second edition teacher’s book 6 - macmillan caribbean · 2017-06-29 · general introduction 4...

CARIBBEAN PRIMARY MATHEMATICS

Second edition

Teacher’s Book 6

Laurie Sealy and Sandra Moore

Macmillan Education4 Crinan Street London N1 9XW A division of Macmillan Publishers LimitedCompanies and representatives throughout the world

ISBN 978-0-230-40126-6Text © Laurie Sealy and Sandra Moore 2014Design and illustration © Macmillan Publishers Limited 2014

First published 2014

All rights reserved; no part of this publication may be reproduced, stored in a retrieval system, transmitted in any form, or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publishers.

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2018 2017 2016 2015 2014 10 9 8 7 6 5 4 3 2 1

General introduction 4

About teaching Mathematics 5

Scope and sequence for level 6 7

About using Teacher’s Book 6 13

Unit by unit: Overview, Teaching suggestions and Sample lesson plans

Unit 1 Number 17

Unit 2 Operations – computation 22

Unit 3 Number concepts / sequences 29

Unit 4 Decimals 35

Unit 5 Operations – multiplication 40

Unit 6 Algebra 45

Unit 7 Operations – division 50

Unit 8 Fractions 55

Unit 9 Working with decimals 61

Unit 10 Percents, fractions and decimals 67

Unit 11 Measures of central tendency 75

Unit 12 Measurement 79

Unit 13 Geometry 85

Unit 14 Perimeter, area and volume 91

Unit 15 Ratio and proportion 97

Unit 16 Problem-solving skills 102

Unit 17 Statistics / sets / integers 108

Unit 18 Consumer / Business Mathematics 114

Final review self-assessments 1–6 118

End-of-Primary Final assessments 1–5 119

Answers to Student’s Book 6 120

Answers to Workbook 6 176

CD-ROM activity directory (list of topics) for level 6 196

Link to Caribbean Teacher Resources website 198

Contents

4 Introduction • Bright Sparks Teacher’s Book 6

General introduction Bright Sparks provides a different kind of Teacher’s Book. Each book gives experienced teachers new and useful ideas, while supporting novice or untrained teachers with concept explanations and lesson details.

Every Teacher’s Book includes that year’s Scope and sequence, all Student’s Book answers, all Workbook answers, a CD-ROM activity directory, and Teacher Resources website link. For every unit in the Student’s Book, the Teacher’s Book provides a breakdown of its objectives, resources for suggested activities, a plan of operation with main ideas and specific lesson ideas, teaching tips, extensions, differentiation ideas, assessment possibilities, and notes for where to use CD-ROM activities. Each unit has a sample lesson plan incorporating all of these elements.

Bright Sparks Teacher’s Books highlight and are built on the following key emphases. These emphases are woven into the suggestions and ideas offered for the various units and lessons throughout the series.

•Problem-solving and developing reasoning skills are essential.

•Teamwork, partner work and social learning lead to deeper understanding.

•Student engagement is essential to student learning.

•Teaching should be concept-based, with lessons built around the ‘big idea’, exploring the concept and then applying it. Procedures are necessary steps but should follow, and not be the focus of, a lesson.

•New information which is linked to prior knowledge is more easily understood and used. Well-prepared teachers break down the topic into manageable parts and help students connect it together once it is well understood.

•The language of Mathematics is semiotic, including both words and symbols. Teachers need to specifically teach this language and how it is used.

•Everyone learns differently. Teaching the same concept using multiple representations (verbal, symbolic, graph/chart, diagram, model, and physical objects) ensures more students learn, and that students have a deeper understanding.

•Differentiating lessons means adjusting the content for all learners, including students with particular needs (e.g. processing, physical limitations, gifted).

•Mental Mathematics is not computation, but, rather is learning to think, to explain one’s thinking to others, to question and to justify one’s answers.

•Task or project-based learning calls on students to use a variety of skills as they identify a problem, explore strategies, choose and then explain their solution.

•Good mathematical games and activities engage students while practising essential skills.

•Assessment is intended to inform teaching and learning. The results of student assessments should be used to identify what concepts or skills need attention and where individual students need support or challenge. Assessment is on-going, process-oriented and takes different forms, including self-assessment, formative and summative assessment.

Teachers will be able to choose that which best suits their own classroom needs from the teaching suggestions and the subject matter in each Teacher’s Book.

5Bright Sparks Teacher’s Book 6 • Introduction

About teaching Mathematics

Steps for lessons Mathematics education research confirms that students construct their understanding of concepts, such as numbers, through real world experiences, and that the beginning of those experiences is with the use of concrete materials.

1 Students first work with concrete materials to model and perfect their understanding.

2 Next, students use visual representations, such as pictures. It is at this point that some of the lessons in the Student’s Book are introduced, and then practised.

3 Once these levels are strong, the symbolic representation may be introduced. The symbolic representation, for example, might be the use of the addition sign and a number sentence.

The order of teaching and learning, then, best begins with the concrete, then moves to the visual or pictorial, and finally to the abstract, or symbols.

Teaching conceptsIn the outline of the units that follow, some suggestions for using activities will be made. However, teachers will come up with many other great ideas. The key is to introduce every new concept with many opportunities to model it using concrete materials and multiple representations.

Research confirms that this process ensures long-term learning. The more familiarity students have with the concrete materials and visual representations, the more easily they will be able later to adapt that learning to more abstract conceptual forms. Manipulatives, then, are not for the teacher to show the class or to do demonstrations. They are for all students to hold and use and physically

manipulate. Students should have time to explore, to model, and to explain to others their understanding. Through their manipulation of the materials and through their discussion, the level of understanding and learning will be made clear.

When there is an obvious gap in student learning, hearing the student describe what he or she is doing can give the teacher clues as to where the gap in understanding lies.

Multiple representationsEven with limited resources, teachers should use different materials and different approaches to teach a single concept. Using multiple representations of one idea, whether concrete, pictorial or symbolic, deepens the conceptual understanding for a student. It also ensures a greater chance that all students, based on their inherent learning abilities, will respond to and understand at least one of the different approaches the teacher offers. On a practical level, if a particular object always represents one idea, then some students who did not understand may stay stuck on that image, but if different objects and approaches are used to represent a concept, then gradually the students see what all the objects have in common.

Peer teaching and teamworkActive learning works best when teachers have clear expectations for the classroom environment when students work in groups, or work with materials. Exploration and discussion can encourage learning, provided these key strategies are conducted within an organised classroom where students know what is expected, how to speak with one another and how to comment when questioning another student’s reasoning. When teaching


students how to have Mathematics conversations, teachers will need to help students learn to show respect for other people’s views and to take turns when talking over their ideas with a partner or in a small group. When reporting back to the whole class, some teachers first help students appoint a group leader. To keep materials sorted and organised, some teachers use colours for groups,

and put the same colour on a bin with Mathematics resources for that group. Each group then has one person whose job it is to return the bin to the shelf. These are a few suggestions of ideas that work well in different classrooms of the region, but each teacher will use what works best based on his or her students and school environment.


Scope and sequence for level 6

Major concept Knowledge and skills

Number / Sequencing / Number conceptsNumber

Use counting skills, read numbers, natural / whole numbers, odd / even, sequencing

Use ordinal numbers

Identify value, place or face value of any digit in a 7-digit number (enrichment: ten and hundred million)

Expand numbers with or without regrouping (including multiplicative or exponential)

Compare and order numbers using <, >, = , with or without decimals or fractions, using appropriate terminology

Identify decimal numbers value, place and expanded forms to 2 (3) decimal places

Recognise Roman numerals used in practical situations (e.g. clocks)

Sequencing Extend patterns involving arithmetic operations, fractions, decimals, square or cube numbers, shapes

Count the interval between two numbers, inclusive or not, solve problems involving consecutive numbers or intervals (e.g. fence poles and spaces)

Number concepts

Identify and use: prime / composite numbers; prime factors; factors; multiples

LCM and HCF (GCF)

Square numbers / square roots

Identify the reciprocal

Compare and exchange decimals and fractions, and place on a number line

Explore integers using a number line

Practical situations using negative integers (temperature)

Explore irrational numbers / repeating decimals

Operations / RelationshipsProperties

Commutative law (×, +)

Associative law (×, +)

Identity elements (×, +) and zero properties

Distributive property of multiplication

Number sense Use properties of numbers to simplify and rearrange equations, break up numbers for mental Mathematics or simplify problem solving approaches

Recognise patterns in a ‘worked problem’ to solve related equations (e.g. halving, doubling)

Restate numbers using the distributive property of ×, with or without regrouping

Estimate reasonable results

Round numbers



Computation Whole numbers

Demonstrate skill with a calculator as an operational tool

Basic operations: mentally add and subtract 1- to 3-digit numbers to 1- to 2-digit numbers

Add and subtract accurately

Multiply and divide by 1- to 3-digit numbers

Use multiplication tables proficiently

Use the four operations in problem solving and explain choice of operation

Convert remainders in division to fractions or decimals

Apply skills to single- and multi-step word problems

Use order of operations (BODMAS)

Solve equations with more than one sign

Fractions Read / write / identify and describe fractions

Illustrate / determine fractions of a whole

Find fractional parts of a given quantity

Identify equivalent fractions

Order and compare fractions

Convert improper fractions / mixed numbers

Add or subtract fractions with or without like denominators, giving answers in simplest form

Simplify fractions and mixed numbers

Use cancellation in multiplying fractions

Multiply or divide fractions, including mixed numbers or fractions and whole numbers, giving the answers in lowest terms

Use fractions in practical situations

Use order of operations with fractions

Decimals Compare and order decimal numbers

Add and subtract numbers with 1 or 2 (3) decimal places

Divide and multiply decimal numbers by multiples of 10

Connect multiplication of decimals to fractions

Multiply and divide up to 3 decimal places by a whole number

Multiply and divide by a decimal number after first multiplying by a power of 10 to change the decimal to a whole number

Solve problems using decimal numbers, including measurement and money situations

Round to the nearest cent or dollar

Use estimation of decimal numbers in practical applications

Convert fractions / decimals / percent



Percent / Percentage

Find the percent of a whole / quantity

Find the total when given a percent

Use percent to solve real world problems

Convert between fraction / percent / decimal to simplify problem- solving situations

Ratio / Proportion

Share a quantity using ratio (partitive proportion) and unequal sharing

Work with proportion, equivalent fractions and scale drawing, understanding the concept of proportion

Solve problems related to direct or indirect proportion (without yet using terms)

Use cross-multiplication (cross products) in problem solving

Work with unit rate (unitary method) in problem solving, practical situations

Relations / Functions / Algebra

Use symbols to represent the unknown

Solve word problems with single variable equations

Use simple formulae and solve problems (e.g. problems involving speed, time and distance)

Recognise algebraic expressions (e.g. ‘10 more than before’ as ‘n + 10’)

Money / Consumer Mathematics

Read and write money to million

State situations using large amounts of money

Discuss uses and value of money, and different types of goods and costs

Understand the idea of savings (value)

Calculate totals using bills and coins

Add, subtract, multiply, divide quantities of money

Use a calculator for tasks involving money

Use percent to calculate hire purchase charges, tax, VAT or in other practical monetary applications

Calculate unit cost / total cost

Total a bill and determine change

Investigate regional currencies

Convert regional and other foreign currencies to local currency and back, using a rate of exchange

Calculate discount

Compare prices for best value

Solve problems with cost price / selling price

Calculate profit and loss / profit and loss percent

Calculate wages, salaries, simple interest, total cost for hire purchase or down payment

Calculate simple interest given a formula



Money / Consumer Mathematics (continued)

Interest for savings / loan

Hire purchase / buying on credit

Use percent in money transactions

Calculate wages and salaries

Discuss the purpose of budgets, taxes

MeasurementTime

Tell time to the minute

Record time in hours and minutes with standard 12 hr or 24 hr clocks and convert to standard time

Calculate using hours and minutes

Compare and convert units of time

Calculate elapsed time (months / days / hours / minutes)

Customary and metric

Be aware of customary units of measurement still used in the region and relate to the SI (metric) system

Develop comparisons for metric and common customary units (e.g. pound and kg) for practical situations

Use metric units of measurement in problem solving and practical situations

Measure and record accurately using different units and tools of measurement

Convert related units of measurement in problem solving (e.g. m to cm)

Apply basic operations (+, −, ×, ÷) to questions using units of measurement

Convert units of measurement where needed in problem solving

Learn the prefixes used in the metric system and its framework

Estimate, measure and record length and solve problems using km, m, cm, mm, including practical situations

Length Measure lengths in one metric unit and convert to another

Use scale to represent distance (e.g. map)

Mass Estimate, measure, record and compare mass using kg and g, mg and tonne

Capacity / Volume

Estimate, measure, record and compare capacity using mℓ and ℓ

Explore and model volume using unit cubes

Calculate volume (using cm3 or m3)

Geometry Identify vertical, horizontal, parallel, perpendicular, intersecting, oblique and curved lines

Describe and name plane (2D) shapes noting the properties, including number of angles and side, edges, vertices, whether open or closed

Classify triangles based on their properties: right-angled, scalene, isosceles, equilateral



Geometry (continued)

Calculate degrees of one angle in a triangle or 4-sided figure

Describe and draw triangles, squares, rectangles, circles

Identify congruent angles in polygons

Name right angles, acute, obtuse (or reflex) angles in practical situations, diagrams and plane figures

Be very familiar with right-angled triangles as 90° angles and point out multiple examples in the environment

Estimate common angles (e.g. construct paper protractors to learn 30°, 45° and 90°)

Use ruler and protractor to measure and draw angles of a given size (OECS)

Calculate degree of angles

Find the measurement of a missing angle in a 3- or 4-sided figure

Calculate degrees of complementary / supplementary angles

Discuss the relationship between interior and exterior angles

Distinguish between plane (2D) figures and solid (3D) shapes

Name and draw cuboids, cubes, spheres, cones, cylinders (pyramids and prisms)

Sort and classify 3D shapes based on their properties

Make models / nets of solid (3D) shapes

Identify and state the number and types of faces (flat or curved) and the number of edges or vertices in solid (3D) shapes

Measure the length of sides to find perimeter in polygons

Calculate the area of triangles, squares, rectangles, (parallelograms, rhombuses) using measurement and formulae

Find the estimated area of irregular shapes using a grid

Find area in practical situations and record using cm2, m2

Compare and contrast perimeter and area of plane shapes of different measurements

Divide a compound shape into parts to find the total area

Find the area of borders

Calculate missing width or length, given the total area

Find the surface area of objects by determining face areas and calculating

State the radius of a circle given the diameter or vice versa

(Use pi and formulae to determine the area or circumference of a circle)



Transformation Create tessellated patterns using plane figures (i.e. ‘tiling a plane’)

Identify and draw lines of symmetry

(Identify and count rotational symmetry)

Use translation / reflection / rotation of plane shapes

(Predict the new positions as a plane shape flips about a line, slides a given distance or makes a 1

4, 12 or 3

4 turn)

Identify coordinate points and name an ordered pair

Data / Statistics / Sets

Describe methods of collecting data

Carry out a project to gather data (survey, questionnaire or interview)

Collect, organise, construct and display data using a frequency table (tally chart), pictograph, bar, block (or line) graph, using scale

Interpret data, apply the data to problems and compare information to draw conclusions.

Use a circle graph (pie chart) divided into degrees, percent, (minutes) or fractions and interpret the data

Discuss measures of central tendency and find the mode and mean of a set of data

(Determine the range and median)

Find the mean (average) of up to 5 numbers or find the total or any missing number from the series, using calculation

Describe, identify, sort, classify and list the elements in a set

Work with the concepts of subset, union and intersection of sets

Use Venn diagrams to display and compare sets and in problem solving

Interpret information shown in a graph or table

Probability Connect to probability as prediction in determining if outcomes are fair, likely, etc.


About using Teacher’s Book 6

How to get the most from the Student’s Books and Teacher’s Books The Scope and sequence, Table of contents and Student’s Book index outline the concepts covered throughout the school year. Some topics are ‘Enrichment’ within the units, where not every syllabus in the region includes that concept. These topics may still be used to extend student learning or for advanced students, or may be left out. Teachers will feel free to use the Bright Sparks materials that best fit their local needs and their particular class in any given year.

The Student’s Books are used to supplement Mathematics instruction and provide an opportunity for examples and written practice to follow practical activities. After first learning the mathematical concepts through active participation, manipulation of concrete models, discussion, problem posing and a variety of approaches, students can then be guided through the exercises in the Student’s Book that supplement that learning. The examples and explanations support classroom teaching and provide prompts as students work independently, while the features (Hint, Challenge, etc.) push student thinking and help students to remember.

Teachers and students sometimes face overcrowding, limited space, high noise levels from adjoining classes, multi-age classrooms, limited resources and varied support. It is challenging to try to find materials, time and even space to carry out practical activities. This is the reality for many teachers in the region. The Teachers Book gives some ideas and suggestions for lessons and games that help students become interested and involved, without using commercial products that are hard to obtain. Teachers will gradually collect

additional ideas for activities and games from colleagues, workshops and through Internet or book research. Students easily become excited and engaged when involved in activities and when they see Mathematics as fun, even as they solidify their learning. As teachers, we hope to keep this enthusiasm kindled and build the confidence that comes with their new skills and knowledge.

Teachers may need to be confident voices in favour of students doing activities or games in Mathematics, rather than following the pencil and paper approaches that are sometimes expected. Research and experience prove that student engagement – being interested and actively involved, including in classroom discussions – leads to better problem solvers with stronger mathematical thinking skills. Teachers may need to be strong advocates for change, helping to educate parents and others who are not yet aware of these proven approaches. Teamwork and problem-solving skills develop through these types of classroom strategies.

Homework Homework is an integral part of the mathematics programme for this year group. The exercises in Bright Sparks Workbook 6 are very tightly linked to the class lessons, and are designed to be used as support homework. Each workbook exercise states which Student’s Book 6 lesson it supplements. Teachers without access to the workbooks, however, will wish to make up questions similar to those done in class, or assign part of a class exercise for homework. Because the homework in this year group is such a key part of learning, a tip that may be useful for teachers is this: Assign a student each week as the ‘Absent Student


Assistant’, with the task of writing out the homework for any absent student on a given day (and the page number of the lesson covered in class). It works well if teachers provide a half-page form that can be filled in with the absent student’s name and page number details by the assistant. When the absent student returns, he or she will more easily be able to fill in any learning gaps.

How to use the Unit Check and summary, Assessments 1–9 and Workbook Self-checks The underlying objective – assessing Mathematics learning – needs to be kept in the forefront. The Unit Check can be used as a tool for self-assessment, introducing students to the opportunity to check themselves and see how well they are learning. You may also wish to use it as a formative assessment tool, to chart student progress in concept areas and plan further teaching. The longer Assessments 1–9 give a broader overview of learning in more than one concept area. In Workbook 6 there are ‘Self-checks’. These reviews work well for student self-assessment, if students are taught how to use the results to recognise their strengths and their needs.

Final Assessments 1–5 The overall tests at the end of each of the Upper Primary levels can be used for both summative and formative assessment. They are designed to summarise the learning gained, and also may be used to diagnose areas needing further support. At the end of Book 6, these final assessments acquire greater importance, as most students will in this year sit end-of-Primary national examinations. These final assessments mix the wording of the test questions, and model the different formats students might encounter. They combine reasoning/problem solving with numeracy and calculation, to give a full overview of the curriculum. As students review their efforts

after the practise tests, teachers will aim to build the students’ confidence as well as their test-taking skill, to enable their true understanding to come through on the examination day.

Informal assessmentTeachers will, as usual, carry out informal checks on student progress. Such informal assessment can be approached systematically, deciding in advance what concepts or skills will be assessed, and for which students, on a weekly basis. In this way, teachers can build up notes on every student, checking for strong or weak understanding and identifying students who have trouble formulating or articulating their reasoning.

Mathematical language / Teaching English language learnersThe language of Mathematics needs to be discretely taught. Students may enter school with a home use of language somewhat different from the Standard English expected in school. In Belize, in particular, and in some other parts of the region, there are students entering school for whom English is not their first language. Techniques to help students’ transition to English while learning the language of Mathematics include:

•specifically teaching the Mathematics vocabulary (e.g. ‘times’, ‘length’), particularly those words which carry more than one meaning (e.g. ‘match’, ‘line’)

•using hand movements to demonstrate while teaching

•using activities that include singing, modelling, describing, and hand-held objects

•encouraging visual learning, such as using sketches in problem solving to show what we know and what we need to find out


•‘revoicing’ – where teachers encourage students to speak in class, and then repeat back what the student is trying to say, using the appropriate language / mathematical terms.

Focus on the ‘big idea’The Teacher’s Books offer suggestions for ideas to engage students and point out correlating activities on the CD-ROM or Teacher Resources website. The Workbooks, in addition, state which specific lesson in the Student’s Book each practice exercise supports. The Teacher’s Books address the key teaching points, the main concepts or the ‘big ideas’

mathematically. In addition, the common misconceptions, or weak areas that are sometimes seen, will be noted, with ideas for bridging these gaps before they become deterrents to learning. As before, units will have ideas for differentiating lessons to reach all learners, teaching tips, ideas for resources (which teachers will supplement with their own activities), extension ideas and sample lesson plans. In the variety of ideas offered, our aim is to support all teachers, novice or veteran, making the teacher’s guides valuable and useful tools that will help teachers enhance student learning.


UNIT BY UNIT: Overview, Teaching suggestions and Sample lesson plans

to think for themselves. The tips, examples, Hints and other features in the Bright Sparks series help teachers to also become more confident. Maths truly is fun to use, to teach and learn, and when teachers project this attitude, treating each student as a bright spark with the capacity to do well and enjoy maths; students will rise to that expectation.

Before teaching the unit, become thoroughly familiar with the Student’s Book explanations, examples, feature boxes, Challenges, activities, the Workbook support, specific CD-ROM activities for the unit and the extras on the teacher resource website.

There is a lot available for teachers, ready to use, or to be modified depending on a particular class. After each lesson, reflect on how well it worked, what student skills need more practice, what games or activities went well and helped students learn, and what did not work well. Use this information for further planning. Prepared teachers with a well-designed lesson plan, supplementary activities, ready resources and a back-up plan, help students make better progress.

Lessons should continue to be based on practical activities, diagrams, models and word problems which connect mathematics to the real world. Use student discussion frequently, to help students refine their reasoning and to give you insights into their learning. Provide opportunities for students to question, and plan in advance what probing questions you will ask that will help bring out the main ideas of the lesson. Give students time for individual work, where you can observe and check with them one-on-one. Include the essential wrap-up at the end of the lesson, using discussion and questions for students to review, recount and remember, even as you informally assess their learning and make adjustments to your plan for the next level of teaching.

An additional note about teaching and planningWhen teachers love mathematics and let it show, students know it. This love is especially noticeable with number sense and reasoning. Teachers who are not confident seem to rely rigidly on procedure, and do not encourage students

17BrightSparksTeacher’sBook6•Unit1

OVERVIEW OF UNITObjectives / OutcomesAt the end of this unit students should be able to:

•Use counting skills, identify real, natural or whole numbers, odd / even, ordinal numbers

•Read, write, name and use small to large numbers

•Identify value, place or face value of any digit in a 7-digit number (enrichment: to ten and hundred millions)

•Expand numbers with or without regrouping

•Recognise expanded form as an expression of the value of each digit

•Compare and order numbers

•Use number sense and numeracy skills to manipulate and simplify equations

•Use properties of numbers to simplify and rearrange equations, break up numbers for mental Mathematics or simplify problem-solving approaches

•Use the associative and commutative properties (+, ×) to simplify problems

RESOURCESManipulatives to demonstrate thousands/hundreds/tens/ones or paper drawings of thousand cubes, hundred squares, ten strips and single squares; scrap paper; markers; counters (bottle tops, etc.); several abacuses (purchased or hand-made) if available.

Teaching the content of the unitIt is surprising how many students still have difficulty with the key skills of identifying, naming and writing large numbers. Even more students still need practice stating the number aloud correctly, which is tightly connected to their understanding of value. Rather than skip Partner Activities, or practical work, thinking it ‘too easy’ at this stage, teachers are encouraged to use these opportunities to informally assess students’ skills and provide support when needed.

Plan of operationThe following are suggestions which may be helpful as you create lessons for this unit.

Most topics have a corresponding practice exercise in the Workbook. Specific CD-ROM activities are noted by the relevant lessons below.

UNIT 1 Number

Place value and counting skills, Value, Expanded form, Short cuts in mental Maths, Expanded numbers with regrouping

18 Unit 6 • Bright Sparks Teacher’s Book 6

Place value and counting skills (1.1)

Use the introduction at the beginning of the unit to talk about what students expect to learn in Mathematics in this school year.

Opening activityHave partners choose and write down a number between 2 and 15. Their task is to use that number to count as high as possible in 3 minutes (timed), either orally or in writing. Have students compare results, and discuss reasons for the differences. Move on to the Partner Activity to give oral practice in naming large numbers, and move through the class informally assessing student skill and understanding. If students need additional practice, opportunities can be created throughout the school week to name large numbers, and compare numbers.

Review place value, stressing the importance of first saying the number aloud (or mentally), as this approach helps us ‘hear’ the place value. Note the difference between ‘face’ value, which is simply stating the name of the digit (numeral), as compared to place value, which is far more significant. Write 30 491, 34 091, 30 941 and 30 914, and have students talk about the differences in value depending on where a particular digit falls, in this case the changing place value of ‘4’. Include identification of place value and value in numbers in the millions, and have students give practical examples where these large numbers, including with money, might be used.

Each of the three written exercises may first be completed orally, to practise naming numbers. Exercises 1.1 A and B are complementary written activities for place value. Exercise 1.1 C is practice in expressing numbers in words or figures. See ‘Teaching tips’ below, regarding commas in numbers.

Use with WB Unit 1, Ex 1–4

Note: While working in this unit, practise and assess students’ use of ordinal numbers through practical activities.

Value (1.2)

Value is the essence of understanding numbers. It is key to numeracy, to number sense and problem solving. If students are not strong at this point in stating value, it is strongly recommended that teachers provide activities with manipulatives, games or similar practical activities to strengthen the concept of value before moving on. To pre-check understanding, have students give examples with 4-digit numbers similar to the questions in the Partner Activity, stating the value, and demonstrating with models or manipulatives. (An example of a model could be drawings of thousand blocks, hundred squares, ten strips and ones, while a manipulative might be these actual materials.) Verbally, then move on to challenge students with numbers you write on the board, up to millions. Connect to practical examples of value. When ready, have students then write the values in exercise 1.2 A and check their progress. Stress that the value tells us what the digit is worth, or what the whole number is worth when we put the value of each digit together. Move on to exercise 1.2 B which reinforces this concept.


Expanded form (1.3)

Explain that expanded form is simply writing the value for each digit in a number. If an abacus is available, have students set up different numbers, for their peers to state the value of each row and of the whole number. Review exercise 1.2 B, where they wrote the value of each digit simply. Then have students discuss the different forms of writing the value of each digit shown in the expanded form examples in the tinted box. Ask partners to compare and contrast, stating what is the

19Bright Sparks Teacher’s Book 6 • Unit 1

same (the value of each digit is expressed) and what is different (the way the value is presented) with the three examples. Give students exercises 1.3 A and B first.

Use with WB Unit 1, Ex 7

Alternate forms for writing expanded form in exercises 1.3 C and D may be a separate lesson. Note that the brackets confirm the value of each digit, when said aloud. In exercise 1.3 D, have students underline each phrase, such as: 54 326 = 5 × 10 000 + 4 × 1000 + 3 × 100 + 2 × 10 + 6 × 1. Using this hint (see Hint box), helps students ‘see’ the values more easily, while allowing teachers to informally check their understanding.


Exercise 1.3 E is a review of Number. Teachers may wish to use it to assess students’ understanding overall, as students work on their own without an introduction. However, for some classes the larger numbers in the last section may require teacher support.

Use with WB Unit 1 Ex 10

Integrated learning

Connect to Science and current events in space exploration. For example, scientists found a ‘blue’ planet ‘only’ 63 light years away (about 600 trillion km). Is that distance considered near or far in space terms?

Short cuts in mental Maths (1.4)

See sample lesson plan below on inverse operations.


Expanded numbers with regrouping (1.5)

One of the key points for this lesson is the concept that both sides of an equal sign must have the same value. Remind students of the previous ‘short cut’ lesson, which also emphasized this concept, and discuss the example in the tinted box. Discuss the gold box, noting regrouping

must occur, and that it can be checked by using the value. It helps some students to visualise the regrouping if they cross out the regrouped number, as shown in the example. Draw their attention to the ‘Remember’ box, which always show key concepts. Exercise 1.5A is the first level of regrouping and is fairly straightforward. Check student understanding before moving to exercise 1.5 B.


Discuss the example box above exercise 1.5 B, using the hint. To regroup these expanded numbers, students must first understand the different forms of presenting expanded numbers, as practised earlier. The last part of the exercise builds on the ‘short cut’ skill from the previous lesson.


Unit 1 check and summary

The Summary and Assessment for this unit is combined with that of Unit 2.

2.

Teaching tips•There are different protocols in regard

to both the use of commas in numbers, and the use of the word ‘and’. The most commonly accepted approach is used here. If this approach different from the local syllabus, teachers will follow their local standard, yet may wish to take the opportunity to explain to students the different approaches, preparing them for alternates they might encounter as they move on to secondary school and the wider world.

•Challenge boxes are intended to extend student thinking, particularly those students who are ready for something harder or more complex. Challenge questions are not generally included in the marks for an exercise, although some teachers use them for bonus points.


•To differentiate for students with gaps in understanding place value, if a classroom computer is available, the CD-ROM from the younger books, level 3 or level 4, block A, may be used independently by individuals or small groups.

Expansion / Extension of unitAs a next step, have students continue exploring large numbers used for distances in space, for example the distance to three objects (moons, planets, etc.). Compare to large numbers used with digital data, finding examples for these (e.g. a terabyte is one trillion bytes).

Sample lesson planUNIT 11.4 Short cuts in mental Maths

Objectives •Expand numbers with or without

regrouping

•Recognise expanded form as an expression of the value of each digit

•Use number sense and numeracy skills to manipulate and simplify equations

•Use the associative and commutative properties (+, ×) to simplify problems

Engaging the students’ interest / ConnectionsGive each pair of students 48 counters (buttons, bottle tops, etc.), and the task of finding as many ways as they can to use all the counters and group them in equal groups of different numbers. (2 sets of 24, 4 sets of 12, etc.) Have partners come to the board and record all the different combinations once. Have students compare and contrast the equations shown, and introduce the idea that 12 × 4 becomes 6 × 8 if the first number is halved and the second number is doubled. Encourage students to find more patterns.

Teaching the lessonMathematical languageGuide students to name the commutative property, where the numbers change position in an addition or multiplication equation (2 × 6 = 12 and 6 × 2 = 12), and review the associative property shown in the Remember box. Ask why changing to 8 × 12 in the example was helpful.

Practical activity / Developing the lesson•See if any students have a similar short cut they could share with the class, which

helps them make a multiplication calculation simpler.

•Talk about the first example in the tinted box. Ask why it is easier to multiply 70 by 10 than 35 by 20.

RESOURCESScrap paper, a large supply of counters; markers and a hard-backed journal (optional).


•Give additional oral examples, such as: 40 × 12, 25 × 30 or 30 × 8.

•Ask students about the example of breaking two numbers into three, in the example 2 × 12 × 4 and then rearranging them to keep the same value (commutative property).

•Explore the possibility of doubling and halving, stressing that both must take place in the same question if it is to keep the same value. Have students all come up with examples where it worked. For example, 18 × 5 = 9 × 10.

•Before students start the exercise, discuss the Challenge and Hint box. Rather than explain why the hint works, leave it for students to sort out and explain on their own. Make sure all students know where to write their paragraph, should they get a chance to do the Challenge.

•Have students do exercise 1.4.

Assign as homework Workbook 6, Unit 1 Exercise 11.

Differentiating for different learning styles •Have students write a few words after each question 1-10 to say what approach was

taken, in working towards the solution. This action helps learners who particularly need to have each step clearly set out.

•To challenge mental acuity, have students choose two ways to manipulate any of the questions 11-20. For example, question 20 could be 25 × 4 × 3 = 100 × 3 = 300, or 5 × 5 × 12 = 5 × 60 = 300, or 25 × 2 × 6 = 50 × 6 = 300, etc. Share results with the class to open up the thinking of students whose thinking is too limited.

Extension activities Start a Maths Journal in a hard-sided book. Mark out one section for short cuts and Hints. Write a page with a summary of what was covered in this lesson.

AssessmentUsing exercise 1.4, all students should be able to be assessed with questions 1–10. Questions 11–20 may be assessed orally, checking that students used Mental maths skills. Check in with each one as he or she is working, and ask what short cut are being used and why. Make informal notes about their understanding and mental maths skill.

Summary of key points Both sides of an equal sign must always have the same value.

We can use the properties of multiplication to manipulate the numbers and make the equation simpler.

Short cuts also include doubling one number while halving the other, or multiplying by a simpler number and adjusting the final result (as was done in the Challenge). When we work with numbers in these ways we are using number sense, and building our problem-solving skills.

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Text12

Unit2•BrightSparksTeacher’sBook6


•Identity elements (×, +) and zero properties

•Mentally add / subtract 1- to 3-digit numbers to / from 1- and 2-digit numbers

•Add and subtract accurately, including large numbers

•Read and write money to million

•State situations using large amounts of money

RESOURCESSmall counters such as buttons, bottle tops, shells or seeds; straws or other manipulatives to represent large numbers (if needed for slower learners); individual whiteboards and dry erase markers (if available) or scrap paper and markers, calculators.

Teaching the content of the unitThe main idea of this unit is to take existing skills in calculation and move forward: to use reasoning as a habit, always thinking whether answers make sense; and to work quickly and efficiently, taking advantage of mental mathematics and short cuts when

possible. The focus this year is problem solving and real life mathematics, using calculation as a tool. For that task, first we check that the ‘tools’ are ready. This unit reviews these tools of calculation. You may wish to have students first take ‘Assessment 1’ as a pre-test, to determine how much extra practice will be needed.

To introduce the unit, have students work in pairs and create a real life problem that needs to use an operation to solve (+, –, ×, ÷). (Challenge some to use two operations in their problem). Tell them they have five minutes, and should write it out and check it. Have a few students share, and ask the class if the problems were real life, day-to-day examples. If some students were unsuccessful in creating a problem, assure them that they will catch on more quickly as the unit progresses. The purpose of the task was for students to make connections and see the purpose of mathematics, as well as to refresh mathematical language regarding ‘operations’ used in calculations.


Most topics have a corresponding practice exercise in the Workbook.

CD-ROM activities are noted by the relevant lessons below.

Rounding and estimating (2.1)

The unit begins with several opportunities for discussion, stressing the idea that

UNIT 2 Operations – computation

Rounding and estimating, Adding mentally, Addition with regrouping, Subtracting mentally, Regrouping, Subtracting large numbers, Subtraction with zeros, Multiplication: review, Division: review, Multiplication and division speed testsUnits 1 and 2 check and summary, Assessment


solutions must make sense, keeping in mind the value. In this lesson, and at frequent opportunities, help students see that estimating what an answer should be through rounding before the calculation, is a mental mathematics skill with lifelong benefits, as well as in exams. Show examples related to knowing how much you will need to pay when shopping, etc. Work through the key lesson points orally, including the Partner Activity and exercise. Discuss the differences we note when a large number is rounded to a small place value (question 19 or question 28 versus question 4 or question 7). Finally, have students write the exercise independently, and check understanding. See Teaching tips below for differentiation.

There is a CD-ROM activity for this lesson.


Adding mentally (2.2)

The Remember box sums up a simple tool for adding mentally, which is often neglected. To practise this skill, exercise 2.2 A (questions 3–20) is useful as a discussion point, where students in pairs can look for and explain how these ‘friendly’ numbers fit together. Another approach is to have students write a few words or numbers for each, to explain/show what they first combined. For exercise 2.2 B, have students estimate the answer and write it before they begin. Remind them to carefully set out the columns when writing the numbers, aligning the decimal points for the money questions, as many errors are made due to this one simple task. At the end, have students circle all of their estimates which were close to their final answers. Encourage students to double check their estimate and their calculation, if the results disagree. Use this opportunity to discuss how much difference we might expect to find, and what would account for the differences (e.g. which place

was rounded). See Teaching tips below regarding the word problem.


Addition with regrouping (2.3)

Teachers may wish to have students work out the answers to exercise 2.3 A on their own at the start of the lesson, assessing whether additional practice or explanation is needed. If so, first go back to the foundational concept of regrouping, using manipulatives (e.g. straw bundles with hundreds, tens and ones, or place value slot charts). Only after the concept has been re-taught, will teachers move to the algorithm and the steps for regrouping shown in the example. See Teaching tips below.

To open exercise 2.3 B, have students discuss their own experiences going to a cinema. Ask what they think a ‘four-plex’ or a ‘concession stand’ mean, compare their ticket prices to those shown, and note the expenses the owner must pay. Have students then work in pairs to solve the problems, and each write the results. Exercise 2.3 C and D provide additional practice.

Use with WB Unit 2, Ex 3 with 2.3C, and Ex 4 for 2.3 A–D

Subtracting mentally (2.4)

Orally review subtraction by starting with what students can do mentally. Ask the class to give you at least three ways to mentally approach 50 – 7, and note these on the board. For example, counting backwards, taking away 10 and adding back 3, take away 5 first ‘because that is easier’ and then take off 2 more, etc. Move on then to 50 – 26 and again ask for mental maths approaches (see example 1). Ask for volunteers to explain the points in the Hint box, in their own words, or with examples. Provide small counters, for students who are unable to think of a way to solve mentally at first. Discourage all use of paper or pencil for this lesson.


Homework Challenge: ‘When 3__2 902 is rounded to the nearest hundred thousand, the answer is 400 000. Which missing numerals could make this statement true?’

Regrouping (2.5)

Partner ActivityPair a stronger student with a weaker student, as they discuss example 2. Informally check in with partners and answer any questions. Pose questions about the concept of regrouping (e.g. why the ‘4’ is now ‘14’) to different pairs. Have students do the first four questions, and check results. If needed, follow with whole group re-teaching of the concept of regrouping for subtraction, before students move on.


Subtracting large numbers (2.6)

There are two main concepts in this lesson. One is reading, writing and saying large numbers, including large sums of money, and connecting these to real-life situations. The other concept is subtracting large numbers. Focus first on the photograph of the large crowd, and similar experiences students may have had. Ask what type of event this might be (e.g. Calypso Junior Monarch, Interschool Sports, International Match). Use the example box and practise saying the numbers aloud. Compare the numbers (one is almost six times larger), and have students write on a scrap paper or individual whiteboards some additional large numbers you call out. Have students write each number and hold it up, even if different from their peers. Before students start the exercise, note that regrouping for small numbers and large numbers takes place in the same way, one place value column at a time.

Challenge homework: ‘Find out the world’s current population and note the date.’

Subtraction with zeros (2.7)

Write on the board 99 999 and 100 000. Ask the difference between the numbers (1). Have two volunteers write ‘– 39 582’ below each number, and then ask two new volunteers to solve. Have students discuss and compare the results, noting the difference between the two new answers (1). Ask which was easier and why. The main idea here is that the concept of regrouping does not change, but the procedure for subtracting large numbers when there are zeros requires a little more care. Have students write down the numbers in the example and subtract it on their own, and then compare their steps to the example. Assist those students needing additional support, while others do the exercise on their own. In a subsequent lesson, review exercise 2.7 A, with particular attention to the money problems in questions 8 and 9. Continue with exercise 2.7 B, and discuss the results to the word problems in questions 7–10.


Multiplication: review (2.8)

See sample lesson plan below.

Use with WB Unit 2, Ex. 7

Division: review (2.9)

The expository text box uses student language to describe division. Ask for volunteers to take a sentence at a time and expand on the idea or give examples. Reinforce the relationship between multiplication and division by asking students why several questions in exercise 2.9 A are multiplication, when the lesson is about division. Then, for each one, have students state two sentences to demonstrate this relationship. For example, ‘36 divided by 4 is 9, and 9 × 4 is equal to 36’.

If students are still not strong and quick with multiplication tables, teachers may wish to move on to lesson 2.10 first, and


provide activities to build up these skills quickly, before coming back to exercise 2.9 B.


Multiplication and division speed tests (2.10)

This lesson provides practice in quickly using basic multiplication and division, in a timed setting. How much time is provided will depend on the students’ development.

How teachers use the five speed tests is also flexible. For some classes, it may be a quick successful lesson; with the activity below as a reward. For those at the other extreme, the speed tests may need to be adapted, perhaps by students first just doing the multiplication in one test, and then the division.

Game to reinforce skill, and provide practice to all levelsTeacher Resource website – Level 6 Games/Activities page ‘Products in a queue’ for a game that can be done by small groups in class to build speed with multiplication.


Units 1 and 2 check and summary

The Summary box may be used before a lesson, may be used to summarise after a specific lesson, or for review prior to the Unit Check. It may also be used for study before the mixed-concept Assessments, Reviews and tests.

Assessment 1

This two-part test covering Number and basic operations may be used as a pre-test (to check understanding prior to the unit) and post-test (after the unit to assess progress), or may simply be used as part of a periodic assessment plan scheduled every few weeks.

Teaching tips•Kinaesthetic learners might more easily

round and estimate if they are taught to touch one finger to the ‘place’ to be rounded, and with the other hand count the digits which will become zeros.

•At this stage in the school year, it is useful to have students write a sentence to specifically answer the main question in a word problem. For example, for exercise 2.2 question 15, students might write: ‘The total number of tickets sold that day was ___ .’ The main purpose is to help students develop the habit of identifying what exactly they are being asked and to respond precisely. Slight differences in wording are fine, if the key question is accurately answered.

•When students make errors in calculations of large numbers, check to see if the errors stem from a misunderstanding of the concept, or of how to use written forms of calculation. If the latter, you may wish to have students estimate the result, and then show you what they think they must do. Make sure students understand the value of each digit in the various places (e.g. ‘3’ and ‘7’ in the tens column is really ‘30’ and ‘70’, so the answer would show in the hundreds place).

•Engage students with the activities described in word problems by creating connections to their own experiences (e.g. in exercise 2.3 B visiting a cinema). When students are fully engaged, and see the purpose for the maths, research shows they are more successful. It may take fewer than 5 minutes of lesson time to have this discussion, and give far greater success.


Expansion / Extension of unit

Calculator activity

Students need to develop skill in using calculators quickly and well. Have students work in pairs to add or subtract several large numbers, recording both the numbers and the results.

