a

The student is able to: • rewrite a decimal as a fraction and a fraction as a decimal.

Skills

Resources • calculator

Language • multiply • fraction • decimal

• converting fractions to decimals • converting decimals to fractions

Memory Masters (N4.3)

Teac he r

• The focus for this unit is multiplication of a whole number 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

Notes

ew i ev Pr

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

w ww

• Set students to work converting the fractions in the workbook to decimals. This will give students a general introduction to the next activity. • Ask students how they might work out the related fraction to the decimals in Question 4. • Set students to complete the task, with help if required. • Question 5 may be completed after teacher-directed discussion.

Challenge

• Students are required to find out whether either of the two paths can be traced using a continuous line that does not trace over the same line more than once and which returns to the starting point. • Suggest that students first find out whether the network is traversable with a continuous line before finding whether it can return to its starting point. • Show all attempts. (Use a coloured pencil for each different attempt.)

. te

m . u

© R. I . C.Publ i cat i ons What to do •f orr evi ew pur posesonl y• • Revise with students that decimals and fractions are interrelated. By dividing the numerator by the denominator we can obtain the relevant decimal. 2 • Using a calculator, practise this with the following examples: 1/5 4/8 /6

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 66 – 67. • 154 • New Wave Maths Book G – Teachers Guide

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

Unit 34–2

Student page 101

Outcomes N4.3, C&D4.4, C&D4.1

Skills • chance • interpret data • record data

Indicators

Resources

Language

The student is able to: • describe information from diagrams which may include arrow diagrams, tree diagrams, Venn diagrams or Carroll diagrams. • order probability devices from the one most likely to the one least likely to produce an outcome.

• calculator • card for spinners • coloured pencils • toothpick, or paper clip for spinner • spinner blackline (page 212)

• divide • table • data • range, median, mean, mode • probability • segment • central point

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

Notes

Memory Masters (N4.3)

Teac he r

ew i ev Pr

• 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 (C&D4.4, C&D4.1) Warm up

w ww

e.g. The data are bimodal having 8 and 9 occurring the most frequently.

. te

m . u

• Review range, mean, median and mode as statistical measures.

• Find the range, mean, median and mode for the set of enrolment figures provided. • Probability is a means of determining the likelihood, or chance, of an event taking place. If I have a six-sided die in my hand and I throw it, what number is likely to come up? The chance is 1 in 6 of any of the six numbers showing. This is the case each time the die is thrown. The probability then of a 3 coming up is 1 in 6 or 1:6. • Construct spinners using stiff card and a toothpick or paperclip as the spinning device. See page 212 of this document.

What to do

o c . che e r o t r s super

• Students are to complete the tables provided to show possible outcomes (number of different colours), possible success (determined by the proportion of the spinner showing the colour; e.g. red is 50%); probability (red is 1:2 as it is 50%). • Use the spinner to make up to 40 spins to check your findings.

Challenge • Students use their knowledge of mean and median to make their determination. Checking back through previous activities finding mean and median may assist. For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 150 – 151. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 155 •

Unit 34–3

Student page 102

Outcomes

Indicators

N4.3, N4.2

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

Skills

Resources • calculator

• subtract • factors • simpler terms • number sentences

• multiplying • renaming numbers • analysing

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.

Main Activity (N4.2)

w ww

. te

m . u

• Numbers are made up of component parts, with each having a relation of 10 to its adjacent place. It is possible to break numbers into these component parts to simplify multiplication. For example, 64 x 7 = (60 + 4) x 7 = (60 x 7 ) + (4 x 7) = 420 + 28 = 448 or 64 x 7 = (64 x 10) – (64 x 3) = 640 – 192 = 448 • This is an example of the distributive property of multiplication over addition. • Work through these examples on the blackboard/whiteboard with the class, using both methods. 38 x 8; 61 x 9; 53 x 4 Each example may be shown as: 38 x 8 = (30 + 8) x 8 = (30 x 8) + (8 x 8) = 240 + 64 = 304 or (38 x 10) – (38 x 2) = 380 – 76 = 304 61 x 9 = (60 + 1) x 9 = (60 x 9) + (1 x 9) = 540 + 9 = 549 or (61 x 10) – (61 x 1) = 610 – 61= 549 53 x 4 = (50 + 3) x 4 = (50 x 4) + (3 x 4) = 200 + 122 = 212 or (53 x 10) – (53 x 6) = 530 – 368 = 212

What to do

ew i ev Pr

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

Warm up

Language

o c . che e r o t r s super

• Students complete sets in workbook. • Assist as required.

Challenge • Students are to provide their reasoning as to which method is the simplest – this may change depending on examples. • Suggest to students that they provide examples to support their argument.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 82 – 83. • 156 • New Wave Maths Book G – Teachers Guide

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

Unit 34—Answers

Student pages 100 – 102

Unit 34–1

1. (a) 8 (b) 9 (c) 5 (d) 3 (e) 4 (f) 7 (g) 7 (h) 5 (i) 6 (j) 9 (b) 91 (c) 26.2• (d) 672.38 2. (a) 132 (e) 7.9 (f) 49.37 3. (a) 6 (b) 29 (c) 26 (d) 28.8 4. (a) (c)

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

(b)

(d)

ew i ev Pr

Teac he r

1. (a) 848 (b) 244 (c) 644 (d) 444 (e) 963 (f) 824 (g) 636 (h) 999 (i) 682 (j) 868 2820 (c) 3536 (d) 35 560 2. (a) 1023 (b) (e) 33 500 (f) 52 200 3. (a) 0.3 (i) 5.27 4. (a) 4/10 = 1/5 (i) 1963/100 (b) 0.5 (j) 1.67 (b) 37/100 (j) 2615/1000 (c) 0.4 (k) 3.83 (c) 8/10 = 4/5 = 2123/200 (d) 0.5 (l) 0.5 (d) 1/ (k) 1/2 4 (e) 0.15 (m) 0.9 (e) 23/100 (l) 14/5 (f) 0.8 (n) 3.6 (f) 11/5 (m) 32/5 (g) 1.8 (o) 7.125 3 (g) 12 /5 (n) 3/4 (h) 4.3 (p) 3.05 (h) 51/4 (o) 61/4 5. Teacher check (p) 9/10 Challenge B

Unit 34–2

Challenge Median is the central point by definition. Mean is the average.

. te

• Students work in pairs to make up fractions, swap work and convert to decimals.

Consolidation 34–2 • Students develop their own spinners and discuss the chance involved.

Consolidation 34–3

m . u

w ww

1. Teacher check 2. (a) 499 (b) 59 (c) 274 (d) 763 (e) 467 (f) 297 3. (a) (20 + 7) x 6 = (20 x 6) + (7 x 6) = 120 + 42 = 162 (b) (40 + 3) x 9 = (40 x 9) + (3 x 9) = 360 + 27 = 387 (c) (30 + 8) x 7 = (30 x 7) + (8 x 7) = 210 + 56 = 266 (d) (50 + 7) x 5 = (50 x 5) + (7 x 5) = 250 + 35 = 285 4. (a) 48 x 8 = (48 x 10) – (48 x 2) = 480 – 96 = 384 (b) 64 x 9 = (64 x 10) – (64 x 1) = 640 – 64 = 576 (c) 37 x 8 = (37 x 10) – (37 x 2) = 370 – 74 = 296 (d) 53 x 9 = (53 x 10) – (53 x 1) = 530 – 53 = 477 5. (a) 552 (b) 228 (c) 747 (d) 322 (e) 380 Challenge Teacher check

• Provide students with further examples to complete.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 157 •

Unit 35–1

Student page 103

Outcomes

Indicators

WM4.2, WM4.3, S4.3

Skills

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

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

Resources • ruler • pencil • mirror, mira (optional)

Language • greater than >, less than < • multiply, add • two-dimensional shape • distorted grid • scale • number multiply

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

Notes

What to do

• This activity is designed for students working collaboratively in groups. As they will need to discuss their opinions and ideas, allow enough time so that they do not feel rushed and for ideas to evolve. Investigative tasks such as these are a good opportunity for students to ‘take a risk’ with maths. • When completing investigative tasks, some students may be more successful in mixed-ability groups rather than in same-ability groups. • Some groups will be able to work independently while others may need guidance. The stimulus questions below may prompt such groups. – What do you notice about the cards? – What colour will the number ‘16’ Numero card be? – What do you notice about the coloured circles within the rectangle? (They are symmetrical.) Students can use a mirror to investigate the symmetrical properties of the cards. – Consider different ways that the 16 dots can be represented on the card. – What do we know about the number 16? (It is even, it is higher than 10, its factors are 1, 16, 8, 2 and 4.) – Look at the other cards. What are the main structures of the designs of the spots? • Note: Students should notice that the patterns within the rectangle are symmetrical. They should continue this when designing their number ‘16’ Numero card. • Although the Numero cards have a limit of three dots wide and five dots high, the new number ‘16’ card can have different dimensions such as 4 x 4, 8 x 2 etc. • 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 time to discuss and evaluate its ability to problem-solve and its success as a group. A ‘group’ or ‘self ’ assessment form could be completed. This information will be helpful for creating groups for future investigative or open-ended tasks.

ew i ev Pr

Teac he r

Main Activity (WM4.2, WM4.3, S4.3)

w ww

. te

m . u

o c . che e r o t r s super

• 158 • New Wave Maths Book G – Teachers Guide

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

Unit 35–2

Student page 104

Outcomes N4.1a, N4.3, N4.1b

Skills • subtracting fractions • adding fractions

Indicators

Resources

Language

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

• calculator • tangram pieces (pages 221 – 224)

• round • nearest tenth • divide, subtract • fraction • simplify • approximately

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

Notes

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

ew i ev Pr

Teac he r

• The focus for this unit is rounding of decimal numbers to the nearest tenth.