Challenge – have students work in pairs. Each student prepares a list of 10 multiplication or subtraction questions similar to the speed tests. Students will exchange lists, and one will use the calculator while the other will mentally solve the questions. See who is faster.

Sample lesson planUNIT 2 2.8 Multiplication review

Objectives (for the specific lesson)•Use the commutative and associative

properties of multiplication

•Recognise multiplication as the inverse of division

•Understand the connection between multiplication and repeated addition

•Demonstrate skill mentally multiplying factors to 12 quickly (15 for Enrichment)

Engaging the students’ interest / ConnectionsGive pairs of students the lower product ‘bingo’ cards and a handful of counters. The teacher or a volunteer will call out multiplication facts in mixed order, with products up to 60 (checking each one on a list after it is called), with very little wait time between each. One student in each pair will cover the product if they have it, while the other will write the matching multiplication fact on a scratch paper. When a row, column or diagonal (of 5) is attained and covered, the pair will say ‘bingo’. The students must verify which facts they used and that the correct products were covered. Carry out three rounds, each round a little faster. Winners might receive simple rewards (e.g. lead the line to lunch break, extra PE time, etc.). At a later time, switch to the higher product cards to add challenge and/or increase the speed to advance skill level. (Allow approximately 5 minutes set up and 10 minutes for the activity.)

Teaching the lessonMathematical languageAsk students for words associated with multiplication (product, factor, repeated addition, etc.). Review the tinted lesson box and discuss the meaning and examples of

RESOURCESA container of dice, small counters (e.g. buttons); ‘bingo’ cards with 5 rows and 5 columns and random products between 0 and 60, or between 63 and 144.


commutative and associative properties, stating why they are useful.

Practical activity / Developing the lesson•Explore the different ways people might say 5 × 7, and whether these differences

matter. For example, 5 times 7; 7, five times; 5 sets of 7 or 7 sets of 5; five 7s or seven 5s, 5 groups of 7, etc. Some Caribbean territories are very specific in how they wish students to state these multiplication facts. Mathematically the products are the same.

•Have students write a paragraph, as in exercise 2.8 A, comparing multiplication to addition (5 minutes), and discuss key points that arise after a few students have shared.

•Question why it is useful to be very quick with basic multiplication facts, guiding students to recognise the value of not having to spend time working out these facts, when trying to focus on how to solve problems. Ask for tips for what to do when unsure, and then show some examples, such as: a) reverse the order (commutative), b) break up the number (e.g. 12 × 9 could be the sum of 6 × 9 twice, or it could be 12 × 10 – 9), c) use repeated addition if needed.

•Practise multiplying quickly, using the speed tests in exercise 2.8 B. Some teachers will photocopy the page, or have students copy the tables prior to writing the answers. Some teachers may have students write just the question number, write the answers during the test, and only write the equations for the ones they get wrong. The main point is that students not write the question and the answer during the speed check, or it throws off an analysis of how quickly and accurately they can multiply mentally.

•For Test 1, give students a generous amount of time; enough to allow even the slowest students some success. Each subsequent speed test should have less time. You may wish to do several on the first day, and save some for a subsequent day, allowing students to practise on their own before the next round.

•The Workbook exercise also will allow further practice.

Use WB Unit 2, Ex 7

Differentiating for different learning styles Activity for small groups to practise multiplication at different skill levels.

a) Basic level, one person rolls two dice, and the others give the product.

b) Mid-level, one person rolls two dice, another person rolls two dice, and the two sums are multiplied.

c) Challenge level, one person rolls one die and the other person rolls 2–3 dice.

Students find the product of the face value represented by the dots (1–6), for each order/arrangement of those dice. For example, 3 × 24 for one die with 3 dots, and two dice where one shows two and the other shows four.

Extension activities Set up an Activity Box with either the higher value ‘bingo’ cards or the lower value cards, sets of multiplication facts with products to match (glued to pieces of card) and counters. Students may ‘play’ on their own quietly in small groups, for practice and fun.


AssessmentMake informal notes during the activities as to which students are relying on their peers to provide the answers, which ‘facts’ are often wrong, etc. In addition, check the speed test results and talk to the students one-on-one to plan how they will improve. Help each student prepare a goal list of the multiplication facts they still need to master, and plan how they will work towards those goals.

Summary of key points We can use the properties of multiplication, or repeated addition to assist us to find the products.

Practice helps build speed in basic multiplication.

We need skill and accuracy in multiplication, in order to focus on higher thinking skills in word problems.



•Find the LCM and HCF (GCF)

•Find square numbers / square roots and basic cubed numbers

•Extend patterns involving arithmetic operations, squared or cubed numbers, shapes

•Expand numbers with or without regrouping (including multiplicative or exponential)

•Solve problems involving consecutive numbers or intervals

•Identify and use: prime / composite numbers; prime factors; factors; multiples

RESOURCESSmall counters of the same size and shape (e.g. bottle tops); playing cards; calculators; markers and scratch or drawing paper; game boards for divisibility game; dice; small cube-shaped blocks (e.g. 25 children’s alphabet blocks).

Teaching the content of the unitThe purpose of this unit is to thoroughly familiarise students with the various forms

of number, and to provide opportunities to use these concepts and forms in different ways. The focus of the unit is primarily to build a strong foundation for reasoning. We want to encourage students to use a variety of strategies this year, and to be comfortable experimenting when they are not sure, building their confidence and success.

Towards this purpose, offer students a quick game to check their prior learning. For example, put each number concept/word on separate small cut out papers, and the corresponding phrase or example on other papers. Pass these out to students, and have them find the person that has the paper connected to theirs. (Alternatively put one list on one side of the board, the phrases on the other, and invite volunteers to draw in the matching lines.) To differentiate for a slower developing class, each pair of papers can be cut as in jigsaw puzzle pieces, so only the matching word and phrase will fit together.




Square and cubed numbers (3.1)

Using arrays to model square numbers helps students with strong spatial

UNIT 3 Number concepts / sequences

Square and cubed numbers, Exponents, Patterns, Order and value, Numbers: review, Factors, Multiples, Divisibility, Prime factors, Unit 3 check and summary


reasoning to visualise square and then cubed numbers. Start with a practical activity forming arrays with counters for each number multiplied by itself from 1 to 15, followed by a diagram for each array. Have students discuss the example with the purple cube, and then draw 53. If small cube blocks are available, use these to model several cubed numbers. Discuss how these models help us picture these kinds of numbers and help to explain their names. Encourage students to memorise square numbers to 152 and cubed numbers to 103, in order to recognise them quickly when they come up in patterns.



Exponents (3.2)

Include the mathematical language box, and the terms base and exponent in the lesson. Encourage students to recall other places they may have seen exponents used (e.g. in the formula πr2, or in algebraic expressions such as ab2). Discuss how the use of exponents makes writing mathematics ideas or numbers simpler.

ActivityProvide calculators for the Activity, and allow students time to explore the square and square root functions on their own. Calculator practice must increase skill but should also provide students a chance to become comfortable exploring, trying out patterns and ideas on the calculator, without concern for doing something wrong.



Patterns (3.3)

See sample lesson plan below. For classes where re-teaching is needed, you may wish to hold back this lesson until after ‘3.5 Numbers: review’ below.



Order and value (3.4)

Practical introductionProvide all pairs of students with four playing cards, two even and two odd (any suit). A face card can be assigned a value of 0, and an ace can be worth 1. Give partners time to experiment with these cards to build different number combinations and record all possible results. One purpose of this activity is to reinforce the concept of place value.

For a challenge, see also Teacher Resource website – Standard 6 Enrichment page ‘Working with number puzzles’.


Numbers: review (3.5)

The main focus of this lesson is to re-examine types of numbers learnt discretely, and make connections and associations when reviewing them together. In addition, word problems based on consecutive numbers are practised.

To start, again provide students opportunities to distinguish between types of numbers: odd and even numbers, natural (counting) numbers and whole numbers, prime numbers, fractions, decimal numbers, ordinals. Then review with more details to be sure all students are well familiar with each. In particular, review prime numbers, and question whether 1 is prime. Review the terms simple fractions and decimal fractions, noting that all of each lie between 0 and 1 on a number line.

Ask students to use any method to try to find the solution to example 1 (guess and check is acceptable). Discuss why there could be many solutions. Have students use calculators, if available, to find additional combinations. Give example 2 with the words ‘consecutive whole numbers’, and discuss the importance of being precise. Again let students use any approach to solve. After discussing their efforts, have partners work through the solution in


their text, and practise a new question (e.g. question 3) with the methodology suggested. Finally, challenge students to extend the approach and give a solution with only odd or only even numbers. Work exercise 3.5 in class and discuss the results.


Factors (3.6)

The main idea is for students to see how simple and straightforward it is to identify factors – the numbers you multiply together to form a product. To demonstrate, put five products on the board, and have students list under each one all the combinations they can think of for two numbers multiplied together to result in that product. For each set of factors, have volunteers list them in order, first filling in ‘1’ on one end and the product on the other. Ask if any are missing. Stress that listing in order makes it easier to compare sets of factors, find what is in common, and locate the greatest/highest common factor.

The Discuss box gives a good example of the problems they will encounter where knowing the HCF/GCF is the key to solving the problem. In this case, 12 is common to both 96 and 120.

(Note: we find both ‘HCF’ and ‘GCF’ used in different Caribbean syllabuses, so include both terms. Teachers will wish to acquaint students with both, but will use whichever is expected locally.)



Multiples (3.7)

Most students have no difficulty understanding the term ‘multiple’ as ‘the ‘answers you get when you multiply’. Looking at the diagram, one could say ‘15’ is the product, and that is also correct. In this lesson, however, we are noting the multiples of 3 or the multiples of 5. Help students remember: multiples are the same

size or larger than a factor, and when we compare multiples to find the LCM, we are looking for the smallest multiple in common. LCM is also a useful tool for problem solving. Teachers will want to use questions 4 and 5 in exercise 3.7 B, to strengthen this skill. For further mental practice, give short challenges such as ‘Two numbers have a sum of 18 and a difference of 2. What is their LCM? What is their HCF/GCF?’ (Note: as with HCF/GCF, both the term ‘lowest common multiple’ and the term ‘least common multiple’ are used in regional syllabuses. Teachers will wish to acquaint students with both terms, but use the term preferred locally.)

The Activity on the next page gives students a chance to practise multiples and common multiples. Simple game boards can be made, similar to the one shown. Help students see that they are not multiplying the numbers shown on the dice, but are thinking of the multiples of each number and looking for a multiple held in common. If a 4 and 5 are thrown, the first common multiple of 4 and 5 is 20, but if this is not on the game board, they look to find the next common multiple (40), and so on. The Activity develops mental maths skill, as students practise more than one operation.



Divisibility (3.8)

Discuss what students already know about divisibility rules, and use the lesson box to help students make the important connection between divisibility and multiples/number patterns. The Partner Activity may be used to help students express mathematical reasoning, orally or in writing. Note that each prime number has just two factors (itself and 1), and challenge students to count aloud in prime numbers as high as they can go. Exercise 3.8 B is a review of factors, prime


numbers, multiples, and commonality, and the Challenge question again shows how LCM can be used in problem solving. See Teaching tip below.



Prime factors (3.9)

ActivityThe calculator activity gives practice with divisibility, engages students and provides social learning. In addition, the calculator is used as a natural tool, and not as the focus of a lesson. The activity works well as a warm up to exploring prime factors.

In the lesson on prime factors, two common models are shown. In example 2, note that some teachers have students continue dividing until the final result is ‘1’. Instead, we have chosen to use the model where the last number is prime (as some students have been known to erroneously include the ‘1’ in the answer). In the factor tree model, note that the middle levels may have different factors (e.g. 4 × 9 may be 2 × 18 or 3 × 12), but the final row should have the same prime factors listed (even if the order is different). Guide students to present their final answers from smallest to largest, which will let them know whether to use exponents in their final answer.



The Summary should be reviewed with the idea of reinforcing how the various forms of number are all connected. The Unit Check may be used for continuous assessment.

Teaching tips•The Challenge after exercise 3.8 B

works very well as a practical activity, particularly when students work in pairs. Provide small bags and counters, and have students experiment, using trial and error, to find a solution that comes out even. The practical work tends to stay in students memories longer, and using LCM or HCF/GCF in problem solving is a very useful tool that needs an extra push to help students remember to use it.

•Explain that any number that is not prime is called a ‘composite’ number. Think of real-life examples of things that are made of more than two parts or elements. For example, orange juice compared to fruit punch, or in Science, an atom compared to a molecule.

• Some students feel a sense of satisfaction in breaking a large number down to its most basic parts, its prime factors. There is a sense of order and logic in doing so, which brings satisfaction to these future engineers and mathematicians.

Expansion / Extension of unitIT – Explore through research how prime factors are used in encryption, and why they are useful.


Sample lesson planUNIT 33.3 Patterns

Objectives •Extend patterns involving arithmetic

operations, square or cube numbers, fractions, decimal numbers, shapes

•Extend patterns involving number intervals

Engaging the students’ interest / ConnectionsWithout saying why, draw the following pattern as a warm up on the board:

→→ ↓ →→→ ↓↓ →→→→ ↓↓↓ →→→→→ ↓______________ Ask what pattern they see, what comes next, and how they know.

Teaching the lessonMathematical languageHave students name the different forms of number which are shown in exercises 3.3 A. They should be able to identify whole numbers, fractions, decimal numbers, ascending and descending series of numbers, cubed numbers, square numbers and money.

Practical activity / Developing the lesson•Explore some patterns that the students themselves create. Start them off by using a

number pattern with one operation (e.g. 5, 8, 11…). Ask what is the first step when a pattern needs to be extended – check the type of number, the interval or sequence with the pattern showing, and what makes sense as next steps. Emphasise once a pattern is extended, it has to make sense and fit the overall pattern. In this case, 3 is added each time, and for visual learners a small ‘+3’ could be inserted between each number in the pattern. Have students generate a few similar patterns and write their suggestions (ascending or descending).

•Have students work with partners to identify the pattern in each question of exercise 3.3 A, orally stating what comes next. Have students write the results independently, and then go over the patterns as a whole class, asking each time. ‘How do you know?’

•Either in the same lesson or the next day, move to patterns with two operations, such as +5, –2 (e.g. 7, 12, 10, 15, 13, 18, 16, 21…) and write ‘+5’ or ‘–2’ below and between each number in the sequence. See the Hint box next to exercise 3.3 B. Have students generate some patterns with two operations, and check understanding.


•Ask students to spot any patterns with operations in exercises 3.3 B or 3.3 C. (The questions in exercise 3.3 B all have operations except question 3, which is descending square numbers. Exercise 3.3 C question 1 are factors of 9, question 2 are cubed numbers, question 8 is a pattern of variables, but all others have operations.) Have students complete these two exercises, and compare their results to a partner. Encourage students to try the Challenges.

•The Workbook exercise provides further practice with patterns.

Differentiating for different learning styles For students struggling with patterns, use concrete objects to create patterns for students to extend. Include shapes to start off, and move on to using counters to represent the numbers shown in the simpler patterns in the exercise. Manipulating the objects that match the numbers can assist some students to better ‘see’ the patterns.

Logic challenge for quicker students

Give the following problem, and encourage ‘guess and check’ as a strategy: ‘Micah is 9 years older than twice the age of Jerome. Jerome is five years younger than Sean, who is eight years younger than Micah. What is the order of boys, youngest to oldest?’ (Jerome, Sean, Micah) For those ready for a greater challenge, add: ‘What is the age of each boy?’ (Jerome is 4, Sean is 9 and Micah is 17.)

Extension activities There is a CD-ROM activity to practise patterns.

Challenge

a) Have students work with partners to identify the pattern in each question of exercise 3.3 A, orally stating what comes next.

b) Have students make up patterns such as this: ‘What is the sixth number in the pattern that starts 1, 8, 27, 64…’

AssessmentCheck students informally as they describe the patterns in exercises 3.3 B and C to a partner and question their reasoning if it is off. Choose a mixed group of ten questions from the three exercises to retain as a student mark.

Summary of key points Thinking about patterns and number puzzles helps develop reasoning, and assists us to more easily solve problems in everyday Mathematics.

Patterns can be formed and extended using different operations or number forms.



•Identify and write decimal numbers value, place and expanded forms to 3 decimal places

•Write and say the names of numbers with decimal fractions using words or figures

•Show decimal numbers as fractions with denominator multiples of ten

•Compare and order decimal numbers using <, >, = using appropriate terminology

•Convert decimals to simple fractions

•Add and subtract decimals, including word problems

•Round decimal numbers to the nearest whole number, tenths or hundredths

•Round money to the nearest dollar

•Discuss uses and value of money, different types of goods and costs, importance of savings.

•Calculate totals using bills and coins, if needed for review

RESOURCESCounters; colours and drawing paper.

Teaching the content of the unitVisual models at the start of the unit are highly recommended to make very clear the connection between decimals and whole numbers. An effective model can be as simple as ‘1’ paper, cut into ten strips for tenths, and then each tenth cut again ten times to lay out hundredths. This 10-minute activity makes an strong impression, and can be referred back to later for students who need the reminder. Overall, this unit is a quick review of the value of decimals, and their use in practical problems, including the skill of rounding, which again links back to their relationship to whole numbers.




Place value (4.1)

Begin with models to show the relationship of decimals to whole numbers, such as the one suggested above. Make clear that place value is simply a tool for recording this relationship. Stimulate discussion about the pattern in the place value columns of the whole numbers, compared to that of decimal fractions (no ‘ones’). Help students note the first place value in decimals is tenths, and why, connecting again to the model. Have

UNIT 4 Decimals

Place value, Value, Comparing decimals, Rounding decimal numbers, Problem solving, Adding and subtracting decimal numbers, Problem solving with decimals, Assessment 2


them state why we sometimes call decimal numbers ‘decimal fractions’ – noting each is a fraction of just one whole (with denominators all multiples of 10). The emphasis remains on the concept, with the written work after.


Value (4.2)

Take the time to emphasise the value of each whole number, as well as the value of each decimal fraction. While it seems very simple, understanding of this concept continues to be an observed weakness seen in regional examinations. Both practical and written practice is worthwhile to catch any misunderstandings at this time.


Comparing decimals (4.3)

Before aligning decimal points and comparing fractions in written form, have students work mentally and orally to compare decimals and whole number / decimals and decide what is reasonable. Work up progressively, first comparing two whole numbers, or just two decimals, then two numbers that have both whole numbers and decimals, and then more than two such numbers. In the written form, aligning the decimals in a column helps students to easily see which whole numbers are larger, and therefore which numbers are larger. See Teaching tips below.


Rounding decimal numbers (4.4)

The two main concepts are that rounding to the nearest whole number involves the tenths place, regardless of how many decimal places might be in a number, and that rounding money to the nearest dollar means rounding cents and leaving just dollars. Practical activities can easily be created, using shopping advertisements. As a challenge for those who are advanced, give the following:

‘Rounded to the nearest thousand, the number is 5000. The sum of its digits is 15. Rounded to the nearest hundred, the number is 5100. Two of its digits are the same. What is the number?’ (answer 5145)



Problem solving (4.5)


Adding and subtracting decimal numbers (4.6)

The main idea for students to remember is to align place values by using the decimal point and not the numeral on the right. The operations are straightforward, but the single most common error for this type of question in examinations is when students carry out the operations using the wrong digits because they have failed to align the decimal points. Exercise 4.6 C may be taught as a separate ‘mini-lesson’, linked to practical activities, and reinforcing the idea that ‘faster’ is represented by smaller numbers.


Problem solving with decimals (4.7)

This lesson is rich in possibilities for discussion. Before working out solutions, each of the questions may be read aloud, and students should be encouraged to connect their own experiences with the themes of the word problems. Once students are engaged in the topic, the mathematics has a real-life meaning and is more easily remembered and solved. Question 2 includes the information in question 1, to reach the total of 200 trees. In question 3, the shop owners gave $876.25, ask students what other types of businesses the ‘other business owners’ might run. Questions 3 and 4 are linked for the total cost. In question 5, clarification of the word ‘exceed’ may be needed.

See Expansion below for ideas of how to integrate this lesson with other subjects.


Assessment 2

This is a two-part test covering all of the concepts practised in Unit 1 through Unit 4.

Teaching tips•Using graph paper (grid paper) for

comparing fractions is useful for students who have difficulty keeping large numbers aligned in their place value columns.

•When students compare, add or subtract whole numbers with decimals, have them first write the decimal point for each of the numbers, and then fill in the numbers around the decimal points. Many errors are made by not aligning, and this tip helps reduce these common errors.

• In questions with money where the total is less than a dollar, if coins are not used or rarely used locally, have students simply work out the questions as if the cent sign were a dollar sign (e.g. 4.5 question 7).

Expansion / Extension of unitMany of the problem-solving pages focus on a theme. The theme in this unit, in exercise 4.7, is the Environmental Club, and the idea of planting trees as a service for the community. Several Social Studies or Science themes may be connected to this topic, and could be included in an Integrated Unit. For example, the baobab tree may traditionally be a site for cultural gatherings, storytelling and events. Tree planting along a shoreline may help prevent soil or beach erosion. Fruit trees encourage self-sustaining agriculture for communities, or may introduce the idea of bartering and trade. Government agencies and NGOs have grants to support local projects, and how these grants are paid for opens a new window into Economics. In Language Arts, the sound of the wind in the casuarinas may be depicted in poetry or prose.


Sample lesson planUNIT 44.5 Problem solving

Objectives •Write and say the names of numbers

with decimal fractions using words or figures

•Compare and order decimal numbers using appropriate terminology

•Add and subtract decimals, including word problems

•Round decimal numbers to the nearest whole number, tenths or hundredths

•Round money to the nearest dollar

•Discuss uses and value of money, different types of goods and costs

•Calculate totals using bills and coins, if needed for review

Engaging the students’ interest / ConnectionsAsk for three volunteers to do a long jump, and two volunteers to measure. Take the class outside, mark a line in the yard, and have one student run and jump from that line. Have two students use the metre sticks and measure the distance of the jump to two decimal places, while other students record the result. Repeat with two more jumps. Once back in class, write the three measurements on the board in a row. Have students compare the numbers to what they saw in terms of which jump was longest. Ask what else they notice.

Teaching the lessonMathematical languageIn this lesson, decimal numbers/decimal fractions/decimals are interchangeable terms. Estimation involves rounding to the nearest whole number, or dollar.

Practical activity / Developing the lesson•Write the three distances recorded in the jump on the board aligning the decimal

points. Have students say what they notice about the three numbers, then order them, compare them, and round them.

•Look at question 1, and similarly discuss, write, compare, and round Kokab’s three jumps.

•Move to question 6 and have students compare the two speeds and state which run

RESOURCESMetre sticks (at least two); play money for small group; several items with price stickers or advertisements with a variety of prices.


took more time. Bring out the point that more distance in the jump is the best jump, but less time in the running is the fastest run. Ask why it is important to think about this when we are solving word problems.

•Move to a discussion of money, rounding and estimating. Using the advertisement or items with price stickers, choose three items to buy. Ask what would happen if you are in the store, and choose three items, but at the cashier you do not have enough money. Have students discuss their personal experiences or those of people they know where this has happened. This discussion opens up the real need to round and estimate the total to check if there is enough before proceeding to the cashier.

•Look at question 7 and round to the nearest dollar. Have students mentally work out what each pencil might cost. (See Teaching tips above.)

•Do several oral examples of rounding money, estimating totals, or estimating distances.

•If students need help, they might work with partners and complete the first five questions. After these are checked, some students may continue independently. Accomplished students may solve all ten problems and the Challenge on their own.

•Remind all students to re-read the key question of the word problem, make sure they have answered it, and put the appropriate word, symbol or measurement in the answers.

Use with WB Unit 4, Ex 5, if this exercise was not used previously.

Differentiating for different learning styles •If money questions prove difficult for some students, set up group tasks with paper

‘money’, cards with money totals and realistic prices with adult-like scenarios where they can trade or make change. Work with the group to build practical skills, and gradually have them take on some other money tasks (e.g. recording the lunch purchases).

•Money challenge for advanced students: Mona bought a scarf for $18 and paid $6 less than three times that price for a shirt. What currency notes/bills should she use to pay her total of both scarf and shirt? What will be her change?

Extension activities Challenge students to do a small project related to Caribbean currencies. Some examples are: a) which Caribbean currency has a coin that is a fraction? b) which Caribbean currency includes no coins? c) Which Caribbean currencies use notes for $1, or use coins for $1?

AssessmentThe main concept areas include rounding, estimating, comparing and mentally dividing a rounded whole. You may wish to isolate one or more of these concepts, and assess on that basis. You may also wish to take the opportunity to see how well students are identifying the key question and answering it, as a problem-solving skill.

Summary of key points Estimating and rounding are useful skills when working with decimals in problem solving. To round cents to the nearest dollar, round up 50 or more cents. To compare decimal numbers, for example in distance or speed, we can more easily compare in written form if the decimal points are aligned.

40

Text12



•Use the distributive property of multiplication


•Multiply 1- to 3-digit numbers

•Use multiplication tables proficiently

•Use operations in problem solving

•Apply reasoning skills

RESOURCESCounters such as buttons, shells or seeds; calculators to check work; scrap paper and pens to make grids.

Teaching the content of the unitThis unit is a straight forward review of multiplication by 1- to 3-digit numbers. It starts with single-digit multiplication, both simple calculations and word problems, and moves on to remind students of steps in an algorithm which may be used for more than one digit.

Teachers will note the use of more than one approach, as two similar but different examples are placed side by

side in exercises 5.2 and 5.3. There are many different algorithms which may be used successfully, and these two are most commonly used in the region. Students will be able to compare and contrast, and choose what they best understand.

We place an emphasis on learning to use the distributive property, and to ‘break down’ numbers, or manipulate them mentally, through which students build reasoning and mental Maths skills for life.





Introduce the topic by having students mentally add 126 three times, and once the total is reached, to add another 126 to that total. Next ask two students to solve the same question in writing. if neither writes 4 × 126, ask for more volunteers until it is shown. To further emphasise the point, have students add mentally 326 three times, then add 326 to the answer, and then again. Have them write out question 1, and compare answers. Ask which was more likely to be an error, the multi-step or multiplying. Have students continue with the first half of the exercise and see if all are on track with both the

UNIT 5 Operations – multiplication

Multiplication: review, Multiplication by 2-digit numbers, Multiplication by 3-digit numbers, Multiplying multiples of 10, Using the distributive property to solve mentally, Working with multiplication, Units 4 and 5 check and summary


concept and the procedures. If time, assign one of each of the word problems to small groups, have them complete it and explain to the class afterwards what they did, why, and how they know it is correct.


Multiplication by 2-digit numbers (5.2)

Have partners compare and contrast the two examples shown in the tinted box. Ask for clarification of various parts of the two similar algorithms shown, and discuss why their approaches might be different. As students begin with the first ten, remind them to keep the numbers clearly spaced apart, particularly as they will have to show regrouping. Check progress before students proceed to complete the exercise.



Multiplication by 3-digit numbers (5.3)

If possible, put up several examples of algorithms used in different countries for students to discuss and compare 3-digit multiplication approaches. examples you might consider showing are lattice multiplication, partial products or grid multiplication, Russian Peasant (double and halve), as well as the two shown in the text. Of these, grid/partial products is the most valuable for reinforcing value and place value, showing the partial products achieved without regrouping. 346 × 432 looks like this on the grid:

× 4 3 2

3 120 000 9000 600

4 16 000 1200 80

6 2400 180 12

All of the partial products are added to reach the total of 149 472. Show students that the lesson is not about the procedure, although it is important to be careful. Rather, success comes from carefully multiplying each value. Exercise 5.3 A

follows this introduction. Exercise 5.3 B and the three exercises in the Workbook provide additional practice, either to use with this lesson, or to support weaker students, or to use later for revision. The word problems in exercise 5.3 B are best worked out in class, and then discussed.



Multiplying multiples of 10 (5.4)

Give students many opportunities to multiply multiples of ten mentally, using examples from the classroom, money or from their own creativity. The simplest approach to assure success, after the concept is understood, is to have students multiply the numbers that are not 0 (shown in purple) separately from the zeros (shown in blue). In this way, if an additional 0 is generated through their multiplication, they are not confused by it.


Using the distributive property to solve mentally (5.5)



Working with multiplication (5.6)

This type of problem, sometimes called a ‘worked problem’ gives many opportunities to apply logic and reasoning skills. All of the questions should be solved through recognising the place value, doubling, halving or in some other way manipulating the stated calculations – not by calculating what is shown. For example, in 5.6 question 7, 46 is the same, but instead of 221, there is 442, which is 221 doubled. Therefore, the answer should also be doubled. After a thorough discussion of the ‘Discuss’ box, you may wish to have students work with partners orally describing what is the same, what is different, and how the differences affect the result, in each case. Ensure that students do not just calculate and


lose the valuable learning and essential skill building. You may wish to present a template for each question where students complete phrases such as these: What is the same…. What is different…. How does that affect the answer…. What is the new answer…. Alternatively, students could write a paragraph about each question, explaining how it relates to the original worked problem, and what they did to solve it, in every case.



The Summary includes both the basic decimal unit, with rounding, estimating and adding, as well as the multiplication review.

Teaching tips•The Challenge for the ‘worked

problems’ in 5.6 has half the total and half one of the factors, so the answer when divided is the same factor (221). Have students experiment using multiplication and division with halving and doubling numbers, and create challenges for their peers.

•Give students practice with calculators, as they explore doubling and halving, or other approaches to worked problems.

•Have them check one another’s multiplication, or check their own after completion, using calculators.

Expansion / Extension of unitHave students create Challenge questions with 1- to 3-digit numbers multiplying large numbers. They can estimate the number of places they think the answer might have, and then use calculators to check.

IT – Challenge students to find out the symbol used for multiplication that is not ‘×’.

Sample lesson planUNIT 55.5 Using the distributive property to solve mentally

Objectives •Use the distributive property of

multiplication


•Apply reasoning skills

RESOURCESSmall counters (e.g. buttons or beads) for each small group; scrap paper and coloured pencils/markers.


Engaging the students’ interest / ConnectionsChallenge small groups to demonstrate with counters the following, however they think it could be done: 5 × 7 = (5 × __ ) + (5 × __ ). In response, students could put 5 sets of 3 and 5 sets of 4, or, 5 sets of 5 and 5 sets of 2, or, 5 sets of 6 and 1 set of 5. In each case, they are breaking up one number and using the distributive property. It is visually very clear, as they use the counters and try breaking it up a different way, or simply lay out the array for 5 × 7, that the total remains the same.

Teaching the lessonMathematical languageUse the words ‘distributive property’, after the buttons/beads were distributed in different ways.

Practical activity / Developing the lessonThe main focus of this lesson is building mental skill at manipulating numbers. It is a skill that comes quickly to some students, and comes with practice to most. It is important to remove the fear of being wrong, to encourage experimentation, and assure students there is often more than one way to reach the right answer.

•Spend time discussing example 1. Why is it easier to make 14 into 10 + 4? 14 × 11 may not be easy for most to solve mentally, but 10 × 11, or 4 × 11 is easy. Adding the two results mentally is also not difficult. Let students know they can break up the 110, and just add the 100 to 44, and then put back on the 10. Assure students that when we can make it simpler to solve, we can work more quickly and confidently and still be right.

•Explore example 2, and have students explain in their own words what was done. Show that 12 × 20 is the same as 12 × 2 × 10, mentally. Bring in the Hint box to this example. Why is it simpler to multiply 12 × 30? Can we make it 12 × 3 × 10, mentally? Why do we take away one ‘12’ from the final answer?

•Reinforce the concept that what is done to one side of an equation must be done to the other. If we make it 12 × 30, we have added an extra 12. To go back to 12 × 29, we must take 12 off the answer as well.

•Ask students to turn to their partner and explain example 3, while listening in to ensure understanding. Then have volunteers explain example 3 to the whole class. Show how the distributive property was used, but the second part was broken down as well. These are steps that take place mentally (e.g. when multiplying 42 × 5), before finally putting the products together.

Differentiating for different learning styles The written out distribution of factors in this lesson is to represent what takes place mentally. The purpose is to show the steps for those that need to see as well as hear. For most students, this breakdown in writing will not be needed as they continue. For some students, the orderly breakdown using the distributive property is the support they need.

The Challenge question is to help students learn to be concise and articulate when describing what they do mathematically.


Extension activities Have students create an equation where the distributive property can be used.

AssessmentWorkbook Unit 5, Exercise 7 has just the breakdown of factors in equations, using the distributive property, and is very useful to check if students understand this skill. In the Student’s Book, exercise 5.5 questions 7–10 also check this skill, as well as requiring the final answer.

Summary of key points Numbers can be broken down using the distributive property, and multiplied mentally.

Reasoning and logic are needed to check that both sides of the equation have the same value.

We can use the commutative, associative and distributive properties to manipulate numbers and more easily find the solution mentally.



•Use symbols to represent the unknown

•Solve word problems with single variable equations

•Use simple formulae and solve problems (e.g. problems involving speed, time and distance)

•Solve problems using strategies

•Recognise algebraic expressions (e.g. ‘10 more than before’ as ‘n + 10’)

•Use order of operations

•Solve equations with more than one sign

•Use order of operations with fractions

RESOURCESCalculators; cards for Order of operations game (for each group playing the game: four of each number from 1–15, and two of each number up to 30, written on small pieces of card).

Teaching the content of the unitStudents have been doing Algebra for years, in different forms but without the terms. For example, the box is a variable

in this simple equation: 2 + □ = 3. Unit 6 takes what they already know about solving simple equations, as outlined in the Hint box in section 6.1, and puts it in forms they will see as they move on to higher mathematics. In particular, students are given extra practice in choosing the operations (signs) that will balance both sides of an equation. Order of operations is reviewed and students are given several exercises to build these skills in the Student’s Book, Workbook and on the CD-ROM.




Overview (6.1)

Review the terms ‘symbol’, ‘the unknown’, ‘sign’ and ‘equation’, linking each to concepts they already know. Be clear that an ‘equation’ is a number sentence that includes an equal sign. Discuss exercises 6.1 A and 6.1 B, and talk about how to solve each one mentally, after identifying the operation and its inverse. After emphasising how straightforward these simple equations are, show how it can be written out, as in the example to the right of exercise 6.1 A. Let students identify the steps in this simple equation, which will be

UNIT 6 Algebra

Overview, Order of operations, Working with operations, Substitution, Mental Mathematics practice, Strategy – making a list


useful to remember for more complicated questions. The Challenge includes two operations. A valuable point is made here, with 4 first subtracted from both sides. There is more practice on this concept later in the textbook.



Order of operations (6.2)

It is helpful for students to know that many people, possibly even the ones who help them with their homework, are not aware of the rules for order of operations. Start the lesson by putting an equation on the board or discussing the one at the beginning of the introduction. Encourage students to say why people might say two possible answers, and let them know that this is the real problem – that there has to be a rule of order so different answers are not given. Draw attention to the ‘Remember’ box. Then go through all of the questions in question 1–10 and have students say what to do first, what to do second, and then how to solve (without giving away the answers). The questions become progressively more difficult, so be sure to give students adequate time to do all, even if more than one lesson time is needed. Questions 11–15 may be done by partners, and followed up orally, with presenters explaining the steps and defending their reasoning.

Note: BODMAS / BOMDAS / PENDAS and other acronyms are often used to remember order of operations. However, they can be misleading for some students. Multiplication and division carry equal weight, yet some students interpret the letters to mean one always comes before the other. For this reason, in the explanation box, ‘MD’ are on the same line (conducted left to right, whichever comes first), as are ‘AS’ (again left to right). See the game activity below under Extension.

Group challenge – do the calculator activity after exercise 6.2.



Working with operations (6.3)

Some of the questions are simple operations, while others include order of operations. There are some students, however, who find this type of ‘puzzle’ very difficult. ‘Guess and check’ strategy can be used by these students, which may boost their confidence and success.

See also Teacher Resource website – Level 6 Arithmetic page ‘Putting in signs to make equations true’ worksheet.



Substitution (6.4)

An activity with a code is a popular way to start this lesson. Students understand that each symbol in a code stands for a letter or number, and can easily then transition to the algebraic substitutions in the lesson.

Again review the term ‘unknown’, noting that we insert a ‘value’ to match from the key box, and then solve. Explain the difference between an ‘equation’, where there is an equal sign, and an ‘expression’, where there is no equal sign. Review exponents, reminding students that the power tells us how many times that figure is multiplied by itself.

Teachers may wish to make exercise 6.4 C, questions 11–20, a separate lesson about algebraic expressions. First, set up several scenarios that can be described using an expression. For example, draw five groups of three triangles. Circle one set and draw an arrow to it with ‘x = ’. Then say the expression ‘Five times more than x’, and the answer expression ‘5x’. Students will need to be assured that there will be no solution as there is with an equation – they are just writing an expression that reflects the phrase or sentences shown.


Encourage students to try the problems in the Challenge box, as preparation for what they might find on examinations, and as problem puzzles they will enjoy solving.

See also Teacher Resource website – Level 6 Enrichment page ‘Simple algebra’.

There is a CD-ROM Activity for exercise 6.4 C.


Mental Mathematics practice (6.5)

This exercise helps build reasoning skill. Students need to be precise, and consider what comes first, and what changes as information is added. Each is an equation, therefore there is a clear answer. Students may be encouraged to use ‘n’ to stand for the unknown (e.g. question 1, n = $85).


Strategy – making a list (6.6)


Teaching tips• Some students may find the verbal

problem puzzles (e.g. in exercise 6.5), order of operations or substitution of signs exercises very challenging. These appear fairly simple, but some higher level thinking is required. Breaking down the lessons into smaller parts, assigning just a few questions at any one time, and giving many opportunities to review the work while students are working, are suggestions that may ensure all students gain these skills.

•For advanced students, have them make up a two-step code, with an encryption sheet and a secret message. (For example, each letter of the alphabet could be represented by a number that is a multiple of 5, then subtract 2, with ‘a’ beginning at ‘5 – 2’.)

Expansion / Extension of unit

Game to practise order of operations

Three to five players play in one group. Shuffle the number cards, take out one and put it in the centre face up. Students then are dealt two cards each. If anyone can make an equation to match the card in the centre, they win the round, take the centre card and set it to one side, take new cards for themselves and put a new card in the centre from the card deck. If no winner, another card will be dealt to each of the players. Students should be encouraged to use order of operation, and must show how all three cards can be used in an equation to match the centre card in order to win. If there are still no winners, another round of cards may be dealt. Decide in advance either how long students may play, or how many rounds will be played before checking to see who has won the most rounds. To mix it up, students may turn in their cards after two or three rounds and start again with two new cards to each player.

Challenge: Write an equation for each of the following.

a) Teri picked up 29 pieces of driftwood. Along with Sami, a total of 60 pieces were collected. How many did Sami collect? (e.g. 29 + n = 60)

b) The total cost for four movie tickets was $2 less than $70. What did each ticket cost? (e.g. 4x = $70 – $2 or 4x + $2 = $70)


Sample lesson planUNIT 66.6 Making a List – strategy for problem solving

Objectives •Recognise and use problem solving

strategies

•Use reasoning skills

Engaging the students’ interest / ConnectionsDramatise the word problem. Write the problem on the board and have a student read it aloud. At this point, keep textbooks closed. Have six volunteers each hold a ribbon or large coloured paper. Choose two ‘directors’ to physically move the students around at the front of the room to show each possible pair combination. Have all other students record the results. Allow students to ‘direct’ the set, without instruction. If they move students randomly, they will learn from this process and appreciate the ordered approach that follows.

Teaching the lessonMathematical languageUse the word ‘random’ to express actions without order or without a pattern. An ‘organised’ list, then, is the opposite, and approaches the task systematically.

Practical activity / Developing the lesson•Still without opening the textbooks, have partners or students independently

practise making different combinations using the squares of paper. After several minutes, have students write the total number of different pairs they made, and cover this total.

•Explore in discussion how to go about the task in an organised way, asking if any of the students used a system. Guide students to suggest that starting with just one colour, and pairing it with all of the other colours in turn, is a good first step.

•Taking the next colour, again systematically pair it with the remaining colours. After writing this step and drawing lines, ask students why there are fewer answers this time. Guide them to acknowledge the first set of pairs would have already included the first and second colours together.

•Have students continue through the rest of the problem on their own, writing letters for the colours and drawing lines to form the remaining pairs.

•Count the results, and discuss any differences. Students who have fewer total pairs need to be given time to find out which ones they left out, and why.

•Give students time to check the original total they had made earlier and covered over.

•Discuss why an organised list is a better approach than randomly making pairs.

RESOURCESScrap paper; small squares of coloured paper if available (6 colours); ruler; one ribbon or large paper in each of six colours.


Differentiating for different learning styles Use the coloured paper squares for those students who have difficulty visualising the organised list. Scrap paper with the coloured papers glued on and lines connecting them, can be used to make it very clear. Enough coloured squares should be available so students can make each possible colour combination on the scrap papers, and see all of them at one time.

The coloured squares may also be useful for students who cannot quite understand that each round does not need to include the coloured pairs previously used. By seeing the actual coloured papers, they may more easily accept that the pair does not need to be recorded again.

Extension activities Have small groups choose five or six lunch items from the school canteen, and make a chart showing the number of days two items could be chosen without repeating the same combinations. Similarly, several topics could be chosen, such as sports/games for PE, or activities for Art. Have students make posters with their results, include an organised list as shown in the textbook, and a sentence explaining their result.

AssessmentView student practice, and informally evaluate their reasoning in discussions and through direct oral questions.

The Challenge question in Workbook Unit 7, after Exercise 2, uses the organised list strategy.

Summary of key points An organised list is a strategy for problem solving that is useful in keeping track of different possible combinations.

50 Unit7•BrightSparksTeacher’sBook6


•Explain and use division as the inverse of multiplication

•Use different forms for division, including mentally

•Apply divisibility rules

•Convert remainders in division to fractions or decimals

•Multiply and divide by 1- to 3-digit numbers

•Use multiplication tables proficiently

•Use four operations in problem solving and explain the choice of operation

•Apply skills to single- and multi-step word problems

RESOURCESSmall counters; containers for the counters such as cups; colours and scrap paper; calculators.

Teaching the content of the unitAs a review of division, the basic concepts of divisibility, ‘fact families’ showing the inverses, word problems with the key words that indicate division, division algorithms (‘long’ form to show where

there might be errors in understanding) and ways to deal with remainders are all covered. Teachers and students will have a chance to find any aspects of division that need extra support, and to practise areas where it is needed.




Division: review (7.1)

Simple divisionThere are four key concepts covered in this simple division exercise. Each has proved to be problematic for some students in this year group, despite its seeming simplicity. It is recommended to assess students’ understanding of each of these four points both orally and in writing, before moving on. The word problems lend themselves well to acting out the scenarios, using 8 cups and at least 200 counters of various types.