• Students 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

• Revise subtraction of fractions to show the working out for the four different examples used.

What to do

Challenge

. te

m . u

w ww

• Where there is like denominators and no regrouping required (Exercise 3), whole numbers may be subtracted, fractions subtracted and the whole number and fraction answers added together. For example, 33/5 – 21/5 = (3 – 2) + (3/5 – 1/5) = 1 + 2/5 = 12/5 • Work through two or three different examples with the whole class before setting students to finish. • Use the diagrams to shade the fraction of each flavour of ice-cream used. • If all flavours were combined, how many equivalents of the full container would have been used? Encourage students to shade the fractional equivalents on the blank diagrams. • Suggest students look for logical combinations; e.g. 1/4 and 3/4. • Use the coffee shop example as a problem-solving exercise for students to share their solutions with the class. Students may work in groups.

o c . che e r o t r s super

• Students use the seven tangram pieces to construct a large rectangle. Draw all attempts and keep notes of how each attempt was made.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 64 – 65. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 159 •

Unit 35–3

Student page 105

Outcomes

Indicators

N4.3, M4.2

The student is able to: • use straightforward timetables and programs with both 12- and 24hour times.

Skills

Resources • calculator • coloured pencils

Language • subtract • calendar • day, week, month, year

• reading a calendar

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.

Main Activity (M4.2)

ew i ev Pr

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

• The calendar activity is best attempted either as a whole-class activity with the teacher directing or as individual work with students completing the work independently. • In the latter case, it will best suit the teacher to share the answers as a whole-class activity.

w ww

Challenge

• Draw a square around a block of four dates. • Add the dates in opposite corners. • Try some other 2 x 2 squares. What do you notice? • Try 3 x 3 squares. Test rectangles. Write about your observations.

. te

m . u

• Discuss with the students months of the year and their names.

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 120 – 121. • 160 • New Wave Maths Book G – Teachers Guide

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

Unit 35—Answers

Student pages 103 – 105

Unit 35–1 1. (a) Possible answers include:

(b) Teacher check

1. (a) 4.9 (b) 2.1 (c) 6.6 (d) 3.2 (e) 6.2 (f) 19.9 (g) 27.0 (h) 18.2 (i) 6.2 (j) 0.9 (b) 74 (c) 253 (d) 207 2. (a) 78 (e) 194.7 (f) 2.85 3. (a) 31/4 (b) 22/7 (c) 11/3 (d) 22/5 1 3 2 (e) 2 /4 (f) 5 /5 (g) 2 /5 (h) 32/3 3 (i) 2 /7 4. (a) Vanilla Choc. Straw. Bubble. Coffee 3 1 7 1 1/2 /4 /4 /8 /8

r o e t s Bo r e p ok u S (b) 21/2 5. (a) 32 – 33 Partly used – 33 (b) Teacher check Challenge

New cartons – 35

ew i ev Pr

Teac he r

Unit 35–2

. te

• Look at a pack of regular playing cards, considering their symmetry. Note: Pay special attention to the picture cards.

Consolidation 35–2 • Encourage students to make up their own word problems similar to the coffee shop problem.

Consolidation 35–3

m . u

w ww

1. Teacher check 2. (a) 407 (b) 537 (c) 2254 (d) 4882 (e) 3569 (f) $5.32 3. (a) – (e) Answers will vary (f) July – December (g) 14 weeks 2 days Challenge Teacher check

• Research to find out how many days old each student is.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 161 •

Unit 36–1

Student page 106

Outcomes

Indicators

N4.3, N4.1b

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

Skills • writing fractions

Memory Masters (N4.3)

Resources • calculator • coloured pencils

• multiply • diagrams • sums • path • vertically, horizontally • cell

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

Notes

ew i ev Pr

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

• The focus for this unit is subtraction of multiples of 10 less than 1000 from multiples of 10 less than 1000.

Teac he r

Language

• Draw four squares divided into fourths on the blackboard/whiteboard. Ask a student to show three-fourths of the total.This may be done by colouring three whole squares of the four or by colouring three out of the four squares within each of the squares—in each as 12 out of 16 fourths were shaded — 12/16 or 3/4 of 4 = 3. • Repeat for 1/8 of three rectangles; 3/5 of four rectangles. A mental approach which may be useful here is find 1/5 or 5 in order to work out 2/5.

w ww

• Work with students to complete the activity in their workbook. Ensure that the students understand that the final answer is a fraction of the original total. • Note: Students can handle rectangular regions better than circular regions when trying fraction questions, so some students may need extra assistance to complete this activity.

Challenge

. te

m . u

What to do

o c . che e r o t r s super

• Students are encouraged to use coloured pencils to show the different paths they used to solve the problem. • Show all attempts and keep notes to explain what was attempted.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 62 – 63. • 162 • New Wave Maths Book G – Teachers Guide

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

Unit 36–2

Student page 107

Outcomes

Indicators

N4.3, C&D4.3

Skills • collating • measuring • recording • interpreting

Resources

Language

• calculator

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

• divide • measure • mass • nearest • kilogram • record, tally • total • range • average

Notes

Memory Masters (N4.3)

Teac he r Number (N4.3)

ew i ev Pr

• The focus for this unit is multiplication of a whole number less than 10 by a multiple of 10 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) Warm up

w ww

The most frequently occurring score and there may be more than one mode; e.g. The data are bimodal, having 8 and 9 occurring the most frequently.

What to do

. te

m . u

• Revise mean, mode, median and range with students:

o c . che e r o t r s super

• Ask students to bring their school bag inside and pack it as if to go home at the end of the day. • One at a time, students bring their school bag to the front of the class for weighing. • Call out the weight to the class. Other students record a mark in the correct place on their table. • Repeat until all school bags have been weighed. • Students work independently to find the range, median, mean and mode.

Challenge • What number will give you the same result if you multiply it by 3 or add 16 to it? • Students are to explore their number knowledge to find the answer to the problem. • Keep records of attempts and notes of reasoning for sharing.

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

New Wave Maths Book G – Teachers Guide • 163 •

Unit 36–3

Student page 108

Outcomes

Indicators

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 • writing fractions

Memory Masters (N4.3)

Resources

• subtract • diagrams • sums • fractions • number • symbol

• calculator

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

ew i ev Pr

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

• The focus for this unit is division of a multiple of 10 less than 1000 by a multiple of 10 less than 100.

Teac he r

Language

• Work with students to complete the activity in their workbook. Ensure that the students understand that the final answer is a fraction of the original total.

w ww

Challenge

. te

m . u

What to do

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 62 – 63. • 164 • New Wave Maths Book G – Teachers Guide

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

Unit 36—Answers

Student pages 106 – 108

Unit 36–1

1. (a) 660 (b) 640 (c) 880 (d) 260 (e) 700 (f) 500 (g) 820 (h) 690 (i) 930 (j) 480 (b) 149.1 (c) $1.08 (d) $1.16 2. (a) 11.6 (e) $6.82 (f) $0.08 3. Teacher check Challenge 8 8 x 3 = 24 8 + 16 = 24

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

ew i ev Pr

Teac he r

1. (a) 120 (b) 120 (c) 320 (d) 360 (e) 210 (f) 270 (g) 400 (h) 220 (i) 140 (j) 110 2. (a) $260.40 (b) $297.85 (c) $610.50 (d) $4259.25 (e) $215.44 (f) $2009.00 3. (a) 31/3 (b) 6 (c) 2 (d) 41/2 (e) 22/5 (f) 14/10 = 12/5 (g) 11/2 (h) 51/4 Challenge Not possible

Unit 36–2

. te

• Students develop their own diagrams to show various fractions.

Consolidation 36–2 • Students develop a list of data which could be collected and used to find range, median, mean and mode.

Consolidation 36–3

m . u

w ww

1. (a) 3 (b) 9 (c) 6 (d) 7 (e) 8 (f) 2 (g) 3 (h) 8 (i) 7 (j) 5 2. (a) $44.54 (b) $4.81 (c) $6.66 (d) $77.63 (e) $25.65 (f) $43.14 3. (a) 2/3 (f) 2/5 (b) 11/2 (g) 3/10 (c) 1 (h) 1/2 3 (d) /4 (i) 12/3 (e) 21/2 (j) 3/5 Challenge Possible solutions are: 4 + 4/4 (4 4+ 4) + (4 4+ 4) + 4/4

• Students develop their own diagrams to show various fractions.

o c . che e r o t r s super

(4 + 4) – (4 4+ 4 +

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

/4)

New Wave Maths Book G – Teachers Guide • 165 •

Unit 37–1

Student page 109

Outcomes

Indicators

N4.2, N4.3, S4.1

Skills • reading grids

Memory Masters (N4.2)

Resources • calculator

• subtract • grid • code • decipher

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

ew i ev Pr

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

• The focus for this unit is completion of number sentences working with brackets and multiple operations (rule of order).

Teac he r

Language

w ww

What to do

• Give the students practice at finding points on the displayed grid then ask them to complete Exercise 3 in their workbook. • Exercise 4 uses the same grid code but students now compile coded students’ names or a message using the grids.