Long division (7.2)

Emphasise for students the place value as they divide, but note that we only work with one place at a time. Review of the algorithm is helpful, and the steps are shown in colour to help students keep on

UNIT 7 Operations – division

Division: review, Long division, Dividing money, Working with remainders, Division with 3-digit divisors, Units 6 and 7 check and summary, Assessment 3


track, but the focus of the lesson is not to be about procedure, but about division. Some may question teaching the long form of division. There are two main reasons it is valuable. This form allows students to more easily see the divisor and dividend, keeping track of place value, a frequent source of error. In addition, understanding this form of division is a lifelong skill, when calculators are not available. To cement the practice and skill, many of the exercises in this unit instruct students to use the long form.



Dividing money (7.3)

When dividing money, and later with other division questions with decimal numbers, it is beneficial for students to have the habit of writing the decimal point in the answer before they begin to do the division. With this habit, they are less likely to forget to put it in the answer, and more likely to place it in the correct position.

Working with remainders (7.4)

Each of the three examples has a slightly different approach to remainders. Have students act out, or sketch on scrap paper, each of these examples, and then discuss why the remainder is treated in the way it is, for each case. Emphasise that the answers always need to be sensible, and we rely on reason to help us consider the options for dealing with remainders. Note that questions that ask ‘On average, how many…’ tell us that not every day will be the same, so generally we take the number, or round it, and leave off the remainder (e.g. Workbook Exercise 3 question 1).

Exercises 7.4 B and 7.4 C provide practice and revision of the previous exercises.

See Teaching tips below.


Division with 3-digit divisors (7.5)




Use the Summary box to help students remember the key points for these two units. Follow with the Unit Check as regular assessment.

Assessment 3

Have students do Part 1 and Part 2 on separate occasions. Talk about Part 1 before students begin, asking about question 4, what ‘Youth Volunteers’ do, and whether they have ever done service to the community. Point out that question 7 is a ‘worked problem’, asking them how they would find those answers (not calculating, but looking for the patterns in the numbers shown).

For Part 2, in question 7 explain what a ‘commission’ is, and discuss the strategy they might use in question 10.

After marking and evaluating the results, take time to go through the overall Assessments with the class, with emphasis where there were the greatest misunderstandings. Re-teach and provide additional practice either to the whole class or to small groups, based on the results.

Teaching tips•The Challenges need to reach mid-

range students who benefit from stretching their thinking. Challenges also need to push the thinking of the quicker students when working independently, and therefore no prepping before the Challenge should be given. For example, with the Challenge after exercise 7.4 C, let students discover on their own that they will need to change the kg to g.


•For the students still struggling with 2-digit divisors, it makes more sense to give them extra practice with 3- to 5-digit dividends, and gradually become successful, rather than move on to 3-digit divisors.

• If students are not successful with division because they are weak in their multiplication tables, teachers might allow some opportunities when they are allowed to use their multiplication tables chart, in order to practise actual division concepts. It is a crutch, but allows them to ‘walk’ through the division process and gain valuable practice while they continue to develop their multiplication skills.

Expansion / Extension of unitHave students create a division question where the only criteria is the remainder is not important, and can be dropped from the answer.

Sample lesson planUNIT 77.5 Division with 3-digit divisors

Objectives •Explain and use division as the inverse

of multiplication

•Use different forms for division, including mentally

•Apply divisibility rules

•Use reasoning when resolving remainders

•Multiply and divide by 1- to 3-digit numbers

Engaging the students’ interest / ConnectionsPlay a ten minute game revising divisors, quotients and fact families: Give students ‘bingo’ cards, and some counters to cover the numbers. A volunteer will call out a single number randomly from a sheet which lists numbers from 2 to 48. Students may cover two numbers that are in the same multiplication/division fact family as the called number. For example, if ‘6’ is called, students may cover 2 and 12, or 3 and 18, or 4 and

RESOURCESBingo: Prepare a class set of ‘bingo’ cards, each with five rows and five columns, and random numbers between 2 and 48 written in the squares; on one sheet of paper, write all numbers from 2 through 50 in random order; counters such as buttons.For Extension, provide access to newspapers or the Internet for research about Caribbean Carnival-type celebrations.


24, or 5 and 30, or 6 and 36. If 48 is called, they may cover 2 and 24, or 3 and 16, 4 and 12, or 6 and 8.

Teaching the lessonMathematical languageReview ‘divisor’, ‘dividend’ and ‘quotient’.

Practical activity / Developing the lesson•Ask students what main question is asked in the word problem

•Discuss the word problem and make reasonable estimates as a group. With the larger numbers, it is even more important for students to make rough estimates for what would be a reasonable answer. If their workings give an answer which does not fit well with the estimate, it triggers the awareness of the need to go back and check the answer again.

•Discuss the algorithm example thoroughly, noting the three colours used in each step. Emphasise the rounded estimate in each step that helps choose a number to multiply.

•Ask whether the answer sentence matches the main question of the problem.

•Question why the remainder is not needed.

•Have students complete questions 1–7, and then check as a group, or with partners.

•Have students complete questions 8–12 independently (to use for assessment).

•Discuss question 13 as a group. Ask how we can simplify the question and form an estimate to solve it more easily (take off 3 zeros from each number). Solve this question together. Discuss the meaning of ‘average price’, which simply means some tickets were higher, some lower, and the remainder is not needed.

•The Workbook exercise allows further practice.

Differentiating for different learning styles •Students who are still struggling with division will need opportunities to work one-

on-one with the teacher or with peer tutors. For visual learners, use manipulatives and ask students to work out how many groups of 3 can be formed from 96. Then ask what operation they used (3 × __ = 96 or 96 ÷ 3 = 32). Move to a similar question with a 2-digit divisor, such as asking and sketching out how many groups of 32 could be found in 960. When these tasks are easily done, transition to the written form.

•For students who have difficulty with the algorithm, as they work on a question ask questions and observe, in order to spot where the difficulty lies. Often simply watching the student work out a question on his/her own is enough to spot the misunderstanding and correct it, after which the student can be observed as he/she tries a few more questions.

•Other techniques for struggling learners include giving fewer questions at one time (then checking, assisting and praising the effort, without pressure to finish all), or to provide tools such as a multiplication chart, or manipulatives (e.g. abacus). For students struggling with the concept, students can first work with smaller numbers until they are ready to move on to larger numbers.


Extension activities The picture of Carnival celebrations will be familiar to students in many islands (even if called ‘Kadooment’ in Cropover, or ‘Bacchanal’, ‘Caribana’, ‘Junkanoo’, or by another name). Have students investigate facts related to these celebrations, share their own experiences, and integrate to Social Studies cultural festivals.

AssessmentUse questions 8–12 for assessment purposes, as these include straight division by a 3-digit divisor, one with a 0, and word problems.

Summary of key points Dividing by a 3-digit divisor, it is especially important to first estimate what a reasonable answer might be. The steps for dividing are the same as used earlier with 1- or 2-digit divisors.

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BrightSparksTeacher’sBook6•Unit8


•Read / write / identify and describe fractions

•Illustrate / determine fractions of a whole

•Find fractional parts of a given quantity

•Identify equivalent fractions

•Order and compare fractions

•Convert improper fractions / mixed numbers

•Add or subtract fractions with or without like denominators, giving answers in simplest form

•Simplify fractions and mixed numbers

•Use cancellation in multiplying fractions

•Multiply or divide fractions, including mixed numbers or fractions and whole numbers, giving the answers in lowest terms

•Use fractions in practical situations

RESOURCESDice; small squares of paper with tape or sticky notes; scrap paper and colours for diagrams and sketches.

Teaching the content of the unitThis unit offers a thorough review of fractions, including estimation and rounding, all four operations, problem solving, strategies, short cuts and hints. All students will not need review in all topics; therefore, the first lesson is a pre-test to check existing knowledge and skills.




Fractions: review (8.1)

Assign this lesson as a pre-test to check what skills are strong and what areas need practice or review. Students will need a block of time without distractions, and should hear no instructions prior to the exercise. After it is completed, teachers

UNIT 8 Fractions

Fractions: review, Estimating and rounding fractions, Simple fractions: review, Equivalent fractions, Simplifying equivalent fractions, Improper fractions and mixed numbers: review, Addition of fractions: review, Subtraction of fractions: review, Adding and subtracting mixed numbers, Picture adding and subtracting mixed numbers, Multiplying fractions: review, Multiplying fractions with cancelling, Multiplying mixed numbers, Word problems, Problem solving with ‘of ’, Division of fractions: review, Problem solving, Unit 8 check and summary


will want to analyse the results, and keep a record of which fraction topics will need extra attention for each student. Students should also be encouraged to keep a record of topics they need to improve (see note below under Extension). Teachers might add an extra incentive for students to focus their best effort, such as offering a reward of a ‘Homework Pass’ that could be used at any time during this unit (student choice) for the high achiever(s).

Estimating and rounding fractions (8.2)



Simple fractions: review (8.3)

Exercise 8.3 reviews fractions at the most basic level, writing decimal fractions as common fractions and vice versa, and adding fractions with same denominators. Students recently practised HCF/GCF, but may still need to be reminded that factors are the same size or smaller than the numbers shown.


Equivalent fractions (8.4)

Use the Activity on the previous page to assist students to fully grasp that equivalent fractions do not change their value. Engage students in discussing the final two questions of the activity. In the tinted box for exercise 8.4, students can observe the difference across the equal sign, and match it (e.g. 3

5 = 12□ the 3 is four

times greater to become 12, so the 5 must become four times greater). Alternatively, they can use cross multiplication (cross products). Orally discuss questions 1–10. In every case, compare the change observed across the equal sign, and state it aloud (e.g. in question 5 the 48 becomes 12 so the numerator over it is four times less). Remind students they can go left to right or right to left. Encourage students to use

logic and reasoning in the problem solving, and to sketch it if they wish.



Simplifying equivalent fractions (8.5)

Reducing fractions is an essential skill that is practised here in a basic form. Students need to identify the GCF/HCF and reduce the fractions in questions 7–16. Check that all students are able to do this step successfully. Encourage students to sketch question 17 and explain their results.

Extra Challenge: ‘What fraction equivalent to 15

39 has prime numbers for both the numerator and the denominator?’


Improper fractions and mixed numbers: review (8.6)

This lesson is a basic overview of improper fractions and mixed numbers, including comparisons. Include diagrams in the introduction and check that students understand improper fractions conceptually, and can match a diagram or model to the symbol for an improper fraction or mixed number.



Addition of fractions: review (8.7)

The methodologies described in the lesson box are commonly used, but not the only approaches. Most important is that students understand that there always must be a common denominator for addition and subtraction of fractions. If this lesson is used for the whole class, the extra challenge below can be added to keep the advanced students engaged.

Extra Challenge: ‘One third of a number added to a quarter of 12 is 4. What is the number?’



Subtraction of fractions: review (8.8)

Students who need the review will find a basic lesson gradually increasing in difficulty. Teachers may wish to remind students of LCM, noting it will be the same size or larger than the numerators shown. Encourage students to review the ‘Hint’ box and explain to a partner why the hint is useful.


Adding and subtracting mixed numbers (8.9)

This lesson combines the several previous review lessons, and adds hints for combing whole numbers when adding, or dealing with whole numbers when subtracting mixed numbers.


Picture adding and subtracting mixed numbers (8.10)

Sketching these subtraction questions reinforces the work for some students and is a valuable support to others. this practice makes the connection between real world and written examples in subtracting fractions. Discuss the ‘Picture it’ box, and sketch a cake with 1

4 and 112 to

confirm it matches as 13. Talk about how

much is left, and what would be the total if there were three cakes (22

3 not used). Teach students to mentally subtract a fraction from a whole number, reinforcing with diagrams rather than equations to ensure they can picture the concept.

For exercise 8.10 B, have students work with a partner to identify the key question, solve it and write the answer to the key question in questions 1 to 3. Assign questions 4–7 and the Challenge for homework. Encourage students to use diagrams or equations in each question.

Multiplying fractions; review (8.11)

Review the tinted lesson, ‘Think’ and ‘Challenge’ boxes with the whole class.

Have students solve the questions mentally and orally.


Multiplying fractions with cancelling (8.12)

Thoroughly go through the ways we can cancel as described in the tinted box, and the need for cancelling. Assure students if we do not fully cancel before working the equation, that we can reduce the answer and end up with the same result. Ask why fully cancelling is more efficient (cancelling completely allows us to work with smaller numbers when we multiply). Have students complete exercises A and B, and check progress.

Multiplying mixed numbers (8.13)

Make the distinction between multiplying a fraction by a whole number (where a denominator of 1 is all that is needed), and multiplying by mixed numbers (where they must first be changed to mixed numbers).


Word problems (8.14)

All students should do this lesson, applying fraction skills to real-life situations. Discuss with students who has shopped at the market or supermarket, and any prices of agricultural goods they are aware of in their locality. The lesson may be integrated with social studies or with the agriculture curriculum. A discussion of value when purchasing and selling goods can easily be linked to this lesson.


Problem solving with ‘of’ (8.15)

Students can discuss any diving they might do, ocean conservancy or if they have any collections, to connect with the word problem topic. The emphasis is to use reasoning, with a key skill just being able to interpret and understand the word problem.


Division of fractions: review (8.16)

The model for division of fractions is useful, as the answer is not always what students expect (see the ‘Picture it’ box). By practising with different model diagrams, students are more able to predict when their answer will be a whole number. The procedures of changing mixed numbers first to improper fractions, or of multiplying by the reciprocal, are useful. More important, however, is that students understand just what they are doing when they divide by a fraction. Spend some time discussing the pattern in questions 11–13.

Exercise 8.16 B provides ten more questions for calculation, and ten more word problems using division of fractions or mixed numbers.



These ten word problems based on real-world situations can be completed by advanced students on their own, or with partners for students who need extra support. Go through the questions together to reinforce the need for reasonable answers.



Review the summary points, and complete the Unit Check. Compare the final score of the overview in the Unit Check with that of the pre-test at the start of the unit.

Teaching tips• Students who need extra support may

find fraction manipulatives helpful, first modelling the equation, and then using the written form. Diagrams and models are also useful for visual learners.

•Bright Sparks Student’s Book 5 has an extensive review of all fraction work, for students who need to go back and re-learn the basic skills. The CD-ROM for that level has many fraction-based activities.

•Fractions, as a topic, often has a wide range of skill levels at the beginning of the unit. Teachers will want to address the weak understandings of students who need it, while not slowing down the progress of the students that already have these skills. Challenges, activities, and models help keep advanced students engaged and on track.

Expansion / Extension of unitStudents may start a study journal, keeping track of goals for improvement in each topic of mathematics.

Challenge

Make a circle graph to represent the following information: At the animal shelter 1

3 of the animals are large dogs, 16

are small dogs, 14 are cats, 1

8 are rabbits and the rest are farm animals. What fraction of the animals represents the farm animals? (answer: 1

8)


Sample lesson planUNIT 88.2 Estimating and rounding fractions

Objectives •Round fractions to 0, 1

2 or 1

•Round simple fractions and mixed numbers

•Estimate totals when adding or subtracting rounded fractions or mixed numbers

•Use estimation and rounding in problem solving

Engaging the students’ interest / ConnectionsAsk for volunteers to remind the class what we must remember for rounding numbers. Then ask how this is similar or different when rounding fractions. Let them know they will have a game to practise this immediately after the introduction.

Teaching the lessonMathematical languageRemind students of the terms numerator, denominator.

Practical activity / Developing the lesson•Review the information in the introduction for 8.2 ‘Fractions and Estimation’.

Discuss the examples, and guide students to notice that when the numerator and denominator are very far apart, it will round to 0, when they are very close together, it will usually round to 1 whole (this will not be true of thirds). When they are unsure, they can make a small fraction into an equivalent fraction with larger numbers, to make it more obvious how it should be rounded. To get better at this skill, they will next play the dice game ‘Estimation Game’.

•Estimation Game: Put students in teams of four, two on each team. Each pair of students takes a turn rolling 2 dice. They put the smaller number as the numerator and the larger number as the denominator. The other team rolls the dice and forms a fraction, and then the teams compare the two fractions. The teams round the fractions to 0, 1 or 1

2, and the team with the larger choice wins a point. After ten rounds, students check which team has gained the most points. Of the possible combinations, note that 1

3 and 23 both round to 1

2 (as do 26 and 4

6). If students are not yet strong in rounding or estimating fractions, playing in groups of four gives them a chance to make decisions together. One or two students who are particularly strong in fraction skills might serve as roaming referees.

•Have the class discuss questions 1–5 and state why each should be placed nearest to 0, 1

2 or 1.

RESOURCESDice (two for each pair of students); scrap paper; marker; small papers with tape or sticky notes.


•If needed, practise estimation with the practical activity noted below.

•Guide students to understand why the fractions and mixed numbers in questions 6–11 should be rounded, and then mentally added or subtracted. Write the results.

•Discuss the word problem in question 12 and help students mentally count how many halves are in 5 wholes. Have students similarly approach the next two word problems, and then discuss their results.

•There is a Workbook exercise to provide further practice (Unit 8, Exercise 1).

Differentiating for different learning styles In the dice game, have students place one die above the other, to visually represent the fraction, and support students who learn more easily visually.

The practical activity below can be used for quicker students to compare and order all of the fractions created.

Extension activities Practical activity: Either with a small group who need more help, or as a whole class, have half write a number between 1 and 20 on the top half of a small paper. Have students underline the number, and fold the paper below the underline. Mix the papers, and then hand them out to the remaining students who are to write any number between 20 and 100. Open the papers and have students bring each up and place it where it best belongs on a number line sketched on the board (0 on one side, 1 on the other, and 1

2 in the middle). This activity can also be used for estimating and comparing fractions.

AssessmentCheck students’ reasoning as they round and estimate in the activities.

Summary of key points Rounding fractions to 0, 1

2 or 1 allows us to estimate the total in addition and subtraction.

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•Divide and multiply decimal numbers by multiples of 10

•Multiply and divide up to 3 decimal places by a whole number

•Multiply and divide by a decimal number after first multiplying by a power of 10 to change the decimal to a whole number

•Solve problems using decimal numbers, including money situations

•Compare prices for best value

•Discuss uses and value of money, and different types of goods and costs

•Round to the nearest cent or dollar

•Use estimation of decimal numbers in practical applications

RESOURCESExercise books for writing (e.g. Maths journal); calculators; picture of dinosaur; access to information sources for student research; scrap paper and colours.

Teaching the content of the unitIn this unit, students will multiply and divide decimal numbers, convert to

fractions, round and estimate, and problem solve, including work with money. They earlier reviewed place value, addition and subtraction, so should be ready to manipulate decimals in more complex ways.

To introduce the unit, have students do a ‘quick write’, where they are given five minutes to write everything they can about decimals. Start them off with: ‘The important thing to remember about decimal is …’





The introduction shows the most common approach to multiplying decimals, where the key point is to multiply as if they were whole numbers, and then count decimal places in order to place the decimal point correctly in the answer. The exercise may be taught over two days, if needed. At the other level, if students are advanced, teachers may wish to have students work out questions 11–25, and use questions 1–10 for oral work.

In exercise 9.1 B the theme of the word problems links well to social studies or an agriculture curriculum, as well as use of

Multiplication: review, Multiplying, Estimates, Working with estimates, Problem solving, Multiplying by multiples of 10, Multiplying money, Dividing decimal numbers, Division with remainders, Division written as a fraction, Dividing decimals by multiples of 10, Dividing by a decimal number, Unit 9 check and summary, Assessment 4

UNIT 9 Working with decimals


money. The ideas or terms of rate, average production, measurement in litres, and earnings are all interwoven in the word problems, and can be understood from context. Read through the questions orally either before or after students work on them together, drawing connections to student experience, adding to their knowledge of farming, and checking that the solutions are reasonable.



Multiplying (9.2)

The lesson provides further practice with slightly larger numbers, both straight calculations and word problems. The latter can be linked to a social studies ‘world cultures’ theme, as each question depicts a cultural item or theme from Japan.


Estimates (9.3)

Emphasise that whole numbers rather than decimals are used for estimation, and therefore it is important to first round to the nearest whole number. This exercise, where the decimal point is placed in the correct position in the answer, has proven to be very useful in developing reasoning when working with decimals. Questions 7 and 9, may prove particularly challenging. If possible, allow students to check their results with a calculator, and if different, write that answer next to their own.

Working with estimates (9.4)

The main idea of the lesson is when rounding money, it is sometimes wise to overestimate, rather than underestimate. Introduce these terms to the students, and give them opportunities to share examples from their own experience. The tinted lesson box also refreshes the concept of breaking up the number with the distributive property, in order to solve it mentally. Both of these are life skills that students will need to use as they

gradually make more purchases on their own. Questions 13 and 14 address the idea of overestimation, while question 15 is best overestimated and rounded to $300 × 60. Teachers may wish to allow students to check their estimates with a calculator, with the main purpose of checking whether students have the correct number of whole numbers in their answers.

The word problems in this exercise, as well as the entire next lesson, are themed around the concept of dinosaurs. The theme continues to engage many students and dinosaur variations of type and time period offer many mathematical opportunities.




Multiplying by multiples of 10 (9.6)

Mathematicians will make it very clear that we do not ‘move the decimal point’ in questions of this type. At the same time, students are more successful if they count the number of places that change, and have this information guide the final answer. For this reason we state that the decimal point ‘appears’ to move, in the final answer. See Teaching tips below. This lesson is an essential step to master, prior to later attempting division with decimals.

The word problems are everyday situations using money. After students complete these questions, talk through each one, and check that all students understand.


Multiplying money (9.7)

There are a few situations where students may see more than two decimal places for money. These are usually financial institutions. Start the lesson with a practical example, and several oral opportunities to round to the nearest cent. After the students practise calculations, and work through the word problems,


discuss the five money situations described in questions 11–15.


Dividing decimal numbers (9.8)

See Teaching tips below. The main idea in this lesson is that the numbers with decimals are all divided by whole numbers. In this case, no special work is needed. Students will place the decimal point for the quotient directly above the decimal point in the dividend, and then divide as usual. Discuss the theme of the word problems, engaging students in sharing their similar experiences if they have visited an animal sanctuary.


Division with remainders (9.9)

The next seven lessons centre on division of decimals. Each takes one small part of the concept to develop and practise. Not all students will need to go fully through each exercise. Assess progress and make adjustments based on the class skill level.

The main idea in this lesson is to add additional zeros to the right of the decimal number if needed to complete the division. See Teaching tips below.



Division written as a fraction (9.10)

Exercise 9.10 A gives students a chance to recognise the fraction bar as a division symbol, and to practise the strategy of converting a fraction to a decimal through division.

Exercise 9.10 B consists of word problems. Throughout the region there are itinerant traders who bring produce by sea or by land to markets where they either sell it to vendors, or sell it straight to customers. These valuable workers are the theme of these word problems. The scenarios in the questions are real-life situations which students might encounter. Prices may vary

throughout the region, but the marketing lesson with money is common to all. Teachers may wish to use some questions as whole class discussions, and others as partner work followed by class review.


Dividing decimals by multiples of 10 (9.11)

The ‘Remember’ box has the main teaching point of the lesson. From the examples, make sure students all understand that dividing by multiples of 10 will result in fewer whole numbers (fewer digits to the left of the decimal point), whereas multiplying by multiples of 10 will result in more whole numbers (more digits to the left of the decimal point). With this skill, they should be able to quickly check their work on an on-going basis. See Teaching tips below regarding looped arrows.


Dividing by a decimal number (9.12)

The skill in this lesson is the one where most errors are made. After the introduction and initial practice, have students complete exercise 9.12 A, check all work and resolve any misunderstandings. Have students do the first five questions in exercise 9.12 B, again check, and if there are no difficulties allow students to continue. Review the word problems as a class. As extra practice, have students use calculators to practise division by decimal numbers.

The mathematical language box presents an opportunity for students to become aware of both repeating decimals and irrational numbers. They may have seen these types of numbers before, and the language box is simply to explain these types of numbers.

Exercise 9.12 C gives practice dividing decimal numbers, through word problems. It includes the different skills, and wraps up the unit. The theme of the word


problems is an Art class, which students should find familiar as they sort out the key questions and reasonable answers.



Review the Summary points in detail, having students explain each one.

The Unit Check may be used for assessment.

Assessment 4

Both Part 1 and Part 2 include work involving fractions and involving decimals.

Teaching tips•Have students write the decimal point

for the answer, before they begin to divide. This tip helps reduce errors in placing the decimal wrongly, or leaving it out accidentally.

•When multiplying or dividing decimals by multiples of 10, some students need the support that comes from drawing in the looped arrows (shown in green in the introduction to lesson 9.6). One dip for each 0 helps these students keep track of what they need to do and be accurate. After more practice, they will be able to do this step mentally.

•From lesson 9.9 onwards, help students to understand that they should no longer write a remainder in the answer when dividing. They should either resolve the remainder by adding zeros to the decimal number and continuing the division, or they should convert the remainder to a fraction and reduce it to lowest terms.

Expansion / Extension of unitThere are connections in this unit to social studies (geography and world cultures), to Science (plate tectonics, dinosaurs, Pangea) and to an agriculture curriculum (raising dairy cows, selling produce), as well as money questions related to consumer mathematics.

IT – Use web tools and information sources to investigate additional facts about dinosaurs for group or individual projects.


Sample lesson planUNIT 99.5 Problem solving

Objectives •Estimate reasonable results

•Round numbers

•Compare decimal numbers

•Multiply decimal numbers

•Solve problems using decimal numbers

•Use estimation of decimal numbers in practical applications

Engaging the students’ interest / ConnectionsSketch on the board or project an image of one of the dinosaurs described in the word problems. Use this picture to stimulate discussion, and encourage students to add their own prior knowledge of the topic.

Teaching the lessonMathematical languageTime periods in the earth’s history have names, one of which is ‘Triassic’. Each of these represents several million years.

Practical activity / Developing the lessonClassroom discussion•Explore what facts students know about dinosaurs and their extinction. Talk about

whether students would have liked to live when dinosaurs still existed, and why. Which dinosaurs would they have wanted to live near?

•Go through questions 1 and 2 with the whole class, accepting reasonable answers for why estimates are needed. Ask students how we can know the weight and size of dinosaurs, when they lived so long ago (fossil record).

•Question 3 links to social studies, and student understanding of world geography. encourage students to imagine living in this period. Ask if they know why the continents drifted apart (shifting of tectonic plates). Question 4 is a challenge question, although not difficult. Give students time to work with a partner, and then share their results.

•Read through questions 5 and 6 together. Explain the hint suggested in question 5 more thoroughly for students who are not yet sure. Discuss the differences in size of the dinosaurs, and identify the key question in each problem. Have students work in pairs to solve each question, checking to see that they have reasonably answered each one. Written exercise: Follow the discussion and partner work with each student independently writing a sentence answer for each word problem.

RESOURCESA picture of one of the dinosaurs described to sketch or project electronically; access to research tools for Extension projects.


Differentiating for different learning styles The reading level of this exercise is higher, due to the terms of the theme. Encourage students to just use the first initial of any names they have difficulty reading or pronouncing, as they will still be able to understand and solve the problem.

Extension activities Project work: students may work independently or in small groups looking at: a) other time periods in the earth’s historical timeline, b) details about dinosaurs not mentioned, c) charts comparing the height or weight of several dinosaurs, d) Challenge: the size of a model if it is made in proportion to the height and length of a given dinosaur, e) any other topic related to the theme.

AssessmentReview student final answer sentences, checking for reasoning.

Summary of key points We use estimation and rounding when working with large numbers, and in problem solving.

We can use mental maths strategies when multiplying decimals, as well as written calculation.

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•Convert fractions / decimals / percent

•Use fractions in practical situations

•Solve problems using decimal number

•Find the percent of a whole / quantity

•Find the total when given a percent

•Use percent to solve real-world problems

•Convert between fraction / percent / decimal to simplify problem-solving situations

•Work with unit rate (unitary method) in problem solving, practical situations

•Calculate discount

•Restate numbers using the distributive property of ×, with or without regrouping

•Compare prices for best value


•Understand the idea of savings (value)

•Discuss the purpose of budgets, taxes

•Use percent to calculate tax and VAT or in other practical monetary applications

•Use a circle graph and interpret the data

RESOURCESHundred square template for students needing support; scrap paper and colours.

Teaching the content of the unitUnderstanding how percents, fractions and decimals are related is an essential skill for this year group. Understanding that relationship allows them to be interchanged in problem solving, leading to more efficient and successful solutions. The conceptual relationship is often addressed in examinations, and it is a useful real-life skill in consumer situations. This unit will review each of the three topic areas, and practise the skills starting at the basic level and moving to the more complex.




Percents (10.1)

This lesson introduces the concept of percent. Visual learners may benefit from working with shaded hundred squares which represent the percentages noted

UNIT 10 Percents, fractions and decimals

Percents, Percent and decimals practice, Percent, fractions and decimals: review, Changing fractions to percents, Percent, decimal and fraction equivalents, Working with percents, Forming percents, Finding the total, Discount, Tax and VAT, Working out solutions mentally, Problem solving, Comparing percents using circle graphs


in the tinted lesson box. Emphasise (if needed through practical demonstrations) that 0.45, 45

100 and 45% all have the same value. The first ten questions may be completed orally. Workbook exercise 1 for this unit may be used as a pre-test to check prior learning.

Note: As the lessons progress, the three key elements need to be continually identified until students are able to do so themselves. These include the percent, the total, and the number being used in the question. In problem solving, one of these three elements will be absent, and students will need to recognise what they need to find, and develop skills for finding the missing element.



Percents and decimals practice (10.2)

Review the basic concepts in the Remember box. Emphasise working out the percent mentally, whenever possible, when working with money. Review the strategy box, and recommend using decimals when finding a percent of a total with money. See Teachings tips below. Help students see how to use mental Maths strategies by talking through the steps in questions 13 and 14.


Percents, fractions and decimals: review (10.3)

Provide practical activities with manipulatives or paper models if needed to have students thoroughly understand this basic lesson. Have students explain to a partner each of the three dots in the tinted box, saying what is important about each point. In questions 1–10, remind students the denominator in the middle box will need 100 as a denominator. In questions 11–20, draw attention to the percents

greater than 100, and the decimal numbers with whole numbers, as in both there is more than 100%. Continue to build mental strategies, for example with question 20 since it states 6 of 40, the same percent will apply to 3 of 20. 100 is divisible by 20 five times, so 3 × 5 gives 15%.


Changing fractions to percents (10.4)

Students earlier practised dividing with fractions to change them to decimal numbers (see ‘Hint’ box and ‘Strategy’ box b). They practised equivalent fractions (see ‘a’). The third approach is multiplying the fraction by 100% (see ‘c’). Having three strategies to choose from, allows students to use the strategy that best fits the numbers in a given question. For example, question 1 works well with ‘Strategy’ box ‘a’ because 20 easily converts to 100, while question 2 works well with ‘b’, dividing the numerator by the denominator. Questions which are multiples of 5, 10, 20 or 25 easily work with ‘c’, multiplying by 100%.


Percent, decimal and fraction equivalents (10.5)

Have a class conversation about the pattern in the ‘Discuss’ box. Then have students complete the graph accurately and memorise it. Memorising it creates a tool which saves a lot of time for students, once they understand the concept. In problem solving, there are numerous opportunities where interchanging fraction, decimal, or percent will greatly simplify the process and help students find the correct solution. Students may learn a few at a time in groups (e.g. the eighths, the tenths, or 1

3 and 23). They may

practise with games (see dominoes under Activity for 10.12 below), and with extra opportunities to show what they know, in the workbook exercise below and the teacher website downloadable worksheet.


See also Teacher Resource website – Level 6 Numeracy page ‘Substituting percent, fraction, decimal’.



Working with percents (10.6)

Examine and compare the two examples in the ‘Discuss’ box. In the first, there are only whole numbers for the money. In the second, there are dollars and cents. Guide students to understand that they have a choice as to which strategy to use, and that it makes sense in this case to use the decimal form for the second example.

In each of the questions in exercise 10.6 A, the work can be made simpler. The first few have a hint listed. Talk through these three questions with students and ensure all understand. Draw attention in question 6a to the fact that they are given the fraction for girls, but the question is about boys. See Teacher tips below.


Solving mentally (10.6 B–C) The hints in the introduction, as well as the practice in the two exercises, are very important for building mental problem solving skills. Teachers will need to talk through the mental steps involved to help students learn what clues to look for and what details are important. For example, to find 300% of a number, we can simply first multiply it by 3, and then make it a percent. See Teaching tips below.


Forming percents (10.7)

This concept is one of the most frequently used aspects of work with percents. It should also be one of the most familiar, as students regularly convert their marks to percents. The example in the tinted box shows one strategy, and reminds students that they may use the other two strategies they learnt earlier as well. Again, it is

learning to recognise which questions work more easily and quickly with a given strategy. Encourage all students to try at least one of the Challenge questions.

There is a CD-ROM activity for this lesson. (Challenge)


Finding the total (10.8)

The tinted lesson box shows two approaches to the same problem. See Teaching tip below regarding teaching more than one approach (using multiple representations).

As in all word problems, have students first state what they know, and what they need to find out. Have students work out the question using both approaches, rather than just discussing each method. For the first, have them fold a page of their exercise book inward to make two columns. As they do the example in the introduction, make sure each step is included, including the words in the three steps of the unitary method. Then have students work out question 1 and question 5 below the example, again showing both approaches for each question. Remind students to exchange fractions, decimals or percents if it will make the problem easier to solve.

In the exercise, students may also use other strategies, such as sketch/diagram. For example, in question 1, 75% is 3

4, which is equal to 45 questions. In a sketch, each 14 would be 15, and the total obviously 60. In question 2, students learnt to find 10% mentally, and in question 3, that 50% is half. In question 8, 121

2% is 18, and

a diagram could help find the total In question 9, 20% mentally could be turned to 10%, then 100% (or 20% × 5). In each case, students may use the approaches shown, or may be encouraged to use their developing skills to solve many of the questions mentally.


Emphasise the last point of the lesson box, where the percent of the total is used to check the answer.


Discount (10.9)

Use advertisements listing discounts to start the class discussion on the topic. Have students say what they know about the topic, and guide them to discuss the differences in the two types of discounts shown in the posters on the right side (one is a percent discount, and one is a dollar discount). Question 7 could be used to stimulate class discussion on this difference. Many students should be ready to understand and use the short cut idea. Rather than using two steps, where the discount is found, and then subtracted from the total (the step students often fail to take in exams!), the short cut has students mentally work out what percent of the total is needed. If it is 25% off, then the sale price is 75% of the original price. (Note: for question 3, we accept $10 off the total price or $10 off the price of each set.)


Tax and VAT (10.10)

Use this topic to discuss how countries pay for development within a country (including customs fees, sales or income tax, tourism or export taxes, and VAT – value added tax on goods and service). Note that there are wide differences between countries in the region in regard to VAT or sales tax, ranging from 0% to 19% or more. In the example, ask why Tanya chose to change the percent to a decimal, rather than a fraction when she worked it out (it is simpler to use decimals with money / 8 is not a factor of 18 / etc.). Ask if any student can think of additional ways to solve this example (e.g.108% of $18 / or change 8% of 18 to 4% of 9, since 4 goes into 100 easily / or other reasonable approaches).

The first five questions are VAT, and

the second five questions are tax. Have students write out the steps in each case for this exercise, to visually check understanding. Students may use any approach for the Challenge.


Working out solutions mentally (10.11)



Activity: Give students practice interchanging percent/decimal/fraction in a game, as a prelude to the problem solving lesson. Make domino cards with pieces of card and a marker. For slower learners, use three markers (one colour for fractions, one for decimals, etc.). At the end of each domino, write a fraction, decimal or percent from the chart in lesson 10.5. As a challenge, include multiples or larger equivalent fractions, which students will need to reduce in order to match. Students may play as for regular dominoes, in small groups, and will become more adept and quicker at matching and interchanging as they play.

This problem-solving lesson reviews the many concepts of the unit. It may be used for revision, for group practice, independent work or partner work, either all at once or in parts. Encourage students to use the different strategies and short cuts they have learnt. Additionally, have students write an answer sentence or phrase to respond to the main question of each word problem.

Comparing percents using circle graphs (10.13)

Emphasise the point that using percentages allows us to compare, even when the totals are different. Give the example of tests which have different total marks. If a student got all right except 5 on a test of 25 questions, it is different than the same number right on a test of 8 questions. Similarly, the two circle graphs have


different totals. Jordan’s family spends 10% on transport each year, and Joseph’s spends 11%, which is very close, but in dollars this represents $12 500 as compared to $9020. Discuss other differences they see, and guide students to recognise the value of using percent to compare. Many of the percents shown are similar, aside from food and housing. Discuss what might make these differences (paying a mortgage versus renting, size of homes, different countries or currencies, whether it is in town or in the country, etc.).

Have students work with a partner to discuss and solve the questions. Some are open response, with no one right answer. See Extension idea below.


Teaching tips•Discuss mental strategies that could

be applied in exercise 10.2: question 3, 20% is 1

5 , and 15th of $50 is easily seen

to be $10. question 6, 25% is 14 , and 1

4 of $48 is easily seen to be $12. Question 8 is 90%, so just take 10% off the total.

•Have students approach all of the questions as a detective, looking for what can be done to make the questions simpler.

•There is additional practice with percent/fraction/decimal exchanges on the CD-ROM for Bright Sparks Student’s Book 5.

•Advanced students can be taught to mentally combine steps. For example, exercise 10.6 C question 13, the company increased earnings by 10%, which means they kept their original 100% and gained 10% more, so the total can be multiplied by 110% in one step.

•Multiple representations: Empirical evidence suggests that students who learn more than one representation of a concept have a greater chance of successfully mastering the skill. By offering more than one approach, as in exercise 10.8, students may either use both, depending on the problem, or use whichever approach they understand and are comfortable applying.

Expansion / Extension of unitHave students work out a personal budget, based on how they spend their spending money (it could be real, or imagined). Students might list regular expenses they might have (bus fare, lunch or snack), as well as savings towards things they would like to buy, savings for the future, and money for small items they buy just for fun. They then would work out a monthly total, how much for each item, and create a circle graph showing what percent of their total goes to each item.

Challenge: A tablet computer cost $486 after a discount of $54.20 and a customs tax of $29.32. Write a plan for finding the original cost, show the solution, and how it was reached.


Sample lesson planUNIT 1010.11 Working out solutions mentally

Objectives •Use the distributive property of

multiplication to break up numbers and more easily solve questions mentally

•Use reasoning skills

•Find the price with or before VAT or tax

•Use mental strategies for problem solving

•Find the percent of a whole / quantity

•Find the total when given a percent

•Use percent to solve real-world problems


•Use percent to calculate tax and VAT or in other practical monetary applications

Engaging the students’ interest / ConnectionsStart with the Challenge at the end of the lesson. Read it aloud, and give three full minutes of silence for students to think through, sketch or work out the problem. Then ask volunteers to explain their reasoning and steps. A volunteer might suggest sketching, where $8.25 is equal to 1

5 (20%) on a fraction bar (e.g. rectangle with 55),

and therefore all of the fifths are also equal to $8.25. Another might work it out as 20% of n = $8.25, using the approaches in lesson 10.8. Other students may have other approaches. Check that the reasoning makes sense, and the answer is correct.

Teaching the lessonMathematical languageVAT is ‘Value Added Tax’, and is placed on the price of goods or services in some countries. Goods are items that we buy, while services are things that people do, such as barbers cutting hair.

The ‘original price’ is the price before VAT or tax is added. The ‘selling price’ is the price the buyer pays.

Practical activity / Developing the lesson•Teaching students to work out mentally is challenging. One approach is ‘meta-

RESOURCESScrap paper; colours; hundred squares if needed for slower students.


cognition’, where the thinking process is described aloud with the teacher modelling the thought process, to help students learn to analyse and take logical steps. The steps described in the tinted lesson box show metacognition. Read through them slowly or have an advanced student do so. After each line, pause and reinforce the thinking if necessary.

•Since we can easily find 10%, working out 15% is relatively easy to do mentally, if we break up the 15% to 10% and 5%. By finding 10% first, and then taking half of that and adding it to the first result, we can mentally find 15% of most numbers.

•Similarly, in the ‘Discuss’ box, there are two other examples employing metacognition to model the thinking process for students. In the first, in order to find 6%, we first find 1%. Finding 1% (or 10%) is a skill that all students should be able to master at this age. In point a, the result for 1% is simply multiplied by 6.

•Point b breaks up the 6% to 1% and 5%. Using the step for finding 5% that was just practised, the result for 1% is added on, to show a total of 6%.

•Writing a paragraph to explain their thinking will be difficult for some students. Give them the opportunity first to do so aloud, perhaps to a partner, and it will become easier for them to then articulate their thoughts in writing.

•Question d is where creativity or challenge will be most evident. Have students discuss these options with a partner, rather than whole group, or at most in a small group. We want to have all students participating. Some possible options are shown below (students may create others).

412% – find 5%, find 1%, take off half of the 1% result and subtract it from the

5% result.

12% – find 10%, add on a fifth of that. Or, find 10%, find 1%, add two of the 1% results to the 10% result.

1712% – find 10%, double it for 20%, find 5%, take half of it (21

2%), subtract it from the 20%.

19% – find 10%, double it for 20%, find 1%, subtract that from the 20%.

•Some students are afraid to experiment in this way, and need encouragement. Through assurance that there is more than one way to find the right answer, most fear is taken off, which allows students to build their mental maths skills.

•Exercise 10.11: Before students start to work on this task, teach the short cut. This approach is not hard for most students, especially once they have learnt to find 15% mentally, and it makes sense to them to find 100%, 10% and 5% and add the results mentally.

•If students are unable to do the work mentally, they should still work out the VAT and tax in questions 1 and 2, and show their written work. For question 1, they can show: 115% of $44 = 1.15 × 44 = ____ or they may write it as a fraction 115

100 × 441 .

Differentiating for different learning styles Talking about the thinking process is helpful for some students, but others need to see or work with materials to best build these skills, even though ultimately they will do them mentally.


For individual students who need further assistance, use a hundred-square template and shade in different colours the percents described above.

Extension activities Find out what the VAT or sales tax is in your country, if any. Ask three shopkeepers, or other adults their opinion about VAT or sales tax, asking them to say at least one good thing and one difficult thing about it.

AssessmentInformally check each student by making notes as they explain to the group, or by pointing to a question and asking him or her to explain what to do mentally, checking their reasoning.

Summary of key points We can calculate the selling price with VAT or sales tax mentally, by breaking up the percent and then finding the total.