Challenge

. te

m . u

• Draw a four quadrant grid on the blackboard/whiteboard. Explain to students that there are four quadrants separated by the two axes. The horizontal axis is commonly called the X axis and the vertical axis the Y axis. To assist in finding points within the quadrants those to the left of centre on the X axis and those below the centre on the Y axis are generally given a negative value. These have been shown in red. The values of the points in the top right quadrant are then both positive. The bottom left positive X and negative Y. The top left negative X and positive Y; and the bottom left both negative values. Other discriminators, such as shaded numbers, are used first then negative values may be used. • The game of battleships and submarines helps develop understanding. • Coordinates are always read along the X axis first and then the Y axis.

o c . che e r o t r s super

• Students use the grid code to make a coded recording of their full name.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 2 – 3. • 166 • New Wave Maths Book G – Teachers Guide

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

Unit 37–2

Student page 110

Outcomes

Indicators

N4.1a, N4.3

The student is able to: • plan sequences of calculations using a calculator memory facility.

Skills

Resources

Language • convert • minutes, hours • multiply, add • fraction • ratio • surfaces

• calculator • 2-cm cubes

• adding • using a calculator

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

Notes

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

ew i ev Pr

Teac he r

• The focus for this unit is the conversion of hours to minutes and minutes to hours.

• Students 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

• Students take out their calculators and practise using the memory function to find the total cost of these items: 3 x $5.24 + 4 x $2.76 + 5 x $1.97 and 6 x $7.56 + 2 x $3.19 + $4.56 + 3 x $2.89

What to do

. te

Challenge

m . u

w ww

• Explain to students that shoppers often use a calculator to keep a running tally of the total of their purchases to ensure they stay within their budget. • To do this, shoppers need to use the memory function to avoid losing their total. • Students use their calculator’s memory function to find the total cost of the shopping list given. • Students use their calculators to find the total cost of each set of items, writing this cost next to the items. • When completed, subtotals should be added, with a progressive total written in the space provided. • Does the final total equal the original? If not, calculations should be rechecked.

o c . che e r o t r s super

• Distribute four 2-cm cubes to each student and ask them to arrange the cubes so that a surface area of 18 is showing. Surfaces under the arrangement are classified as showing. • Show all attempts and keep notes on how you make your arrangements.

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

New Wave Maths Book G – Teachers Guide • 167 •

Unit 37–3

Student page 111

Outcomes

Indicators

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

The student is able to: • use straightforward timetables and programs with both 12- and 24hour times

Skills • converting times • adding times

Resources • calculator • TV guide

Language • convert • divide • dollars, cents • timetable • 12-hour time • 24-hour time

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) Warm up

• Display a copy of a TV guide for students. • Discuss that when we want to record a TV program we need to set the timer on our video recorder. • Select a number of programs from the guide and ask students what time would need to be set on the video recorder, remembering that they operate in 24-hour time. • When happy that students are displaying a level of understanding, set them to work on the activity in the workbook.

ew i ev Pr

Teac he r

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

w ww

• Students complete exercise 3 in the workbook. • Exercises 4 and 5 require students to ascertain particular programs within a category and to find the total time. • Share their answers.

Challenge

. te

m . u

What to do

o c . che e r o t r s super

• Students are to arrange the numbers 1 – 8 inclusively in the eight circles so that no two consecutive numbers are directly connected. • Record all attempts and keep notes of activities.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 116 – 117. • 168 • New Wave Maths Book G – Teachers Guide

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

Unit 37—Answers

Student pages 109 – 111

Unit 37–1

1. (a) 72 mins (b) 297 mins (c) 100 mins (d) 132 mins (e) 225 mins (f) 12 h 4 h 44 mins (g) 9 h 20 mins (h) 7 h 7 mins (i) (j) 3 h 2. (a) 466.44 (b) 2259.4 (c) 442.33 231.242 (f) 310.644 (d) 2833.46 (e) 3. 1. $ 8.40 $ 8.40 11. $ 31.80 $ 111.58 2. $ 3.60 $ 12.00 12. $ 11.64 $ 123.22 3. $ 3.25 $ 15.25 13. $ 43.35 $ 166.57 4. $ 11.25 $ 26.50 14. $ 7.71 $ 174.28 5. $ 11.16 $ 37.66 15. $ 5.07 $ 179.35 6. $ 8.86 $ 46.52 16. $ 3.57 $ 182.92 7. $ 6.30 $ 52.82 17. $ 5.06 $ 187.98 8. $ 9.80 $ 62.62 18. $ 6.42 $ 194.40 9. $ 7.80 $ 70.42 19. $ 0.99 $ 195.39 10. $ 9.36 $ 79.78 20. $ 2.56 $ 197.95 $197.95 $197.95 Challenge Teacher check

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

ew i ev Pr

Teac he r

1. (a) 67 (b) 56 (c) 56 (d) 40 (e) 78 (f) 97 (g) 78 (h) 83 (i) 103 (j) 106 $2.92 (c) $5.29 (d) $5.16 2. (a) $7.14 (b) (e) $4.48 (f) $22.18 3. (a) ROBERT (b) COLLEEN (c) ROCHELLE (d) BRETT 4. Teacher check Challenge Teacher check

Unit 37–2

. te

• Use maps or atlases to find specific locations to give practice in using x and y points.

Consolidation 37–2 • Students select items from a shopping catalogue to make their own shopping lists.

Consolidation 37–3

m . u

w ww

1. (a) $4298 (b) $70.01 (c) $86 (d) $47.29 (e) $0.06 (f) 6 98 700c (g) 6400c (h) 9700c (i) 8c (j) 66c $0.17 (c) $5.97 (d) 73.8 2. (a) 0.36 (b) (e) 93.7 (f) 1.15 3. (a) 5.30 (b) 9.00 (c) 11.30 (d) 14.00 (e) 16.30 (f) 17.30 (g) 19.00 (h) 20.30 4. (a) 2 hours 5. (a) 31/2 hours 6. (a) Answers will vary Challenge 4 6

• Students can repeat the activity using a current TV guide.

o c . che e r o t r s super 2

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

New Wave Maths Book G – Teachers Guide • 169 •

Unit 38–1

Student page 112

Outcomes

Indicators

N4.3, N4.1b

The student is able to: • use judgments of length to estimate the position of fractions on a number line.

Skills • rounding fractions • writing fractions

Memory Masters (N4.3)

Resources • calculator • fraction chart (page 211)

• subtract • arrange • mixed numbers, whole number • fractions • number line • round • nearest • squares • diagram

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

Notes

ew i ev Pr

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

Teac he r

Language

w ww

• Show students the fraction chart, or distribute fraction charts among the students. Use the fraction chart to arrange the fractions and mixed numbers on the number line provided. • Discuss placement of fractions with whole class. • Rounding of fractions is undertaken in the same manner as rounding whole numbers and decimals. If the numerator is less than half the denominator the fraction is rounded down. If the numerator is half or greater the fraction is rounded up. • Work through these with the class: 23/8, 41/3, 32/5, 27/8, 42/3, 34/8, 51/2. • Set class to work to complete the task. • Ask students to round the laps run by Brett and Mitchell over five days to find the approximate total number of laps run by each.

. te

Challenge

m . u

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• What to do • Ask students to suggest reasons why we round. What is the purpose of rounding? • Use a washing line. Ask two students to hold the ends, while other students peg numbers on the line in the correct order.

o c . che e r o t r s super

• Students are to find all the squares that they can in the diagram. • Keep records and notes showing how the squares are found and counted.

• 170 • New Wave Maths Book G – Teachers Guide

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

Unit 38–2

Student page 113

Outcomes

Indicators

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

Resources

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

Skills • problem-solving

Language • descending order • divide

• calculator • coloured pencils • 2-cm cubes

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

Notes

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

ew i ev Pr

Teac he r

• The focus for this unit is ordering 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) Warm up

• Display on blackboard/whiteboard or using an overhead a very simple family tree with grandparents, mother and father and two children. Grandmother A + Grandfather A Grandmother B + Grandfather B

Father

Child 1

Child 2

. te

m . u

w ww

• 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 possible routes taken between A and D. alternative routes shown by lines

o c . che e r o t r s super

• Read the directions to students from their workbook activity for completion of the arrow diagram.

What to do

• Suggest to students that is may be easier to see the combinations they choose by using a different coloured pencil for each combination. • All the combinations must include one shirt, one coat, one pair of trousers and one pair of shoes.

Challenge • Distribute the 2-cm cubes to individual students. • Students show all possible combinations and explain why the combination is correct or incorrect. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 171 •

Unit 38–3

Student page 114

Outcomes

Indicators

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

The student is able to: • use place value to read, write, say and interpret large whole numbers, oral or written. • read and order dates in a time line.

Skills • completing a table • ordering

Memory Masters (N4.3)

Resources • calculator

• subtract • time line • decade • century • chronological order

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

ew i ev Pr

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 1000 to a whole number less than 10.

Teac he r

Language

• Ask students to explain what a decade and a century are. • A decade in time starts from 1 and goes to 0; e.g. 21–30; 61–70. • Consider language—‘deca’ = 10, decagon etc. • A century in time also starts from 1 and goes to 0; e.g. 1701–1800, 1401–1500. • When students write the decade that a year falls in they need to write the starting year and final year; e.g. 1601–1610. If writing the century, the same applies 1701–1800.

w ww

• Students should write both the decade and the century in the spaces provided. • When decades and centuries have been entered students then write the order of events, starting with 1 for the first explorer to voyage to Australia. Where two explorers arrived in the same decade, take the original list of year of voyage to ascertain correct order.

Challenge

. te

m . u

What to do

o c . che e r o t r s super

• Use the Internet to explore the Julian and Gregorian calendars.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 54 – 55. • 172 • New Wave Maths Book G – Teachers Guide

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

Unit 38—Answers

Student pages 112 – 114

Unit 38–1 1. 3 x 33, 90 + 9, 100 – 1, 198 ÷ 2 8.3 (c) 3.5 (d) 2.7 2. (a) 0.3 (b) (e) 0.41 (f) 0.28 3.