•Discuss measures of central tendency and find the mode and mean of a set of data

•Determine the range and median

•Find the mean (average) of up to 5 numbers or find the total or any missing number from the series, using calculation

RESOURCESCounters, including a large number of beads/buttons/beans; containers for the counters such as cups; colours; scrap paper, markers and shop paper.

Teaching the content of the unitThis short unit reviews the important concept of finding the mean, commonly referred to as the ‘average’. Related concepts of range, mode, and median are also described. Working with real-life examples generally include two out of three elements. These elements include the numbers themselves, the total and the mean.

To introduce the unit and the concept of means, tell a short story:

There was a family with seven children. Every time their Aunt Neda came to visit,

she would bring silver dollars for her nieces and nephews. She always put the silver dollars in a bowl, and each child would reach in and take a handful. As the younger children got older, they realised there was a problem with the way the gift was shared. What was it?

(Students will realise that all students did not receive the same number of coins.) After discussion, ask how the coins might be shared more fairly.




Overview (11.1)

Activity: choose a group of four to five students, and have them each write their ages on the board, along with the ages of their siblings. Ask students to look at the ages and tell you the average age for all. Because of the variation of ages, they will not be able to just look and do so. Introduce the concept of mean, which they will practise in this unit.

Leave the ages on the board to come back to later, and explore the information in the tinted lesson box. Teach students, as a first step, to put the numbers in order. This step allows the range and the mode to be quickly obvious, and allows a short step to

UNIT 11 Measures of central tendency

Overview, Working with means, Problem solving, Units 10 and 11 check and summary, Assessment 5


find the mode. Find the median, mode and mean of the examples given. Then come back to the ages listed in the introduction, and find the mean. The exercise may be completed independently



Working with means (11.2)




Discuss the chart on the right, explaining the event that caused these extinctions. Ask why non-avian dinosaurs show a survival rate of 0%. Why do they think frogs managed a 100% survival rate (speculate). Question 1 uses the chart to practise finding the mean.

Questions 2 and 4 practise finding the other elements when the mean is given, while in question 3 students must analyse what they know, combine facts, and find the mean. The Challenge presents a new skill. When two means are combined we must first go back to the totals, combine these, and then find the new mean.



The Summary reviews key points from both Unit 10 Percents, and Unit 11 Measures of Central Tendency. Review and revise these points with the class, prior to the Unit Check.

Assessment 5

Both Part 1 and Part 2 include a full range of questions from the units previously studied. Encourage students to use the mental mathematics skills, the short cuts, a variety of strategies, and above all good reasoning skills, as they complete and check their answers.

Teaching tips• In common usage, we hear the term

‘average’. For example, what is the average price for a lunch, if the six prices are …’ The mathematical term is ‘mean’. Teachers will want to help students recognise that both terms have the same meaning.

•Encourage all students to explore and experiment as they try strategies and ideas. Use the Partner Activity after Assessment 5 below to engage students and encourage exploration.

Expansion / Extension of unitIdea for research – Before the change to Coordinated Universal Time (UTC), it was fairly common to hear the term ‘Greenwich Mean Time’, or GMT. You might hear a beep on the radio, indicating a new hour, linked to this time standard. Investigate why the word ‘mean’ is part of the name ‘Greenwich Mean Time’, and explain it in a short paragraph.


Sample lesson planUNIT 1111.2 Working with means

Objectives •Discuss measures of central tendency

and find the mode and mean of a set of data

•Determine the range and median

•Find the mean (average) of up to five numbers or find the total or any missing number from the series, using calculation

Engaging the students’ interest / ConnectionsPractical activityGive students a simple task with different results – ask them to add the date (day and month) when they were born, and take that number of counters. Students then get in groups of five and compare their numbers. Each group should determine the range, median, and mode for their group and record the result on a shop paper. Putting all of their group’s counters together on a table, have them divide the counters into five piles, which will represent the mean for their group. Compare the results recorded by all the groups. Have a representative from each group explain to the class what they did, what their total was, their mean and how they handled any remainder.

Teaching the lessonMathematical languageStudents should understand and use the terms mean, mode, range and median, as explained in the introduction for lesson 11.1.

Practical activity / Developing the lesson•Practical activity: A familiar task – fill a jar with tamarind balls, table tennis balls or

similar objects about that size. Have ten or more students write their estimation for the total number of balls in the jar on scrap papers. Have volunteers put all the estimates in order, and as a class find the range, the mode, the median and then calculate the mean.

•Discuss the three bullet points and the terms total, average/mean, and numbers from the introduction to lesson 11.2. Ask which of these elements they had in their practical activity (they had the numbers and the total, and had to find the mean).

•Explore example 1 and example 2, stating which two of the three elements are present in each one, and which is missing. in example 1, there are 5 scores, so they must multiply by the mean to get the total (the opposite of what would have been done to find the mean). In example 2, we know the mean and we know some of her scores, so must find the total first to then find the missing number. In this example

RESOURCESA large number of counters, such as buttons, beads, or beans (they need not all be the same); one large shop paper for all groups to write result; one colour marker for each group; for the Extension activity one large jar, filled with tamarind balls, or other similar items.


there are two steps, finding the total, then subtracting the numbers we know to find the missing number.

•This process becomes quite simple with practice, but may sound complicated orally. Give students many opportunities with problems such as these, including the exercise below and the Workbook exercise.

Differentiating for different learning styles Use beads to allow students to demonstrate questions like example 2, where there is a number missing. As students literally see the total, and set aside groups of beads to stand for the numbers they do know, it is very easy for them to recognise that the missing number is the beads left over.

Extension activities Challenge: Have students work with a partner and create three word problems. Each word problem should have a different one of the three elements missing.

AssessmentExercise 11.2 questions 1–4 may be used to assess progress.

Summary of key points The mean is the average when a set of numbers is shared equally. To find the mean, we add the numbers and divide by the number of numbers. If a number is not known, by subtracting from the total the numbers we do know, we can find the missing number.

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Text12



•Tell time to the minute

•Record time in hours and minutes with standard 12-hour or 24-hour clocks and convert to standard time

•Calculate using hours and minutes

•Compare and convert units of time

•Calculate elapsed time (months / days / hours / minutes)

•Be aware of customary units of measurement still used in the region and relate to the SI (metric) system

•Develop comparisons for metric and common customary units (e.g. pound and kg) for practical situations

•Use metric units of measurement in problem solving and practical situations

•Measure and record accurately using different units and tools of measurement

•Convert related units of measurement in problem solving (e.g. m to cm)

•Apply basic operations (+, −, ×, ÷) to questions using units of measurement

•Convert units of measurement where needed in problem solving

•Learn the prefixes used in the metric system and its framework

•Estimate, measure and record length and solve problems using km, m, cm, mm, including practical situations

•Measure lengths in one metric unit and convert to another

•Use scale to represent distance (e.g. map)

•Estimate, measure, record and compare mass using kg and g, mg and tonne

•Estimate, measure, record and compare capacity using mℓ and ℓ

RESOURCESRulers, metres sticks; eye dropper, graduated containers; balance scales; containers of various sizes; items from the environment to measure; paper for diagrams and colours; thermometer if available.

Teaching the content of the unitThis unit provides a full review of measurement, including units of length, mass and capacity, with the international standard of the metric system as well as some customary (imperial) units of measurement. Temperature and time are also explored. Teachers will help students

UNIT 12 Measurement

The metric system, Units of length, Units of capacity, Units of mass, Problem solving, Enrichment: Customary and metric measurement, Enrichment: Temperature, Time, Calculating elapsed time, Practice with time, Working backwards, Exact answers or estimates? Unit 12 check and summary


learn to be accurate and precise, and to recognise incidences where estimation is sufficient.

To introduce the unit, start with a simple practical activity. Have students measure and compare the widest expanse they can make when stretching and spreading the fingers of one hand. Measure with the tip of the little finger at 0, towards the tip of the thumb on the other side. Have students compare their results. Ask what other similar measurements they have seen (e.g. wingspan of a bird).




The metric system (12.1)

This lesson reviews the relationship between units of metric measurement. The prefixes milli-, centi- and deci- are the three units smaller than the main unit (in this case metre), while the most commonly used prefix greater than metre is kilo-. While we teach students to recognise the full system, only some of the units are commonly used and listed in the regional syllabuses. The chart in exercise 12.1 A shows the most common units of measurement for length, mass and capacity, each in a separate colour. Students will need to learn km, m, cm and mm for length, kg and g for mass (and tonne), and ℓ and mℓ for capacity. A second main idea of this lesson is to build a logical sense of the metric relationships, which are then checked in exercise 12.1 C and in Workbook Unit 12 Exercise 1, question 11–15.


Units of length (12.2)

See sample lesson plan below. Use with WB Unit 12, Ex 2

Units of capacity (12.3)

The discussion box helps build the spatial concept of the two common units of capacity. Teachers will wish to add additional practical examples that students can see and touch. The precise conversion of metric measurements follows. In particular, discuss the differences between question 4 and question 8.

Make connections to Science, where metric units of capacity are used, and Medicine, where liquid medications are dispensed using these units. If available, have students estimate the size of 1–5 drops of water from an eyedropper, and then measure this quantity using a graduated container. Use larger containers and have students pour water to estimate a litre, a half-litre, and a quarter litre. Students will then measure their estimates, using a graduated container, and converting the latter two to 500 mℓ and 250 mℓ, often see in real-life situations (such as water bottles and small juice packs).


Units of mass (12.4)

Most people think of these units as measurements for ‘weight’, but the mathematical term students need to learn is ‘mass’. (The meaning is slightly different, in that weight is determined by gravity, but mass is constant even in outer space.) Provide practical activities with digital or other scales to help students learn to be precise and record accurate measurements. Both exercises and the Challenge help students connect these units of measurement with real-life situations. Note: the tonne is shown as 1000 kg. They may also see ‘ton’, which is 2000 pounds, marked on some heavy vehicles (the two are not the same). If this is true locally,


challenge a few students to find out the difference between ton and tonne.



The fifteen questions in the exercise include measurements of distance, mass and capacity, in everyday situations. In many questions, there is more than one unit of measurement, and students will need to be reminded that they must make all units of measurement match before they can solve the problem. Teachers may wish to assign some questions in class, for assessment, and some of the problems for homework practice.

Enrichment: Customary and metric measurement (12.6)

If customary units are not on the local syllabus, teachers may opt to include the lesson as Enrichment and just for information, but not for assessment. Some regional syllabuses do include customary units, and for these schools the lesson is assessed as for the other work. Begin the lesson with discussion of what students already know. Help students separate the terms they know or have used in their communities into metric or imperial (customary) measures. See if some students recall additional measures not listed, and what they measure (e.g. gill for liquid, furlong at a horse race, league or nautical mile at sea, rod for measuring out a length to dig for ground provisions).


Enrichment: Temperature (12.7)

Temperature is generally forecast in degrees Celsius (°C) throughout the Caribbean region, although on overseas television students may hear it stated in Fahrenheit (°F). In this lesson, students have the opportunity to discuss temperature variations from the far north to the equator. Use diagrams and examples to show that 0 degrees is the point of freezing, and very cold temperatures are lower than 0, and represented with the minus

sign. Encourage students to use a vertical number line, and diagram the questions as they work in small groups.

Use with WB Unit 12, Ex 5 Challenge

Time (12.8)

The first exercise is a basic review of time. Check student work for any students who have any gaps in this knowledge. The second exercise is noted as Enrichment since most but not all Caribbean syllabuses include 24-hour time. Let students understand that this form of time is used internationally, including on plane tickets, so is useful for all to understand. Discuss the explanation, and if students are proficient, have them complete the Challenge. Have students be very careful converting times between noon and midnight.

Note: While most countries in the Western hemisphere use the same format for 24-hour time (__ __ : __ __ ), students may occasionally see it written with a full stop rather than a colon.

The third exercise should also be simple review, converting other units of time. Check for understanding, and re-teach concepts if needed.

Use with WB Unit 12, Ex 6 and 8 (and Enrichment Ex 7)

Calculating elapsed time (12.9)

This lesson is useful for students who still have some fears of experimenting, and worry about getting questions wrong, since it discusses more than one approach to solve the problem. In the first bullet, the mental steps are each slowly explained in a simple, straight-forward way. The response to each mental step is shown at the right. This approach is very useful for many students, particularly those who are not strong calculating units of time. In explaining the calculation approach, ensure students understand that one hour regroups to 60 minutes. Students should be free to choose whichever approach


makes most sense to them, even though the first exercise is written showing the subtraction algorithm.

Challenge: ‘There are five time zones between Holland and Antigua. A plane leaves Amsterdam at 8 am and flies for 7 hours to Antigua. What is the local time when it arrives?’

Practice with time (12.10)

Prior to the exercise, have students do a quick revision with a partner of the key points of the previous exercise. Exercise 12.10 may be used as an overall assessment, particularly the word problems in questions 14–20.


Working backwards (12.11)

Begin the lesson with a personal story of what you did that day, and how long each event took. For example, ‘I reached the classroom at 7:45. Before that, I spent ten minutes in the office, 24 minutes travelling to school, and 6 minutes waiting for the bus outside my house. Right before that, I spent 20 minutes eating breakfast. At what time did I start my breakfast?’ Have students explore on their own how they might solve it, and ask a few volunteers to share their thinking.

Read the problem in the blue box, and discuss the ‘Picture it’ box. Emphasise the importance of first listing what is known. Divide the class in half, with one group following the first approach (gold box, top right), and the other half following the second approach. Share the results and discuss which they preferred. Students will have different preferences.

Exercises 12.11 A, 12.11 B and the Challenge offer five opportunities for students to use these strategies. Whether singly or all at once, arrange for students to complete all five.

Exact answers or estimates? (12.12)

In this exercise and discussion, students are building logical thinking, and reasoning skills. Discuss why the exact answer is important in some of the questions. Have students on their own come up with additional situations where an exact answer is, or is not, needed.


Review the summary box together, and encourage students to use it for revision. The Unit Check may be used for ongoing assessment.

Teaching tips•After measuring the width of the hand

span, have students mark a spot on the wall, place the tips their fingers there, stretch both arms as wide as possible, and mark the tips of the fingers on the other side with chalk. Use a measuring tape to measure their arm span.

• Students need to keep measurement in the physical realm, and not complete the unit solely based on theoretical pencil and paper descriptions. If they are to become skilled at estimating the mass, or comparing lengths, or judging a rough idea of capacity, these skills will come from actually manipulating physical objects and using tools of measurement.

Expansion / Extension of unitInvestigate units of measurement, including temperature, that are used in Science and Social Studies.

Challenge project – Using a coat hanger, create a working balance and demonstrate to the class how it works.


Sample lesson planUNIT 1212.2 Units of length

Objectives •Recognise the common units for

length (m, km, cm and mm)

•Measure using units of length

•Compare and convert metric units of length

•Make realistic comparisons between objects and standard measurements

Engaging the students’ interest / ConnectionsActivity: One student puts one hand near, but not touching, the bottom of the metre stick (‘0’ end down), while another person holds it up. The metre stick is suddenly dropped, and the first student catches it. Measure the place on the stick where it was caught. Have different students try, and compare the distance each student’s stick dropped before it was caught.

Teaching the lessonMathematical languageUnits of measurement are the named units such as centimetre or litre. They correspond to specific measurements, fairly closely replicated in the tools of measurement we use (such as a metre stick).

Practical activity / Developing the lesson•Practical activity: Have students identify a mm, a cm and a m, on measuring tools.

•Explore situations where we might use km.

•Guide a discussion to help students recognise why they must have just one unit of measurement when solving problems. Discuss the example of Alex’s birdhouse in the tinted lesson box, and give other examples from the classroom where operations cannot take place until one or the other measurement is converted.

•The points on the right side of the tinted lesson box are very useful for revision.

•Complete exercise 12.2 A orally.

•Practical activity: Have students find some part of their body that is just one centimetre (e.g. length or width of a particular fingernail), one millimetre and one metre. Make a poster with drawings and labels for these units of length (save for later additions of units of mass and capacity).

•Review multiplying and dividing by units of 10 before students work out conversions in exercise 12.2 B. Emphasise that changing from large units to small units will give us more, and changing from small units to large will give us fewer.

RESOURCESMeasuring tape, rulers, metre sticks; pencil; small items to measure in mm.


•In exercise 12.2 C, students can complete questions 1–7 independently. Discuss each of the questions 8–10 as a whole class, comparing the three different means of getting to school – canoe, minibus, and walking – and the distances that each student travels.

•The Workbook exercise will allow further practice.

Differentiating for different learning styles Provide practical activities where students measure items in and around the classroom, using measuring tools.

Have students put a ruler and a metre stick side by side, aligned by the 0. Have them compare the unit marks for mm, cm and m.

Extension activities While the present generation is becoming adept at using metric measurement, many older people still use imperial units of measurement. Have students investigate and compare both, and come up with easy to remember links. For example, about 21

2 cm is equal to 1 inch (5 cm ≈ 2 inches), and 1 m = 39 inches, which is approximately 1 yard or 3 feet.

AssessmentExercises 12.2 B and 12.2 C may be used to assess progress in classwork.

Summary of key points The common units of measurement for length are metre, kilometre, centimetre and millimetre.

To convert from one unit to another, we multiply or divide by units of 10.

To solve problems that have more than one unit of measurement, first one must be converted to a common unit of measurement.

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•Identify vertical, horizontal, parallel, perpendicular, intersecting, oblique and curved lines

•Describe and name plane (2D) shapes noting the properties, including number of angles and sides, edges, and vertices, whether open or closed

•Classify triangles based on their properties: right-angled, scalene, isosceles, equilateral

•Calculate degrees of one angle in a triangle or four-sided figure

•Describe and draw triangles, squares, rectangles, circles

•Identify congruent angles in polygons

•Name right angles, acute, obtuse (or reflex) angles in practical situations, diagrams and plane figures

•Be very familiar with right-angled triangles as 90° angles and point out multiple examples in the environment

•Estimate common angles (e.g. construct paper protractors to learn 30°, 45° and 90°)

•Use ruler and protractor to measure and draw angles of a given size (OECS)

•Calculate degree of angles

•Find the measurement of a missing angle in a three- or four-sided figure

•Calculate degrees of complementary / supplementary angles

•Discuss the relationship between interior and exterior angles

•Distinguish between plane (2D) figures and solid (3D) shapes

•Name and draw cuboids, cube, sphere, cone, cylinder (pyramids and prisms)

•Sort and classify 3D shapes based on their properties

•Make models / nets of solid (3D) shapes

•Identify and state the number and types of faces (flat or curved) and the number of edges or vertices in solid (3D) shapes

•Create tessellated patterns using plane figures (i.e. ‘tiling a plane’)

•Identify and draw lines of symmetry

•Enrichment – Identify and count rotational symmetry)

•Use translation / reflection / rotation of plane shapes

•Enrichment – Predict the new positions as a plane shape flips about a line, slides a given distance or makes a turn

UNIT 13 Geometry

Lines and angles: review, Measuring angles, Calculating angles, Angles in triangles and quadrilaterals, Investigating practical applications, Plane (2D) shapes, Transformations, Symmetry, Tessellations, Lines of symmetry, Solid figures, Nets, Unit 13 check and summary, Assessment 6


RESOURCESProtractors, compasses if available; rulers; card and scissors; graph (grid) paper; colours.

Teaching the content of the unitThis unit reviews basic terms and concepts of geometry, and expands that learning to include additional ideas and applications. Most lessons will need some practical aspect, so students both create geometric figures and interpret them.




Lines and angles: review (13.1)

This lesson is a basic review with definitions and examples, which some students may need to revise before advancing.

In the first exercise, make it clear that we are looking at the lines in the two dimensional drawing, and not a 3D shape. The second ‘Mathematical language’ box reviews types of angles and their precise definitions.

Have students stand and use their arms, or their arms and a partner’s arms, to demonstrate parallel lines, types of angles, etc. Both of the written exercises may be useful to judge the strength of the students’ foundational understanding.

There are two CD-ROM activities for this lesson.


Measuring angles (13.2)

Practical lesson: Teach students that an angle is a measure of turn, and discuss what this means. Use a protractor, to explore several types of angles. As a class, create two paper protractors, the first for right angles and 45 degree angles, following the steps described. For the second, students will explore on their own how it could be created. Once they have had time to do so, confirm that it is first folded in half and then each half is folded in thirds, to show 30 degrees. Have students use a ruler and pencil to darken the fold lines. Have students put their initials on the paper protractors, and then complete exercise 13.2 A.

If a real protractor is available, teach students how to use it, as shown in the ‘Enrichment’ box and exercise 13.2 B. Have students take care to place the centre hole or dot on the vertex of the angle, and align one arm (ray) with the horizontal line at the bottom of the protractor. As with a ruler, if it is not aligned exactly, the results will not be accurate.

See Extension below, regarding drawing angles with a protractor.

Calculating angles (13.3)

These exercises should be approached in the same way as a word problem, first checking what is known, and then what is unknown. In some cases, there are several unknowns, and they will need to be able to identify these and have a strategy. For example, in exercise 13.3 B question 1, there are five unknowns (x), and we know they are on a straight angle (180 degrees). Orally, have students state what is known, and what is unknown, for every question in each exercise. See Teaching tip below.

Challenge: Teach students to recognise complementary and supplementary angles. Have students look at question 4 and question 7 in exercise 13.3 A. Because both angles together form 90 degrees, they can


be called complementary angles. Similarly, question 1, question 2 and question 3 are angles that together form 180 degrees, and these are examples of supplementary angles. The Workbook exercises 2–5 give practice in drawing and calculating angles.



Angles in triangles and quadrilaterals (13.4)

Have students discuss the top box regarding circles and quadrilaterals, and then do the Partner Activity, exploring what they think might be true for the triangle and why. Provide students with card and scissors, to show their ideas. With the second card, have students copy either question 3 or question 5, cut it out, then cut it down the middle, and prove the two new triangles can form a rectangle or square. Discuss their results.

Challenge riddle: ‘I am an angle in a triangle that also has a right triangle and an angle half as big as I am. How many degrees is my angle?’ (answer: 60°)

Investigating practical applications (13.5)

Carry out the Activity ‘Transferring energy’. Help students understand that the angle of the ruler changes the results. Give all students time to explore this concept, and at the end of the lesson, ask volunteers to try to explain it.

Homework: Have students bring in sketches or pictures of buildings with unusual angles (architecture), pictures or drawings of bridges and the angles used to create them, or a sketch of some other place in the environment where they observed angles. The sketches should be well presented, labelled, and ready for display.

Plane (2D) shapes (13.6)

Activity ‘Bisecting a line’: If compasses are available, have students carry out the two drawings, bisecting a line and drawing a

circle. Use the prompts in the ‘Discuss’ box to encourage students to connect this skill to actual usage.

The lesson on plane (2D) shapes should be an easy review, although some students may need to learn the words parallelogram, rhombus and trapezium. (Note: In North America, a trapezium is termed a trapezoid.)

PolygonsHave students copy the diagrams in exercise 13.6 B and use rulers to draw in the diagonals. Discuss the example, and have students look for and name each new shape they create. Give time for students to compare results. The Challenge is an early chance with tessellations, which will come up again later in the unit. Remind students who try the Challenge that each polygon they choose to use must be the same size each time they use it (e.g. the red triangles), therefore they may wish to make them first from card, and trace these.



Transformations (13.7)

Have students use graph (grid) paper for this lesson.

Working with transformations helps build logic and reasoning skills. It may not be familiar to some teachers of Primary Mathematics, but is helpful to students for seeing patterns and in spatial development. The three forms include reflection (as you would see in a mirror) where the image is flipped and reversed, translation, where the object slides the same distance precisely, and rotation, which is similar to the movement of a hand on a clock.

Give students opportunities with each form of transformation, moving a shape they made on the graph paper. After practice movements, have students trace the card, complete a transformation, and


trace the new position. Labels and arrows can then be added, and students can explain what they did to a partner.



Symmetry (13.8)



Tessellations (13.9)

Practical lesson – Download the shapes on the teacher resource website page, and the explanation of tessellations. Have students create a page fully covered by a tessellated pattern, and colour it. Link the lesson with Art, even as students build their spatial reasoning. See Teaching tips below. The workbook exercise 8 also provides space for creating tessellations.

See also Teacher Resource website – Level 6 Challenge page ‘Tessellations’,


Lines of symmetry (13.10)

This review lesson should be straight-forward for most students, but check if there are any who seem unsure. Note that it is the outline of the figure, and not the pattern inside the figure, which we are looking at when putting in the lines of symmetry. In the exercise, question 8 and question 9, students should copy the diagrams with a good amount of empty space on the opposite side of the dotted line. They may turn the figure so the line is vertical.


Solid figures (13.11)

The introductory lesson for 3D solid shapes may precede that of 2D shapes, if preferred.

This lesson is again a review, with the real life images loosely related to the 3D shapes named. Check student understanding. Review vertices, edges and faces, noting

that faces can be flat or curved. The Mathematical language box gives the accepted definition for an edge (two faces meet), and a vertex (three or more edges meet). Students may need to be taught that the plural of vertex is vertices.

Extra Challenge: ‘What is the name of a triangular pyramid that has only equilateral triangles?’ (answer: tetrahedron)



Picturing solid shapes – nets (13.12)

A lesson on nets should always include a practical activity. If students are very familiar with the cube, shown in the introductory tinted lesson box, give opportunities for students to create nets of other 3D shapes. See Teacher tips below. The workbook provides space for forming the net of a cone and a triangular pyramid.




Review the Summary with students, expanding on any points as needed. The Unit Check may be used for classroom assessment.

Assessment 6

Measurement, time, geometry are the main topics covered in this two-part assessment.

Teaching tips• In the lesson on calculating angles, as

in other work with geometry, taking the time to say aloud what is known, and what is unknown, helps students who are auditory learners to better understand the visual diagrams. It also helps students with processing difficulty when there is a pattern involved with approaching each problem, making it very clear what they need to then do to solve it.


• Some students with different learning styles may be particularly drawn to the lesson with tessellating patterns. There is constancy in these repeating patterns with which they can be very successful. At the same time, students who are very focused on calculations can be quite challenged by this skill building lesson.

• It is hard for some students to mentally picture the solid shape made by a net, unless they have first had several hands-on experiences. This idea may be hard for visual learners to grasp, as to them it is quite obvious. Teachers will want to help students see that individuals perceive the world differently, even as teachers try to ensure they use a variety of teaching approaches.

Expansion / Extension of unitHave students use a ruler and a protractor to measure and draw angles of a given size on graph (grid) paper. Demonstrate

with a 90 degree angle, which students are familiar with, and have them align the horizontal line with a line on the graph paper. Help students place the protractor and ruler, and draw the second arm (ray) for the precise angle. Have students then try to create a 75 degree angle and a 15 degree angle.

Have students compile a list of professions that need to regularly use the skills of this unit, and create a class collage or poster. Examples might include textile designer, engineer, etc.

Challenge: Following Unit 13 Exercise 11 in the workbook is a puzzle to challenge students to think ‘outside the box’. It will prove challenging to many students (and adults). The solution is in the workbook answers at the end of this book.

Create a model of an unusual solid (3D) shape, such as a tetrahedron or a pentagonal pyramid.

Sample lesson planUNIT 1313.8 Symmetry

Objectives•Identify and draw lines of symmetry

•Enrichment – Identify and count rotational symmetry

Engaging the students’ interest / ConnectionsGive students a piece of card and ask them to create a shape of any kind they wish, about half the size of one of their hands. Let them know they will be using this card a little later in the lesson.

RESOURCESGraph (grid) paper; rulers; small pieces of card and scissors; pencils and colours.


Teaching the lessonMathematical languageRevise the terms congruent (which the shapes were in the previous exercise), and symmetry. Diagram A in the example shows a line of symmetry which can be compared to the line of reflection in the previous exercise. In diagram B, there is no perfect reflection.

Practical activity / Developing the lesson•Have students use graph (grid) paper for this lesson.

•Practical activity – Have students draw a large box on graph (grid) paper. and draw a dotted line through the centre of the box. Have them turn the paper so the line is horizontal, and draw a complicated shape with straight lines and angles that begins and ends on the dotted line. Students then must match that shape (reflect it) on the other side of the line. When this is completed correctly, the dotted line will be a line of symmetry.

•Exercise 13.8 A may be done for homework.

•Rotational symmetry, in exercise 13.8 B, only occurs when the turn brings the image to exactly reproduce itself in a new position. In the drawing shown, there will be four times when the image is repeated exactly, as the shape turns. In this example, the shape must turn 90 degrees for each repeat of the pattern.

•Have students notice hubcaps on cars, or other items in the environment that show rotational symmetry.

•Give several practical demonstrations. including taking shapes cut from card and turning them on the grid paper, counting each time the image is exactly reproduced (e.g. four times for a square, five times for a regular pentagon, etc.).

•Have students look closely at question 4 and decide which shape a, b or c is just a quarter turn of the original shape.

Differentiating for different learning styles Kinaesthetic learners more easily understand this concept when several practical examples have been carried out. Put a dot in one corner of the shape, and turn it on the graph paper, holding the pencil in the centre. Each fractional turn is more obvious in this way.

Extension activities There is a CD-ROM activity for this lesson.

AssessmentDemonstrate rotational symmetry and ask individual students how many times the image is repeated in one full rotation. Keep a record of which students have grasped the concepts.

Summary of key points Symmetry, where one half of a figure exactly matches the other half, is closely related to reflection. A line of symmetry divides a figure exactly with both sides mirroring the same size and shape. Rotational symmetry is where the image is reproduced at certain points as a figure is rotated.

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•Measure the length of sides to find perimeter in polygons

•Calculate the area of triangles, squares, rectangles, (Enrichment – parallelogram, rhombus) using measurement and formulae

•Find the estimated area of irregular shapes using a grid

•Find area in practical situations and record using cm2, m2

•Compare and contrast perimeter and area of plane shapes of different measurements

•Divide a compound shape into parts to find the total area

•Find the area of borders

•Calculate missing width or length, given the total area

•Find the surface area of objects by determining face areas and calculating

•State the radius of a circle given the diameter or vice versa

•Enrichment – Use pi and formulae to determine the area or circumference of a circle

•Explore and model volume using unit cubes

•Calculate volume (using cm3 or m3)

RESOURCESRulers; scrap paper and colours; small pieces of card; scissors; compasses if available; pieces of string at least 35 cm long; small cube blocks (e.g. alphabet blocks) or squares of treats (marshmallow/fudge/guava); toothpicks and paper plates; popped popcorn; bowl; paper and tape for the Extension activity.

Teaching the content of the unitThe unit moves from a basic level reviewing each topic, perimeter, area and volume, to more complex problems in each concept area. Practical activities should be part of each lesson, to help build spatial skill, and help students connect the concept to the written representation of the concept.




Perimeter: review (14.1)

Let students know that the brief review of perimeter in the tinted lesson box applies to real world examples as well as diagrams of polygons. Encourage them to give examples from the immediate

UNIT 14 Perimeter, area, and volume

Perimeter: review, Area: review, Area: right-angled triangles, Area: triangles, parallelograms and rhombuses, Area of compound shapes, Exploring area, Working with circles, Surface area, Volume, Unit 14 check and summary, Assessment 7


environment that represent perimeter (e.g. fence around the school yard). Review the two examples, with special note of example 2, where the measurement of one side is missing. Complete exercise 14.1 A orally as a whole class, sketching some examples on the board. Have students complete exercise 14.1 B independently. Discuss questions 5–9 together, sketching the word problems.

Working with perimeters This successful ‘toothpick’ idea has helped many students. Using the gold box top right, students extract the key information. Since the length is twice the width, students imagine two toothpicks for the length, and one for the width. It is easy to visualise six toothpicks altogether, as they sketch it. Students mentally solve 48 m in 6 parts is 8 m for each ‘toothpick’, which makes the length 16 m and the width 8 m. Have students work with partners and solve the five word problems, and the Challenge.


Area: review (14.2)

The review includes a comparison of perimeter to area, and an overview of area as the covering of the surface of a flat shape using square units. It revises using formulae for the area of a rectangle and square, and estimation for the area of irregular shapes by using a grid.

In lesson 14.2 B, the problem-solving questions are based around the theme of the Peace Banner. As students work out the questions in area, integrate the topic to social studies and the theme of nations working together.

Homework Challenge: ‘The number of square metres in a square matches the number of metres in its perimeter. How long is each side?’ (answer: 4 m)


Area: right-angled triangles (14.3)

If this has not yet been done, carry out a brief practical lesson where students fold squares or rectangles to create right-angled triangles. Actually folding and measuring the paper, calculating the area of the square or rectangle and then calculating the area of the right-angled triangle (half the previous total) is more meaningful to all students than simply seeing the diagram and discussing it or just learning the formular. The short practical activity will also positively affect how well they remember later to take 1

2 of the base × height. Have students sketch the three triangles in the exercise, and then work out the area.


Area: triangles, parallelograms and rhombuses (14.4)

Emphasise that for triangles that are not right-angled, the height is still measured at a right angle to the base, and height is represented by a dashed line. Note that the line for height is sometimes outside the triangle itself (green triangle on the right side of the example).

The area of a parallelogram or rhombus also uses a similar dashed line for the height, with the difference being that the area is not divided by two. If students are not familiar with the terms rhombus or parallelogram, they can think of them as a square and a rectangle with no right-angled corners – ‘tilted over’ – as students might say. Students may have been taught at a younger age that there was a shape called a ‘diamond’. Mathematically, this shape (when all sides are the same length) is a rhombus.



Area of compound shapes (14.5)

Logical thinking and exploration are both a part of work with compound shapes.


Students will need to think about how to break up the shape into regular polygons, in order to calculate the area. Exploration plays a part, since many shapes offer more than one possible approach. While the first example is fairly clear, discuss with the class why both of the approaches shown in the second example will give the same answer. Then challenge the class to find a third approach not shown (taking the complete rectangle without the missing corner, and then subtracting the area of the corner). If they are not yet able to discover it on their own, wait, and tell the class they will come back to it another day. Students will need to use rulers for the first exercise. Have students sketch the drawings, and put their measurements on the sketches, along with the lines they used to ‘break up’ each compound shape.

The second part of the exercise is a key lesson in reasoning. Using the example in the ‘Picture it’ box, students may be given a chance to work it out on their own or with a partner. Then, go through it with the class, showing how they must find the area of the rectangle and subtract it from the total to find the full circle, and then divide by 2 to find the area of one semi-circle. The four questions in the exercise each require logical reasoning and several steps. Some students may need to work with a partner, while others will enjoy the challenge.



Exploring area (14.6)

The skill in example 1 is ‘seeing’ the larger shape for a compound shape, and imagining a piece has been removed. This skill was suggested as a third option in the discussion above for the second example in 14.5. Students have described these diagrams as ones where you ‘make it a whole, and then take a bite out it’.

Example 2 is another common situation, where students are asked for the area of

just a border, picture frame, walkway or in this case, pool deck. The area of the ‘inside’ (picture, flower garden, pool, etc.) will be subtracted from the area of the whole structure (deck and pool together). See Teaching tips below.

In example 3, students use what they know, to find the missing measurements. It exercises basic numeracy and reasoning skills. Encourage students to explain to a partner what was done and why. See Teaching tips below.

The two exercises and the Challenge that follow are all very important for skill building. Teachers may wish to spread the exercises over more than one lesson, ensuring there is time to analyse, discuss and evaluate students’ answers.

Use with WB Unit 14, Ex 4 and 8

Working with circles (14.7)

Note: in a few regional territories, this lesson may be just for Enrichment.

Emphasise the vocabulary, particularly the difference between diameter and radius, Teach the symbol for pi, and explain why we say the ‘approximate value’ (See note under Extension, below).

Give practice with compasses, if available, and have students draw the circles in question 3 (they need to first divide the diameter by 2) and question 5. In question 4, after drawing a circle, have students write their measurements for the radius and the diameter.

Activity: Have students draw a circle with a diameter of exactly 10 cm, either with compasses or by tracing an object with that diameter. Students will then very carefully align the string with the circumference of the circle, and cut off the excess. They will measure the string, compare its length to the diameter, and find an approximation of pi.



Surface area (14.8)




Volume (14.9)

Help students make the distinction between volume and capacity, from a practical standpoint as well as by definition.

Practical activity: Use cubic blocks, such as children’s alphabet blocks, or use squares of guava candy, fudge, or marshmallows. In small groups, lay out a pattern of ‘blocks’ 2 × 2 × 2 (use toothpicks if needed to hold the stacks together). Record the dimensions and the volume. Have different groups add either more length, more height or more width, and record the total. Then assign each group a different stack to make, all of different heights, lengths and widths (three groups should do one each of questions 1–3). Have all groups write volume as cm3. Discuss and write the first three questions of the exercise, comparing the results to the group demonstrations. Students can then complete the exercise and Challenge independently. See Extension Activity below.




Review the points in the summary with the class, re-teaching as needed and encouraging students to use the Summary for revision.

The Unit Check can be used for on-going assessment.

Assessment 7

The two-part assessment emphasises the concepts of perimeter and area, including compound shapes, volume, work with circles and problem solving.

Teaching tips•Area of compound shapes often

requires students to break down larger shapes into manageable parts. This skill may not come easily to all students, even with practice. These learners may benefit from using replicas of the shape cut out of paper, with the measurements included, and then cutting the compound shape into each of the smaller parts. Once the area of each smaller part is recorded on the piece, the pieces may be put back together, and the total calculated.

• In complex questions or word problems with area, encourage students to sketch what they know, and put in the measurements. This step allows more students to focus on what they need to find out, and decide on a workable strategy. In questions such as the one with the pool and deck, students can darken in the part of the diagram where they need to find the area.

•When students verbalise their understanding, they are, in effect, teaching someone else that skill. Studies suggest students are better able to retain learning when they have taught those skills to others.

Expansion / Extension of unit•Relate the value of pi described in lesson

14.7, to the idea of irrational numbers, described in the mathematical language box in Unit 9 lesson 12 B. Investigate who discovered pi, as noted in the Challenge for lesson 14.7.

•Activity with Volume – see lesson 14.9, activity using popcorn to explore concepts of volume.

•Write a definition of ‘capacity’, and an example, as compared to ‘volume’ and an example.

•Complete the Challenge after exercise 14.7 C.


Sample lesson planUNIT 1414.8 Surface area

Objectives •Find the surface area of objects

by determining face areas and calculating

Engaging the students’ interest / ConnectionsStudents in groups will take a box and tape it sealed. They will then make a plan for cutting the box open so it lies flat, and looks like the nets they worked with earlier.

Teaching the lessonMathematical languageThe surface area is the total area of each face of a solid shape.

Practical activity / Developing the lesson•On the inside of the box, students will label each face of their net, and then together

decide which faces share the same measurements. Next, they will measure each face, and list on a scrap paper the area of each face. In the middle of the net, students will write the dimensions and area of each labelled face, as in the example box, and calculate and list the total surface area. Display the results of the different boxes on the wall, and have students discuss and compare.

•Have students discuss the surface area of the prism described in the lesson box, and compare it to their work. Guide them to recognise the need to find the area of a triangle, with the height of the triangle shown as 4 cm. Ask why the length 6 cm does not match the height of the triangle. If needed, make a model of this diagram to help students grasp the height of the triangle and the length of one side is different.

•Move on to discuss question 3, finding the height of a square based pyramid. Again, note the height of the triangle is different from the length of the side or base.

•Have students make a net for each of questions 1–5, write the dimensions for each, and then calculate the surface area. See note below.

•The Workbook exercise will allow further practice.

Differentiating for different learning styles Having students sketch a net for each 3D figure gives them the opportunity to mentally and graphically sort out the shape of each face. Later, it gives the teacher the advantage of spotting where errors may be made as students ‘picture’ each face.

RESOURCESSeveral empty boxes of different dimensions, such as cereal boxes, tissue boxes, etc.; tape, scissors; measuring tape/or metre stick and rulers; markers.


Extension activities There is a CD-ROM activity which students can explore independently.

AssessmentCheck student understanding through diagrams, calculations and informal discussions.

Summary of key points To find the surface area of a solid figure, we find the area of each face and combine them to find the total.

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•Share a quantity using ratio (partitive proportion) and unequal sharing

•Work with proportion, equivalent fractions and scale drawing, understanding the concept of proportion

•Solve problems related to direct or indirect proportion (without yet using terms)

•Use cross-multiplication (cross products) in problem solving


RESOURCESScrap paper; colours; counters to demonstrate ratio if needed for small group support; graph or grid paper; rulers.

Teaching the content of the unitThis unit has many opportunities to explore the concepts of ratio and proportion, a topic identified as needing extra support in the upper Primary / lower Secondary years. The emphasis remains on

problem solving, and each lesson builds in difficulty level, with occasional mixed-concept practice pages interspersed.




Overview of ratio (15.1)

Key points: a) the order of the items in the ratio and the order of the numbers in the ratio must match, b) ratios should be reduced just as fractions are reduced, by using common factors. After the introduction, have students work independently, and check their understanding in the review after the exercise. Encourage all students to try each Challenge.


Equivalent ratios (15.2)

Fully discuss the questions and the example in the tinted box. Have students work fairly quickly to complete the first three questions, and check these. Discuss questions 4 and 5, work them out, and ensure students understand before moving on.

Proportion: Proportionality is a key concept area that needs special attention. The lesson box discusses how the word

UNIT 15 Ratio and proportion

Overview of ratio, Equivalent ratios, Using cross products and proportion, Ratio and rate, Unit rate, Enrichment: Scale drawings, Working with a total, Problem solving, Unequal sharing


proportion is used in common speech, and then compares these to Mathematics. Engage students in a full discussion on this topic, having them give examples of proportion, and lack of proportion (e.g. drawing a figure on the board with one leg short and one leg long).

Activity for proportionality: Take students outside, and ask for four volunteers. have them form a square with their arms outstretched. Next ask for eight volunteers. Have them also form a square with their arms outstretched. Compare the first square to the second square (one is twice as long and twice as wide). Have the first set of volunteers join the second set and ask them to form a new square. After they try, ask the rest of the class why it is not working. Starting again, ask for 10 volunteers. Have them form a rectangle and say how many persons for the length and how many for the width (3 × 2). Give the challenge of making a rectangle 2 times larger, and let them work out that both the length and the width will need to be increased by the multiplicative constant. If they make only the length twice as long, for example, ask the rest of the class if the original shape has been kept. If not, the shapes were not in proportion.

Emphasise the meaning of proportion as two equivalent ratios, and review the ratios shown in the example. If students are keen, give examples of direct and indirect proportion, noting they will work with these more in secondary school. Have students work out the exercise independently.