1. 1.814, 1.423, 1.181, 1.163, 1.149, 1.147, 1.127, 1.03, 1.019, 1.002 (b) 0.08 (c) 1.67 (d) 1.7056 2. (a) 0.8 (e) 6.06 (f) 21.2375 3. 24 Challenge Teacher check

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

ew i ev Pr

Teac he r

4. (a) 1 (g) 3 (m) 6 (b) 0 (h) 1 (n) 6 (c) 1 (i) 6 (o) 7 (d) 1 (j) 4 (p) 4 (e) 4 (k) 6 (q) 12 (f) 5 (l) 4 (r) 20 5. (a) 15 (b) 16 (or 17 if always rounded to even) Challenge 25 (if rectangles are also considered squares)

Unit 38–2

. te

• Provide students with further opportunities to round fractions by writing more examples on the board.

Consolidation 38–2 • Students use an arrow diagram to show their family tree.

Consolidation 38–3

m . u

w ww

1. (a) 284 (b) 153 (c) 200 (d) 278 (e) 432 (f) 355 (g) 281 (h) 526 (i) 143 (j) 246 2. (a) 5.07 (b) 2.587 (c) 0.074 (d) 53.24 (e) 45.26 (f) 77.1 3.

• Students make their own time line of a series of events.

o c . che e r o t r s super

Challenge Teacher check

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

New Wave Maths Book G – Teachers Guide • 173 •

Unit 39–1

Student page 115

Outcomes

Indicators

N4.3, S4.1

The student is able to: • use conventional maps to find locations and paths which meet everyday specifications, such as the closest post office or the safest route.

Skills • reading a map • calculating • problem-solving

Memory Masters (N4.3)

Resources • calculator • coloured pencils • local road map

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 (S4.1) Warm up

Notes

ew i ev Pr

Number (N4.3)

w ww

• The map in the workbook shows a network of roads joining the towns that a delivery driver visits. • Students are to find a route that visits each town once only and does not backtrack on any road. The route taken needs to be the shortest available. • Use coloured pencils to show different routes to assist in determining the shortest. • Calculate the number of kilometres covered in this route. • How many different routes are there? • Calculate the time taken to cover the route if the average speed is 75 km per hour and an average of 35 minutes is spent at each stop. • Calculate the time the driver needs to start in order to finish at 7.00 p.m. • Share their findings with the class.

. te

m . u

• Discuss features on a large map of Western Australia.

Challenge

• add • economy • speed

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

• The focus for this unit is subtraction of a whole number less than 100 from a multiple of 10 less than 1 000.

What to do

Language

o c . che e r o t r s super

• Use a road map to design a paper route for your area, using your house as the starting point. Make sure you do not have to double back over any roads.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 10 – 11. • 174 • New Wave Maths Book G – Teachers Guide

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

Unit 39–2

Student page 116

Outcomes

Indicators

N4.3, N4.1a

The student is able to: • understand the multiplicative nature of the relationship between places for whole numbers; i.e. as they move from right to left, each place is 10 times the one before. • ‘count’ in decimal fractions.

Skills • adding • recording • ordering

Resources

Language • add • column • first • number • subsequent • place value • symbol

• calculator

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

Notes

Memory Masters (N4.3)

Teac he r

ew i ev Pr

• The focus for this unit is multiplication of a whole number 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

© R. I . C.Publ i cat i ons orr evi ew pur posesonl y• What to• dof • Discuss with students the relationship between place value columns. Ask what they think would happen if you added 10 to a number. (The tens column increases by 1.) • What happens if 1000 is added to a number? (The thousands column increases by 1.)

w ww

Challenge

• Students are to explore the possibilities to find the answer to the problem. • Show all attempts and keep records of how each attempt was made.

. te

m . u

• Complete the exercise in the workbook and answer the questions as class discussion. • Exercise 4 requires students to find the order relationship between the sets of numbers. Revise greater than > and less than< symbols before asking students to complete the exercise.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 175 •

Unit 39–3

Student page 117

Outcomes

Indicators

N4.3, M4.2

The student is able to: • compare and order length, capacity and mass measurements provided in common standard units.

Skills

Resources • calculator • variety of 2-D shapes

• calculating • problem-solving

Memory Masters (N4.3)

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 (M4.2)

ew i ev Pr

Teac he r

• subtract • minute, second, hour, day, week, month, year • shape • rectangular prism

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

Warm up

Language

• Discuss with students the relationship between units of time – seconds, minutes, hours, days, weeks, months and years.

w ww

• Work with the whole class to complete the relationships in the workbook. • Use this information and a calculator (if required) to find the answers to the time sums in the workbook. • Discuss answers as a whole class.

Challenge

. te

• Students are to discover the shapes required to make a rectangular prism, then draw a rectangular prism. • Keep notes of all activities to discover the shapes. • Keep a record of all drawing attempts.

m . u

What to do

o c . che e r o t r s super

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 114 – 115. • 176 • New Wave Maths Book G – Teachers Guide

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

Unit 39—Answers

Student pages 115 – 117

Unit 39–1

1. (a) 804 (b) 406 (c) 603 (d) 606 (e) 604 (f) 808 (g) 909 (h) 408 (i) 206 (j) 606 2. (a) 63/6 = 61/2 (d) 35/10 = 31/2 (b) 43/8 (e) 53/12 = 51/4 1 (c) 8 /7 (f) 46/10 = 43/5 3.

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

(a) Same as number being added. (b) Teacher check 4. (a) > > (f) = > (b) > < (g) > > (c) < < (h) > < (d) > < (i) > < (e) < > (j) > <

ew i ev Pr

Teac he r

1. (a) 224 (b) 115 (c) 146 (d) 245 (e) 138 (f) 212 (g) 126 (h) 208 (i) 119 (j) 117 2. (a) 6/5 = 11/5 (d) 9/7 = 12/7 (b) 8/6 = 11/3 (e) 4/3 = 11/3 5 1 (c) /4 = 1 /4 (f) 8/5 = 13/5 3. Friendmantle, Ferth, Fidland, Kuchea, Dindin, Mindoon, Doodyay, Gortham, The Ponds, York, Sheverley, Crookton, Armydale (a) 513 km (b) Yes. Longer. Teacher check explanation. (c) 13 hrs 50 mins (d) 5.10 a.m. Challenge Teacher check

Unit 39–2

Challenge 6

. te

• Use a local map to find the best route to deliver the local paper to all householders.

Consolidation 39–2 • Students work in pairs to develop sets of numbers. Swap work and use >, < and = to show order relationship.

Consolidation 39–3

m . u

w ww

1. 2 x 8 x 2 x 2, 4 x 4 x 4, (other answers are possible) 2. (a) 41/7 (d) 71/9 4 1 (b) 3 /12 = 3 /4 (e) 82/4 = 81/2 65/10 = 61/2 (c) 61/5 (f) 3. (a) 60 secs (b) 60 mins = 3600 secs (c) 24 hours = 1440 mins = 86 400 secs (d) 7 days = 1680 hrs = 10 080 mins = 604 800 secs (e) 12 mths = 52 wks = 365 days = 8760 hrs 4. (a) 131 mins (b) 192.3 secs (c) 112 hrs (d) 222 mins (e) 385 mins (f) 205 (g) 157 Challenge rectangles and squares

• Students complete a table to find how many seconds minutes hours days weeks months years they have been alive.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 177 •

Unit 40–1

Student page 118

Outcomes

Indicators The student is able to: • fill in number sequences involving constant multiplication or division.

N4.1a, N4.3, N4.4

Skills

Resources • calculator • coloured pencils

• completing patterns • calculating • using a calculator

Language • round • nearest hundredth • subtract • number pattern • expanded form • total value • power • network • continuous

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

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 (N4.4) Warm Up

• Number patterns provide an interesting challenge to determine the sequence that has been used to create the pattern. In some cases more than one pattern may be used. • Write the following patterns on the blackboard/whiteboard and ask students to solve them as a whole class. 3, 6, 12, 24, _, _ (doubling) 4, 8, 12, 16, _, _ (multiples of 4) 3, 4, 6, 8, 12, 12, 24, 16, _, _ (combination of the above two sequences)

ew i ev Pr

Teac he r

• The focus for this unit is rounding of decimal numbers to the nearest hundredth.

w ww

• Ask students to complete the patterns shown in their workbook. • Share their solutions with the whole class. • Exercise 4 is a different form of a pattern. Each number is raised to a power. The pattern is formed by multiplying the number by itself by the number of times shown by the power it is raised to; e.g. 82 is 8 x 8—raised to the fifth power; e.g. 85 is 8 x 8 x 8 x 8 x 8. • Complete the exercise.

Challenge

. te

m . u

What to do

o c . che e r o t r s super

• Students should use coloured pencils to show each attempt at passing over the network with a continuous line. • Keep notes explaining each path attempted.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 88 – 91. • 178 • New Wave Maths Book G – Teachers Guide

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

Unit 40–2

Student page 119

Outcomes N4.2, N4.3, C&D4.1

Skills • tossing a coin • tallying • recording • problem-solving

Indicators

Resources

Language

The student is able to: • order probability devices from the one most likely to the one least likely to produce an outcome.

• calculator • isometric grid paper (page 202) • coloured pencils • coins • counters • tin

• estimation • number sentences • divide • table • Pascal’s triangle • mathematician • probabilities

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

Notes

Memory Masters (N4.2)

Teac he r

ew i ev Pr

• The focus for this unit is completion of number sentences working with brackets and multiple operations (distributive property).