Homework Challenge: ‘A small box has 18 blue tiles and 51 green tiles. What is the ratio of blue tiles to green tiles in lowest terms?’ (answer: 6 : 7) ‘If I have five boxes of tiles, how many of each tile do I have?’ (answer: 90 blue, 255 green)



Using cross products and proportion (15.3)



Ratio and rate (15.4)

Students will be familiar with the concept of rate, even if using different words to describe it. Discuss the ‘Think’ box, and the example in the lesson box, and make sure students understand that both sides of the ratios increase the same number of times (the same rate). In the example, both sides are 15 times larger in the answer.

As with proportion and cross products, teachings students to label the items being compared will significantly reduce the number of student errors.


Unit rate (15.5)

A unit rate brings one side down to a unit, to 1. While not all rates can be reduced to 1 whole, all of the questions or problems on this page can be. Build mental skills, working through the lesson example, and a few of the word problems. Stress that students who can solve the questions mentally (by finding the value of 1 unit), will be able to work quickly and efficiently. At the same time, show students how they can write the workings either as a fraction or as a decimal.

Another aspect of unit rate is shown in the lesson and word problems for exercise 15.5 B. The tinted lesson box shows the thinking behind these problems. At first the total number of days for just 1 is determined through multiplying, and then that total is distributed amongst the number for the question in the word problem. This type of question may be approached mentally, by thinking through the steps just described, using reasoning. Students may also learn one of several written methods, as shown.


In 15.5 C the word problems are mixed and cover the several concepts just practised. Whether as a whole or a few at a time, this exercise is useful to assess how well students are progressing. The Challenge may be for bonus points.


Enrichment: Scale drawings (15.6)

Investigation of scale drawings is a chance to use proportion, in practical ways. Discuss places where scale drawings are used, and the lesson box ideas. Have students use their ruler and graph paper and create scaled drawings as described in the exercise.

For a challenge, see also Teacher Resource website – Standard 6 Challenges page ‘Scale and proportion’.

Working with a total (15.7)

Teachers are encouraged to put the emphasis on reasoning, as students think through the word problems. In the example with cherries, for example, four cherries at a time are coming out of a set of 80. Students will see right away that 80 is 20 times greater than 4. Using what they learnt earlier, they can mentally multiply the 1 by 20, and then increase the 3 twenty times for the final answer. The methods shown are those often seen in the region, however teaching students to rely on reasoning skills is more effective.

In ratios with three quantities, students will need to decide how many parts there will be altogether, and divide this into the total to find the value of each share. This value can then be applied to each of the parts, whether there are two or three or more.

Exercise 15.7 C mixes the different concepts, and can be used to check student understanding.



This lesson includes unit rate, proportion, work with ratio, opportunities to use cross products, to find the total or the value of shares in ratio, and each other concept covered in this unit. The 15 word problems may be used one at a time as starters for the lesson, or several questions at once as review, for assessment, or for revision.

Unequal sharing (15.9)

This lesson resolves the question of what to do when something is not shared equally. It presents a useful life lesson that can be applied to other situations, for example when a profit is to be shared by two or more persons, after the expenses are taken out. The tinted lesson box shows students an approach that is more effective than the guess and check strategy.


Teaching tips•Equivalent ratios: When students are

reducing the ratios, they may find it easier do so in several steps if it helps them to find the lowest equivalent ratios. Encourage them to keep reducing until they can do so no more. Gradually they will learn to reduce more efficiently using larger numbers.

•To make the question of ratio or unequal sharing easier for kinaesthetic learners, use counters and lay them out in the same proportion as in some of the word problems. Model it with these students, and then have the students model problems on their own.


Sample lesson planUNIT 155.3 Using cross products and proportion

Objectives •Solve problems related to proportion


Engaging the students’ interest / ConnectionsOrally run through several scenarios where the responses must stay in proportion. Recipes make good examples: ‘A recipe calls for 3 cups of oats to make 24 biscuits. 11

2 cups of oats in the same recipe will make how many biscuits?’

Teaching the lessonMathematical languageA proportion exists where two ratios are equivalent.

Cross multiplication, also called cross products, is a tool to check if ratios are equivalent. It can be used to find a missing number in equivalent ratios.

Practical activity / Developing the lesson•Say aloud or write on the board several examples of two ratios, asking students

to put up a hand if they hear proportions, and put their arms down if the ratios are not proportions. For example: 3 : 8 and 2 : 3 are not equivalent ratios so are not proportions. 2 : 5 and 8 to 20 are equivalent ratios, and proportional.

•Discuss the example in the lesson box. Note that the working is carefully labelled. Writing the labels significantly reduces errors, and helps students understand what they need to find out.

•The top-right and bottom-left numbers should have the same product as the top- left and bottom-right. When there is an unknown, one product is found, and then divided by the third given number.

•In some cases, it is simple to use equivalent fractions to find the unknown. In practising the strategy, however, these simple numbers are used at first to help students easily see if they are correct.

RESOURCESScrap paper; pencils; ruler; recipe.

Expansion / Extension of unitChallenge: ‘A small box has 18 blue tiles and 51 green tiles. A large box holds 575 tiles in the same proportion. How many green tiles are in the large box? Show two ways to find a solution.’ (answer: 425)


•All student work should be labelled, before it is checked.

•Have students work out the five questions, showing cross products that match the example.

•Discuss all results thoroughly. Ask who understands and/or likes this strategy. Talk about who thinks the strategy will be useful to them, and the types of word problems where they might use it.

Differentiating for different learning styles Some students will have trouble learning how to label and set out the problems in order to use cross products. For these students, a framework for the first few will need to be set out, with labels included.

Encourage students to try the Challenge questions, and then work it out together .

Extension activities Have students bring in a favourite simple recipe. They might ask a parent or grandparent, and should then write down the list of ingredients, and how many servings (portions) it makes. Give students several minutes of class time to compare their recipes, finding what is the same and what is different. Ask for volunteers to organise the recipes, and prepare them to be scanned and saved, or put into a class scrapbook.

AssessmentCheck student work with questions 1–3.

Summary of key pointsTo be a proportion, the ratios must be equivalent.

Cross products is a strategy or tool to help find the missing element in a proportion.

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•Work with proportion, equivalent fractions and scale drawing, understanding the concept of proportion

•Solve problems related to direct or indirect proportion


•Calculate unit cost / total cost

•Total a bill and determine change

RESOURCESScrap paper and colours; calculator if available; advertisements and/or tins and bottles for the shopping extension.

Teaching the content of the unitStrategies for mentally solving problems, as well as written methods, will be combined in this unit to help all students find the tools to develop their problem solving skills. Students will use diagrams, formulae and organised lists as problem solving strategies.




Unit rates (16.1)




Combining unit rates (16.2)

Part of what makes these types of questions challenging is there is nothing obvious with the numbers in the question to guide us to a solution. The key is first to find out the unit rate for one minute. When both have a common measurement – the rate for one minute – they can be compared, combined and worked. Students need to apply both understanding and skill, which is one reason this type of question is so often found on standardised tests.

To build the conceptual basis, using the example, help students break down what they know, and say how much of the tank each hose alone would fill in one minute. Then help them see that together the two hoses would fill the tank the same amount as both 1-minute quantities added together. These two steps can be demonstrated and are fairly clear with practice.

UNIT 16 Problem-solving skills

Unit rates, Combining unit rates, Using diagrams, Using formulae, Looking for links, Listing details, Calendar problems, Poles and spaces, Problem-solving practice, Units 15 and 16 check and summary, Assessment 8


Next determine how much of the whole tank has been filled in that minute, and for this a diagram is useful. Draw a large circle, divide it into 12ths, and shade 5. 5

12 represents how much of the tank is filled by both hoses in one minute. Now have students try to put another shaded block of 5

12 with another colour in the circle. This represents two minutes. Left over in the circle there are 2

12. Shade this portion a third colour. Ask how many 5

12ths fit in this last piece? 2

12 ÷ 512 = 2

5. Only a fraction of the 5

12 will fit. Combine the results, noting that each refers to how many minutes. The answer is 22

5 minutes.

After the concept is well-understood, students can use the shorter version shown after the example. Please note, some students are taught to simply invert the combined unit rate for one minute ( 5

12), to find the answer. This approach does not help students understand the concept of what they are doing and why.


Using diagrams (16.3)

Spatial and visual learners particularly benefit from using diagrams with word problems. While formulae are useful, learning to use diagrams helps students with their understanding of units of time, speed and distance. Simple diagrams check their understanding. What is markedly helpful is for students to see the two units of measurement they are using both shown on the grid. In the example, both the distance and the time end at the same point. Using the grid, they can visually notice at what point they are looking for the answer.

This lesson is particularly useful for assessment of problem-solving development.


Using formulae (16.4)

Take time to fully discuss the ideas, examples and the ‘Think’ and ‘Remember’ boxes. When students write the formulae in their notes, the use of colour (as shown by the large arrow) is an easy technique to help them remember the three units.

Have students work independently on the questions. See Teaching tips below.

Enrichment: Pythagorean TheoremThe box describes the basics of the formula and how it is used. Have students try it as a challenge, and find the height of the tree.

For a challenge, see also Teacher Resource website – Standard 6 Arithmetic page ‘Using formulae in problem solving’.



Looking for links (16.5)

As students get closer to their national examinations, continue to encourage students to explore, experiment and put together what they have learnt to make problem solving simpler. In addition, specifically teach the pitfalls they may encounter. For example, in the first part of the example, students may be misled and try to divide 36 by 4. Help them see why the wording needs to be examined closely: 1

3 of a number = 36, not 13 of 36.

In the second part of the example, some students get lost trying to find a percent of 15 that is equal to 8. Instead, they simply need to see that the 45 is 3 times greater than 15, so 8 also becomes 3 times greater. Have students work out the ten questions on their own, giving one on one support where needed, and allowing peers to assist once they are finished (without giving away the answers).

For a challenge, see also Teacher Resource website – Standard 6 Numeracy page ‘Working with rates’.



Listing details (16.6)

Labels, listing what is known, and sorting what needs to be found out are all part of the strategy practised in this lesson. Encourage students to begin each time by recording the person who has the simplest fact (e.g. Jake and 24). They may also assign a letter to each person, and set it out algebraically.

Calendar problems (16.7)

The list strategy works well with calendar problems. As the days in the months vary, some students find it simplest to set out each part of the question, and add the result.

The term ‘inclusive’ indicates both the starting and ending units of time are counted. If the term ‘inclusive’ is not present, or the word ‘between’ is used, students need to very carefully ready the wording to assess the question.

See Teaching tips below regarding leap years.


Poles and spaces (16.8)

These are a type of problem that must be analysed in a particular way, rather than a mathematical concept in itself. These types of questions are always best graphed out in a diagram or a sketch. The key concern is again a question of inclusion – are both the beginning and end points counted, or are the spaces between the posts what is significant? Work through the tinted lesson box example, helping all students draw a simple diagram. Work through at least the first two questions as a whole class, asking volunteers to help sketch. Give students the task of completing question 3 and question 4 on their own, and then discuss the results and diagrams. Have students work out questions 5–6 on their own, and question 7 (where the layout is a circle) as a challenge.


Problem-solving practice (16.9)

The word problems cover several concept areas, with the purpose of pausing to check for errors, misunderstandings or areas of weakness in problem solving. For this exercise, have students not only solve the problems, but also set out for each one what is known, what they need to find out, show the strategy or explain the working, and finish with a sentence or phrase that answers the key question of the problem. As teachers analyse the student work, they will make notes for each student as to skills still needed.


Discuss the Summary, answering any lingering questions or re-teaching if needed. Use the Unit Check to measure continued progress.

Assessment 8The two-part assessment includes ratio and proportion, rate and unit rate, unequal sharing, use of formulae, calendar and a variety of word problems where different skills and strategies can be applied.

Teaching tips• If a question arises about the use of the

term ‘average’, explain that when we do something there are some moments slower and some faster, but the overall speed is the average speed.

•Note: leap years are divisible by 4 and not divisible by 100, with the exception of years divisible by 400 (2000).

Expansion / Extension of unitChallenge Homework: May has two clear bottles. One holds 5 ℓ and one holds 3 ℓ. Work out the steps she will need to take in order to get just 4 ℓ of water (without estimating). Solution: there will be 4 steps. Fill the 5 ℓ bottle, pour from it into the 3 ℓ bottle (leaves 2 ℓ in the bottle), empty


the 3 ℓ bottle, pour the 2 ℓ left into the 3 ℓ bottle. Fill the 5 ℓ bottle. Pour from it into the 3ℓ bottle (needs just 1 ℓ to fill). 4 ℓ will be left in the 5 ℓ bottle.

Organised list challenge: ‘Ricardo takes on his trip 2 pairs of pants, khaki and black,

along with 3 shirts in green, blue and yellow, and 2 hats, one a cap and one with a brim. How many different outfits can he make wearing pants, shirt and hat each time?’

Sample lesson planUNIT 1616.1 Unit rates

Objectives•Use strategies to solve word problems

•Solve problems using unit rate

•Use diagrams to visualise and solve word problems

Engaging the students’ interest / ConnectionsChallenge question: ‘With the help of a baker and a shop owner, Class 6B created a record long sandwich. They cut 22 servings from a section 176 cm long, and still had 1

3 of the sandwich left. How long was the sandwich altogether?’ Divide the class in two, giving each group shop paper and markers.

Have half of the class explore how many servings there would be, and have the other half of the class explore how long each serving would be. The first group might show 22 servings = 2

3 so 11 servings = 13, and just add 22 and 11 to find 33 servings. While the

second group might show that 22 servings = 176 cm, so 1 serving = 176 ÷ 22, or 8 cm per serving. Together they can find out the length of the sandwich.

Teaching the lessonMathematical languageThe main focus of this lesson is for students to use numeracy and reasoning skills to more easily solve problems involving unit rate. They may be able to solve some problems with what they know about fractions, and about percents that easily divide 100, but will also learn the skill of finding a unit rate.

Practical activity / Developing the lesson•Discuss times when a class might have a fund-raising activity, connecting to any the

class itself has done.

•Review the information in the example, clearly setting out what is known and what is needed.

•Discuss the advantage of finding 1%, to then find 100%.

•Explore the short cut, and why it is helpful to look for percents easily related to 100.

RESOURCESShop paper and markers; rulers, advertisements and/or tins and bottles for the shopping extension.


•Compare the two methods shown in ‘step 3’, the first line showing what we need to do, and the second line showing calculations, and make sure students are able to understand the methods are two representations of the same concept.

•Pose questions based on some of the word problems in the exercise.

•Have students work on their own if able, or with a partner if needed, to solve the questions.

•Discuss the results, and whether each is reasonable. Find some questions where students used different approaches, and have them explain what they did to their peers, and why, and answer any questions.

The second page of the lesson has a strategy that has proved to be very popular for students and teachers:

Draw an upright rectangle on the board, and divide it into thirds using dotted lines. Shade two of the thirds, and next to them show 2

3 = 650 mℓ. Explain if 23 = 650, than 1

3 is equal to half that, and write 325 in each of the shaded parts.

Ask students to be logical: if each of these 13s is 325 mℓ, what is the other (non-

shaded) 13? Write 325 mℓ in that section. Have students explain they can add all the

thirds to find the total.

Note: although a working is shown in the example, we would encourage students to mentally solve these questions, reasoning out the steps and drawing diagrams/sketches.

•Have students draw a sketch for each of the word problems in questions 1–6, to embed this strategy in their ‘mental toolbox’.

•Check that the drawings are logical, and represent the information in the problem accurately.

•There are three Workbook exercises for additional practice.

Differentiating for different learning styles Challenge: 75% of the students brought their ticket money for the visit to ___, which totalled $514.50. How much money was needed altogether? Write a paragraph describing what is known, what is needed, and at least two different ways to solve the problem. Finding opportunities to ask students to use more than one way to solve a problem, challenges all students and has several positive effects.

Simple frameworks to learn the skills are in the Workbook Unit 16 Exercise 1 questions 1–2, and very simple and straightforward word problems for these skills are included in exercises 2 and 3.

Extension activities Shopping – when different stores or shops sell the same item for different prices, we need to be able to tell which is the best deal. Give several examples from advertisements or make up examples to write on the board (e.g. 4 for $10, or 6 for $14). Guide student discussion to recognise that it is necessary to find the cost of just one item, in order to work out the best deal. Similarly, two different-sized tins or bottles of the same item may be compared. To work out which is the better buy, students have to work out the price per unit (per mℓ, ounce, gram, etc.).


AssessmentCheck student work and sketches, as well as solutions, to ensure understanding.

Summary of key points The unit rate is based on one part or one percent. Finding the unit rate is a useful problem solving skill, particularly with money in shopping comparisons.

Sketches or diagrams are useful strategies when a part is known and the total needs to be determined.

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•Interpret information shown in a graph or table

•Describe methods of collecting data

•Carry out project to gather data (survey, questionnaire or interview)

•Collect, organise, construct and display data using a frequency table (tally chart), pictograph, bar, block (or line) graph, using scale

•Interpret data, apply the data to problems and compare information to draw conclusions

•Use a circle graph (pie chart) divided into degrees, percent, (minutes) or fractions and interpret the data

•Connect to probability as prediction in determining if outcomes are fair, likely, etc.

•Describe, identify, sort, classify and list the elements in a set

•Work with the concepts of subset, union and intersection of sets

•Use Venn diagrams to display and compare sets and in problem solving

•Explore integers using a number line

•Practical situations using negative integers (temperature)

•Explore irrational numbers / repeating decimals

•Demonstrate skill with a calculator as an operational tool

•Recognise Roman numerals used in practical situations (e.g. clocks)

RESOURCESCalculators, if available; circle template; colours and ruler; two dice per pair of students; graph (grid) paper; examples with Roman numerals.

Teaching the content of the unitThis unit includes several opportunities for students to analyse, to interpret, to classify, list, or display information. It includes several Enrichment sections related to set theory, integers and graphs, which are included in some regional syllabuses as preliminary work prior to secondary school mathematics.




UNIT 17 Statistics / sets / integers

Overview, Graphs, Pictographs, Circle graphs, Patterns and probability, Enrichment: Sets, Venn diagrams, Enrichment: Union and intersection of sets, Enrichment: Integers, Enrichment: Adding and subtracting integers without a number line, Enrichment: Comparing integers, Enrichment: Multiplying and dividing integers, Using a calculator with integers, Enrichment: Coordinates, Unit 17 check and summary


Overview (17.1)

Use this lesson to reinforce the skills of collecting, describing, organizing, displaying and interpreting data. Check that all local syllabus points are covered as students carry out the group activities and discussions.


Graphs (17.2)

Encourage students to find graphs in their environment, including those for keeping track of tests scores, or those found in newspapers or online. Contrast and compare the graphs. The graph shown in the exercise should be very familiar to students by this point. Check, however, that students always notice first what the gradation is in the y-axis (vertical left side), and know how to handle a situation where the bar falls between markings. Encourage students to make a circle graph, as shown in the Challenge.

The line graphs are included as Enrichment because they are not included in all regional syllabuses. However, they are not at all difficult, and are useful in understanding direct or indirect proportions (e.g. where the speed increases and the distance increases in the same amount of time, or where the speed increases and the time decreases for one set distance). In this Enrichment exercise, students compare how the same information looks when displayed on a line graph. Generally, we would not use this type of data for this type of graph, but it is useful in this instance as students recognise the dot and the top of each bar match.

See Teaching tips below.

There is a CD-ROM activity for this lesson. (Enrichment: Line graphs)


Pictographs (17.3)

The main idea for a pictograph is to know what each figure represents. The

steps they need to consider when turning the pictograph information into a bar graph are all noted. Ensure students read through these sentences before they begin.

Encourage all students to do the Challenge, and write out their responses to ‘a’ and ‘c’.

Circle graphs (17.4)




Patterns and probability (17.5)

Introduce the lesson with a quick game: Have all students write on a scrap paper the numbers 2 through 12. With a partner, they take turns rolling two dice. For each turn they may cross one of their numbers. If the number has already been crossed, they hand the dice back to their partner. The game continues for 10 rolls each. Partners should compare results, notice any patterns, and discuss these with other teams.

The many activities available for probability are very useful to help students recognise patterns and trends. The activities have the added advantage of engaging all of the students, as there is an element of surprise, with the results unknown in advance.

Discuss the concept of predicting, described at the beginning of the lesson, and the term ‘probability’, including the word from which it is derived (probably). Ensure students understand that probability is always represented with a fraction or decimal between 0 and 1, where 0 means never and 1 is sure. Make up examples where students say if the result is sure, likely, not likely or never.

Ensure students understand the full lesson, including the formula for finding probability and stating probability. If possible, do some of the activities outside,


such as the skipping activity displayed. This activity works well in groups of four, with one recorder, one jumper and two turners, at any one time. Carry out activities in class to enhance this learning. Exercise 17.5 may be worked out independently, with results that vary.

See Extension activity below.

See also Teacher Resource website – Standard 6 Games / Activities page ‘Probability’.



Enrichment: Sets (17.6)

The concept of sets is a review, as are Venn diagrams. Not all regional syllabuses include sets, although most include Venn diagrams. The lessons about sets are useful for all students, but should only be used for assessment purposes if included in the local syllabus. Remind them of the meaning of the curly brackets, which include the members of the set, and that a set with no members is called an empty set. Discuss the lesson, how to interpret what is in each of the circles in the Venn diagram, and what is in the box but not the circles.

See Extension activity below.

Venn diagrams (17.7)

The first exercise gives practice in interpreting information displayed in the Venn diagram. The second exercise gives students a chance to create a Venn diagram on their own. Check that students put a box around the diagram, and that the numbers within the three circles or in the box match the total exactly. The latter point is a frequent error seen by students when creating their own diagrams.

Include subsets after the following Enrichment lesson. The workbook exercise gives students a chance to fill in a Venn diagram already created for them.

Use with WB Unit 17, Enrichment: Ex 6

Enrichment: Union and intersection of sets (17.8)

Union and intersection of sets is fairly straightforward, once students remember the symbols, and their meanings. See Teacher tips below.

The key point for subsets is that everything in it is also in the larger set. The emphasis should not be on the vocabulary, but on giving the examples from real life or numbers.

Enrichment: Integers (17.9)

The first part of this particular lesson about integers should be shared with all students, as exposure to the world around them. They will hear on television where the temperature is ‘below 0’, and will have been exposed to the concept of being ‘in debt’. The concept represented on the number line is that these types of situations can be represented on the left side, with numbers starting below 0 and counting outwards in sequence, –1, –2, and so on. Even if you will not be including the sections using operations with integers, you may still wish to do the first part of section 17.9, and perhaps exercise 17.11 A where numbers are compared.


Use with WB Unit 17, Enrichment: Ex 7–8

Enrichment lessons: Integers (17.9, second half); Adding and subtracting integers (17.10); Comparing integers (17.11); Multiplying and dividing integers (17.12)

These Enrichment lessons all include information about how to add and subtract positive and negative integers, using a number line or using the guidelines, and about how to multiply or divide positive and negative integers.

Discuss questions 11–13 in lesson 17.10, to give a real-life context to integers, and come up with similar examples from the students’ experiences.


Display a number line across the front of the room, with 0 in the centre, and 25 negative integers to the left, and 1–25 to the right. Use the number line as students go over the exercises they have complete and give the result. Have students make and display a poster in the classroom with the guidelines for adding, subtracting, multiplying or dividing integers.

Practical activity: Put a thermometer in the freezer of the school canteen for a short time. Read the result when it comes out. Find out how many degrees below zero are recorded on the thermometer.

Use with WB Unit 17, Enrichment: Ex 9 and 12

Using a calculator with integers (17.13)

Provide students or partners each with a calculator. Go through the lesson in the tinted box, allowing students time to explore and experiment on their own, using positive and negative integers. Compare calculators to see if the function keys are the same, and if the sequence of steps needed is the same.

Give a string of four or five integers, both positive and negative, for students to order. Guide their understanding that a large number, when it is a negative integer, actually has a very low value, as it is far from 0. for example –215 is smaller than 3. Students have the opportunity to put several integers in order in the workbook exercise 10 questions 9–10.

Have students work in small groups to do the Activity.

Use with WB Unit 17, Enrichment: Ex 10–11

Enrichment: Coordinates (17.14)

Review what students remember about coordinates. Remind them that an ‘ordered pair’ has the format (__, __ ), with the first number indicating the direction to the right horizontally (‘across the field’) or x-axis, and the second number indicating the direction upwards vertically (‘up the tree’) or y-axis. Show them that this work

they have done all occupies just the top right section of a full Cartesian plane.

Draw a large diagram on the board showing all four quadrants, with the origin (0, 0) labelled, and the x- and y-axes. Carry out the lesson as described, using the graph to demonstrate. Ask questions 1–5, and conduct as a whole group class. Show that the negative numbers are to the left or down.

Note: If students did not do the enrichment with the integers, they would not be practising the Activity listed in lesson 17.14 A.

Exercise 17.14 B may be done by all students. Then do the Practical Activity below, followed by the Activity in the text.

Practical Activity: Have students clear their desks, and then stand around the perimeter of the room. Arrange the desks in neat rows and columns. The back (SW) corner of the layout will stand for (0, 0). The desks north from that point represent the y axis, and the desks to the right represent the x-axis. It may help to run a ribbon or rope across these desks to make it very clear they are ‘0’. Next, place objects randomly on several of the desks in the room, and have students name the position where each object can be found. For example, put a bat on the desk four columns to the right and three rows up, and name it (4, 3).


Use with WB Unit 17, Enrichment: Ex 13


The Unit Check only assesses the information that all students should know, and does not include calculations with integers.


Teaching tips•Graphs are representations of

information, usually displayed in ways that match specific purposes, such as comparisons, or changes over time. In the Primary years, what is most vital is that students are exposed to a variety of types of graphs, learning to read and interpret the displayed information.

•To remember the symbols for union and intersection: We can say that the ∪ symbol can be compared to a bowl that includes everything (union), and that the ∩ symbol is slippery to try to stand on, so the members have to hold hands with a member in common from the other set, so as not to slip off.

Expansion / Extension of unitChallenge: Create a Venn diagram to represent the following information. ‘In our class of 36 students, one third have a brother. 4 have a sister and 12 have both a brother and a sister. Put a box around the Venn diagram, and label each oval. Show how many students have no siblings (brothers or sisters).’

Activity: Students work in small groups and roll two dice 36 times, recording the results. Each group discusses and compares their results, and develops some conclusions. They then discuss their results and conclusions with another group. Challenge: Which number shows up least often? (7) Which numbers show up most often? Why? (one answer – some numbers can be formed with more combinations)

Sample lesson planUNIT 1717.4 Circle graphs

Objectives •Interpret data, apply the data to

problems and compare information to draw conclusions

•Use a circle graph (pie chart) divided into degrees, percent, (minutes) or fractions and interpret the data

•Interpret information shown in a graph or table

•Demonstrate skill with a calculator as an operational tool

Engaging the students’ interest / ConnectionsWithout prior discussion, have students write on a small slip of paper the name of a __ they like, and collect these in a box (a sport, song, movie, book, etc.). Pull only eight names from the box, list them on the board, and tell students whether or not their choice is there, they are to choose from the list something that they like. Use tally marks and record student choices. Have students work in groups of two to four, give them a paper with a circle template, and ask them to make a circle graph with the

RESOURCESSheets for each small group with a circle template; something to trace to make a circle or compasses; calculators if available; coloured pencils; ruler; poster paper.


results. Students will have to decide which choice needs a greater portion of the circle, etc. Provide no guidance, and set the time at 10 minutes. Then discuss the challenges and their thinking. (Note: we would not expect complete and accurate graphs at this point, as the point is for students to explore what is important in creating a circle graph.)

Teaching the lessonMathematical languageThe information in a pie chart, or circle graph, may be shown with percent, with fractions, with degrees as on a clock, or just with numbers. In every case, the total of the figures is equal to one whole.

Practical activity / Developing the lesson•Help students notice that the circle graph on ‘Favourite subjects’ uses percent, and

the sections are proportional to the size of the percents. Ask why this is a helpful aspect of circle graphs (makes comparison much easier).

•Explore the questions in the first exercise, which enhance the students ability to interpret what they see on the graph.

•Point out the questions where the percent is to be converted to fractions. Point out the questions where the percent is to be converted to degrees, reminding students there are 360 degrees in a circle. Ask why changing the English and the Computer studies percents to fractions makes it simpler to work with these figures and convert them to degrees. Have students notice that the graph is based on a total of 120 students.

•Although the circle graph is the theme, the lesson has students use their skill in converting degrees of a circle, fractions, percents and decimals. Have students use calculators to check their work in exercise 17.4 A, if these are available. They should write their calculator answer next to their original answer, if different.

•Exercise 17.4 B includes a variety of units on the graph itself, but the total of all together is still equal to one whole. Note that keeping these units in proportion allows the figures to be compared, and if they were different, comparisons would be impossible.

•Have students notice that the graph is based on a total of $15 000.

Differentiating for different learning styles For struggling learners, revise circle graphs with fewer options and with all of the same unit used (e.g. all as percents, or all as fractions).

Extension activities Give students time to make a decision about the units to be used in their original graph from the opening exercise, and to complete their graphs with appropriate labels, and with each of the eight sections shaded. Compare the results, putting the graphs side by side. In some, the order will be different, or the unit used, but the proportional size of each shaded section should match.

AssessmentUse all of exercise 17.4 B.

Summary of key points Pie charts / circle graphs are based on one whole. They may be broken down into degrees, percents, fractions or a combination of all. We can use pie charts / circle graphs to compare information based on a total.

114 Unit18•BrightSparksTeacher’sBook6


•Solve problems with cost price / selling price

•Calculate profit and loss / profit and loss percent

•Calculate wages, salaries, simple interest, total cost for hire purchase or down payment

•Calculate simple interest given a formula

•Interest for savings / loan

•Hire purchase / buying on credit

•Use percent in money transactions

•Calculate wages and salaries

•Discuss the purpose of budgets, taxes

•Calculate unit cost / total cost


•Investigate regional currencies

•Convert regional and other foreign currencies to local currency and back, using a rate of exchange


•Understand the idea of savings (value)

RESOURCESCalculators; rulers and scrap paper; graph (grid) paper; advertisements of prices including hire purchase; newspaper or online listings of bank currency rates of exchange.

Teaching the content of the unitStudents will have multiple opportunities to build their knowledge and skill in consumer topics in this unit. Let students know that most errors occur in this topic due to lack of care with details.




Profit and loss (18.1)

Revise the vocabulary terms profit, loss, cost price and selling price. Discuss the examples in the lesson box, and find both the profit per bottle as well as the profit per case. Help students understand that some word problems will ask about individual items while others will ask about a large set. After students work through

UNIT 18 Consumer / Business Mathematics

Profit and loss, Profit and loss as a percent, Problem solving, Simple interest, Value added tax, Hire purchase – buying on credit, Foreign exchange, Unit 18 check and summary, Assessment 9


the word problems, discuss them as a class and check student understanding.


Profit and loss as a percent (18.2)

Discuss the example, and bring out the main idea that converting the profit or loss to a percent will allow it to be compared. In the example, both individuals made the same profit in dollars, but the costs varied. Emphasise that conversion to percent showed a wide difference between the profits earned. See Teaching tips below.

Exercise 18.2 B provides opportunities for students to work with others, and to discuss the details in each question.




The ten word problems in this exercise are very useful for building skill and understanding of profit and loss. Teachers may wish to use them in part, or all at once. Students may do them in groups, and then share their results with the whole class, or may work on them independently.

Homework Challenge: Each day for four days, Jay and Jordan doubled the sales of their homemade fudge. It cost them $16 to make the fudge the first day, and their sales totalled $32. a) What percent profit did they earn the first day? b) If they made twice as much fudge on the second day, what was their percent profit for that day? c) How much did they earn in total sales on the third day? d) How much more was their total sales on the fourth day, compared to the first day?

Simple interest (18.4)

Teach students that there are two kinds of interest, simple and compound. For now, they will just work with simple, but they may learn about compound in the enrichment box. Guide students to recognise the terms in the formula, and work out the word problems.

See also Teacher Resource website – Level 6 Money page ‘Savings and interest’.



Value added tax (18.5)

VAT is just one type of revenue building by a country, as noted earlier. In this exercise, a VAT of 15% is included in each price. To find the original price, students need to learn that the price shown represents 115% of the original price, and that the original price will be less. Some students find this ‘going backwards’ step very challenging.


Hire purchase – buying on credit (18.6)


For a challenge, see also Teacher Resource website – Standard 6 Money page ‘Personal budget’.


Foreign exchange (18.7)

Foreign exchange is a regular part of Caribbean life as students get older. They will be exposed to different currencies as they travel within the region, as they are in contact with family members within the region or overseas, and as they interact in businesses and with tourists. The mathematical perspective is treating each foreign exchange transaction as a proportion. The rate of exchange is a ratio, and the money compared to it will be an equivalent ratio. It is recommended to build these skills slowly, and as students become competent with the basic level, to gradually move to the more complex. When equivalent ratios are set out, have students label both side. Guide students to notice that when the amount of money given is the same currency as the side of the rate of exchange that shows $1, it is a simple matter of multiplying. When the currency is the other side of the equal sign, the amount is divided.


Show students how the strategy of cross products is a useful tool for exchanging currency, and have them set out exercise 18.7 B in this form.

Exercises 18.7 C practises these skills at a more complex level.

Exercise 18.7 D gives a full page of word problems on the key concept areas of this unit. In every question, students will practise real-life situations that they might encounter, and use the skills from this and earlier units.



Review the Summary with students and re-teach any parts that students need help understanding. The Unit Check may be used as a class assessment.

Assessment 9

The assessment reviews graphs and consumer mathematics, with an enrichment section assessing understanding of Venn diagrams.

Teaching tips•A phrase that helps some students

remember how to find the profit or loss percent, is ‘profit or loss, over cost’.

•For many currencies, the rate of exchange changes on a daily basis. Have students check the newspaper, or the display in the bank, and notice changes to currencies.

Expansion / Extension of unitChallenge question – how many different combinations of local currency (notes or notes and coins) can be used to represent $58.

Hire purchase: Bring in advertisements that show the cash price, and the hire purchase payment price. Either ask at the store or check online to find out the interest rate due for hire purchase at that particular store. Calculate how much would be paid over one year for an item purchased by hire purchase, as compared to cash.

Sample lesson planUNIT 1818.6 Hire purchase – buying on credit

Objectives •Calculate total cost for hire purchase

or down payment

•Understand hire purchase / buying on credit

•Discuss the purpose of budgets



Engaging the students’ interest / ConnectionsShow an advertisement for an item to be purchased using hire purchase (‘on credit’). Discuss the advantages and disadvantages of buying in this way, having students write the pros and cons on the board.

RESOURCESAdvertisements showing prices of items sold by hire purchase; calculators if available.


Teaching the lessonMathematical language‘Hire purchase’ is also called ‘buying on credit’. It means to pay a very small part of the cost as a ‘down payment’ (sometimes no down payment at all), with the balance of the cost paid in small amounts weekly or monthly.

Practical activity / Developing the lesson•Discuss the uses and value of money, and the different kinds of goods people buy,

and how they get the money to buy these things. Expand on the idea discussed earlier, of having a budget, and help students understand how hire purchase may help some people to have the things they need and stay within a budget. By paying a small amount each week or each month, buyers can plan out how to get the things they need before they have saved the full cost.

•Explore the downside of hire purchase – that in the end the buyer pays much, much more for the item, and that in some cases the item breaks or wears out even before all of the payments have been made.

•Work through the first two questions in small groups, and compare results.

•Have students then work with a partner and complete question 3 and question 4. Point out they must find the total price in one question, and the down payment in the other.

•Have students complete questions 5–8 independently, and assess their understanding with these four questions.

•Give extra credit for all students who attempt the Challenge question: one point if they attempted the question, and two points if they get it right.

•Workbook Unit 18, Exercise 6 gives two additional questions for practice.

Differentiating for different learning styles Some students may need a framework where they can fill in the name of the item, the down payment, the number of weeks or months for payments, and then the total cost.

Extension activities After each group, or pair of students, works out a solution in questions 1–4 above, they should use a calculator to check the answers. Using calculators is a skill that all students should be strong in by the end of this school year.

Mini project: In groups have students work out how much money would be needed each month for all the goods and services for a family of four. They should talk with family and other adults, and work out costs for housing, food, clothing, medical, and needed goods and services. Give the groups time each day for a week to discuss the information they are collecting, make decisions together for what makes sense, and finally to make a small poster with their results. When each group shares its results, emphasise that no one group is ‘right’, and just like any community, different people will find different objects more important.

AssessmentUse questions 5–8 to assess understanding.

Summary of key points Hire purchase is a method of buying goods with the purchase price paid over a period of time. The price is higher with hire purchase than it would be if cash were paid.

118

Workbook Final Review Self-Assessments 1–6Review 1–3 are multiple choice.

Review 4–6 are open response.

Some teachers will use the student self-assessment Reviews 1–6 periodically spread across the terms to help students see what they already know and what they still need to learn. Others may choose to use these tests as short checks that students can do on their own, prior to taking the full practice tests in class.

Workbook Final Review Self-Assessments 1–6•BrightSparksTeacher’sBook6

119

Student’s Book End-of-Primary AssessmentsPart A is multiple choice

Part B is open response

Each test should be timed, and should take place in an examination-like setting, as much as possible given the classroom environment and school schedule. Students should prepare their materials ahead of time, including scrap paper, pencils, and rulers.

All should know that they must be very quiet, not move from their seat unless it is very urgent, should stay focussed, and respect their peers by staying very quiet when they are done, so no one feels hurried or rushed. Students who practise this setting from time to time during the year may be more relaxed and better able to perform at their best when the national examinations occur.

Teachers should ensure there are opportunities to talk about the questions after the test, to find the ‘tricky’ part, discuss alternative approaches that all reach the correct solution, and allow peers to explain their understanding. When these ‘maths talks’ take place, the tests become teaching tools, ideally suited to serve as a focal point for greater understanding and confidence, and not a competitive exercise.

Final Assessment 1

Final Assessment 2

Final Assessment 3

Final Assessment 4

Final Assessment 5

BrightSparksTeacher’sBook6•Student’s Book End-of-Primary Assessments

120 Answers • Bright Sparks Teacher’s Book 6

Answers to Student’s Book 69 sixty-five thousand

10 eight hundred eighty-four thousand, and ten

11 12 040

12 211

13 6717

14 32 468

15 84 005

16 11 000

17 56 804

18 48 012

19 100 000

20 329 200

21 200 318

22 950 000

23 16 700

24 13 103

25 2 001 011

1.2 Value

Partner Activity:

a 80

b 1000

c 200 000

d 70 000

e 200

1.2 A

1 9

2 800

3 90

4 10 000

5 5

6 6000

7 600

8 30 000

9 50

10 300 000

11 5

12 3000

13 800

14 20 000

15 500

Challenge: c

1.2 B

1 8749 = 8000 + 700 + 40 + 9

2 56 = 50 + 6

3 2020 = 2000 + 20

4 6080 = 6000 + 80

5 61340 = 60000 + 1000 + 300 + 40

6 76000 = 70000 + 6000

7 4270 = 4000 + 200 + 70

8 815 600 = 800 000 + 10000 + 5000 + 600

UNIT 1: Number and place value1.1 Place value and counting skills

1.1 A

1 tens

2 hundreds

3 tens

4 thousands

5 hundreds

6 tens

7 ten thousands

8 ones

9 thousands

10 hundreds

1.1 B

1 2 2 9

2 2 8 7 0

3 7 6 3 0

4 4 2 4 8 7

5 8 6 7 9 0

6 1 3 8 4 7

7 9 6 0 6 1

8 2 0 0 0 0 0

9 1 2 0 6 2 0

10 6 8 9 7 4 6 8

1.1 C

1 five hundred forty-nine

2 two thousand six hundred (or twenty-six hundred)

3 eighteen thousand two hundred

4 sixty-seven thousand eight hundred (and) thirty

5 seven hundred forty-two

6 forty-nine thousand, one hundred (and) thirteen

7 fourteen thousand

8 three million, four hundred fifty thousand

121Bright Sparks Teacher’s Book 6 • Answers

9 20 860 = 20 000 + 800 + 60

10 950 210 = 900 000 + 50 000 + 200 + 10

1.3 Expanded form

1.3 A

1 851

2 9432

3 3024

4 840

5 7060

6 12 825

7 67 900

8 52 000

9 524 999

10 804 625

1.3 B

1 942 = 900 + 40 + 2

2 3579 = 3000 + 500 + 70 + 9

3 1200 = 1000 + 200

4 24 000 = 20 000 + 4000

5 15 500 = 10 000 + 5000 + 500

6 30 068 = 30 000 + 60 + 8

7 156 000 = 100 000 + 503000 + 6000

8 255 000 = 200 000 + 50 000 + 5000

9 862 080 = 800 000 + 60 000 + 2000 + 80

10 943 762 = 900 000 + 40 000 + 3000 + 700 + 60 + 2

1.3 C

1 1234 = (1 × 1000) + (2 × 100) + (3 × 10) + (4 × 1)

2 459 = (4 × 100) + (5 × 10) + (9 × 1)

3 21 600 = (2 × 10 000) + (1 × 1000) + (6 × 100)

4 32 055 = (3 × 10 000) + (2 × 1000) + (5 × 10) + (5 × 1)

5 157 648 = (1 × 100 000) + (5 × 10 000) + (7 × 1000) + (6 × 100) + (4 × 10) + (8 × 1)

1.3 D

1 4380 = 4 × 1000 + 3 × 100 + 8 × 10

2 662 = 6 × 100 + 6 × 10 + 2 × 1

3 45 480 = 4 × 10 000 + 5 × 1000 + 4 × 100 + 8 × 10

4 24 701 = 2 × 10 000 + 4 × 1000 + 7 × 100 + 1 × 1

5 336 009 = 3 × 100 000 + 3 × 10 000 + 6 × 1000 + 9 × 1

1.3 E

1 8542

2 20 460

3 967 000

4 84 002

5 200 907

6 406 040

7 90 300

8 18 506

9 460 212

10 58 050

11 400

12 20 000

13 8000

14 90

15 4000

16 100 000

17 6

18 6684, 22 647, 32 557, 57 739, 773 890

19 790, 7902, 79 002, 709 002, 790 002

20 50 000, 50 609, 50 709, 57 809, 50 909

21 fifty-nine million

22 one hundred forty-nine billion, six million

23 a 9842

b 2489

24 0

1.4 Short cuts in mental Maths

1 10

2 30

3 8

4 6

5 10

6 11

7 10

8 6

9 8

10 12

11 = 64 (8 × 8)

12 = 81 (9 × 9)

13 = 84 (7 × 12)

14 = 112 (14 × 8)

15 = 132 (11 × 12)

16 = 210 (3 × 7 × 10)

17 = 540 (6 × 9 × 10)

18 = 240 (12 × 2 × 10)

19 = 360 (10 × 4 × 9)

20 = 300 (25 × 4 × 3)

Challenge:

Check paragraphs for reasoning.