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&D4.1) Warm up

What to do

w ww

m . u

• Discuss chance processes and probability with the class. ‘In a bag I have two red counters and one yellow counter. What is the probability of me choosing the yellow counter?’ (1 in 3 or 1/3). ‘What is the probability of choosing a red counter?’ (2 in 3 or 2/3). Ask a student to draw one counter out, 20 times. Replace the counter after each draw. Keep a tally to see if your predictions are correct. • Pascal, a French mathematician, discovered a mathematical solution to determining the probability of throwing either a head or a tail when flipping a coin. His solution is known as Pascal’s triangle.

• With one coin there is an equal chance of throwing a head or a tail, that is 1:1 or 1/2. • There are several patterns embedded in the triangle.Their patterns may be used to extend the table. If students are to extend the grid table it is suggested that they use isometric grid paper. • Answer the questions on the page using Pascal’s triangle and your calculator. The class may be best divided into small groups. • Share their findings.

. te

Challenge

o c . che e r o t r s super

• Students are to explore ways of completing a star in the octagon without lifting their pencil during the drawing. • Write a description of how the star was drawn. • Use different coloured pencils to show each attempt.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 150 – 151. R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 179 •

Unit 40–3

Student page 120

Outcomes

Indicators

N4.2, N4.3, N4.4

The student is able to: • describe a sequence sufficiently for a peer to reproduce it.

Skills • completing a table • investigating • recording

Memory Masters (N4.2)

Resources • calculator • various shaped pyramids • pencil

• multiply • investigate • table • draw • relationship • faces, vertices, edges • pyramids • triangles, square, pentagonal, hexagonal, octagonal • patterns

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

ew i ev Pr

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

• The focus for this unit is working with the distributive property of operations – the order of completion of operations.

Teac he r

Language

w ww

• Record the number of faces (not including the base), vertices and edges on the table provided in the workbook. • Students are to draw each pyramid shape in their workbook. • Investigate the relationships that emerge from the faces, vertices and edges. • Write an explanation of each of the patterns found between the relationships. • Share the findings of each group with the whole class.

Challenge

. te

m . u

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• What to do • Display or distribute pyramids among groups of students so they can investigate the faces, vertices and edges. • The relationship between vertices, faces and edges was discovered by Euler. He also worked on topology or networks.

o c . che e r o t r s super

• Students are to write and record all their workings in finding the solution to this problem. • Share their solutions with the class.

For a relevant assessment activity refer to RIC-0087 Maths Assessment – Level 4 pages 52 – 53. • 180 • New Wave Maths Book G – Teachers Guide

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

Unit 40—Answers

Student pages 118 – 120

Unit 40–1

1. (a) 157 (b) 136 (c) 136 (d) 122 (e) 103 (f) 181 (g) 112 (h) 22 (i) 30 (j) 22 (b) 7/10 (c) 11/9 (d) 11/5 2. (a) 9/10 (e) 6/7 (f) 8/9 3. 4.

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

5. Teacher check 6. The total of 2 touching triangles = the triangle below. 7. Add along the row Challenge

ew i ev Pr

Teac he r

1. (a) 47.65 (b) 6.28 (c) 0.93 (d) 5.28 (e) 16.24 (f) 2.38 (g) 5.68 (h) 17.29 (i) 19.82 (j) 0.21 85/6 (c) 62/3 (d) 41/10 2. (a) 35/8 (b) (e) 73/7 (f) 52/3 3. (a) 32, 64, 128 (double) (b) 10, 12, 13 (+1, +2) (c) 8, 12, 10 (2, 4, 6 = multiples of 2 3, 6, 9 multiples of 3) (d) 60, 53, 46 (–7) (e) 125, 216, 343 (cube consecutive numbers) 4. (a) 2187 (b) 256 (c) 4096 (d) 3125 (e) 1296 (f) 32 768 Challenge

Unit 40–2

. te

• Students develop their own patterns. Swap with a partner to solve.

Consolidation 40–2 • Research, using the Internet, to find out about Pascal.

Consolidation 40–3

m . u

w ww

1. (a) 38 (b) 27 (c) 40 (d) 62 (e) 48 (f) 48 (g) 120 (h) 24 (i) 27 (j) 40 5 2. (a) 2/6 = 1/3 (d) /10 = 1/2 6 /12 = 1/2 (b) 4/8 = 1/2 (e) 3 1 2 (c) /9 = /3 (f) /10 = 1/5 3.

• Students use a 50c coin to find faces, vertices and edges.

o c . che e r o t r s super

Number of faces = number of sides of base shape Number of vertices = number of sides of base shape + 1 Number of edges = number of sides of base shape x 2 Challenge 400 Rings 1 2 3 4 5 6 7 8 9 10 Guests 1 3 5 7 9 11 13 15 17 19 Rings 11 12 13 14 15 16 17 18 19 20 Guests 21 23 25 27 29 31 33 35 37 39 total = 400

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

New Wave Maths Book G – Teachers Guide • 181 •

Teac he r

ew i ev Pr

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

w ww

. te

m . u

o c . che e r o t r s super

• 182 • New Wave Maths Book G – Teachers Guide

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

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

ew i ev Pr

Teac he r

Additional Activities

w ww

. te

Contents

m . u

o c . che e r o t r s super

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

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

New Wave Maths Book G – Teachers Guide • 183 •

Space Activities S4.1

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

ew i ev Pr

Teac he r

1. Use local community maps in activities to locate places and features within the neighbourhood. 2. Students work in pairs to give unambiguous instructions for moving and locating objects in their class or school environment. Encourage students to use directions such as: left, right, north, south, east, west, 90° turn, 45° turn, 180° turn etc. S4.2 1. Provide recycled materials for students to use to make miniature models of real-life objects. For example, students could make a model of their bedroom and the items in it, then use these models to rearrange the furniture to its best position. This will require students to ensure models are to scale, at least with each other. 2. Students draw simple and complicated 3-D models, including the use of solid lines for what they can see and dotted lines for the parts of the models that can not be seen. S4.3 1. Make a simple shape with dimensions less than 2 cm. Put your initials or name on one side. Use the 2-cm grid paper on page 201 of New Wave Maths Teachers Guide and describe how you would move your shape from one square to another by flipping, sliding and rotating. Working with a partner, see if you can find another way to move between the two squares. Which was the easier? Repeat this activity several times. Change partners and find out if they have any different moves. 2. Use a centre point grid on page 200 of New Wave Maths Teachers Guide to create a pattern. Draw a simple or complex pattern in one quadrant of the grid, then rotate or reflect the pattern and repeat it in each of the other three quadrants. State whether you have rotated or reflected the pattern each time.

w ww

. te

m . u

3. From a supply of magazines, newspapers or old copies of the Yellow Pages®, find a number of logos or trademarks. Cut them out. Check them for symmetry or design – reflectional or rotational. Make your own logo or trademark with at least one line of symmetry. Make a logo for your favourite sport with a line of reflectional symmetry. Design a logo for your school with a line of rotational symmetry. S4.4 1. Use Venn diagrams on page 219 of New Wave Maths Teachers Guide to classify three-dimensional shapes on attributes. Classifications may look at: • odd and even vertices; • plane and non-plane faces; • prisms, pyramids and others; • regular and irregular; • square faces, triangular faces and other faces; or • students organise their own classifications.

o c . che e r o t r s super

2. Students are to construct a bridge to span a gap of 15 cm. Students may use one of two sets of materials: (a) 40 straws and 20 dressmaking pins; or (b) 10 sheets of newspaper and masking tape. Bridges may not be attached to the surface on which they stand. Bridges must be able to support a minimum of 100 g mass suspended from the midpoint of the bridge.

3. Investigate shapes in buildings, structures, play equipment and other parts of the environment. Give reasons why certain shapes are used in particular situations; for example, the triangle for rigidity. • 184 • New Wave Maths Book G – Teachers Guide

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

Measurement Activities M4.1 1. Students compare the capacity of a variety of containers, using common attributes and sensible statements. For example, a 1-litre milk carton will hold less than a 2-litre ice-cream container because the carton is smaller than the container. 2. Provide students with various real-life situations in which they are required to distinguish the difference between perimeter and area. For example, new fencing around the school would require the knowledge of the perimeter of the school. New lawn for a particular area within the school grounds would require the knowledge of the area. M4.2 1. Within the limits of the school premises, mark out an area 100 m x 100 m – one hectare. If space does not allow, mark out either an area 100 m x 50 m or 50 m x 50 m to give students an idea of 0.5 or 0.25 of a hectare and, therefore, some indication of the actual size of a hectare.

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

ew i ev Pr

Teac he r

2. Using kitchen or bathroom scales, measure the mass of a variety of selected or given objects. Estimate the mass first. Have students record their results on a chart. Possible objects: container of nails; child; brick; large stone; or containers of sand, rice, stones or leaves. 3. Set up a weather station to measure wind speed and direction, temperature, rainfall and possibly humidity. Commercial weather stations are available for purchase. Take readings each day and plot temperature means and monthly rainfall on graphs.

M4.3 1. Class trips/excursions should be used to estimate distance and check odometer readings to find actual distances.

2. Measure students’ steps to find how long their step is, then use this to measure the distance around the school oval. Extend this to measure one kilometre. Accuracy may be checked by using a trundle wheel to measure one kilometre accurately.

w ww

3. Collect a variety of containers to compare capacities. (a) Compare 2 L soft drink bottles, 2 L cordial bottles and 2 L ice-cream containers. (b) Compare 1 L milk cartons, 1 L standard measuring containers and 1 L soft drink bottles. (c) Compare 500 mL containers with each other. (d) Discuss the actual capacity of each of the containers, making suggestions as to why they change.