1.5 Expanded numbers with regrouping

1.5 A

1 10 ones

2 12 ones

3 16 tens

4 18 hundreds

5 2 tens

6 5 hundreds

7 5 thousands

8 0 hundreds

9 8 tens

10 16 thousands

1.5 B

1 8

2 7

3 13

4 89

5 36

6 78

7 17

8 15

9 86

10 28

11 65

12 80

13 800

14 8000

15 28

UNIT 2: Operations – computation2.1 Rounding and estimating

Partner Activity:

1 $4 + $7 = $11

2 $43 + $59 = $102 or $40 + $60 = $100

3 $541 + $633 = $1174 or $540 + $630 = $1170

4 $56 + $72 = $128 or $60 + $70 = $130

5 40 + 60 + 900 = 1000

6 1600 + 6200 + 100 + 2000 = 9900

2.1

1 800 000

2 800 000

3 100 000

4 200 000

5 90 000

6 55 000

7 15 000

19 397 000

20 420 000

21 6100

22 900

23 55 480

24 5500

25 6610

8 630 000

9 480 000

10 640 000

11 5000

12 13 000

13 2000

14 450 000

15 680 000

16 26 000

17 5900

18 64 600

26 21 350

27 67 050

28 537 230

29 $91

30 $1

31 $896

32 $52

33 $780

34 $67

35 $5932

2.2 Adding mentally

2.2 A

1 7

2 26

3 100

4 90

5 90

6 40

7 60

8 300

9 100

10 1000

2.2 B

1 278

2 687

3 686

4 2486

5 335

6 899

7 769

8 2888

9 $359.49

10 $7799.15

11 $2927.95

12 77999

13 919 878

14 799 999

15 400 tickets were sold.

2.3 Addition with regrouping

2.3 A

1 594

2 793

3 1818

4 8157

5 10 693

6 7139

7 6807

8 9161

9 21 909

10 19 901

2.3 B

1 196

2 298

3 494

4 $13.50

5 $70

6 $319.23

7 a $2213.95

b $1900.30


2.3 C

1 912

2 6329

3 10 351

4 21 568

5 267

6 12 074

7 24 379

8 40 100

9 $119.55

10 $527.81

2.3 D

1 $980.92

2 8596

3 85 794

4 $813.95

5 $8948.60

6 68 006

7 $92.55

8 134 985

9 110 443

10 17 177

11 16 009

12 25 647

13 49 959 km

14 No, he only has 24 138 points.

15 We must earn $1168.25 to match what we spent.

2.4 Subtracting mentally

1 2

2 7

3 43

4 4

5 15

6 13

7 24

8 8

9 35

10 39

11 26

12 76

2.5 Regrouping

1 760

2 59 954

3 45 770

4 18 953

5 41 109

6 9611

7 11 568

8 20 089

2.6 Subtracting large numbers

1 34 811

2 8106

3 219 968

4 18 519

5 11 038

6 $27.11

7 $75.20

8 $4587.40

9 a 220 000 000

b It is more.

10 a It increased by 1 580 357 000.

b 20 years

2.7 Subtraction with zeros

2.7 A

1 155

2 392

3 2708

4 7749

5 20 225

6 44 729

7 4943 tickets

8 $12 450

9 $12.35

10 $20 600

2.7 B

1 177

2 105

3 17 944

4 2626

5 109 950

6 41 426

7 911 050

8 $4135

9 11 700

10 1346

2.8 Multiplication: review

2.8 A

The paragraph should compare addition and multiplication, perhaps explaining that we generally use both to find a total, or that multiplication is a form of repeated addition. Students should make reference to the associative and commutative properties that both operations share.

2.8 B

Test 1

1 32

2 30

3 27

4 16

5 10

6 42

7 54

8 48

9 60

10 33

Test 2

1 100

2 28

3 21

4 30

5 64

6 56

7 36

8 12

9 63

10 110


Test 3

1 132

2 80

3 49

4 9

5 48

6 108

7 90

8 12

9 66

10 88

Test 4

1 32

2 88

3 72

4 35

5 60

6 70

7 45

8 24

9 24

10 36

Test 5

1 144

2 18

3 132

4 20

5 15

6 48

7 77

8 40

9 81

10 50

Test 6

1 14

2 16

3 36

4 28

5 45

6 72

7 121

8 24

9 84

10 96

Test 7

1 22

2 108

3 36

4 60

5 54

6 56

7 36

8 99

9 63

10 96

Test 8

1 42

2 84

3 44

4 16

5 77

6 72

7 80

8 84

9 44

10 72

2.9 Division: review

2.9 A

1 9

2 7

3 7

4 5

5 9

6 4

7 3

8 4

9 4

10 12

2.9 B

Test 1

1 3

2 4

3 11

4 10

5 12

6 8

7 8

8 12

9 10

10 12

11 11

12 7

13 10

14 9

15 9

Test 2

1 8

2 1

3 11

4 4

5 9

6 9

7 7

8 5

9 6

10 4

11 10

12 7

13 7

14 12

15 8

Test 3

1 2

2 11

3 4

4 12

5 9

6 6

7 4

8 12

9 5

10 7

11 3

12 10

13 4

14 3

15 2


2.10 Multiplication and division speed tests

Test 1

1 30

2 7

3 24

4 4

5 27

6 2

7 64

8 12

9 100

10 4

Test 2

1 5

2 30

3 7

4 32

5 9

6 25

7 4

8 24

9 9

10 132

Test 3

1 45

2 36

3 7

4 8

5 9

6 20

7 33

8 8

9 49

10 12

Test 4

1 8

2 28

3 6

4 4

5 8

6 9

7 11

8 12

9 6

10 11

Test 5

1 6

2 110

3 33

4 40 km

5 12

6 8

7 3

8 24

9 24 km

10 104 beads

Units 1 and 2 check and summaryCheck

1 C

2 B

3 D

4 A

5 D

6 D

7 C

8 B

9 B

10 D

Assessment 1Part 1

1 a 9

b 600

c 8

d 10 000

e 36

f 12

g 23

h 76

i 10 000

j 100

2 a hundreds

b thousands

c tens

d hundred thousands

3 a 6000

b 800

c 900 000

d 500

4 a 11 309

b 200 416

c 9003

d 720 602

5 a six thousand, five hundred seventy

b twenty-four thousand, one hundred

c six hundred eighty-seven thousand, three hundred (and) two

d four hundred thousand, seven hundred and eleven

6 a 1050, 1500, 1501, 5100

b 4802, 48 002, 48 020, 48 022

c 804, 8004, 80 400, 800 040

d 30 395, 300 395, 303 952, 309 352


Part 2

1 2

2 2457

3 a 10

b 10

c 6

d 4

e 60

f 500

4 a $57 b $683 c $3091

5 a 85 000 b 903 000 c 17 000

6 a 49 500 b 904 100 c 100 400

7 $416.93

8 27 810

9 $8104

10 1126

11 25 773

12 $1100.26

13 a 249 sticks b 90 cm

14 2532

15 a Approximately 11 900 or 12 000

b Approximately $104 or $100

c Approximately $2400

UNIT 3: Number concepts / sequences3.1 Square and cubed numbers

3.1 A

1 Check that students have drawn arrays set out as squares with 5 rows of 5 columns, 6 rows and 6 columns and 9 rows and 9 columns.

2 2 × 2 = 4 6 × 6 = 36 10 × 10 = 100

3 × 3 = 9 7 × 7 = 49 11 × 11 = 121

4 × 4 = 16 8 × 8 = 64 2 × 12 = 144

5 × 5 = 25 9 × 9 = 81

3 Have students check partners orally.

3.1 B

1 1

2 8

3 27

4 64

5 125

6 216

7 343

8 512

9 729

10 1000

3.2 Exponents

1 34

2 53

3 62

4 25

5 103

6 85

7 93

8 112

9 44

10 75

3.3 Patterns

3.3 A

1 678, 789

2 39, 37

3 60, 72

4 10, 5

5 65, 71

6 48, 96

7 1 (or 44), 11

4 (or 54)

8 4, 8

9 4 × 4, 5 × 5

10 60¢, 65¢

11 60, 48

12 123, 2

13 100, 1000

14 80, 160

15 35, 41

16 90, 85

17 25, 36

18 66, 77

19 121, 144

20 f6, g7

21 rectangle with two rows of three boxes

22 three sets of double notes

23 three triangles pointing right


24 shape matching the first in the series (arrows face W and N)

25 one big green dot

3.3 B

1 $2.50

2 5, 0

3 16

4 7

5 23

6 24

7 36

8 55

9 10

10 4.0

3.3 C

1 72

2 64

3 100

4 $3.95

5 $5.00

6 288

7 12.5 or 1212

8 a + b + c + d

9 18

10 12 (4

8 reduced. The other fraction in the pattern was reduced, so this answer is also expected to be reduced.)

3.4 Order and value

1 4578

2 8754

3 8754

4 4587

5 4578

6 8745

3.5 Numbers: review

1 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

2 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

3 27 ÷ 3 = 9, so the numbers are 8, 9 and 10

4 50 ÷ 5 = 10, so the numbers are 8, 9, 10, 11, 12

5 45 ÷ 3 = 9, the odd number before is 7 and the odd number after 9 is 11: answer: 7, 9, 11

3.6 Factors

3.6 A

1 10 = {1, 2, 5 , 10}

2 16 = {1 , 2, 4, 8, 16 }

3 20 = {1, 2, 4 , 5 , 10, 20 }

4 18 = {1, 2, 3, 6, 9, 18}

5 28 = {1, 2, 4, 7, 14, 28}

6 36 = {1, 2, 3, 4, 6, 9, 12, 18, 36}

7 64 = {1, 2, 4, 8, 16, 32, 64}

8 50 = {1, 2, 5, 10, 25, 50}

9 48 = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}

10 60 = {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}

11 12 = {1, 2, 3, 4, 6, 12}

12 120 = {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120} (a common error is to leave out 24)

13 72 = {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}

14 90 = {1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90}

15 42 = {1, 2, 3, 6, 7, 14, 21, 42}

16 15 = {1, 3, 5, 15}

17 25 = {1, 5, 25}

18 40 = {1, 2, 4, 5, 8, 10, 20, 40}

19 54 = {1, 2, 3, 6, 9, 18, 27, 54}

20 63 = {1, 3, 7, 9, 21, 63}

3.6 B

Common factors are in italics:

1 16 = {1, 2, 4, 8, 16}

20 = {1, 2, 4, 5, 10, 20}

HCF/GCF = 4

2 24 = {1, 2, 3, 4, 6, 8, 12, 24}

40 = {1, 2, 4, 5, 8, 10, 20, 40}

HCF/GCF = 8

3 45 = {1, 3, 5, 9, 15, 45}

27 = {1, 3, 9, 27}

HCF/GCF = 9

4 75 = {1, 3, 5, 15, 25, 75}

30 = {1, 2, 3, 5, 6, 10, 15, 30}

HCF/GCF = 15

5 36 = {1, 2, 3, 4, 6, 9, 12, 36}

54 = {1, 2, 3, 6, 9, 18, 27, 54}

HCF/GCF = 9

Discuss:

120 = {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120}


96 = {1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96}

24 is the HCF/GCF of 96 and 120, so 12 cm squares will fit with no overlaps.

3.7 Multiples

3.7 A

1 11, 22, 33, 44, 55

2 9, 18, 27, 36

3 3, 6, 9, 12, 15, 18, 21, 24, 27

4 2, 4, 6, 8, 10, 12, 14, 16, 18

5 4, 8, 12, 16, 20, 24, 28

3.7 B

Common multiples are bold:

1 3, 6, 9, 12, 15, 18, 21, 24

7, 14, 21, 28, 35, 42, 48, 54

2 5, 10, 15, 20, 25

4, 8, 12, 16, 20

3 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

3, 6, 9, 12, 15, 18, 21, 24, 27, 30

4 10, 20, 30, 40

8, 16, 24, 32, 40

In 40 minutes they will both complete a pot at the same time.

5 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30

5, 10, 15, 20, 25, 30

3, 6, 9, 12, 15, 18, 21, 24, 27, 30

The LCM of 2, 3, and 5 is 30.

The smallest possible class size is 30 students.

3.8 Divisibility

3.8 A

(Divisibility questions imply no remainder and just whole numbers.)

1 17, 23, 5

2 24, 10, 28, 42, 18, 36

3 24, 42, 18, 39, 36

4 18, 36

5 10, 5

6 24, 42, 18, 36

3.8 B

1 68 = {1, 2, 4, 17, 34, 68)

2 56 = {1, 2, 4, 14, 28, 56}

3 24 = {1, 2, 3, 4, 6, 8, 12, 24}

36 = {1, 2, 3, 4, 6, 9, 12, 36}

4 16 = {1, 2, 4, 8, 16}

40 = {1, 2, 4, 5, 8, 10, 20, 40}

5 35 = {1, 5, 7, 35}

28 = {1, 2, 4, 7, 14, 28}

HCF/GCF = 7

6 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

7 LCM = 40

8 LCM = 120

9 LCM = 30

10 LCM = 204

Challenge:

LCM of 5, 6 and 7 = 210

The least number of biscuits at the beginning was 210.

3.9 Prime factors

1 24 = 2 × 2 × 2 × 3

2 45 = 3 × 3 × 5

3 40 = 2 × 2 × 2 × 5

4 30 = 2 × 3 × 5

5 60 = 3 × 2 × 2 × 5

6 50 = 2 × 5 × 5 (or 2 × 52)

7 81 = 3 × 3 × 3 × 3 (or 34)

8 64 = 2 × 2 × 2 × 2 × 2 × 2 (or 26)

9 72 = 2 × 2 × 2 × 3 × 3 (or 23 × 32)

10 84 = 2 × 2 × 3 × 7 (or 22 × 3 × 7)

Questions 11–12: the order of the factors may vary, as may the initial factors chosen for 12 (e.g. 3 × 4). The final answer should match as shown, with prime factors listed in order.

11 96

12 × 8

2 × 6 × 2 × 4

2 × 2 × 3 × 2 × 2 × 2

Answer 96 = 2 × 2 × 2 × 2 × 2 × 3


12 210

10 × 21

2 × 5 × 3 × 7

Answer 210 = 2 × 3 × 5 × 7

UNIT 3 check and summaryCheck

1 D

2 B

3 A

4 D

5 A

6 B

7 D

8 C

9 A

10 D

Unit 4: Decimals4.1 Place value of whole and decimal numbers

1 2

2 1

3 5

4 7

5 thousandths

6 seven tenths, 0.7

7 eighteen hundredths, 0.18

8 five hundredths, 0.05

9 three hundred twenty-six thousandths, 0.326

10 twenty-six thousandths, 0.026

11 310

12 76100

13 141000

14 7100

15 291000

4.2 Value

4.2 A

1 8100

2 700

3 50

4 41000

5 110

4.2 B

1 810

2 10

3 3100

4 400

5 7

6 2000

7 500

8 41000

9 810

10 30

11 hundreds

12 tenths

13 hundreds

14 hundredths

15 tenths

16 thousands

17 tens

18 hundreds

19 tenths

20 thousandths

21 six tenths

22 twenty-one, and five tenths

23 one hundred (and) thirty-five, and eight hundredths

24 sixty-six, and sixty-six hundredths

25 one hundred (and) twenty, and seven thousandths

26 31

27 8 110 or 8.1

28 219 310 or 219.3

29 7260 410 or 7260.4

30 89 1881000 or 89.118

4.3 Comparing decimals

1 240.7 > 204.7

2 1200 > 120.4

3 12.65 < 120.65

4 13.06 = 13.060

5 6008.4 > 6008.14

6 20.55, 21.5, 210.5

7 36.67, 360.06, 360.6

8 1200.292, 1208.2, 1257.25

9 16.05, 16.058, 16.5, 16.57


10 20.47, 200.7, 204.7, 240.7

11 0.6, 0.06, 0.006

12 6340.5, 634,89, 634.809

13 $550, $500.75, $59.50, $50.95

14 170.7, 17.7, 17.07, 17.007

15 Kerry 7.03 sec, John-Paul 7.08 sec, Akil 7.4 sec, Kamal 7.54 sec

4.4 Rounding decimal numbers

1 891

2 253

3 1267

4 85 734

5 36.0

6 125.8

7 260.5

8 60.0

9 $64

10 $71

11 $127

12 $590

4.5 Problem solving

1 1.8 m

2 $26 + $39 = $65

3 Yes, rounded up $27 + $2 = $29

4 $487 + $13 = $500; I need to save $500.

5 He earned $2060 per month.

6 a 63 sec

b He was faster at the Commonwealth Games.

7 It is about 20¢ per pencil. (or $0.20)

8 a $11

b $10.55

9 2041 km

10 It is about 19 (or 20) points.

Challenge:

I have less than $30, because in both cases the prices were rounded down ($84 + $86).

4.6 Adding and subtracting decimal numbers

4.6 A

1 4408.1

2 565.51

3 1800.5

4 3571.9

5 1948.25

6 3450.55

7 1815

8 27 305.4

9 4980.672

10 2595.79

4.6 B

1 283.56

2 298.87

3 421.14

4 3052.16

5 0.183

6 0.014

7 16.705

8 535.07

9 5859.156

10 4666.52

11 $157.57

12 $15.95

13 $65.49

14 $148.59

15 $1863.01

4.6 C

1 87.1 sec

2 171.98 sec

3 1.2 seconds faster

4 1.9 seconds faster

5 The first leg was the slowest.

4.7 Problem solving with decimals

1 $262.50

2 $512.25

3 $2173.05

4 $55.64

5 $76.95

6 152.98 m

7 $50

8 $9.06

Assessment 2Part 1

1 a 9

b 27

c 10

2 8

3 169

4 a 34

b 26

c 52

d 124

5 a h15

b 78

c 32

d 36

e 15

f 14

g 0.01


h 49

i 64

j (regular hexagon)

k (arrow points left)

6 a 1. 2. 3

b 0. 1, 2

7 a 8631

b 1368

c 8631

d 8316

8 a 8, 9, 10

b 15, 17, 19

c 8, 10, 12, 14, 16

9 a 1, 2, 13, 26

b 1, 2, 4, 5, 10, 20, 25, 50, 100

10 a 1, 3, 9

b 12

c 5

Part 2

1 a 12, 24, 36, 48, 60

b 20

2 24 potatoes

3 a 2, 3, 5, 7, 11

b 23 × 5 (or 2 × 2 × 2 × 5)

c 22 × 3 × 5 (or 2 × 2 × 3 × 5)

4 a 7

b 4

c ten thousands

d hundredths

e 3

f 3000

g 50

h 100 000

5 a eight hundredths

b five thousand sixteen and twenty-five hundredths

6 610 or 0.6

7 a 815.32

b 9512.002

8 408.21

9 a 329.08 < 3290.08

b 1600.1 = 1600.10

c 2119.5 < 21 195.5

10 a 253.7, 302.05, 302.15, 648.3

b 6004.92, 649.32, 649.23, 604.9

11 a $241

b $1287

12 a 3596.5

b 3996.7

13 $22 + $125 = $147 (or $20 + $125 = $145)

14 Yes

15 621.57

16 1331.29

17 10 048.73

18 82 057.31

UNIT 5: Operations – multiplication5.1 Multiplication: review

1 1630

2 2682

3 18 328

4 21 654

5 10 416

6 37 112

7 23 709

8 102 496

9 231 273

10 389 648

11 There were 456 people altogether.

12 Yes

13 There were 312 hot dog sausages altogether.

14 $2000

15 The hike was 3.04 km long.

5.2 Multiplication by 2-digit numbers

1 1782

2 4725

3 $97.32

4 85 260

5 83 646

6 106 290

7 $541.50

8 393 624

9 77 130

10 1 834 724

11 22 176

12 618 478

13 4 377 105

14 $998.20

15 $22 783.25

16 522

17 180 chairs

18 $5143.20


5.3 Multiplication by 3-digit numbers

5.3 A

1 46 464

2 207 684

3 63 525

4 277 986

5 193 024

6 159 140

7 258 408

8 264 519

9 340 336

10 549 978

11 1 659 600

12 1 032 837

5.3 B

1 $75.04

2 131 652

3 174 846

4 113 316

5 159 478

6 933 225

7 127 260

8 82 550

9 3 108 420

10 $1926.21

11 381 276

12 1 286 358

13 a 4900 cones were sold.

b $1575

14 a $14 400

b $1840

c Yes

15 a 3750 m (Note: each lap is the distance one time, not to and from)

b $637.50

c $483.30

5.4 Multiplying multiples of 10

1 24 000

2 600 000

3 2 000 000

4 24 000

5 300 000

6 540 000

7 490 000

8 720 000

9 125 000

10 a 15 000 copies

b $1200

c $980

d $840

Challenge:

a 5 400 000 b 68 000

5.5 Using the distributive property to solve mentally

1 5 × (6 + 3) = (5 × 6) + (5 × 3) = 45

2 3 × 37 = (3 × 30) + (3 × 7) = 111

3 8 × 51 = (50 × 8) + (1 × 8) = 408

4 15 × 16 = 15 × (10 + 6)

= (15 × 10) + (15 × 6)

= 150 + 90

= 240

5 56 × 14 = 56 × (10 + 4)

= (56 × 10) + (56 × 4)

= 560 + (50 × 4) + (6 × 4)

= 560 + 200 + 24

= 784

6 95 × 18 = (90 × 18) + (5 × 18)

= (90 × 10) + (90 × 8) + (5 × 10) + (5 × 8)

= 900 + 720 + 50 + 40

= 1710

7 76 × 48 = (76 × 40) + (76 × 8) = 3648

8 527 × 34 = (500 × 34) + (20 × 34) + (7 × 34) = 17918

9 91 × 972 = (91 × 900) + (91 × 70) + (91 × 2) = 88452

10 8 × 505 = (8 × 500) + (8 × 5) = 4040

Challenge:

Check students’ reasoning.

5.6 Working with multiplication

1 10 166

2 10 166

3 221

4 46

5 8840

6 1326

7 20 332 (double factor, double product)

8 20 332 (double other factor, double product)


9 23 (quotient divided by doubled divisor)

10 110.5 or 11012 (quotient divided by

doubled divisor)

11 243 (ten times less)

12 24 300 (ten times more)

13 1215 (half the factor, half the product)

14 27

15 54 (half the divisor, half the quotient)

Challenge:

221 The dividend is half the product of the first worked problem, and the divisor is half of one of the factors. Therefore, the result matches the other factor.

Units 4 and 5 check Check

1 D

2 B

3 C

4 D

5 D

6 C

7 B

8 D

9 A

10 A

UNIT 6: Algebra6.1 Overview

6.1 A

1 8

2 5

3 7

4 7

5 12

6.1 B

1 15

2 8

3 24

4 50

5 36

Challenge:

m ÷ 8 + 2 = 42

m ÷ 8 + 2 – 2 = 42 – 2

m ÷ 8 = 40

m = 5

6.2 Order of operations

1 80 – 20 = 60

2 4 + 6 = 10

3 45 – 3 = 42

4 12 + 12 = 24

5 2 × 6 = 12

6 2 + 12 = 14

7 24 – 12 = 12

8 200 × 40 = 8000

9 10 + 4 = 14

10 5 + 25 = 30

11 False, because you do what is inside the brackets first.

12 False, 36 ÷ 9 is 4, and 3 × 7 is 21, so it would be 4 – 21, which is not 7.

13 False, you do 5 ÷ 5 first, so it is 25 + 1, which is not 6.

14 False, 90 – 5 + 5 is not 30.

15 True, 124 is 3, add 12 is 15.

6.3 Working with operations

1 ÷ , ×

2 – , ×

3 × , ×

4 + , +

5 – , +

6 × ,

7 – , × , ×

8 + , ×

9 × , ×

10 + , + , + , ×

6.4 Substitution

6.4 A

1 7

2 8

3 12

4 19

5 15

6 11

7 22

8 5

9 12

10 17


6.4 B

1 11

2 30

3 36

4 1

5 30

6 11

7 70

8 0

9 50

10 4

6.4 C

1 n = 3

2 p = 4

3 m = 16

4 a = 6

5 b = 2

6 t = 0

7 k = 8

8 y = 9

9 a = 3

10 q = 6

11 5x

12 y – 16

13 6(a – 2)

14 (60 ÷ 6) + b or b + (60 ÷ 6)

15 w + 19

16 x – 12 = 50

17 20 – y

18 x ÷ 8

19 (y – 1) ÷ 2

20 3 (2n) Note: you may wish to accept 3(n + n). Other letters may be used for the variable.

Challenge:

a 2(31 + 1) = n; their grandmother is 64 years old.

b 2 × 25 – 10 = n; Mr Browne has 40 goats.

6.5 Mental Mathematics practice

1 n + 15 = 100 n = $85

2 n – 3 = 1 n = 14 years old

3 (17 – 5) – 2 = n n = 10 years old (may omit brackets)

4 10 × 6 = 12n n = 5 chairs

5 (64 ÷ 2) ÷ 8 + 3 = n n = 7

6 √2(52 × 2) n = 10

7 12

8 384

9 halve

10 n ÷ 3 = 20 × 10 n = 600

UNIT 7: Operations – division 7.1 Division: review

1 3, 5, 9

2 5

3 none

4 2, 5

5 3

6 2, 3, 5, 6

7 3

8 3, 9

9 2, 3, 6

10 3, 5, 9

11 a 96 ÷ 3 = 32

96 ÷ 32 = 3

3 × 32 = 96

32 × 3 = 96

b 180 ÷ 30 = 6

180 ÷ 6 = 30

30 × 6 = 180

6 × 30 = 180

c 272 ÷ 16 = 17

272 ÷ 17 = 16

16 × 17 = 272

17 × 16 = 272

12 2

13 9

14 13

15 7

16 41

17 14

18 110

19 7

20 28

21 56

22 a 200

b 5

c 4


7.2 Long division

In this exercise, accept remainders, or answers with the remainders converted to decimals or fractions.

1 79 r 3, 79.6 or 79 35

2 89

3 76 r 1, 76.11 or 76 19

4 22

5 61

6 280

7 22

8 16 r 3, 16.2 or 1615

9 83 r 1, 83.0833 or 83 112

10 56 r 5, 56.125 or 5618

11 19 r 3, 19.6 or 1935

12 90

13 35 r 4, 35.5 or 3512

14 125 r 5, 125.55 or 12559

15 47 r 4, 47.22 or 4729

16 306 r 8, 306.66 or 30623

17 a About 54 of each group

b 163 butterflies

7.3 Dividing money

1 $6.02

2 $27.95

3 $36.50

4 $4.40

5 $10.04

6 $0.60

7.4 Working with remainders

7.4 A

1 12 weeks. Similar to example 1.

2 134 bottles. Similar to example 1.

3 They need 40 more cherries. Similar to example 3.

4 He spent about 8 hours a day. Similar to example 1.

5 Each hiker got 2.5 ℓ. Similar to example 2.

6 a 4 shelves are needed. Similar to example 1.

b They need 7 more books. Similar to example 3.

7 She could complete 3 arrangements. Similar to example 1, but the remainder is dropped.

8 They need 14 more litres (7 more 2 ℓ bottles). Similar to example 3.

7.4 B

1 1595

2 801

3 540

4 886 r 1

5 11

6 18

7 170

8 21 r 1

9 35

10 $160.25

11 63

12 58

13 4108

14 1375

15 8123

7.4 C

1 $2.40

2 13

3 $0.13

4 13

5 55 r 1, 55.025 or 55 140

6 26 r 20, 26.4 or 26 25

7 502

8 $0.42

9 a 4 shows

b 15 shows

c $2.50

10 38

11 $0.38 or 38¢

12 $2.10

13 $0.92 or 92¢

14 Yes

Challenge:

187.5 g were needed per sheet. (Note: the question specifies kg and then asks for the answer in grams.)


7.5 Division with 3-digit divisors

1 211

2 303

3 406

4 321

5 56 r 3

6 804 r 4

7 403 r 5

8 210 r 50

9 250 r 20

10 178

11 26 bands

12 60 hours

13 $25


1 B

2 A

3 C

4 C

5 C

6 A

7 D

8 B

9 C

10 D

Assessment 3 Part 1

1 $3438

2 $497 000

3 $75 600

4 $27 000 ($13 500 each day for two days)

5 a 26 208

b 19 360

c $46 250

d $4426.80

e 1 187 106

f 324 120

g 1 200 000

h 3 000 000

i $1225

j $312.50

6 a 50

b 40

c 700

d 10

e 5

f 4

7 a 421

b 15 156

c 12 630

d 2526

e 36

f 72

g 7578

h 30 312

i 151 560

j 151 560

Part 2

1 a 3

b 17

c 25

d 20

e 14

f 8

g 100

h 120

i 21

j 25

2 a 4

b 2

3 a ×, –

b – , ×

c ÷ , ÷

d ×, –

e ×, –

4 a 4

b 5

c 18

d 50

e 5


5 9 weeks

6 a 7 arrangements

b 8 flowers

7 $194.57

8 28 bookmarks

9 a 604

b $75

c 180

10 10 days

UNIT 8: Fractions8.1 Fractions: review

1 815

2 1

3 1625

4 12

5 13

6 298

7 416

8 34

9 57

10 59

11 >

12 <

13 =

14 158

15 3 124

16 313

17 8 124

18 320

19 732

20 6

21 27

22 12

23 12

24 16

25 18

8.2 Estimating and rounding fractions

1 0

2 1

3 12

4 0

5 12

6 2

7 1

8 2

9 4 or 4 12

10 2

11 7

12 10

13 yes

14 Less. Example explanation: The two 12 ℓ

and the 2 ℓ make 3 ℓ. All that is left is 34 ℓ, so the total is less than 4 ℓ.

15 Approximately 2

8.3 Simple fractions: review

For questions 1–10, the denominators may be stated as a multiple of 10, without simplifying the fraction.

1 210

2 510

3 42100

4 51100

5 3661000

6 33100

7 4621000

8 69100

9 1281000

10 2100

11 0.7

12 0.3

13 0.26

14 0.75

15 0.25

16 0.12

17 0.5

18 0.095

19 0.09

20 0.004

21 1 (or 44)

22 79

23 35

24 710

25 1115

26 1516

27 1112

28 78

29 1315

30 912 or 3

4

31 HCF/GCF = 2

32 HCF/GCF = 2

33 HCF/GCF = 4

34 HCF/GCF = 2

35 HCF/GCF = 4

8.4 Equivalent fractions

1 5

2 14

3 45

4 8

5 36

6 45

7 8

8 4

9 2

10 45

11 4 hours

12 9 apples

13 25 children

14 5 hours

15 68 = 3

4


8.5 Simplifying equivalent fractions

1 8

2 2

3 1

4 3

5 40

6 1

7 45

8 12

9 23

10 13

11 19

12 711

13 15

14 13

15 45

16 89

17 a 3 pizzas

b 3 pizzas (some would be left, but 2 pizzas would not be sufficient)

c 8 pizzas

8.6 Improper fractions and mixed numbers: review

1 118

2 212

3 147

4 215

5 212

6 53

7 195

8 234

9 525

10 638

11 =

12 <

13 =

14 a 258

b 3 18

8.7 Addition of fractions: review

1 23

2 1315

3 1415

4 11118

5 1 124

6 2940

7 1 740

8 1 320

9 1318

10 11360

8.8 Subtraction of fractions: review

1 14

2 1330

3 29

4 16

5 815

6 120

7 1120

8 112

9 124

10 760

11 112 km

12 1 512 km

13 16 km

14 a 56

b 16

8.9 Adding and subtracting mixed numbers

1 518

2 212

3 678

4 11 720

5 3 712

6 112

7 913

8 15 415

9 1179

10 523

8.10 Picture adding and subtracting mixed numbers

8.10 A

1 12

2 158

3 714

4 916

5 815

6 1223

7 234 – 11

2 = 114

8 412 + 11

2 = 6

9 212 – 1 = 11

2

10 1 – 12 = 1

2

8.10 B

1 514 hours

2 Less. Mr Lee said ‘over’ 13 of the money

was spent on wages.

3 3 hours

4 318 watermelons

5 a 3 cm

b 2 cm


6 a 112 kg

b 312 kg

7 a 110 kg × 2 × 5 for each kitten = 2 kg

altogether in five days

b 712 ℓ

Challenge:

a Sample solution: 34 doubled is 11

2 , 11

2 doubled is 3. 3 divides 9 evenly, and there are four 3

4s in each 3, with the result of 12 sets of 3

4 in 9.

b 45 of a carton yields 3 glasses. 4 full cartons = 4 × 5

5, or 205 in all. There

are 5 sets of 45 in 20

5 so there are 5 sets of 3 glasses of milk, or 15 glasses of milk in all.

8.11 Multiplying fractions: review

1 58

2 1427

3 564

4 77120

5 16 of the box

Challenge:34 were given out, which leaves 1

4 of the box. 1

2 of the 14 remaining in the box were

put on a shelf, and 8 were kept out. These 8 pencils represent 1

8 of the box. There were 64 pencils in the box altogether.

8.12 Multiplying fractions with cancelling

8.12 A

1 3940

2 821

3 215

4 2780

5 736

6 754

8.12 B

1 319

2 12

3 18

4 16

5 34

6 37

7 12

8 35

9 124

10 215

11 112

12 175

8.13 Multiplying mixed numbers

1 334

2 214

3 112

4 1

5 635

6 412

7 1379

8 5

9 5

10 62936

8.14 Word problems

1 a $3

b 412 kg = $3, 3 kg = $2, and

112 kg = $1, so $6 altogether.

2 a $8 × 214 = 18, $5 × 2 = $10. He

paid $28 altogether.

b $134

3 a 3 bags for $2 plus 2 bags at $0.75, 5 bags total $3.50 and weigh 2.5 kg. She paid $1.40 per kilogram overall.

b 12 kg = $0.75, so they paid $1.50 per kg.

4 a 111324 kg

b $51

c $21

d $9.50

5 58 bags (4 bags for every kg, and 2 bags for the 1

2 kg)

Challenge:

a To the nearest cent, $2.72

b 38 of 64 = 24 ears of corn

8.15 Problem solving with ‘of’

1 4 sea fans

2 35 fish

3 15 fish

4 3 broken sand dollars

5 515

6 99 reef fish

7 16 shells

8 223 hours


8.16 Division of fractions: review

8.16 A

1 4

2 2

3 310

4 1 325

5 14

6 9

7 25

8 10

9 223

10 110

11 2

12 6

13 5

14 8 glasses

15 10 buoys

Challenge:

Sample answer: The answers in questions 11–13 answer the question if it is turned around to ask 1

3 of 6, 34 of 8, or 1

2 of 10.

8.16 B

1 223

2 135

3 38

4 83 = 22

3

5 23

6 35

7 20

8 14

9 112

10 1

11 12 kg

12 144 pieces

13 26 bags

14 a 4 strips

b 223 m

15 34 × 12 = 9 or 12 ÷ 3

4 = 9

16 114 kg

17 312

minutes

18 71932

19 123

20 712 mins

8.17 Problem solving

1 $15 500

2 16

3 30 seconds

4 512

5 138 m

6 3112 m

7 13 container

8 20 pieces

9 15 tables

10 214 kg

Unit 8 check and summary Check

1 A

2 A

3 D

4 A

5 C

6 B

7 A

8 C

9 B

10 D

UNIT 9: Working with decimals9.1 Multiplication: review

9.1 A

1 21

2 25.2

3 $21

4 61.6

5 $135

6 9

7 17.6

8 12.4

9 0.25

10 0.078

11 $0.48

12 0.03

13 0.018

14 603.4

15 $1483.20

16 449.78

17 0.035

18 0.0054

19 0.696

20 26.68

21 $27

22 $207

23 $240

24 1561.8 m

25 50.56 m

9.1B

1 422.8 kg

2 722.4 kg for both cows

3 1912.5 ℓ

4 $2.34

5 $2430

6 $52.91


7 $130

8 126.4 ℓ

9 18 868 ℓ

10 181.76 kg

9.2 Multiplying

1 772.8

2 140.07

3 0.08

4 97.6

5 $770.40

6 $68.78

7 $11.10

8 $20 801

9 $1831.20

10 1638.3983

11 139.384

12 27.832

13 $1421

14 8288.37

15 $4822.56

16 $79.92

17 $60.80

18 $102.60

19 $991

20 $5724

9.3 Estimates

1 338.4

2 187.5

3 339.75

4 0.819

5 62.5

6 12.25

7 1.1

8 943.76

9 10.2

10 129

11 307

12 4

13 12.5

14 200.1

9.4 Working with estimates

1 72

2 60

3 110

4 225

5 120

6 36

7 24

8 169

9 1200

10 1000

11 about 115 m

12 about 36 ft tall

13 9 weeks

14 $36

15 $18 000

9.5 Problem solving

A range of suggested answers is given for questions 1 and 6b.

1 About 230 – 260 million years ago

2 Some possible answers: The

information we have is approximate, and changes as new information is discovered. The fossil record gives us approximate dates. It is not possible to be exact in collecting information from so long ago. The period when they lived spanned a long period.

3 About 145 million years

4 The time between Stegosaurus and Tyrannosaurus

5 About 176000 times

6 a Eodromaeus was about 3 times larger.

b Roughly 150–155 million years

c It was about ten times heavier.

9.6 Multiplying by multiples of 10

9.6 A

1 284

2 204

3 629.4

4 3125.5

5 763.64

6 89 790

7 63 612

8 70 550

9 280 651

10 3266

9.6 B

1 95 822

2 4380

3 0.4

4 160

5 730

6 75 400

7 327.9

8 3680

9 215 306

10 9847.03

11 $12

12 $148

13 820

14 $0.98 × 1000 × 3 cases = $2940

15 $119 500

9.7 Multiplying money

1 $64.54

2 $73.02

3 $16320

4 $127.68

5 $0.58

6 $505

7 $460.66

8 $10.35

9 $1.19

10 $10.61

11 $13.75

12 $182.33

13 $61.28


14 $41.23 (Note: based on the company charging for increments less than a minute.)

15 $215.33 (Accept $215.37, if the price is rounded before multiplying by 9.)

9.8 Dividing decimal numbers

1 4.29

2 3.12

3 5.02

4 5.54

5 2.63

6 18.7

7 4.83

8 45.8

9 5.01

10 16.4

11 $18.15

12 $4.50

13 $8.25

14 0.25 kg

15 6.24

9.9 Division with remainders

1 2.52

2 3.65

3 21.55

4 1.7

5 0.65

6 0.275

7 6.35

8 2.16

9 $7.50

10 $15.25

9.10 Division written as a fraction

9.10 A

1 0.2

2 0.5

3 0.25

4 0.75

5 0.6

9.10 B

1 a 0.92 kg

b $4.14

2 By the sack costs less

3 It is 10¢ cheaper per banana when sold by the bunch.

4 About 170 coconuts

5 $11.25

6 1.25 ℓ

7 0.248 kg

8 $15

9 (about) 0.125 kg

9.11 Dividing decimals by multiples of 10

1 2.354

2 3.74972

3 0.15047

4 0.038

5 0.075

6 2.749

7 8.469

8 0.0576

9 0.46

10 0.4005

11 0.364467

12 7.7455

13 0.394

14 29.4

15 0.48

16 0.84

17 0.0699

18 0.1504

19 7.5

20 3.86

9.12 Dividing by a decimal number

9.12 A

1 5

2 50

3 365

4 6.7

5 5.42

9.12 B

1 1.1

2 64.5

3 6.2

4 2.45

5 8.5

6 7.4

7 8 km

8 $3.14

9 $5.40

10 Three times

9.12 C

1 0.9 kg

2 0.06 kg

3 $28.50

4 $1149.20

5 $238

6 No

7 30.5 cm

8 34.5 hours

UNIT 9 check and summaryCheck

1 B

2 D

3 A

4 B

5 D

6 C

7 D

8 A

9 D

10 B


Assessment 4Part 1

1 54.6

2 75.578

3 $37 593.75

4 22.383

5 $37.13

6 $53.29

7 $274.55

8 a 400

b 250

c 600

d 1000

9 About $90

10 Less

11 a 378 (378.0)

b 988 (988.0)

c 950.38

d 3919.5

12 a 350

b 3.689

c 1.2673

d 98436.7

e 0.192486

f 845881

g $1.31

h $7.53

13 250 bags

14 $4.50

15 a 5.05

b 15.4

c 25.1

Part 2

1 This week he earned more.

2 $3.15

3 2 students

4 712

5 1125

6 a 12 b 18 c 5

7 a 6 b 18 c 1

3

8 a 12

b 0

c 1

d 12

9 a About 1512

b About 5

10 About 13 glasses

11 a 14

b 53100

c 234

12 a 0.29

b 0.125

c 2.7

13 HCF/GCF = 18

14 a 916

b 435

15 a 1415

b 112

c 1138

d 3 112

e 32330

f 31720

16 180

17 2 hours

18 a 240 slices

b 40 slices

c 13 of 5kg = 12

3 kg

d 175 g ( 740 kg)

e 5 cups

Challenge:

a Elisa is 10 years old

b 1, 3, 4, 6, 7, 8


UNIT 10: Percents, fractions and decimals10.1 Percents

1 42%

2 36%

3 12%

4 1%

5 84%

6 52%

7 76%

8 29%

9 40%

10 16%

11 74%

12 13%

13 28%

14 90%

15 40%

16 30%

17 8%

18 91%

19 20%

20 22%

21 0.34

22 0.47

23 0.16

24 0.09

25 0.61

26 0.5 (accept 0.50)

27 0.07

28 0.11

29 0.9 (accept 0.90)

30 0.2 (accept 0.20)

10.2 Percents and decimal practice

1 a 0.5 (0.50)

b 0.05

c Answers will vary

2 a 500

b 100

3 $10

4 $6.20

5 $6

6 $12

7 $48

8 $1.35

9 $0.81

10 $0.90

11 $65.62

12 $268.80

13 a 25%

b $51.15

c $17.05

14 a 55%

b $101.25

10.3 Percents, fractions and decimals: review

1 0.35 = 35%

2 0.80 = 80%

3 0.75 = 75%

4 0.25 = 25%

5 0.17 = 17%

6 0.60 = 60%

7 0.04 = 4%

8 0.32 = 32%

9 0.70 = 70%

10 0.73 = 73%

11 0.89

12 1.62

13 425%

14 112

15 150%

16 2%

17 44%

18 36 out of 50

19 61%

20 85%

10.4 Changing fractions to percents

1 65%

2 50%

3 40%

4 80%

5 6212%

6 76%

7 44%

8 40%

9 80%

10 85%

11 3712%

12 25%

10.5 Percent, decimal and fraction equivalents

Fraction Decimal Percent

1 0.5 50%

2 14 25%

3 34 0.75

4 0.2 20%

5 25 40%

6 35 0.6


7 0.8 80%

8 110 10%

9 310 0.3

10 0.7 70%

11 910 90%

12 18 0.125

13 0.375 3712%

14 58 621

2%

15 78 0.875

16 0.3_ 331

3%

17 23 662

3%

18 1 1 (or 1.00)

Challenge:

a 3412%

b 214 , 2.25

c 4 320, 415%

10.6 Working with percents

10.6 A

1 $12.24

2 $18

3 $4.50

4 56%

5 30 words

6 a 3712%

b 12 boys

7 $504 (60% of original price)

8 $12

9 $47.55

10 15 cupcakes

10.6 B

1 860

2 69

3 5.7

4 $8

5 102

6 11

7 45

8 12

9 a 3 students

b Mentally: 25% is 14, 1

4 of 48 is 12, three times that to represent 75%. These students got 36 words right.

10.6 C

1 28

2 440

3 720

4 904

5 $3.60

6 $4

7 250

8 11

9 $300

10 450

11 4.5

12 $24

13 a 110% of $84 000 = $92 400

b $100 800

14 160 students are girls.

10.7 Forming percents

Challenge:

a 80% increase

b 20% increase

1 6623%

2 64%

3 36%

4 92%

5 45%

6 a 90%

b 95%

c 26

7 80%

8 80%

9 a 12 students do not walk to school.

b 25%

10 70%

10.8 Finding the total

1 60 questions

2 200

3 $30

4 80

5 64 medals

6 48 marbles

7 40 cars

8 a 18

b 96 books

9 150

10 56


11 $21

12 $426

10.9 Discount

1 $48

2 $40.50

3 $42.45 total – $10 = $32.45 for discount off total price. Also accept $10 off each pen’s marked price, $9.95 + $12.50 = $22.45

4 Yes

5 $25.31

6 $13.95

7 She would save more with the $10 off discount.

8 $34.65

9 $48

10 $16

10.10 Tax and VAT

1 $5

$55

2 $12

$92

3 $1.75

$11.75

4 $6.30

$48.30

5 $17.28

$113.28

6 $1.80

$31.80

7 $13

$213

8 $3.84

$51.84

9 $2.08

$34.08

10 $2.79

$64.79

Challenge:

a 1078

b 500

10.11 Working out solutions mentally

Discuss:

d Solutions should not be standard calculation algorithms, but should be different mental strategies that students understand.

For 412%, for example, you could easily

find 10%, and half of that is 5%. Find 1%, take half of it, and subtract it from the 5% total. For 171

2%, they

could find 35%, and take half of that total. For 19%, they could find 20%, then find 1% and subtract.

10.11

1 $50.60

$41.40

$73.60

2 Shirt $21.20, sandals $15.90, hat $10.60, dress $47.70

Challenge:

$41.25


1 a $28

b $4

2 84%

3 $154

4 $68

5 $153 600

6 36 are not black belly sheep

7 28 words

8 25%

9 80%

10 18% discount

11 48 questions

12 175

13 $74.75

14 $74.52

10.13 Comparing percents using circle graphs

1 30%, 36%

2 $43 750 Jordan’s, $16 400 Joseph’s

3 Joseph’s

4 Joseph’s

5 One family may have a mortgage, rent may be higher, house sizes may be different, neighbourhood housing costs vary, age of the house, etc.

6 a 5%

b $6250

c $4100


d The percentages are based on different total incomes.

7 Food

8 a $9020

b $12 500

c The type of transportation could be different, e.g. owning a car may have car payments, insurance, gas and maintenance, while public transportation requires fares paid for each family member each time.

UNIT 11: Measures of central tendency11.1 Overview

1 $166

2 12.5

3 12 kg

4 78.4%

5 81.2%

6 $705

7 $7.28

11.2 Working with means

1 8.54

34.3

8.6

9.4

2 Tony

3 a 84

b 18.2

4 2


1 a 90%

b 3313%

c turtles and crocodiles/alligators

d fish and lizards

e frogs and dinosaurs

2 a 168 runs

b 26

3 4.4

4 $7

Challenge:

639


1 A

2 C

3 B

4 A

5 B

6 D

7 B

8 D

9 D

10 C

Assessment 5 Part 1

1 12

2 1564

3 115

4 1

5 2

6 1516

7 157

8 4

9 715

10 35

11 38 of a pizza

12 178 hour

13 36

14 a 28%

b 96%

c 74%

d 125%

15 32

16 16

17 10

18 81

19 78

20 1

21 a 0.22

b 0.37

c 1.56

d 0.98

22 a 0.40 × $5.50 = $2.20

b 0.24 × $18.25 = $4.38

23 a 44%

b $27.16


24 a 0.75 = 75%

b 0.12 = 12%

c 310 = 30%

d 18 = 0.125 = 121

2%

25 a 70%

b 25%

c 75%

d 150%

26 a 35

b 114

c 2150

d 23

27 a 15

b $20

c $128

d $7.50

e 18

f 7

g 10.5 or 1012

h 42

28 c

Part 2

1 $1.40

2 60%

3 56%

4 28 students

5 40%

6 55

7 56

8 20%

9 $150

10 120

11 15%

12 a Yes

b $21.30

c $59 – $44.25 = $14.75

He saved $14.75.

d No

13 $59.40

14 $18.40 per metre

15 $517.00

16 $167.90

17 a $17

b $14.50

18 $93.75

19 a $16.82

b $21.58

c $19.54

20 58

21 $110

22 a $11

b $10.40

c $10

UNIT 12: Measurement12.1 The metric system

Discuss:

a Some reasons for an international system might include: so all people could use the same system, so everyone understands the systems everyone is using, so countries can trade with each other, to make trade more efficient, because it would cost less overall if everyone used one system.

b The system is logical as it uses units of ten, and there is a relationship between the units of measurement for distance, mass and capacity (10 cm × 10 cm × 10 cm cube holds exactly 1 ℓ of water and weighs 1 kg).

12.1 A

Missing elements in the chart:

(answers in bold)

kilometre km 1000 metres

metre 1 m metre

centimetre cm 1

100 metre

millimetre mm 11000 or 0.001 metre

kilogram kg 1000 grams

gram gm 1 gram

litre ℓ 1 litre


millilitre mℓ 11000 litre

12.1 B

1 1

2 1

3 1

4 1

5 1

6 1000

7 1000

8 1000

9 1000

10 1000

12.1 C

1 Gram

2 Litre

3 Kilometre

4 Distance

5 Grams or kilograms

6 Capacity (liquids)

7 Tonne

8 Litre

12.2 Units of length

12.2 A

1 b

2 c

3 Accept all reasonable responses (e.g. the thickness of a small book)

12.2 B

1 1

2 8.9

3 10

4 4.86

5 4.2

6 1.34

7 4.5

8 79.4

9 0.045

10 630

12.2 C

1 Divide

2 Multiply

3 Divide

4 Divide

5 cm

6 m

7 km

8 60 km

9 60 km

10 12.5 km (or 12 500 m)

12.3 Units of capacity

Discuss:

1 Litres, as a tank holds a large quantity

2 ml, such as for cough syrup, eye drops etc.

3 2 litres

12.3

1 0.2

2 8.5

3 2.689

4 6000

5 22 000

6 300 000

7 57 000

8 438

9 8

10 500 mℓ

11 1510 mℓ

12 2 ℓ

13 650 mℓ

14 375 mℓ or 0.375 ℓ

15 50 doses

12.4 Units of mass

12.4 A

1 a $5.45

b $6.40

2 20 000

3 0.75

4 557

5 0.0002

6 250 kg

7 3.4 kg

8 Kate

Challenge:

His answer is reasonable if it is a very big jar that can hold 300 sweets.

12.4 B

1 7.921

2 6280

3 0.104


4 826 000

5 1750

6 0.125 or 18

7 The salsa weighed less.

8 $5.00

9 34000

10 Yes


1 Larger

2 Smaller

3 4.5 km

4 105 cm

5 400 g

6 0.2 kg or 200 g

7 12 ℓ or 500 mℓ

8 Millimetre

9 About 8

10 About 10

11 a 6 cm

b About 0.5 cm

12 125 mℓ

13 484 g

14 180 600 m

12.6 Enrichment: Customary and metric measurement

12.6 A

1 Pounds would be changed to kilograms

2 Quarts would be changed to litres

3 Feet and inches would be changed to centimetres (or metres and centimetres)

4 Pounds would be changed to kilograms

5 Pounds would be changed to kilograms or grams

6 Pints would change to litres and ounces to mℓ

7 Miles would change to kilometres

8 Gallon would change to litres

9 Square yards would change to square metres

10 Teaspoon would change to mℓ

11 Gallon would change to litre

12 Inches would change to centimetres (cm)

13 Pint would change to litre

14 Tablespoon would change to mℓ

15 Ounces would change to grams

12.6 B

1

Length Mass Capacity

centimetre kilogram litre

metre gram millilitre

kilometre2 a

Length Mass Capacity

millimetre pound – c tablespoon – c

inch – c milligram gallon - c

foot – c dry ounce – c

teaspoon - c

mile – c ton – c kilolitre

yard – c tonne cup – c

light year fluid ounce – c

pint – c

quart – c

dessert spoon – c

b Circled words should be: millimetre, light year, milligram, tonne, kilolitre

12.7 Enrichment: Temperature

12.7 A

1 58°C 2 28°C


12.7 B

1 27°C

2 9°C

3 10°C

4 6°C

5 18°C

6 7°C

7 19°C

8 96°C

12.8 Time

12.8 A

1 3:00 a.m.

2 4:00 p.m.

3 12:01 a.m.

4 12:01 p.m.

5 8:15 p.m.

6 6:00 a.m.

7 7:00 p.m.

8 12:15 p.m.

9 11:15 p.m.

10 8:54 a.m.

Enrichment: 24 hour clocks

12.8 B

1 04:50

2 13:30

3 18:20

4 08:29

5 00:56

6 2:28 a.m.

7 12:25 p.m.

8 9:10 p.m.

9 11:16 p.m.

10 3:18 p.m.

Challenge:

1 03:00

2 16:00

3 00:01

4 12:01

5 20:15

6 06:00

7 19:00

8 12:15

9 23:15

10 08:54

12.8 C

1 5 weeks

2 3 days

3 4 hours

4 43200 sec

5 1.5 or 112 years

6 56 days

7 2 years

8 180 minutes

9 10800 seconds

10 84 hours

12.9 Calculating elapsed time

12.9 A

1 4 hrs 03 mins

2 14 mins 23 sec

3 48 mins 00 sec

4 2 hrs 47 mins

5 48 mins 18 sec

6 19 hrs 52 mins

7 4 hrs 55 min

8 6 mins 08 sec

12.9 B

1 a 1 hr 10 mins

b 6:45 p.m.

2 a 3 hrs 18 mins

b 1:05 p.m. (or 13:05)

3 55 mins

4 1 hr 12 mins

12.10 Practice with time

1 75

2 36

3 114

4 19

5 212

6 200

7 116

8 210

9 12

10 14

11 41 mins 44 sec

12 10 hrs 23 mins

13 2 weeks 6 days

14 2 hrs 45 mins

15 8:25 p.m.

16 4:30 p.m.

17 It is before midnight.

18 a 4 hrs 15 mins

b 221 hours

19 It averages to 5 hours per week.

20 Teisha practised 30 minutes (Note: they each practised 30 minutes, while Natasha did an additional 30 minutes, for a total of 11

2 hours dance practice altogether.)

12.11 Working backwards

12.11 A

11:45 a.m. – 2 hr 25 mins = 9:20 a.m. – 3 hours = 6:20 a.m. – 15 mins = 6:05 a.m.

He left home at 6:05 a.m.


12.11 B

1 The hat cost $29.50.

2 She spent 20 minutes hanging ribbons.

3 He left home at 9:15 a.m.

12.12 Exact answers or estimates

1 No, the game just needs squares.

2 No, it does not need to be exact (however, a close estimate is needed).

3 Yes, an exact time is needed.

4 Yes, experiments need exact measurements.

5 No, she does not need to know the exact amount of water.

6 Yes, to practise on a regulation court they need to be exact.

Unit 12 check and summaryCheck up

1 B

2 C

3 A

4 B

5 D

6 B

7 C

8 A

9 C

10 D

UNIT 13: Geometry13.1 Lines and angles review

Partner Activity:

1 Angle FYN

2 Line segment CB

3 Angle XBQ

4 Line KT

5 Points: A, B, C, D, E, F

Line segments: AC, BD, DE, CD

Lines: AB

Rays: BA, BD

Angles: CAB, ABD, BDE, BDC, FDE

13.1 A

1 a Answers to include five of the following:

BC, AD

AB, DC

AE, DG

AD, EG

DC, GH

DG, CH

JK, NP

JN, KP

b Answers to include two of the following:

JP, NK

JP, MN

JP, MK

NK, JP

NK, MP

NK, JM

c Same answers as for b.

2 Student diagrams will vary.

13.1 B

1 Acute angle

2 Obtuse angle

3 Straight angle

4 Reflex angle

5 Right angle

6 Acute angle

7 Reflex angle

8 Right angle

13.2 Measuring angles

Activity:

Approximations with the paper protractor most likely would be combinations of units of 30 degrees, or 45 degrees, with 90 or 180 degrees. Suggested answers might include the following, but should be used for discussion, rather than marking.

13.2 A

1 About 60 degrees

2 About 135 degrees

3 180 degrees

4 About 30 degrees

5 90 degrees

6 About 60 degrees


7 About 150 degrees

8 90 degrees

Enrichment: Using an exact protractor

13.2 B

Help students use an exact protractor to measure the same angles, and discuss the differences between these and their estimates.

13.3 Calculating angles

13.3 A

1 x = 140°

2 x = 90°

3 x = 100°

4 x = 30°

5 x = 45°

6 x = 30°

7 x = 15°

8 x = 90°

9 x = 40°

13.3 B

1 x = 36°

2 x = 310°

3 2x = 50°, so x = 25°

4 x = 40°

5 x = 45°

6 x = 30°

13.4 Angles in triangles and quadrilaterals

Partner Activity:

Check sketches, look for reasonable comparisons to the quadrilateral, and a final labelling with angles totalling 180°.

1 x = 45°

2 x = 90°

3 x = 60°

4 x = 450°

5 x = 15°

13.5 Investigating practical applications

Activity: Transferring energy

Students should discover differences

in the results as they hold the rulers at different angles when launching the coins. Encourage discussion to show where greater energy was transferred.

Activity: Accurate drawings

Check student steps to bisect a line, and to draw a circle, using compasses.

Discuss:

Check lists and guide students to expand their thinking if needed.

13.6 Plane (2D) shapes

13.6 A

1 Square

2 Parallelogram

3 Equilateral triangle

4 Rectangle

5 Isosceles triangle

6 Rhombus

7 Trapezium

8 Right-angled triangle

13.6 B

1 Either two trapeziums or one trapezium and one triangle

2 Answers might include triangle and (irregular) 7-sided shape, or, trapezium and (irregular) hexagon, or two (irregular) pentagons

3 Two right-angled triangles

4 Triangle and quadrilateral, or, triangle and trapezium

5 Triangle and (irregular) pentagon or two trapeziums

6 Two triangles

Challenge:

Students need to have no spaces not covered by their chosen shapes if their work is to be a tessellation, like the example.

13.7 Transformations

Partner Activity:

Check that students have followed the conventions for reflections, translations or rotations.


13.8 Symmetry

13.8 A

Check that student drawings show symmetry along a line.

13.8 B

1 a a to b requires 90° of turn.

a to c requires 180° of turn.

d to c requires 90° of turn.

b It is called rotational symmetry because the figures rotate around a centre point.

2 Students should be able to explain why their activity did or did not work, after checking with a mirror

3 Students should be able to state exactly how their figure slid (e.g. two squares down and three over) and should be able to show with dotted lines how each point of their shape slid on the grid.

4 Figure c

13.9 Tessellations

Activity:

Check student sketches to ensure their understanding of the repeating pattern, with no gaps, as expected in a tessellation.

13.10 Lines of symmetry

1 While a circle has infinite lines of symmetry, the tomato drawing shown here has just one, through the centre stem.

2 One line of symmetry, vertically through the midpoint of the pot

3 One line of symmetry through the vertical centre

4 Two lines of symmetry, one vertical and one horizontal

5 One line of symmetry , same as for question 2

6 One line of symmetry, vertically through the centre of the trident

7 One line of symmetry, vertically through the centre

8–9 Check that both sides of the drawing match exactly, if the figure were folded at the line of symmetry

10 4 lines of symmetry, two diagonals, one vertical and one horizontal

11 3 lines of symmetry, through each point and midway through the line opposite

12 2 lines of symmetry, one vertical and one horizontal

13.11 Solid figures

13.11 A

The objects shown in the pictures in the Student’s book are similar to the figures named below, while not being exact representations. (Note: a triangular pyramid is a ‘tetrahedron’ when all faces are equilateral triangles.)

1 Triangular prism

2 Sphere

3 Cone

4 Cuboid

5 Cylinder

6 Irregular solid figure


8 Cube

9 Cylinder

10 Cone

11 Square-based pyramid


13 Cylinder


15 Triangular prism

13.11 B

1 Cone: 2 faces, 1 edge, 0 vertices by the definition of the coming together of 3 or more edges, but some mathematicians report the point formed by the curved face (apex of the cone) as a vertex, giving the answer of 1 vertex.

2 Cube: 6 faces, 12 edges, 8 vertices


3 Cuboid: 6 faces, 12 edges, 8 vertices

4 Triangular prism: 5 faces, 9 edges, 6 vertices

5 Triangular pyramid: 4 faces, 6 edges, 4 vertices

6 Cylinder: 3 faces (2 flat, 1 curved), 2 edges, 0 vertices

7 Sphere: 1 curved face, 0 edges, 0 vertices

8 Triangular prism: 5 faces, 9 edges, 6 vertices

9 Square-based pyramid: 5 faces, 8 edges, 5 vertices

10 Cylinder: 3 faces (2 flat, 1 curved), 2 edges, 0 vertices

Challenge:

Curved faces – cone, cylinder, or sphere

13.12 Picturing solid shapes – nets

Activity:

Have students check to confirm their predictions after they fold their model net into the cube. Students should discuss and experiment with the different overall patterns of the squares in the net, to try other ways to make a cube.

13.12

1 Square-based pyramid

2 Cuboid

3 Triangular prism

4 Cylinder

5 Rectangles

6 Four

7 Cuboid (square prism)

8 Pyramid

Unit 13 check and summaryCheck

1 D

2 B

3 C

4 B

5 C and D (two marks)

6 A

7 B

8 C

9 4

Assessment 6Part 1

1 a Distance

b Mass

c Capacity

d Very heavy objects

e Kilometres

2 a 0.1

b 1000

c 1000

d 1

e 100

f 4.4

g 0.8

h 12 000

i 8.9

j 0.28

3 a Divide

b Multiply

c 10

d 100

e 1000

4 a Millimetres

b Kilograms

c Grams (or milligrams)

d Litres

e Centimetres

5 a 0.003567

b 8.93

c 1006

d 37 500

e 42 800

f 0.05

g 0.01925


h 0.498

i 18 000

j 0.0029418

6 400 g or 0.4 kg

7 42.1 kg

8 Check student diagrams.

9 31.1°

10 17°

11 1500 kg

12 a 3:00 a.m.

b 11:53 p.m.

c 12:05 p.m.

d 5:30 a.m.

e 6:00 p.m.

13 a 1461 days (36514 × 4)

b 384 hours

c 210 minutes

d 120 sec + (0.75 × 60) = 165 sec

14 9:58 p.m.

15 4:14:55 p.m. or about 4:15 p.m.

16 a 36 hrs 45 mins

b 5 mins 33 sec

c 16 hrs 21 mins

Part 2

1 12:50 p.m.

2 exact measurements

3 a angle

b parallel lines

c point

d perpendicular lines

4 a–f Check students’ drawings.

5 a AG and BH

b AG/DE, AG/CF, BH/DE, BH/FC, DE/FC

c AG/DE, BH/DE

6 a x = 40°

b x = 100°

c x = 30°

d x = 35°

e x = 30°

7 a regular hexagon

b trapezium

c parallelogram

8 a equilateral triangle

b square

9 a (4 lines of symmetry)

b (3 lines of symmetry)

10 a–b (Both sides of each shape should match exactly.)

11 a Faces = 6

Edges = 12

Vertices = 8

b Faces = 3

Edges = 2

Vertices = 0

c Faces = 1 curved face

Edges = 0

Vertices = 0

d (square-based pyramid – also could be interpreted as triangular-based pyramid (including a tetrahedron), with responses 4, 6 and 4)

Faces = 5

Edges = 8

Vertices = 5

e Faces = 2 (1 flat and 1 curved)

Edges = 1

Vertices = 0 or 1 (See note in answers to 13.11 B as to different conventions about vertices.)

12 Triangular pyramid (also accept ‘triangular-based pyramid’ or ‘tetrahedron’)


UNIT 14: Perimeter, area and volume14.1 Perimeter: review

14.1 A

1 10.2 cm

2 19 m

3 30 cm

4 48 m

5 25.5 cm

6 15 m

7 7.5 m

8 5 m

9 2 cm

10 15.5 m

14.1 B

1 a 16 cm

b 20.5 cm (accept 20 cm)

2 a 20 m

b 45 cm

c 19.9 cm

d 22.08 cm

3 11

4 26 m

5 29.5 m

6 25.25 m

7 It is 1 m wide.

8 22 – 7 = 15 ÷ 3 = 5 m, so the free throw area is 5 m wide

9 840 ÷ 4 = 210 m

10 56.25 cm

14.1 C

1 a Length = 80 m

b Width = 40 m

2 a Width = 15 m

b Length = 45 m

3 25 cm

4 Check students’ diagrams.

Ends ‘need 2 branches each’, presumably side by side, so sides are then 4 × 2m, and the total perimeter = 2 + 8 + 2 + 8 = 20 m.

5 25.5 cm

Challenge:

The width of Sahar’s rectangle is twice as long as Sunita’s. The length is then also twice as long, and is 18 cm. The perimeter of Sahar’s rectangle is 9 + 9 + 18 + 18 = 54 cm.

14.2 Area: review

14.2 A

1 120 m²

2 176 cm2

3 63 m2

4 44.1 cm2

5 10 800 cm2 or 1.08 m2

6 15 cm2

7 235 m2

8 227.5 cm2

9 28.8 m2

10 25 000 m2

14.2 B

1 200 ÷ 4 = 50, 502 = 2500 cm2

2 52 500 cm2 or 5.25 m2

3 There were 4200 squares.

4 150 000 cm2 (convert first as 1000 cm × 150 cm)

5 150 000 cm2 ÷ 2500 cm2 = 60 squares

6 There will be 70 sections.

7 384 tiles

8 If he used 384 tiles the first time, and now used 768 tiles, the area would be 784 cm2, since each tile is one centimetre square.

9 12 cm

10 48 cm

11 31 + 2 + 2 = width, 52 + 2 + 2 = length, area = 1960 cm2

12 1600 cm2

13 160 cm

14 400 cm2

15 14 cm

14.2 C

1 Length = 8 cm

2 Width = 5 cm

3 Length = 1713 m

4 Width = 4 cm

5 Length = 25 cm

Partner Activity:

Estimates will be roughly 14–16 cm2, with


the emphasis on students using good reasoning skills.

14.2 D

1 8 cm

2 15.25 m

3 576.7 cm

4 55

5 1 276 500 ÷ 690 = 1850 mm

6 Roughly 15 (+ or – 1)

7 Roughly 13 (+ or – 1)

8 Roughly 11 (+ or – 1)

14.3 Area: right-angled triangles

1 40 cm2

2 15 cm2

3 150 cm2

14.4 Area: triangles, parallelograms and rhombuses

1 80 cm2

2 45 cm2

3 150 cm2

4 44 cm2

5 60 cm2

6 500 mm2

14.5 Area of compound shapes

14.5 A

Students may form the total using different approaches than those shown below.

1 3 × 3 – 1.5 × 1.5 = 6.75 cm2

2 3 × 1 + 2 × 1 + 1 × 1 = 6 cm2

3 3.5 × 1.5 + (3.5 × 1.5 ÷ 2) = 5.25 + 2.625 = 7.875 cm2

4 3 × 3 + (1 × 1 ÷ 2) = 9.5 cm2

5 2 × 4 + 0.5 × 1.5 = 8.75 cm2

6 3 × 4 ÷ 2 = 6 cm2

14.5 B

1 12 cm2

2 180 m2

3 44 m2

4 250 cm2

14.6 Exploring area

14.6 A

Students may use different approaches to reach the same answers.

1 a 20 × 25 – 18 × 16 = 212 m2

b 10 × 14 – (2 × 2 ÷ 2) = 138 cm2

2 a 40 × 50 – 5 × 30 = 1850 cm2

b 15 × 22 + [(22 – 14) × (19 – 15) ÷ 2] = 330 + 16 = 346 cm2

c 16 × 16 – (4 × 3 × 3) = 256 – 36 = 220 m2

d 24 × 22 – 2 × (12 × 18÷ 2) = 528 – 216 = 312 m2

3 275 cm2

4 50 – 24 = 26 m2

5 a 100 m2

b 21 m2

14.6 B

1 7 m2

2 48 × 35 ÷ 2 = 840 cm2

3 900 cm2

4 7 × 2.5 – (3 × 1.25 × 1.25) = 17.5 – 4.6875 = 12.8125 m2

5 25 × 25 – 20 × 20 = 225 cm2

6 (12 × 14) – (8 × 10) = 88 in2

Challenge:

a 0.8 × 1.5 – 0.32 = 0.88 m2

b The area of the wall is 240 × 360 = 86 400 cm2

86 400 – [(360 – 20 – 20) × (240 – 20 – 20)] = the area of the blue border = 22 400 cm2

14.7 Working with circles

1 The radius is half the diameter: 2r = d

2 24 cm

3 8.5 cm

4 Check students’ drawings.

5 A = πr2 = 102 π = 314 cm2

6 C = 2πr = 20 π = 62.8 cm

7 400π = 1256 cm2

8 40π = 125.6 cm


9 625π = 19 621.5 cm2 (Note: diameter is 50 so radius is 25)

10 A = 5024 cm2, C = 251.2 cm; 5024 is greater than 251.2

14.8 Surface area

1 448 cm2

2 56 cm2

3 224 cm2

4 1800 cm2

5 1192 cm2

14.9 Volume

1 72 m3

2 24 m3

3 250 cm3

4 150 m3

5 1000 cm3

6 1500 cm3

7 11 250 cm3

Challenge:

a 900 cm3

b When the rock is placed in the water, the difference shown by the markings on the container will tell him how much water was displaced by the rock. This will tell him the volume of the rock.

Unit 14 check and summaryCheck

1 B

2 A

3 D

4 A

5 B

6 A

7 B

8 D

9 C

10 C

Assessment 7Part 1

1 a 32 m

b 52 cm

c 15 cm

2 a 2x = 15 cm, x = 7.5 cm

b x = 5.5 m

3 72 cm

4 a 2200 m

b 240 000 m2

c 100 000 m2

5 40 000 cm2

6 30 cm

7 70 cm

8 About 12 cm2

9 a 7.5 m

b 16 cm

10 a 56 cm2

b 768 cm2

c 120 cm2

11 a 792 cm 2

b 385 cm2

12 a 9.75 cm2

b 16 cm2

c 3.5 cm2

d 5.25 cm2

e 4.5 cm2

Part 2

1 693 m2

2 134 cm2

3 2.25 m2

4 35 m2

5 303 750 cm3

6 a 31.4 cm

b 78.5 cm2

7 a 1256 m2

b 5256 m2

c 325.6 m

UNIT 15: Ratio and proportion 15.1 Overview of ratio

1 2 : 3, 3 : 2, 2 to 3, or 3 to 2

2 3 : 5, 5 : 3, 3 to 5, or 5 to 3


3 4 : 3, 3 : 4, 4 to 3, 3 to 4

4 a 3 : 2

b 5 : 1

c Star Students to Awards, 5 : 4

d 2 : 1

e Sports Day, 3 : 1

5 a 6 : 7

b 2 : 3

c 2 : 5

d 4 : 4, reduces to 1 : 1

e 5 : 3

f 19 : 27

g 16 : 9

h 25 : 10, reduces to 5 : 2

i $54 : $18, reduces to 3 : 1

j 36 : 108, reduces to 1 : 3

6 Students’ drawings should show eighths, coloured 5 : 3.

7 Students’ drawings should show 8 triangles and 5 squares.

8 Students’ drawings should show 8 squares and 5 triangles.

9 She needs 7 ripe : 3 green, because the salad is for that night.

10 9 : 6 reduces to 3 : 2, and 30 : 42 reduces to 5 : 7. The second bag has a greater proportion of chocolates, which Fetaui likes best.

Challenge:

Kyra would finish 2 cards when Sanjay finishes 6 cards.

When Kyra finishes 5 cards, Sanjay would have finished 15.

15.2 Equivalent ratios

15.2 A

1 a 9 : 8

b 1 : 2

c 9 : 1

d 17 : 7

e 6 : 11

f 13 : 3

2 a =

b ≠

c ≠

d =

e ≠

f =

3 a 18 : 12

b 40 : 72

c 72 : 45

4 a 36 comedies and 24 discovery movies

b 16 discovery movies

5 108 wheat loaves

15.2 B

1 a No

b Yes

c No

d Yes

e No

2 a 4 eggs

b 3 eggs

c 412 cups of flour

d 96 cookies

Challenge:

5 batches of 24 cookies will give 40 bags of 3 cookies.

15.3 Using cross products and proportion

1 35 sheep

2 10 cyclists

3 192 minutes

4 a $20

b $24.30

Challenge:

The map would show the distance as 55 cm.

15.4 Ratio and rate

1 120 pages

2 a 54 tickets

b 144 tickets

3 21 boxes


4 15 fish

5 288 km

6 $15

7 25 people

8 $51.80

9 49 sheep

15.5 Unit rate

15.5 A

1 $2.50

2 $0.75

3 $0.40

4 $4.15

5 $2.97

6 $9.99

7 a $0.55

b $2.75

8 a $0.65

b $6.50

9 a 28 shells

b 140 shells

10 a 15 km

b 10 cm

11 a 400 km/hr

b 5 hours

12 a $6

b $300

15.5 B

1 18 weeks

2 6 days

3 3 m

4 6 packs

5 8 days

6 12 shirts

15.5 C

1 250 words

2 18¢ each

3 9 eggs

4 30 cm

5 126 students from overseas

6 a 24 kites

b four kites

7 208 minutes

8 4 costumes

9 $75

10 a 23 cup

b 112 cups

c 30 potatoes

d 10 cups

Challenge:

a $7.15

b 20 mins

15.6 Enrichment: scale drawings

1 Classrooms 4T and 6G both measure 15m by 20 m, the open area measures 15 m by 7.5 m, and the office measures 7.5 m by 7.5 m.

2 The red triangle is 3.5 cm base and 2.5 cm height.

Check student drawings.

a Twice as large is 7 cm by 5 cm.

b Half as large is 1.75 cm base and 1.25 cm height.

3 Check student drawings.

Challenge:

Check student drawings, using the scale 2 cm : 5 m.

15.7 Working with a total

15.7 A

1 14 and 35

2 15 portraits

3 20 guppies

15.7 B

1 a 56

b 16

2 a $90

b $30

15.7 C

1 $25 older brother and $15 younger brother

2 a $300 was given to his nieces.

b $450 was given to his nephews.


3 44 points

4 a Gran is 60 years old.

b He will be 20 years old.

5 a 15 boys

b 8 : 5

Challenge:

64 cm3 means each side of the cube is 4 cm. If each length is doubled, the new cube is 8 cm × 8 cm × 8 cm with the volume of 512 cm3.


1 24 bags

2 15 oranges

3 $10

4 24 questions

5 12 km

6 $7.20

7 15 days

8 12 shirts

9 5 hours

10 3 hrs 20 mins (or 313 hrs)

11 $2.67

12 3 days

13 2 hours

14 510

15 $1176 for each set of t-shirts in one colour (Hint: 504 ÷ 3 = 168 of each colour shirt needed. Shirts sell 4 to a pack, so 168 ÷ 4 = 42 packs at $28 per pack)

15.9 Unequal sharing

15.9 A

1 Kay gets $11 and Rachel gets $14.

2 One receives $279 and one receives $405

3 The first boy receives 33 plums, and the others get 26 plums each.

15.9 B

1 VJ picked 61 cherries and Bijoux picked 41.

2 The eldest brother earned $12.50 and the others each earned $8.75.

3 Emily has 152 stickers and Liz has 137.

4 Class 4R and 5D earned 46 points, while Class 6J earned 64 points.

5 $17.48 and two sets of $15.86

Unit 16: Problem solving skills16.1 Unit rates

16.1 A

One example of a short cut is shown in brackets after each answer below. Students should check results by multiplying figures from the original question.

1 584 cupcakes (25%, × 4)

2 $1168 (5%, × 20)

3 $18 (8 × the price of 3 slices)

4 120 bags (18 = 15% so 6 = 5%, × 20)

5 $24 (7 = $3.50 means $0.50 per slice, × 48)

6 $325 (8% = $26 so 4% = $13, × 25)

7 $40 (unit rate, 0.80 each, × 50)

8 $700 (28% = $196 so 4% = $28, × 25)

16.1 B

1 $430

2 $564

3 27 students

4 38 = 18, so there are 48 marbles altogether

5 80 sheep

6 25 students

16.2 Combining unit rates

1 157 mins

2 2 811 mins

3 178 mins

16.3 Using diagrams

1 300 km/hr

2 400 km/hr

3 120 km

4 24 km/hr

5 4 hrs


6 9 km

7 325 km

8 54 km

16.4 Using formulae

1 134 hours

2 750 km

3 800 m/hrs or 1313 m per minute

4 112 hrs or 90 mins

5 8 m/sec

6 15 km

7 3.2 km/hr

8 He can complete 40 cm per hour.

Enrichment: The Pythagorean Theorem

A2 + 82 = 102, therefore A2 + 64 = 100 and A2 = 36 so A = 6

The height of the tree is 6 m.

16.5 Looking for links

1 125

2 38

3 36

4 27

5 30 (Hint: start with 6212% = 5

8, 58 is 25 so

18 is 5…)

6 22.5

7 27

8 54

9 10 eighths

10 60

16.6 Listing details

1 a Rani

b 42

2 a $1200

b $112

3 a $0.82

b $0.26

4 a $15

b $12

5 4112 hours

6 8 : 35 a.m.

16.7 Calendar problems

1 48 days

2 96 days

3 23rd December

4 5 months

5 2 leap years

6 1st March 2003

7 244 days (inclusive)

8 a 1st May

b 13th May

c Monday

16.8 Poles and spaces

1 250 m

2 a 15 streetlights

b 210 m

3 a 1200 m

b 401 posts

4 57 m

5 435 cm

6 a 40 m

b 80 m

7 10 poles

16.9 Problem-solving practice

1 a 11 391

b 5236

c 4875

2 a 1339

b 926

3 4819

4 Shivani

5 20 rows of 20 tiles = 400 tiles

6 480 squares

7 4 trips



1 D

2 A

3 C

4 B

5 A

6 A

7 B

8 D

9 C

10 D

Assessment 8Part 1

1 a 3 : 4

b 3 : 1

c 6 : 7

d 4 : 3

e 2 : 1

f 20 : 9 : 6

2 a yes

b yes

3 a 12

b 10

c 10

4 32 comedies and 24 cartoons

5 144

6 6 for $2

7 5

8 12 : 18

9 40

10 15

11 a 12 cup

b 5

c 412 cups

d 60

e 12 cup

12 48

13 168 km

14 36 days

15 12 weeks

16 6 days

17 16 cases

18 3 pages

Part 2

1 25 cm

2 48 books on one shelf and 16 books on the other

3 45

4 Erika earned $125, Jacqueline earned $75 and Ermie earned $100

5 15 years

6 a 27 girls

b 27 : 24, or 9 : 8

7 $7.20

8 6

9 $160

10 $2.56

11 $28 for Mike and $40 for Mike

12 One picked 9 cashews and one picked 14.

13 a $208

b $281.60 (note: 20% of total savings)

14 4 hours

15 32 students

16 21112 seconds

17 440 km/hr

18 51 km

19 72 km/hr

20 260 m per hour

21 She completes 15 cm per hour, or 1 cm in 4 mins (or 0.25 cm/min).

22 27

23 30

24 15

25 $895 (assumes matching side tables)

26 $0.30 or 30¢

27 98 days, inclusive

28 14 cages

Challenge:

a 60 coins

b 90 mins or 112 hrs


Unit 17: Statistics / sets / integers 17.1 Overview

Group Activity

Check for each point under ‘Making a graph’. Give particular attention to the y-axis, checking that the numbers chosen are uniform and reasonable.

17.2 Graphs

1 31 children

2 12 children

3 5 children

4 2 children

5 3 children

6 9 children

Challenge:

a The number of children for comparison

b Check student drawings (or IT graphs) showing the circle graph / pie chart format

Enrichment: Line graphs

Discuss:

The same data is shown in the two graphs, and how it is presented can be compared by students. However, the information in these graphs is not the type that changes over time. Therefore, the bar graph is the better choice for this data.

17.3 Pictographs

Check students’ graphs. The bar graph should show the following totals:

roti and curry 50 people

rice and stew 60

fish and chips 75

salad 35

chicken and chips 60

cheese cutters 45

veggie burgers 50

hot dogs 50

pizza 75

samosas 30

Challenge:

a No. There is no way to know who made the choices in the data given.

You would need to collect two sets of data, and display the data in two similar graphs or with two separate bars for each choice on a bar graph.

b IT or using ruler and graph paper. Have students create a presentation copy for their work portfolios.

c Answers will vary, and may include different food choices (listing what the new choices would be), and some of the same food choices. The data would be different, as the number of students surveyed and their preferences would vary.

Accept all reasonable answers, with explanations, for how the data would be displayed.

17.4 Circle graphs

17.4 A

1 12.5% of 120 = 15 students

2 36 students

3 45 students

4 15

5 18

6 72 degrees

7 135 degrees

8 English

9 Science

10 Maths/Science and Computer Studies/English

11 13

12 Computer studies and Science

13 81

14 30%

15 0.375

17.4 B

1 25%

2 19%

3 $4800


4 15

5 $2850

6 $600

7 Lights

8 Advertisements

9 Canteen supplies and costumes

10 Advertisements and hall

17.5 Patterns and probability

Partner Activity:

Answers will vary. Have alternate pairs of partners check results for their peers for questions 1–4. Check final results. Ensure the final paragraph addresses the key question, and is reasonable.

a – c Answers will vary based on the ten subjects chosen. Check that the probability stated is reasonable, an

17.6–17.7 Enrichment: Sets and Venn diagrams

17.7 A

1 47

2 10

3 5

4 7

5 16

6 4

7 3

8 2

17.7 B

Check students’ diagrams for each of the 9 points, with an extra point awarded for overall format.

17.8 Enrichment: Union and intersection of sets

1 C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

2 D = {2, 4, 6 ,8}

3 E = {1, 3, 5, 7, 9}

4 F is the set of primary colours

5 G is the set of colours of the rainbow

6 H is the set of factors of 32

7 I is the set of the first five multiples of 8

8 {red, yellow, blue}

9 {red, orange, yellow, green, blue, indigo, purple}

10 {8, 16, 32}

11 {1, 2, 4, 8, 16, 24, 32, 40}

12 {2, 4, 6, 8}

13 {1, 2, 3, 4, 5, 6, 7, 8, 9}

14 {1, 3, 5, 7, 9}

15 {1, 2, 3, 4, 5, 6, 7, 8, 9}

17.9–17.10 Enrichment: Integers and Adding and subtracting integers

17.10

1 7

2 0

3 –3

4 3

5 1

6 2

7 –11

8 –7

9 –3

10 –11

11 5 degrees C

12 –2 m – 3 m = –5 m

13 I had $14 in my purse, but I owed $12 and expected to be repaid $5.

$15 – $5 – $8 + $12 = $14

Challenge:

To add a negative number can be thought of as subtracting that number. It is possible to subtract a larger number from a smaller number, but the result will be less than 0.

17.11 Enrichment: Comparing Integers

17.11 A

1 <

2 >

3 >

4 <

5 <

6 =

7 <

8 >

9 >

10 <

17.11 B

1 –4

2 1

3 9

4 2

5 –18

6 21

7 50

8 –13

9 –48

10 –16

11 –36

12 130

13 –275

14 78

15 25

16 50

17 –515

18 215

19 35

20 –50


17.12 Enrichment: Multiplying and dividing integers

1 24

2 –54

3 5

4 75

5 24

6 –3

7 –13.5

8 169

9 9

10 –125

17.13 Using a calculator with integers

Check students’ skills with the calculator, and that their explanations are reasonable.

17.14 Enrichment: Coordinates

17.14 A

1 T

2 S

3 E

4 Z

5 (3, 5)

Group Activity – Grids should be drawn and labelled with precision.

17.14 B

1–4 Check student diagrams.

Activity:

a–i Check students’ graphs against a model.

j Towns planners use a grid for several reasons. Accept all reasonable answers.


1 A

2 B

3 C

4 D

5 B

6 C

7 C

8 D

9 C

10 B

Enrichment: Roman numerals

A and B Check that results are reasonable.

C 1 movie or television show

2 Examples: an introduction or a booklet

3 Olympic

4 examples: CARIFESTA, on examination papers, etc.

D 1 III

2 XVII

3 1XXX

4 VL

5 C

6 CXXV

7 LXXXVIII

8 LIV

9 XXXVI

10 XCIV

bonus – MMXVI

UNIT 18: Consumer / Business Mathematics18.1 Profit and loss

18.1 A

1 $15.85 profit

2 $10.50 profit

3 $18.70 (selling price)

4 $31.35 (cost price)

5 $271 loss

18.1 B

1 $3 profit

2 $33 profit

3 $110160

4 a $0.76

b $27.36

18.2 Profit and loss as a percent

18.2 A

Cost price

Selling price

Profit or loss

Profit/loss percent

1 $2 4% loss

2 $245 $35

3 $3915 3313%

4 $11288.40 $1472.40

5 $51.49 $2.71

18.2 B

1 $510

2 a $240

b 100%


3 a $7.50

b 20%

4 a $8.40

b $2.10

c $210

d 25%

5 a $280.50 costs

b $115.75 profit

c About 41%

6 Answers will vary.


1 $5

2 a $140

b 140%

3 $6 loss per shirt

4 a $15

b 50%

5 a $360

b 20%

c $864

d 40%

6 $500

7 a $1600

b 6623%

8 $0.75

9 a $5160

b 60% loss

10 a $270

b $262.50

18.4 Simple interest

1 a $20

b $112

2 $400

3 $390

4 Jaime

Enrichment: Compound interest

This involves rounding off cents, and applying interest rate at the end of each year to the accumulated totals.