. te

m . u

o c . che e r o t r s super

4. Use a number of different small containers (between 200 mL and 500 mL) and mark on the container the level you think the capacity of the container should be. Use a graduated measuring container to check the accuracy of your estimate.

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?

6. Improvise a spring balance by joining strong elastic bands together to make your own scales. Calibrate by weighing objects of known mass.

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

New Wave Maths Book G – Teachers Guide • 185 •

Measurement Activities Cont. 7. Encourage students to estimate the length of time particular events will take or have taken. Bus trips, TV programs, lesson times and the time taken to walk to or from school are some examples. Each should be checked for accuracy. Many of these activities can be done incidentally.

M4.4a 1. Use 1-metre lengths of dowel joined with plastic tubing or large lumps of modelling clay to make a model of one cubic metre. Estimate how many cubic metre models would fit in your classroom.

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

2. Use a variety of containers with the same base area but different heights – measure both surface area and base area to see what the ratio between surface area and volume is in each case.

Teac he r

ew i ev Pr

3. Use a sheet of paper or card, about A4 size, to see how many different containers you can make without cutting. Measure the surface area and volume of each to find out which is the greatest. No ends are required on models.

M4.4b 1. Use the distorted grid paper on pages 203 – 204 of the New Wave Maths Teachers Guide to allow students opportunities to draw a figure in a specified way. 2. Use small numbers of cubes to build an arrangement. Ask students to increase the size of the arrangement by double or triple the amount, so the new arrangement looks the same as the original, just larger.

w ww

. te

m . u

o c . che e r o t r s super

• 186 • New Wave Maths Book G – Teachers Guide

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

Number Activities N4.1a

Provide opportunities to read scales where each calibration is not labelled; for example, a measuring tape marked in hundredths but labelled in tenths, a scale marked in grams but labelled in kilograms.

N4.1b

Students use paper tape to fold into equal parts and shade a particular number of parts to show fractional amounts. For example, fold the paper tape into five equal parts and shade three of those parts to show 3/5.

N4.2

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

Where possible, give the students as much practice as possible in problem solving. Use the suggested outline on page 241 of New Wave Maths Teachers Guide to attempt problem-solving activities.

Teac he r

Encourage students to create and explain patterns, providing a starting point and a rule for their partners to follow. Check to see if the explanation was clear enough, matching the pattern against the original.

w ww

. te

m . u

N4.4

ew i ev Pr

N4.3 1. Estimation is an essential skill and should be insisted on prior to any calculation. New Wave Maths Teachers Guide provides many opportunities throughout the book for the development of estimation skills. 2. Have students estimate how many people can stand in their classroom—treat this as a record attempt Students describe how this may be checked without filling the room. 3. Estimate, then calculate, how many months, days and hours you have been alive. 4. Estimate, then calculate, how many letters are used in a whole newspaper, library book and/or your class reading book. Describe how you worked out your answer.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 187 •

Teac he r

ew i ev Pr

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

w ww

. te

m . u

o c . che e r o t r s super

• 188 • New Wave Maths Book G – Teachers Guide

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

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

ew i ev Pr

Teac he r

Assessment

w ww

. te

Contents

m . u

o c . che e r o t r s super

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

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

New Wave Maths Book G – Teachers Guide • 189 •

Student Outcomes Working Mathematically

Chance and Data

WM4.1 The student compares the ways in which familiar mathematics is done or used in own and other communities.

C&D4.1 The student places events in order from those least likely to those most likely to happen on the basis of numerical and other information about the events.

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

and paths.

S4.3

S4.4

Number

The student attends to the shape, size and placement of parts when matching, making and drawing things, including making nets of 3-D models which can be seen and handled and using some basic conventions for drawing them.

N4.1a The student reads, writes, says, counts with and compares whole numbers into the millions and decimals (equal number of places).

N4.1b The student reads, writes, says and understands the meaning of fractions and, for readily visualised fractions, estimates their relative size and position on a number line and shows equivalence between them.

The student recognises rotations, reflections and translations in arrangements and patterns and translates, rotates and reflects figures and objects systematically to produce arrangements and patterns.

N4.2

The student understands the meaning, use and connections between the four operations on whole and decimal numbers, and uses this understanding to choose appropriate operations (whole multipliers and divisors) and construct and complete equivalent statements.

N4.3

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.

The student selects, describes and compares figures and objects on the basis of spatial features, using conventional geometric criteria.

w ww

Measurement M4.1

The student selects appropriate attributes, distinguishes perimeter from area and time from elapsed time, and chooses units of a sensible size for the descriptions and comparisons to be made.

M4.2

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.

M4.3

The student uses the known size of familiar things to help make and improve estimates, including centimetres, metres, kilograms, litres and minutes.

. te

m . u

S4.2

ew i ev Pr

Teac he r

WM4.2 The student asks questions to clarify the essential C&D4.2 The student collaborates with peers to plan what mathematical features of a problem and uses data to collect and how to classify, sequence and problem-solving strategies which include those tabulate them to answer particular questions, and based on identifying and organising key information. sees the need to vary methods to answer different questions. WM4.3 The student uses examples to support or refute mathematical conjectures and attempts to make C&D4.3 The student displays frequency and measurement simple modifications of conjectures on the basis of data using simple scales on axes and some examples. grouping, and summarises data with simple fractions; highest, lowest and middle scores; and WM4.4 The student checks, when prompted, that answers means. are roughly as expected and that methods and answers make sense. C&D4.4 The student reads and makes sensible statements about the information provided in tables, diagrams, Space line and bar graphs, fractions and means, and comments on how well the data answer their S4.1 The student uses distance, direction and grids on questions. maps and plans and in descriptions of locations

o c . che e r o t r s super 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 G – Teachers Guide

Extracted from Mathematics Outcomes and Standards Framework – Student Outcome Statements, Education Department of Western Australia 1998.

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

Working Mathematically—Record Sheet

Apply and Verify

Reason Mathematically

Mathematical Strategies

Comment

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

ew i ev Pr

Teac he r

Student Name

Contextualise Mathematics

Outcome Category

w ww

. te

m . u

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 191 •

Space—Record Sheet

Reason Geometrically

Represent Transformations

Comment

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

ew i ev Pr

Teac he r

Student Name

Represent Shape

Represent Location

Outcome Category

w ww

. te

m . u

o c . che e r o t r s super

• 192 • New Wave Maths Book G – Teachers Guide

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

Measurement—Record Sheet

Indirect Measure

Estimate

Direct Measure

Comment

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

ew i ev Pr

Teac he r

Student Name

Understand Units

Outcome Category

w ww

. te

m . u

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 193 •

Chance and Data—Record Sheet

Interpret Data

Summarise and Represent Data

Comment

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

ew i ev Pr

Teac he r

Student Name

Collect and Organise Data

Understand Chance

Outcome Category

w ww

. te

m . u

o c . che e r o t r s super

• 194 • New Wave Maths Book G – Teachers Guide

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

Number—Record Sheet

Reason about Number Patterns

Calculate

Understand Operations

Comment

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

ew i ev Pr

Teac he r

Student Name

Understand Numbers

Outcome Category

w ww

. te

m . u

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 195 •

Proforma This proforma has been provided for you to copy and use with your class. You can either: • select an activity and evaluate the whole class; or • select a small group of students and evaluate their work. The indicators are found on the relevant page in the New Wave Maths Teachers Guide.

1. Photocopy this page. 2. Write the appropriate date, strand, outcome(s) and indicators. 3. Photocopy enough for one per student being assessed. 4. Inform the students they are being assessed on the activity they are about to complete. 5. Students complete the activity in the workbook. 6. Mark the work completed by the student.

r o e t s Bo r e p o u k Mathematics Proforma S 7. Attach the proforma to the appropriate workbook page. 8. Record evaluation as required.

Teac he r

✄

Date

Outcome(s)

ew i ev Pr

Strand

Indicators Demonstrated Needs Further Opportunity

w ww

✄

Mathematics Proforma Strand

. te

Date

m . u

Classroom Teacher

o c . che e r o t r s super

Outcome(s)

Indicators Demonstrated Needs Further Opportunity

Classroom Teacher • 196 • New Wave Maths Book G – Teachers Guide

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

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

ew i ev Pr

Teac he r

Photocopiable Resources

w ww

. te

Contents

m . u

o c . che e r o t r s super

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

Grid Paper...............................................................................................................198 – 204 Number Charts and Cards.................................................................................205 – 208 Place Value Charts.................................................................................................209 – 210 Fraction Chart and Number Line................................................................................. 211 Spinners – Blank................................................................................................................. 212 Calendar – Any year........................................................................................................... 213 Bingo Cards............................................................................................................214 – 217 3-D Model Attribute Table.............................................................................................. 218 Venn diagrams – Blank...................................................................................................... 219 3-D Shapes......................................................................................................................... 220 Tangrams.................................................................................................................221 – 224 Nets..........................................................................................................................225 – 231 Paper Circles..................................................................................................................... 232 Curve Stitch and Line Pattern............................................................................233 – 234 Graph and Table – Blank................................................................................................... 235 New Wave Maths Book G – Teachers Guide • 197 •

Teac he r

ew i ev Pr

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

w ww

o c . che e r o t r s super

1-cm dot grid paper.

. te

m . u

• 198 • New Wave Maths Book G – Teachers Guide

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

Teac he r

ew i ev Pr

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

w ww

o c . che e r o t r s super

1-cm grid paper.

. te

m . u

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

New Wave Maths Book G – Teachers Guide • 199 •

Teac he r

ew i ev Pr

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

w ww

o c . che e r o t r s super

1-cm centre point grid paper.

. te

m . u

• 200 • New Wave Maths Book G – Teachers Guide

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

Teac he r

ew i ev Pr

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

w ww

o c . che e r o t r s super

2-cm grid paper.

. te

m . u

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

New Wave Maths Book G – Teachers Guide • 201 •

Teac he r

ew i ev Pr

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

w ww

o c . che e r o t r s super

1-cm triangle grid paper.

. te

m . u

• 202 • New Wave Maths Book G – Teachers Guide

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

Teac he r

ew i ev Pr

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

w ww

o c . che e r o t r s super

Distorted grid.

. te

m . u

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

New Wave Maths Book G – Teachers Guide • 203 •

Teac he r

ew i ev Pr

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

w ww

o c . che e r o t r s super

Distorted grid.

. te

m . u

• 204 • New Wave Maths Book G – Teachers Guide

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

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 B58 59 60 r 51 52pe 53 54 55 56 57 o ok u S62 63 64 65 66 67 68 69 70 61

ew i ev Pr

Teac he r

41 42 43 44 45 46 47 48 49 50

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

✄

0 – 99 Chart

w ww

m . u

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

30 . te 40 50

31 32 33 34 35 36 37 38 39

o 41 42 43 44 45 46 47 48c 49

. c e 51h 52 53 54 55 s 56t 57r58 59 er o 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 R.I.C. Publications® www.ricpublications.com.au

New Wave Maths Book G – Teachers Guide • 205 •

Basic Facts

8 0 8 16 24 32 40 48 56 64 72 80

9 0 9 18 27 36 45 54 63 72 81 90

10 0 10 20 30 40 50 60 70 80 90 100

X 0 1 2 3 4 5 6 7 8 9 10

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

9 0 9 18 27 36 45 54 63 72 81 90

10 0 10 20 30 40 50 60 70 80 90 100

Teac he r

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

ew i ev Pr

w ww

. te

m . u

✄

X 0 1 2 3 4 5 6 7 8 9 10

o c . che e r o t r s super

• 206 • New Wave Maths Book G – Teachers Guide

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

Number Cards

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

three

ew i ev Pr

Teac he r

one

. te seven

m . u

w ww

Photocopy onto coloured card. Cut out and laminate. You may wish to enlarge to A3.

•f orr evi ew pur posesonl y•

nine o c . che e r o t r s super eight

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

New Wave Maths Book G – Teachers Guide • 207 •

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

ew i ev Pr

Teac he r

Whole Number Lines

w ww

. te

m . u

o c . che e r o t r s super

• 208 • New Wave Maths Book G – Teachers Guide

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

1000

100

•

/10

/100

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

ew i ev Pr

Teac he r

10 000

w ww

. te

/100

m . u

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 209 •

Tens

Ones

•

Tenths

Hundredths

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

ew i ev Pr

✄

Teac he r

Ten thousands Thousands Hundreds

w ww

. te

m . u

Ten thousands Thousands Hundreds

o c . che e r o t r s super

• 210 • New Wave Maths Book G – Teachers Guide

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

Fraction Chart 1/ 10 1/ 9 1/ 8 1/ 6 1/ 5

1/ 3

ew i ev Pr

Teac he r

1/ 4

r o e t s Bo r e p ok u S 1/ 2 1 whole or 1/1

•f or0.25 r evi ew pu r poseson l y• 0.5 0.75 1/ 4

1/ 8

w ww 0.11

1/ 9

0.1

1/ 10

. te

0.17

1/ 6

0.22 2/ 9

0.2

1/ 5

1/ 2

3/ 8

5/ 8

3/ 4

o c . che e r o t r s super 0.33

0.44

1/ 3

0.3 3/ 10

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

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

7/ 8

m . u

7/ 9

0.7 7/ 10

0.8

4/ 5

0.83

5/ 6

0.89 8/ 9

0.9 9/ 10

New Wave Maths Book G – Teachers Guide • 211 •

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

ew i ev Pr

Teac he r

Spinners

w ww

. te

m . u

o c . che e r o t r s super

• 212 • New Wave Maths Book G – Teachers Guide

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

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

ew i ev Pr

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

Teac he r

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

. te

m . u

w ww

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

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

New Wave Maths Book G – Teachers Guide • 213 •

27 100 14

16 100 28

100 90

14 . t18

ew i ev Pr

r o e t s Bo r e p o u 36 S 2 80 60 21 k12 30

100

w ww

o c . che e r o t 30 4 r 6 100 15 3 s super

• 214 • New Wave Maths Book G – Teachers Guide

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

Note: These cards can be cut along the dark, solid lines to make bingo cards.

m . u

Teac he r

100 12

ew i ev Pr

Teac he r

r o e t s Bo r e p o u 80 S18 5 12 9 8k 6

16 . t 21

m . u

w ww

Note: These cards can be cut along the dark, solid lines to make bingo cards.

o c . che e r o t 6 r 2s 4 8 s 2 40 uper

28 100 16

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

New Wave Maths Book G – Teachers Guide • 215 •

r o e t s Bo r e p o u 5 4 35 40 25 k9 S

100

20 . te

o c . che e r o t 100 40r 10 4 2 50 s super

• 216 • New Wave Maths Book G – Teachers Guide

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

Note: These cards can be cut along the dark, solid lines to make bingo cards.

w ww

m . u

ew i ev Pr

Teac he r

r o e t s Bo r e p o u k 25 S 4 25 100 21 25

28 . t 32

Teac he r

ew i ev Pr

m . u

w ww

Note: These cards can be cut along the dark, solid lines to make bingo cards.

o c . che e r o t 35 r 18 32 12 s 12 50 super

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

New Wave Maths Book G – Teachers Guide • 217 •

Teac he r

Attributes

ew i ev Pr

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

3-D Model

o c . che e r o t r s super

• 218 • New Wave Maths Book G – Teachers Guide

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

Note: This table can be used to explore any attributes of 3-D shapes.

w ww

. te

m . u

Teac he r

ew i ev Pr

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

✄

. te

m . u

w ww

Note: These Venn diagrams can be used as required by the students to sort items into categories.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 219 •

Teac he r

ew i ev Pr

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

o c . che e r o t r s super

• 220 • New Wave Maths Book G – Teachers Guide

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

Note: Students list or draw examples of these shapes found in their environment.

w ww

. te

m . u

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

ew i ev Pr

Teac he r

Circular Tangram

. te

m . u

w ww

Enlarge to A3. These circles are used together as the circular tangram.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 221 •

Enlarge to A3. Use this tangram puzzle to make various shapes.

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

ew i ev Pr

Teac he r

Tangram

o c . che e r o t r s super

• 222 • New Wave Maths Book G – Teachers Guide

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

Enlarge to A3. Use this Pythagorean puzzle to make various shapes.

w ww

. te

m . u

•f orr evi ew pur posesonl y•

Teac he r

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

ew i ev Pr

Enlarge to A3. Use this circle puzzle to make various shapes.

Circle Puzzle

. te

m . u

w ww

Enlarge to A3. Use this broken heart puzzle to make various shapes.

•f orr evi ew pur posesonl y•

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 223 •

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

ew i ev Pr

Teac he r

The Magic Egg Puzzle

o c . che e r o t r s super

• 224 • New Wave Maths Book G – Teachers Guide

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

Enlarge to A3. Use this magic egg puzzle to make various shapes.

w ww

. te

m . u

Teac he r

ew i ev Pr

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

. te

m . u

w ww

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

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 225 •

w ww

. te

m . u

o c . che e r o t r s super

• 226 • New Wave Maths Book G – Teachers Guide

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

Cut out these nets to make an enclosed cylinder. Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

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

ew i ev Pr

Teac he r

Cylinder Net

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

ew i ev Pr

Teac he r

Cube Net

. te

m . u

w ww

Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 227 •

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

ew i ev Pr

Teac he r

Square Prism Net

o c . che e r o t r s super

• 228 • New Wave Maths Book G – Teachers Guide

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

Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

w ww

. te

m . u

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

ew i ev Pr

Teac he r

Rectangular Prism Net

. te

m . u

w ww

Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 229 •

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

ew i ev Pr

Teac he r

Triangular Prism Net

o c . che e r o t r s super

• 230 • New Wave Maths Book G – Teachers Guide

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

Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

w ww

. te

m . u

. te

o c . che e r o t r s super

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

m . u

Teac he r

Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

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

•f orr evi ew pur posesonl y•

ew i ev Pr

w ww

Enlarge to A3. Cut along dotted lines and fold along solid lines. Glue tabs to complete the construction.

Triangular Pyramid Net

New Wave Maths Book G – Teachers Guide • 231 •

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

ew i ev Pr

Teac he r

Paper Circles

w ww

. te

m . u

o c . che e r o t r s super

• 232 • New Wave Maths Book G – Teachers Guide

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

Curve Stitch and Line Pattern 10 9 8 7 6 5 4 3 2

ew i ev Pr

Teac he r

r o e t s Bo r e p ok u S 1

1 2 3 4

w ww 10

. te

m . u

o c . che e r o t r s super 10

9 8 7 6 5 4 3 2 1

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

New Wave Maths Book G – Teachers Guide • 233 •

Curve Stitch and Line Pattern 10 9 8 7 6 5 4 3 2

ew i ev Pr

Teac he r

r o e t s Bo r e p ok u S 1

6 5

w ww

. te

10 1

o c . che e r o t r s super 1

4 5

m . u

6 7 8 9 10

6 5 4 3 2 1

• 234 • New Wave Maths Book G – Teachers Guide

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

Teac he r

ew i ev Pr

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

w ww

. te

m . u

o c . che e r o t r s super

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

New Wave Maths Book G – Teachers Guide • 235 •

Teac he r

ew i ev Pr

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

w ww

. te

m . u

o c . che e r o t r s super

• 236 • New Wave Maths Book G – Teachers Guide

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

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

ew i ev Pr

Teac he r

Parent Information

w ww

. te

Contents

m . u

o c . che e r o t r s super

Expectations of Knowledge of Basic Facts................................................................. 238 Primary School Mathematics..............................................................................239 – 240 Problem-solving Strategies............................................................................................. 241 Concrete to Mental......................................................................................................... 242 Mathematical Learning Areas......................................................................................... 243 Homework Policy............................................................................................................. 244

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

New Wave Maths Book G – Teachers Guide • 237 •

Parent Information Expectations of Knowledge of Basic Facts

Year 1

An informal, general introduction to number and combinations.

Year 2 Discovery approach (manipulating concrete material) to finding and learning addition and subtraction facts.

Year 3

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

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.

ew i ev Pr

Teac he r

Consolidate basic sums to 18, and differences taking from 18. Extend recall of basic multiplication and division facts to facts of 6, 7, 8 and 9 times tables.

The following suggestions can be used at home to assist your child in becoming more proficient at gaining automatic recall of the basic number facts. The ideas are not exclusive; many alternatives may be used.

4 5

7 8

‘Snap’ – played with flashcards. Play as for ordinary snap. A variation – write pairs of numbers on cards, or blank playing cards, without operation signs. Child may add, subtract, multiply or divide the pair of numbers to find a matching pair.

w ww

‘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.

. te

Throw two dice then either add, subtract or multiply the two numbers shown.

m . u

o c . che e r o t r s super

Race the calculator. Call out a basic fact, while you work out the answer using the calculator your child attempts to race you to the correct answer, working mentally. ‘Sums, Differences, Products’ (add, take, multiply) – The game is played using a hundred chart. Call a pair of numbers from the basic facts. Your child covers the sum, the difference and the product of the two numbers called on the hundreds chart. Play for a given time—for example, five minutes,—or until all of a set of basic facts have been used. ‘Bingo’ – The game is played as for ordinary bingo. You call a basic fact, use basic number fact sheet, your child covers the correct answer if it is on the card. First to cover the card or a line wins the game. When using board games encourage your child to add onto the total when throwing the die, or add the total of the dice, rather than counting on. ‘Numero’ is one of the best mathematical games available which can be used at home to develop mathematics skills.

• 238 • New Wave Maths Book G – Teachers Guide

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

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

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

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.

ew i ev Pr

Teac he r

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

w ww

2 .5 6 4 1 0 .3 5 1 2 .4 + 5 .7 2 2 1 .0 3 4

. te 5

6 .8 3 4 – 2 .9 1 7 3 .9 1 7

Year 7

Subtract the numerators 4 – 2 = 2 and write the denominator as it appears.

1+1=2 3 3 3

11 + 22 = 33 5

m . u

Year 6

4– 2=2 5 5 5

Add or subtract the numerator. Add or subtract the whole number. Write the denominator as it appears.

o c . che e r o t r s super 3– 1=2 4 4 4

24 – 13 = 11 5 5 5

Consolidation of above.

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

New Wave Maths Book G – Teachers Guide • 239 •

Parent Information Primary School Mathematics Grouping and Sharing – Multiplication and Division Commencing in Year 2, concrete activities are recorded, using the multiplication symbol in a number sentence from late Year 2 or as ready. Number sentence recording of concrete activities is carried on into Year 3. The written algorithm is introduced in its extended form in Year 4, working to the abbreviated form when the student understands the process. Year 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

r o e t s Bo r e p ok u S 3

1 468 –300 168

Then repeat for the tens. Firstly exchange the 100 for 10 tens which gives 16 tens. The 16 tens are then shared among the 3 people. How many tens does each person receive? (5) How many tens were shared altogether? (16) How many tens are left? (1)

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.

This is recorded as:

ew i ev Pr

Teac he r

This is recorded as:

76 x 58 608 + 3800 4408

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)

w ww

$5 4 . 2 7 x 46 3 2 5 .6 2 2 1 7 0 .8 0 $2 4 9 6 . 4 2 Year 7

$4 7 . 6 9 x 64 1 9 0 .7 6 2 8 6 1 .4 0 $3 0 5 2 . 1 6

15 468 –300 168 – 150 18

. te

This is recorded as:

156 468 –300 168 –150 18 – 18 0

m . u

Year 6

2 x 5

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.

0 .5 8 x 69 5 .2 2 + 3 4 .8 0 4 0 .0 2

468

o c . che e r o t r s super It is assumed this process would be modelled with Multibase Arithmetic Blocks (MAB).

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. • 240 • New Wave Maths Book G – Teachers Guide

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

Parent Information Problem-solving Strategies 1

To assist your child in solving mathematical problems, the following strategies may help. 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.

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

Use appropriate computation: addition, subtraction, multiplication, or division, to work out the problem.

ew i ev Pr

Teac he r

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.

w ww

Write or present conclusions clearly for others to be able to check your findings.

. te

m . u

Check results.

o c . che e r o t r s super

Some Problem-solving Activities Around the Home

The following suggestions can be used at home to assist your child to become more proficient in problem-solving. The ideas are not exclusive; many alternatives may be used.

1 2 3 4 5

When planning the next family holiday, include your child in the planning, budgeting, activities, travelling time, itinerary and allow him/her to help solve any problems which may arise. When renovating your home—painting, replacing flooring, fencing, grassing or reticulating the garden—encourage your child to participate in the planning, costing, measuring and evaluation of the budget. When planning a party, include your child in the planning, catering, shopping and cooking. When fertilising your lawn, invite your child to help you work out how much fertiliser will be required for the area. Also work out the cost and the best way to approach the task to ensure even coverage. When planning your family’s next big purchase, encourage your child to help work out a savings plan.

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

New Wave Maths Book G – Teachers Guide • 241 •

Parent Information Concrete to Mental – Including the Calculator

Dear Parent(s) There are a number of different means of completing the four algorithms. Children start by using concrete materials to work through the algorithms to develop understandings. As their knowledge and understanding develop, students move to more abstract means of achieving the solutions to the algorithms. These solutions may be achieved by pencil and paper calculations or by working the solutions mentally (the ultimate aim).

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

During these developmental phases, children will encounter algorithms that are complex, difficult or are a means to another step or the final solution. In such cases, the child should be encouraged to use a calculator to find the solution to the algorithm. The calculator is an invaluable aid in mathematics and its use is to be encouraged from the very beginning of a child’s days at school.

Teac he r

Should you encounter any problems, please contact me. Kind regards

Classroom Teacher

✄

ew i ev Pr

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.

w ww

Dear Parent(s)

There are a number of different means of completing the four algorithms.

. te

m . u

Concrete to Mental – Including the Calculator

o c . che e r o t r s super

Children start by using concrete materials to work through the algorithms to develop understandings. As their knowledge and understanding develop, students move to more abstract means of achieving the solutions to the algorithms. These solutions may be achieved by pencil and paper calculations or by working the solutions mentally (the ultimate aim). During these developmental phases, children will encounter algorithms that are complex, difficult or are a means to another step or the final solution. In such cases, the child should be encouraged to use a calculator to find the solution to the algorithm. The calculator is an invaluable aid in mathematics and its use is to be encouraged from the very beginning of a child’s days at school. Children who have great difficulties in completing algorithms are to be encouraged to use the calculator to find the solutions after first estimating the answer. Estimation skills are essential in showing the development of mathematical knowledge. Should you encounter any problems, please contact me. Kind regards

Classroom Teacher • 242 • New Wave Maths Book G – Teachers Guide

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

Parent Information Mathematical Learning Areas Mathematics comprises of a series of learning areas. These learning areas are outlined for teachers in the Student Outcome Statements document produced by the Education Department. There are seven learning areas, each of which is outlined briefly below.

Appreciating Mathematics Appreciate the role of mathematics in their own and other communities.

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

Working Mathematically

Space

ew i ev Pr

Teac he r

Thinking about ideas, investigating, applying, verifying and reasoning mathematically. In brief, problem-solving.

Knowledge of location (place), shape, transformations (changes), and reasoning geometrically (angles, constructions and other geometrical relationships).

Measurement

Understand units of measure, measure objects using measuring units, estimate measures and calculate measurements.

w ww

m . u

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

. te

o c . che e r o t r s super

Understand symbols and graphs. Represent variation. Solve equations and inequalities.

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

New Wave Maths Book G – Teachers Guide • 243 •

Parent Information Homework Policy

Dear Parent(s) As part of my Homework Policy I encourage students to regularly undertake given exercises in the reinforcement of mathematical concepts learnt at school. These activities will be within the expected competency level of the children; however, there may be times when, due to unforeseen circumstances, your child does encounter difficulties with the homework. Please take the time to assist with the processes involved, but please encourage your child to ‘have a go’.

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

Should you encounter any problems, please contact me. Kind regards

Classroom Teacher

✄

ew i ev Pr

Teac he r

Your encouragement and positive support are crucial to the continued development of your child’s mathematical skills.

w ww

Dear Parent(s)

m . u

Homework Policy

As part of my Homework Policy I encourage students to regularly undertake given exercises in the reinforcement of mathematical concepts learnt at school. These activities will be within the expected competency level of the children; however, there may be times when, due to unforeseen circumstances, your child does encounter difficulties with the homework. Please take the time to assist with the processes involved, but please encourage your child to ‘have a go’.

. te

o c . che e r o t r s super

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

• 244 • New Wave Maths Book G – Teachers Guide