$1040 end first year (4% gained), $1082 end second year (4% of $1040 added to form the new total), $1125 third year, $1170 fourth year, $1217 fifth year, $1278 sixth year, $1342 seventh year, $1409 eighth year, $1479 ninth year, $1553 tenth year, $1646 eleventh year.

Answer: $1745 at the end of 12 years.

18.5 Value added tax

1 $50

2 $400

3 $750

4 $120

5 $41.40

6 $280

7 $78

8 $1400

18.6 Hire purchase – buying on credit

1 $999

2 $344

3 $1559.48

4 $199

5 $166.50

6 $282

7 $180

8 $1107

Challenge:

$84 000

18.7 Foreign exchange

18.7 A

1 TT $110.25

2 US $300

3 TT $312.50

4 GY $8240

5 BB $990

6 JM $2000

7 US $28

8 BB $280 (Note: rate of exchange is on the next page in the Student’s Book.)

9 US $8000

10 XC $14 286 (may accept $14 300 to $15 000)

18.7 B

1 $1368

2 $282

3 $7.50

4 $50

5 $46.53

6 $220

7 $86

8 $29.25

9 $720


18.7 C

1 BB $313.50

2 TT $84

3 XC $67.88

4 BB $30

5 BB $148.15

6 Rayshawn

18.7 D

1 $400

2 $50

3 $96

4 $7

5 $180

6 $175

7 $15

8 $25

9 $22, $24, $32

10 $48.60

11 $40

12 $452.63 (rounded to the nearest cent)

13 $1320

14 $90

15 Yes, 100% is the original price, and 15% is VAT, so $19 × 115% gives the final sale price.

Unit 18 check and summary1 C

2 D

3 B

4 C

5 D

6 A

7 D

8 A

9 D

10 C

Assessment 91 a 1

4

b Fruit salad and patties

c Fish and chips

d 43 degrees (rounded to the nearest degree) or accept approximation, ‘About 40 degrees’

e 24 buyers

f 111 buyers

g 3712%

2 a $3.05 loss

b $42.60 profit

c $393.69 loss

d $17 profit

3 a 36 bags

b $30.50

4 $652.50

5 $86.80

6 20%

7 $150

8 $195

9 $1800

10 a $280

b $320

Enrichment:

1 22 students

2 26 students

3 cricket

4 18 students

5 13 students

6 8 students

7 16 students

8 112 students

9 65 students

10 0 students

Final assessment 1Part A

1 D

2 A

3 D

4 A

5 C

6 B

7 D

8 D

9 C

10 A

11 B

12 B

13 C

14 B

15 A

16 B

17 A

18 D

19 C

20 D

21 D

22 C

23 C

24 A

25 B


Part B

1 5 hundreds + 3 tens + 2 ones

2 Thousands

3 Eighty-nine thousand (and) forty

4 32 561, 31 561, 30 661, 30 561

5 about 5900 to 6000

6 37 938

7 60 616

8 98 328

9 2004

10 5460

11 13

12 13

13 53

14 11

15 a 6

b hundredths

c 6000

16 1547, 547.1, 547.01, 547

17 $178.99

18 $21 900

19 10 604

20 $58 200

21 209 280

22 6250

23 a 321

b 14 445

c 12 840

d 1605

e 28 890

f 28 890

24 8

25 16

26 11

27 a 11

b 8

c 7

d 8

28 34%

29 51100

30 12

31 712

32 7

33 3 310

34 1725

35 2 712

36 a 34 = 0.75 = 75%

b 35 = 0.6 = 60%

37 96%

38 25%

39 $86.25

40 $23

41 8

42 Check students’ drawings – angles less than 90 degrees.

43 a a = 230 degrees

b a = 45 degrees

44 a 3 faces

b 2 edges

c 0 vertices

45 a 48 cm

b 32 m

46 18 m

47 a 138 m2

b 162 cm2

48 3233, 2144, 214.4, 32.3

49 $27.75

50 Four black and white photographs

51 $120

52 3 113 mins

53 150 degrees

54 $40.46

55 $138

56 613

57 6.25 m2

58 $25

59 54 hours



1 C

2 D

3 C

4 D

5 B

6 D

7 C

8 D

9 A

10 A

11 C

12 B

13 D

14 B

15 B

16 A

17 D

18 B

19 A

20 C

21 D

22 C

23 B

24 A

25 A

Part B

1 5035

2 6 × 1000 + 5 × 100 + 2 × 1

3 800

4 Fifty-eight thousand, three hundred (and) forty-two

5 69 320

6 63 629, 63 829, 64 829, 64 839

7 15 600

8 68 791

9 44 013

10 40 150

11 94418 or 944.125

12 8671

13 64

14 12

15 121

16 23 × 5 or 2 × 2 × 2 × 5

17 22 000.09 or 22 000 9100

18 433.06, 4330, 4330.6, 4330.62

19 848.74

20 There are 9 different shirts in all.

21 $7700

22 $344

23 $3.19

24 19 200

25 a 620

b 33 480

c 16 740

d 31 000

e 16 740

f 50

26 29 (3 × 8 is completed first, due to order of operations.)

27 18

28 a 19

b 104

c 25

d 900

29 0.51

30 15

31 1018 32 21

33 5 124

34 5 110

35 34

36 $54.60

37 a 0.625 6212% or 62.5%

b 12 50%

38 $157.50

39 72

40 $90.53 (90.525 round up the cents)

41 20 bags

42 Correct answers will have angles greater than 90° and less than 180°.

43 a 40°

b 80°

44 a 6

b 12

c 8

45 a 266 cm

b 66 cm

46 110.4 m

47 a 144 m2

b 261.5 cm2

48 729 cm3

49 18 times

50 $101.50

51 360 km/ hr

52 32 students


53 $2.90

54 $26.60

55 $600

56 US $210

57 $72

58 160 cm

59 172


1 B

2 C

3 B

4 B

5 D

6 C

7 D

8 A

9 C

10 A

11 B

12 A

13 C

14 C

15 D

16 B

17 A

18 A

19 A

20 B

21 D

22 C

23 C

24 C

25 B

Part B

1 10 000

2 Hundreds

3 One hundred thousand, four hundred sixty

4 91 084

5 7510

6 150 000

7 $2000

8 $282.69

9 10 908

10 5

11 63

12 4, 5, 6, 7, 8

13 2 × 33 or 2 × 3 × 3 × 3

14 a 4

b hundreds

c 3100

15 612.02

16 6.042 > 6.004

17 630.19

18 9 different types

19 384 tins

20 15 200

21 $1662

22 546 000

23 a 32

b 4832

c 30

d 154 624

e 64

f 77 312

24 6

25 35

26 29.25

27 today

28 0.375

29 0.28

30 23

31 3

32 1515

33 23

34 7 712

35 318

36 23 112 kg

37 a 0.333 3313%

b 14 25%

38 $15.20

39 $1240

40 15%

41 $48.50

42 6500 g

43 Answers will measure more than 180° but less than 360°.

44 a 35° b 95°

45 a 1 (1 curved face)

b 0

c 0


46 a x = 2.5 cm

b x = 4 cm

47 a 40 cm

b 400 cm2

48 296 cm2

49 8.9 m2

50 6650 cm2

51 334 cups of rice

52 Shad receives $156 and Shoni receives $194.

53 666 km

54 445 minutes or 4 minutes 48 seconds

55 45

56 $31.20

57 $72

58 $3600

59 $7200

60 $400

Final assessment 4 Part A

1 B

2 A

3 C

4 C

5 D

6 D

7 D

8 C

9 C

10 A

11 A

12 B

13 D

14 C

15 D

16 B

17 A

18 D

19 D

20 C

21 D

22 C

23 D

24 B

25 D

Part B

1 61

2 540

3 80

4 Five hundred and twenty thousand (and) seventy-five

5 11 650

6 $790

7 $6501.98

8 167 184

9 4348

10 5

11 5

12 1

13 2 × 52 or 2 × 5 × 5

14 a 0

b Hundredths

c 200

15 Forty thousand (and) fifteen, and three tenths

16 47.667

17 230.18 > 230.09

18 $870.50

19 $144 300

20 $1629

21 599.3

22 20720

23 30

24 1

25 10

26 43

27 1000

28 282

29 28

30 16.8

31 0._3

32 0.8

33 14

34 1725

35 47

36 913

37 225

38 35

39 5


40 a 0.875 8712% (or 87.5%)

b 1720 85%

41 $375

42 60

43 $281.75

44 80 runs

45 540

46 Check student answers

47 a x = 45°

b x = 60°

48 a 4

b 6

c 4

49 31 cm (rounded to nearest cm)

50 64 m

51 58.5 m2

52 260 m2

53 1 vertical and 1 horizontal line of symmetry

54 a $105

b $140

55 158.4 km

56 $60 000

57 5 adults

58 64 km/hr

59 20

60 $15 loss

61 $1008

62 $1092.50

63 $228

64 864 cm2

65 48 marbles

66 320 loaves


1 B

2 A

3 A

4 C

5 A

6 D

7 D

8 B

9 C

10 B (order of operations apply)

11 A

12 D

13 B

14 C

15 A

16 B

17 D

18 A

19 B

20 C

21 C

22 B

23 C

24 B

25 D

Part B

1 6000

2 10

3 10

4 tens

5 9 000 000

6 Sixty-two thousand, nine hundred (and) six

7 $2.68

8 About $6

9 110

10 23

11 48 (subtract 9)

12 21, 23, 25, 27, 29

13 23 × 37 or 2 × 2 × 2 × 37

14 a 2

b hundred thousands

c 20 000

15 Eight hundred eighty thousand three hundred fifty, and one hundredth


16 320.61

17 4576.59

18 $126 950.36

19 8 different outfits

20 72 bottles

21 46 800

22 17 000

23 $2064

24 91.35

25 1000

26 10

27 $0.06 (rounded to nearest cent)

28 $62 140

29 7

30 41 (order of operations applies)

31 21 (order of operations applies)

32 2

33 About $21 (round to 7 kg at $3)

34 About 40 (estimate 25 into 1000)

35 $3.64

36 150%

37 0.75

38 712

39 29

40 123

41 15

42 5 740

43 a 38 371

2% or 37.5%

b 1.5 150%

44 39.2

45 $410.40

46 $77.80

47 Check that students’ drawings show 90° (right) angle where lines intersect.

48 a p = 50°

b p = 30°

49 a 6

b 12

c 8

50 a x = 5 cm

b x = 15 m

51 136 m2

52 150 m2

53 D

54 25.12 m

55 15 sheep

56 a 58

b $25

57 $2000

58 $890

59 36

60 150%

61 $162.50

62 $460

63 $18 200

64 $1320

65 26

66 5 : 4


Answers to Workbook 612 one hundred twenty thousand, nine

hundred (and) forty-four

13 seven hundred sixty-two thousand, (and) ninety-five

14 two million, seven hundred eighty thousand, three hundred (and) twelve

15 six million, five hundred two thousand, four hundred (and) sixteen

4 Writing numbers

1 11 020

2 916

3 1462

4 22 107

5 672 000

6 71 900

7 400 000

8 48 153

9 511 206

10 8 698 000

11 72 516

12 734 680

5 Value

1 30

2 800

3 6000

4 200

5 80

6 500

7 40 000

8 2000

9 40 000

10 700 000

6 Value

1 100 + 30 + 6

2 700 + 9

3 5000 + 60 + 5

4 60 000 + 9000

5 20 000 + 3000 + 700 + 8

6 40 000 + 2000 + 90 + 1

7 80 000 + 8000 + 300 + 20

8 200 000 + 3000 + 700 + 60

7 Value / expanded form

1 972

2 8321

3 4035

4 950

5 8070

6 23 936

7 78 600

8 63 000

9 746 772

10 903 514

UNIT 1: Number 1 Place value

1 86 tens

2 947 tens

3 819 ones

4 3040 thousands

5 8217 hundreds

6 3712 tens

7 53 586 thousands

8 80 704 hundreds

9 126 709 hundred thousands

10 46 623 ten thousands

2 Place value

HTh TTh Th H T O

1 9 2 4

2 4 0 0 2

3 5 3 5 9 8

4 9 7 8 0 1

5 2 4 9 5 8

6 1 0 7 1 7 2

7 4 1 0 4 0 0

8 5 3 1 7 0 4

3 Writing numbers

1 nine hundred (and) sixty

2 three hundred (and) eighty-nine

3 one thousand four hundred (or fourteen hundred)

4 three thousand (and) forty-six

5 two thousand one hundred (and) fifty

6 twelve thousand, three hundred (and) eighty

7 twenty thousand, five hundred (and) twenty-two

8 nineteen thousand, six hundred (and) seventeen

9 ninety thousand (and) eleven

10 three hundred ninety thousand

11 four hundred eighty thousand (and) three hundred


8 Expanded form

1 5432 = (5 × 1000) + (4 × 100) + (3 × 10) + (2 × 1)

2 360 = (3 × 100) + (6 × 10)

3 40 505 = (4 × 10 000) + (5 × 100) + (5 × 1)

4 14 780 = (1 × 10 000) + (4 × 1000) + (7 × 100) + (8 × 10)

5 218 003 = (2 × 100 000) + (1 × 10 000) + (3 × 1)

9 Expanded numbers

1 2356 = 2 × 1000 + 3 × 100 + 5 × 10 + 6 × 1

2 672 = 6 × 100 + 7 × 10 + 2 × 1

3 21 340 = 2 × 10 000 + 1 × 1000 + 3 × 100 + 4 × 10

4 284 005 = 2 × 100 000 + 8 × 10 000 + 4 × 1000 + 5 × 1

5 364 758 = 3 × 100 000 + 6 × 10 000 + 4 × 1000 + 7 × 100 + 5 × 10 + 8 × 1

6 70 910

7 423 000

8 52 100

9 704 016

10 605 030

10 Number

Review:

1 a 400

b 6000

c 10 000

d 0

2 a Thousands

b Hundred thousands

c Ten thousands

3 6

4 a 7795, 33 758, 43 668, 68 840, 884 901

b 801, 8013, 80 113, 801 113, 810 113

c 61 010, 61 111, 61 810, 68 710, 68 910

5 a 6211, 2612, 2611, 2161

b 911 151, 91 511, 9115, 915

c 414 063, 411 406, 410 463, 41 406

11 Mental Maths

(Note: students may use alternate strategies to those shown. They should be able to explain their reasoning, and attain the correct answer.)

1 24 × 5 = 12 × 10

2 6 × 25 = 3 × 50

3 8 × 15 = 4 × 30

4 70 × 6 = 10 × 7 × 6

5 11 × 50 = 11 × 5 × 10

6 90 × 11 = 10 × 9 × 11

7 86 × 5 = 43 × 10

8 120 × 4 = 4 × 12 × 10

9 80 × 8 = 10 × 8 × 8

10 22 × 6 = 11 × 12

11 16 × 6 = 8 × 12

12 4 x18 = 8 × 9 = 72

13 25 × 16 = 25 × 4 × 4 = 100 × 4 = 400

14 8 × 20 = 8 × 2 × 10 = 160

15 35 × 6 = 70 × 3 = 10 × 7 × 3 = 210

16 4 × 24 = 8 × 12

17 7 × 90 = 7 × 9 × 10 = 630

18 29 × 6 = (30 × 6) − 6

19 19 × 7 = (20 × 7) − 7

20 39 × 8 = (40 × 8) – 8 (= 312)

12 Expanded numbers with regrouping

1 2 tens

2 5 hundreds

3 12 tens

4 7 thousands

5 15 hundreds

6 8 tens

7 18 thousands

8 12 ones

9 315 thousands

10 40 ten thousands

13 More expanded numbers with regrouping

1 790 = 79 × 10

2 4270 = 42 × 100 + 7 × 10


3 4000 = 4000 × 1

4 396 = 3 × 100 + 96 × 1

5 6281 = (5 × 1000) + (12 × 100) + (8 × 10) + (1 × 1)

6 574 = 500 + 60 + 14

7 2758 = 2000 + 600 + 158

8 3259 = 32 × 100 + 59 × 1

9 7964 = (796 × 10) + (4 × 1)

10 8503 = 8 × 1000 + 4 × 100 + 10 × 10 + 3 × 1

UNIT 2: Operations – computation1 Rounding

1 900 000

2 800 000

3 300 000

4 300 000

5 700 000

6 70 000

7 30 000

8 90 000

9 260 000

10 420 000

11 1000

12 80 000

13 9000

14 128 000

15 358 000

16 7000

17 75 700

18 407 100

19 531 200

20 7300

21 680

22 3270

23 33 260

24 4490

25 315 010

26 $68

27 $2

28 $0

29 $930

30 $83

2 Adding mentally

1 1216

2 4080

3 10 000

4 $82 (may accept $80 or $85)

5 $6000

6 52

7 35

8 52

9 400

10 15

11 7000

12 2400

13 127

14 300

15 200

3 Addition

1 1134

2 554

3 7663

4 18 953

5 13 695

6 59 824

7 51 881

8 79 777

4 Addition

1 816

2 3040

3 9361

4 11 383

5 4640

6 10 379

7 7929

8 33 131

9 $14.00

10 $66.15

11 $37 856

12 $255.10

5 Subtraction with regrouping

1 1603

2 41 109

3 30 240

4 47 948

5 56 070

6 21 518

7 6293

8 2808

9 15 833

10 39 954

11 39 010

12 50 322

6 Subtraction with zeros

1 1050 boys

2 $1038

3 $185 375

4 4185

5 56 992

6 5983

7 1182

8 27 081

9 181 194

10 4568

7 Multiplication

1 132

2 80

3 49

4 9

5 48

6 108

7 90

8 12

9 66

10 88

11 14

12 16

13 36

14 28

15 45

16 72

17 121

18 24

19 84

20 96


8 Division

1 5

2 8

3 10

4 6

5 9

6 12

7 9

8 9

9 8

10 11

11 6

12 7

13 12

14 6

15 12

16 7

17 8

18 6

19 4

20 10

9 Mixed multiplication and division

1 9

2 7

3 9

4 11

5 8

6 6

7 4

8 6

9 60

10 9

11 5

12 12

13 6

14 11

15 24

16 12

17 64

18 12

19 60 decorations

20 240 shirts

UNIT 3: Number concepts / sequences1 Square numbers

1 8 rows and 8 columns



4 81

5 100

6 121

7 144

8 169

9 196

10 225

2 Exponents

1 74

2 95

3 103

4 67

5 113

6 122 = 144

7 34 = 81

8 54 = 625

9 43 = 64

10 26 = 64

3 Identifying patterns

1 72, 77

2 1, 213 3 60, 72

4 64, 128

5 0.1, 1

6 25, 36

7 75¢, $1.25

8 350, 3500

9 81, 76

10 36

11 35

12 e + f, f + g

13 414, 21

8

4 Order and value

1 1269

2 9612

3 1269

4 1296

5 9621

5 Consecutive numbers

1 12

2 218, 219, 220

3 70

4 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113

6 Factors

1 {1, 2, 3, 4, 6, 8, 12, 24}

2 {1, 2, 3, 5, 6, 10, 15, 30}

3 {1, 19}

4 {1, 2, 17, 34}

5 {1, 2, 4, 8, 16, 32, 64, 128}

7 Common factors

1 1, 2, 4, 8

2 1, 3, 9


3 2

4 8 items

8 Multiples

1 5, 10, 15, 20, 25, 30, 35

2 6, 12, 18, 24, 30, 36, 42

3 7, 14, 21, 28, 35, 42, 48

4 8, 16, 24, 32, 40, 48, 56

5 15, 30, 45, 60, 75, 90, 105

9 Common multiples

1 (Numbers to be circled are shown here in bold.)

6, 12, 18, 24, 30, 36, 42, 48

8, 16, 24, 32, 40, 48, 56, 64

2 78

3 105

4 40 tamarind balls

5 42 Cubs

10 Divisibility

1 0, 5

2 even numbers

3 divided evenly by 9

4 divided by 3

5 0

11 Factorising

1 a 80

4 × 20

2 × 2 × 2 × 10

2 × 2 × 2 × 2 × 5

b 2) 80

2) 40

2) 20

2) 10

5

c 24 × 5 or 2 × 2 × 2 × 2 × 5

2 Shown below is the preferred order for listing prime factors. However, they may be listed in any order and still be correct.

a 30 = 2 × 3 × 5

b 140 = 2 × 2 × 5 × 7 or 22 × 5 × 7

c 264 = 2 × 2 × 2 × 3 × 11 or 23 × 3 × 11

d 500 = 2 × 2 × 5 × 5 × 5 or 22 × 53

3 Student models will vary. Check that the final result is:

180 = 22 × 32 × 5 or 2 × 2 × 3 × 3 × 5

Challenge:

All multiples of 6 can be divided by both 2 and 3.

UNIT 4: Decimals1 Place value and value

1 7

2 5

3 thousandths

4 3000

5 4100

6 0.3

7 0.1

8 0.25

9 0.07

10 0.432

11 0.006

12 0.029

2 Place value and value

1 210

2 40

3 4100

4 31000

5 200

6 Tens

7 Tenths

8 Ones

9 Hundredths

10 Thousandths

3 Writing with words

1 Nine tenths

2 Four hundred fifty and six tenths

3 Eighty-six and five hundredths

4 Three thousand six hundred twenty and ninety-two thousandths

5 416.81

6 13.011

7 2016.102

8 101 400.11


4 Comparing decimals

1 a 100.5 > 100.05

b 980.6 > 98.06

c 321.804 > 321.8

2 a 0.5, 0.05, 0.005

b $925, $900.25, $92.50, $90.25

c 230.6, 23.6, 23.06, 23.006

5 Rounding

1 35.4 seconds

2 a 58.0 (or 58)

b 347.1

c 482.3

3 a $23

b $51

c $2355

4 3298.06

5 $6330

6 Adding and subtracting

1 897.88

2 3021.85

3 5234.3

4 5884.06

5 5481.8

6 3749.35

7 20 502.979

8 $632.60

9 $254.13

UNIT 5: Operations – multiplication1 Simple multiplication

1 535 pieces

2 27 620

3 20 796

4 18 844

5 761 139

6 412 440

7 355 195

8 485 586

9 170 804

10 561 888

2 Simple multiplication

1 300 meat patties

2 $936 + $1596 = $2532

3 $123

4 279 469

5 147 628

6 565 394

7 506 506

8 $755.72

9 1 354 230

10 $2875.60

3 Multiplying by 2- and 3-digit numbers

1 460 guests could be seated.

2 172 250

3 183 819

4 59 640

5 2 308 468

6 109 967

7 151 191

8 $10 166.57

9 6 561 838

10 1 423 056


1 39 240

2 1 647 198

3 212 976

4 23 232

5 $40 962.60

6 $14 274.90


1 174 624

2 204 078

3 3 420 670

4 a 10 400 m or 10.4 km

b 13 200 m or 13.2 km

c 2800 m or 2.8 km

d $3300

e $29 700

6 Multiplying by multiples of 10

1 360 000

2 4 800 000

3 450 000

4 280 000

5 177 000

6 621 000

7 99 000

8 216 000

9 18 000

10 9 500 000



1 1200 + 240 + 60 + 12 = 1512

2 6

3 50

4 3

5 10

6 20

7 100

8 60

8 Using a worked problem

1 58

2 426

3 3408

4 50

5 12 354

6 12 354

7 213

8 116

UNIT 6: Algebra1 Simple equations

1 21

2 9

3 8

4 12

5 9

6 15

7 16

8 28

2 Order of operations

1 37

2 15

3 33

4 10

5 13

6 3

7 498

8 36

9 5

10 152

3 Operations

1 20 ÷ 5 = 2 × 2

2 6 × 25 = 15 × 10

3 19 − 2 − 2 = 5 × 3

4 25¢ × 4 = $10 ÷ 10

5 16 − 7 × 2 = 8 ÷ 22

6 (32 + 1) + 20 = 65 − 35

7 6 + 9 × 6 = 60

8 80 − 2 × 10 = 7 × 8 + 4

9 23 + 0 = 28 − 4 − 42 or

23 – 0 = 28 − 4 − 42

10 24 ÷ 4 ÷ 2 = 51 ÷ 17

4 Solving for the unknown

1 a 5

b 30

c 8

d 66

e 4

f 3

g 80

2 a h – 20

b m2 + 5

c 7(p + 1) y − $14

d (y − $14) ÷ 2 or 2

e (g – 14) ÷ 3 or g – 143

5 Solving for the unknown

1 11

2 65

3 121

4 30

5 25

6 14

7 17

8 16

9 9

10 15

6 Mental Maths practice

1 15 years old

2 6 years old

3 $70

UNIT 7: Operations – division1 Simple division

1 110

2 112

3 11

4 20

5 16

6 385

7 135

8 46

9 476

10 428

2 Long division algorithm

1 234 r 2, or 234.125, or 23418

2 79 r 4, or 79.__36, or 79 4

11

3 852 r 5, or 852.8_3, or 8525

6

4 298 r 3, or 298.2, or 29815

5 416 r 3, or 416.1__36, 416 3

22

6 532


Challenge:

He can make 10 different combinations.

Student diagrams or methods will vary, but should be shown.

3 Problems

1 He caught on average 8 fish per day.

2 53 sacks were needed.

3 It will take 17 weeks.

4 There will be 7 full boxes.

4 2-digit division

1 92 r 8, or 92 47

2 405 r 5, or 405 532

3 310 r 8, or 310 413

4 $2.27

5 63

6 $24.12

5 3-digit divisors

1 180 bottles (Note: it asked how many can be filled.)

2 10 days (Note: the answer represents full days of work.)

3 304

4 523 r 5

5 186 r 6

6 220

7 234

8 210

9 240 r 8 or 240 4123

10 130 r 50 or 130 562

UNIT 8: Fractions1 Rounding and estimating fractions

1 Fractions should be placed on the number line as follows:

Between 0 and 12:

29 2

637

Between 12 and 1:

59 17

30 34 5

6 1012

78 14

15

2 about 112

3 about 1

4 about 0

5 about 3

6 about 112

2 Simple fractions

1 0.15

2 0.4

3 0.06

4 0.008

5 0.019

6 65100 = 13

20

7 61000 = 3

500

8 1941000 = 97

500

9 310

10 4100 = 1

25

11 710

12 112

13 13

14 1

15 6

3 Equivalent fractions

1 45

2 2036

3 4550

4 2460

5 2128

6 3696

7 832

8 2056

9 3045

10 3048

11 710

12 742

4 Reducing

1 34

2 1213

3 18

4 23

5 27

6 15

7 13

8 1635

9 925

10 1217

11 35

12 78


5 Improper fractions and mixed numbers

1 1110

2 113

3 134

4 123

5 149

6 245

7 113

8 194

9 272

10 a 4220

b 2 110

c 3 cakes

6 Addition

1 1 720

2 119

3 5960

4 1 124

5 1 320

6 1118

7 1 320

8 11324

7 Subtraction

1 415

2 215

3 112

4 34

5 112

6 16

7 124

8 29

9 325

10 813

8 Adding and subtracting mixed numbers

1 712

2 1

3 1178

4 558

5 18 712

6 1538

7 1416

8 923

9 120

10 5 215

9 Multiplying

1 914

2 310

3 532

4 825

5 325

6 3245

7 29

8 512 (Note: asks about books for older children.)

10 Multiplying

1 112

2 49

3 142

4 2

5 25

6 225

7 920

8 155

9 2 34

10 2 14

11 Multiplying

1 a 412 buckets

b 112 buckets

c 3 buckets

2 1012 minutes

3 4 students

4 145 hours (or 108 minutes)

12 Division

1 212

2 2

3 134

4 113

5 623

6 18

7 30 bags can be filled (Note: it asks how many full bags.)

8 12 coloured bulbs

13 Division

1 13

2 213

3 178

4 28

5 44

6 5

7 8

8 14 containers


UNIT 9: Working with decimals1 Simple multiplication

1 0.96

2 0.0497

3 0.072

4 0.01416

5 0.033

6 2.968

7 735 litres

8 5178.24

9 $53.25

10 $349.65

2 Multiplying

1 $985.60

2 5853.844

3 $58.75

4 $0.48

5 24.096

6 $6860.52

7 0.0000012

8 1.44

3 Estimating and rounding

1 41.4

2 67.28

3 21115

4 24.29

5 30.21

6 18

7 342

8 528

9 6

10 46

11 341

12 4059

13 $80

14 Roughly $64 (accept $65)

15 Approximately $250


1 7120

2 0.03

3 3200

4 63 419

5 83 769

6 43 752.8

7 0.48

8 $230.70

9 $90 850

10 $48 361.90

5 Multiplying money

1 $9.63

2 $13.30

3 $3315

4 $107.05

5 $16.07

6 $143

7 $3.51

8 $142.56

9 $11.59

10 $12.05

6 Dividing decimal numbers

1 $3.25

2 6.62

3 1.26

4 11.3

5 6.14

6 0.36

7 $4.50

8 $8.25

7 Division with remainders

1 a 0.8

b 0.625

c 0.125

2 14.5 kg (or 1412 kg)

8 Dividing completely (no remainder)

1 18.5

2 21.5

3 $12.20

4 13.6

9 Dividing by multiples of 10

1 5.96193

2 0.37269

3 21.781

4 0.0798

5 0.2375

6 0.14245

7 52.9

8 0.164

9 0.0732

10 0.75

11 5.5238

12 3.7

13 0.629

14 0.0028

15 40.86

10 Dividing by a decimal number

1 2.1

2 6.2

3 4.26

4 33.2

5 20.3

6 94.2

11 Problem solving

Review:

1 12.4 litres per minute

2 21 cm

3 $217.50

4 $6.35

5 $0.95


11 58

0.625621

2%

12 78

0.875871

2% (87.5%)

13 13

0._3

3313%

14 23

0._6

6623%

4 Problem solving

1 $72

2 1535 or 15.6

3 $95

4 $5.31

5 $849

5 Solving mentally

1 239

2 660

3 760

4 95

5 $5

6 42

7 15.5

8 62

9 11

10 $18

6 Percent of amount

1 $2.50

2 34

3 $45.50

4 40

5 $2.46

6 75

7 26

8 $29.68

9 195

10 $22

11 1893

12 195

7 Forming percent

1 50%

2 35%

3 40%

4 54 crabs

5 4 students

8 Finding the total

(Suggested approaches, solving simply with proportionality, are shown for each. However, accept all reasonable working.)

1 55 20% = 11, 100% = 55

2 60 marks 80% = 48, 100% = (48 ÷ 8) × 10

3 780 45% = 351, 5% = 351 ÷ 9 = 39, 100% = 39 × 20

4 15 servings 6623% = 2

3, 10 = 23 so

15 = 1 whole recipe

UNIT 10: Percents, fractions and decimals1 Percents, fractions and decimals

1 87%

2 29%

3 17%

4 61%

5 0.48

6 0.14

7 0.09

8 0.9

9 50100 = 50%

10 20100 = 20%

11 60100 = 60%

12 6100 = 6%

13 12100 = 12%

14 1 3100 = 103%

2 Percents, fractions and decimals

1 0.8 = 80%

2 0.75 = 75%

3 0.52 = 52%

4 0.45 = 45%

5 0.64 = 64%

6 0.42 = 42%

7 225

8 360%

9 2%

10 19 out of 25

3 Percents, fractions and decimals

1 75%

2 40%

3 80%

4 60%

Fraction Decimal Percent

5 12

0.5 50%

6 14

0.25 25%

7 15

0.2 20%

8 25

0.4 40%

9 18

0.125121

2% (12.5%)

10 38

0.375371

2% (37.5%)


5 160 students 52 students = 20%, 5 × 52 = 100%

6 300 books 258 = 86%, 3 = 1%, 300 = 100%

9 Discount

1 a $8.20

b $12.30

2 $414

3 $50

4 $20

10 VAT and tax

1 a $70

$470

b $10.35

$79.35

c $38

$238

d $1.92

$40.32

2 a $46

b $73.60

3 a $64.50

b $49.67

UNIT 11: Measures of central tendency1 Overview

Review:

1 a 10

b 16

c 19

d 15

e 13.86

2 5.81

3 $3.65

4 a 58%

b 67%

c 49%

e Kareem is improving.

5 154.75 cm or 15434 cm

2 Working with means

1 400 total 80 mean

2 375 total 75 mean

3 90 score 92.2 mean

4 68 score 317 total

3 Problem solving

1 a 1368

b 402

2 a $311.85

b $62.37

c $207.15

d $86.5

3 a The group of three students has the highest top score.

b 3085

UNIT 12: Measurement1 Metric conversions

1 1000 m

2 1 m

3 1000 mm

4 1 kg

5 1 ℓ

6 1 g

7 1000 mg

8 1 km

9 0.001 ℓ

10 10 mm

11 Divide

12 Multiply

13 10

14 Divide, 100

15 Divide

2 Length

1 7.1 m

2 24.1 cm

3 0.099 m

4 6.74 cm

5 1.102 km

6 30.6 mm

7 890 cm

8 1500 g

3 Capacity

1 1 ℓ

2 21 mℓ

3 21000 mℓ

4 1800 mℓ

5 20 mℓ

6 0.604 ℓ

7 600 mℓ

8 1500 mℓ

9 3.375 ℓ

10 125 mℓ


4 Mass

1 2420 g

2 31 700 g

3 7.23 kg

4 12 500 g

5 0.875 kg

6 440 g

Enrichment: 5 Customary measures

pounds your weight

feet height

ounces flour for a cake

miles long distances

tons cargo container

gallons fuel

pints goat milk

yards length of cloth

inches width of a picture

teaspoons cough syrup

Challenge:

17 degrees

6 Time

1 5 : 00 p.m.

2 3 : 00 a.m.

3 7 : 30 p.m.

4 11 : 15 p.m.

5 6 : 10 a.m.

Enrichment: 7 24-hour clocks

1 13 : 00

2 08 : 00

3 12 : 05

4 18 : 35

5 00 : 18

8 Time

1 240 minutes

2 86 days

3 3 years

4 420 seconds

5 30 hours

6 34 day

7 212 years

9 Calculating time

1 7 hrs 29 mins

2 29 hrs 15 mins

3 21 hr 22 mins

4 13 hrs 15 mins

5 67 hrs 4 mins

6 28 hrs 7 mins

7 8 hrs 54 mins

8 9 hrs 5 mins

9 2 hrs 32 mins

10 12 : 21

UNIT 13: Geometry1 Lines

1 GH and any other line, or, AB and any other line

2 CD and EF

3 AB and either CD or EF

2 Lines

1 (straight line)

2 (angle greater than 180 and less than 360 degrees)

3 (angle greater than 90 and less than 180 degrees)

4 (angle less than 90 degrees)

3 Angles

1 68°

2 30°

3 130°

4 Angles

1 x = 90°

2 x = 115°

3 3x = 30°, x = 10°

4 x = 55°

5 x = 30°

6 x = 65°

5 Angles on clocks

1 30 minutes

2 60 degrees

3 210 degrees

4 300 degrees

5 13

6 210 degrees

7 150 degrees

8 20%


6 Diagonals

1 (2 diagonals)

2 (0 diagonals)

3 (9 diagonals)

4 (5 diagonals)

5 (9 diagonals)

6 (2 diagonals)

7 Transformations

1 Check student drawing is correct.

2 Check student drawing is correct.

8 Tessellations

Student answers will vary. All should use two or three polygons and form a repeating pattern without gaps or overlaps.

9 Line symmetry

1 Check students have drawn five lines of symmetry.

2 There are no lines of symmetry.

10 Properties of 3D shapes

1 Faces 5

Edges 9

Vertices 6

Name: triangular prism

2 Faces 6

Edges 12

Vertices 8

Name: cuboid

3 Faces 2

Edges 1

Vertices 0 (by definition of a vertex ‘where 3 or more edges meet’)

Name: cone

4 Faces 6

Edges 12

Vertices 8

Name of shaded face: trapezium (it is also a quadrilateral, and a closed polygon)

11 Nets of 3D shapes

1 (net of a cone: 1 semi-circle and 1 circle)

2 (triangular prism: 2 triangles and 3 rectangles)

Challenge:

From the arrowhead, a line goes up the left column and extends above the top dot the same distance as if to a new dot. The next line goes down, passing through the two dots (middle dot in top row and middle dot in right column), ending to the right of the bottom right dot (as if to a new dot extended out to the right of the bottom right dot). Finally, the fourth line goes across the bottom row and back to the arrowhead.

UNIT 14: Perimeter, area and volume1 Calculating perimeter

1 a 46 cm

b 47.6 m

2 a x = 7 cm

b x = 9.75 cm

c x = 10 m

d 2x = 9

x = 4.5 m or 412 m

2 Perimeter problem solving

1 128 cm

2 22 m

3 a 16 m

b 32 m

4 148 cm

3 Perimeter problem solving

1 12.5 or 1212 m

2 2.25 or 214 cm

3 14 cm

4 10 cm

5 5 m (Note: some students could interpret the problem such that 2 m is for both lengths, and if their reasoning is explained well, an answer of 3 m might be acceptable. However,


students who do not grasp the need for two lengths might also reach that solution.)

4 Sketching area problems

Check that students have sketched the information in each word problem.

1 187 m2

2 a 13 m

b 52 m

3 a 150 cm2

b 234 cm2

c 84 cm2

5 Calculating

1 15.8 cm

2 305 cm

3 728.5 or 72812 mm

4 18.25 or 1814 m

6 Area

1 880 m2

2 312.5 cm2

3 44 cm2

4 97.5 m2

7 Area of compound shapes

1 224 – 56 = 168 cm2

2 7.875 m2

8 Exploring of area

1 63 – 24 = 39 m2

2 64 + 12(2 × 8) = 72 cm2

3 Check student sketches. area = 5812 in2

9 Area of circles

1 the distance around

2 twice (or two times as wide as)

3 94.2 cm

4 153.86 cm2

10 Surface area

area of bottom 80 × 50 = 4000 cm2

area of top 80 × 50 = 4000 cm2

area of long sides 80 × 15 = 1200 (× 2) = 2400 cm2

area of short sides 50 × 15 = 750 (× 2) = 1500 cm2

total surface area = 11 900 cm2

11 Volume

1 343 cm3

2 0.7875 m3

3 90 m3

4 313 m

Challenge:

a 8 m3

b Volume is a measurement of the amount of space taken up by a 3D object, so the volume of the tank is in m3. Capacity is the measurement of what fills the space, so the water it takes to fill the tank is measured in litres.

UNIT 15: Ratio and proportion1 Ratio

Review:

1 2 : 3, 2 to 3, 3 : 2, or 3 to 2 (Note: we do not encourage the use of the horizontal bar to depict ratio, as it has been found to contribute to a misunderstanding of proportionality.)

2 a 2 : 7

b 1 : 10

c Student answers will vary.

3 a 7 : 2

b 3 : 2

c 31 : 11

d 9 : 2

4 Correct results will show 20 equal parts, with 14 in one colour and 6 in another colour.

2 Equivalent ratios

1 2 : 1

2 1 : 2

3 4 : 1

4 11 : 9

5 9 : 1

6 21 : 10

7 =

8 =

9 ≠

10 ≠


3 Equivalent ratios

1 a 5

b 14

c 40

2 60

3 5 drawings

4 a yes

b no

c no

4 Using proportion

1 a 3 cups

b 412 cups

c 144 treats

d 96 treats

2 EC $21.65

3 40 students

4 a 36 litres

b 350 km

5 Rate

1 a 96

b pages minutes

3 : 4

45 : 60

c 30 pages

2 $7.50

3 9 minutes

6 Unit rate

1 $28

2 $1.50

3 $0.75

4 $13

5 $27

7 Unit rate

1 a 15 km

b 40 cm

2 a 425 km/hr

b 6 hours

3 6 days

8 Problem solving

1 15 limes

2 $2

3 a 9 baskets

b 20 baskets

4 212 minutes

Challenge:

a 136 km

b 20 hours

9 Problem solving

1 112 cups

2 18 cups

3 5 tins

4 1 : 1

5 112 cups

10 Working with ratios

1 35 years old (now)

2 a 24 children

b 36 children

3 1445 points

4 a 36 white cars.

b 6 : 5

5 9 days

11 Unequal sharing

1 Logan receives $231

Ajay receives $349

2 First share $250

Each of the other shares $125

3 Neville earned $66

Habib earned $22

Tony earned $11

UNIT 16: Problem-solving skills1 Step 1 : $50.35

Step 2 : 50.35 ÷ 25

Step 3 : ($50.35 ÷ 25) × 100 = $201.40

2 $48 = 40% of my money

$12 = 10% of my money

$120 = 100% of my money

3 20% = $200, so 100% = $1000

The trip costs $1000.

4 9 dogs = 10%, so 90 dogs = 100%

There were 90 dogs at the shelter.


2 Working with units

1 6 = $18.60, 1 = $3.10

5 hot dogs cost $15.50

2 3 = $5.55, 1 = $1.85

5 slices cost $9.25

3 10 = $7.90, 1 = $0.79, 50 = $39.50

10 = $7.90 multiplied by 5, 50 = $39.50

Accept all other reasonable approaches.

For example: 10 = $7.90, 100 = $79.00, divide by 2, 50 = $39.50

4 4 = $3, 1 = $0.75 Tante May

5 = $4, 1 = $0.80 Tante Dee

Tante May had the lowest price.

3 Working with units

1 3 fish = 6%, 1 fish = 2%, 50 fish = 100%

There were 50 fish altogether.

2 108 trees = 27%, 4 trees = 1%, 400 trees = 100%

There were 400 trees before the storm.

3 13 = 9, one whole cake = 27 slices

4 78 = $91, 1

8 = $13, 88 = $104

Altogether I have $104

5 14 = 12 teachers, 4

4 = 48 teachers

The school has 48 teachers.

4 Combining unit rates

1 In one minute Avi peeled 15 of the bag.

In one minute Tom peeled 14 of the

bag.

15 + 1

4 = 920 of a bag peeled in one

minute, so together it took them 229

minutes.

2 3000 km in 6 hours is 500 km/hr

3 560 km/hr

5 Sketching word problems

Sketches will vary. The answer is 12 km.

6 Using formulae

1 70 km/hr

2 260 km

3 3 km ÷ 4 km/hr = 45 mins or 34 hour

4 10 km = 100 min, 1 km = 10 min, 6 km = 60 minutes

Average speed is 6 km/hr

5 34 hr × 6 km/hr = 41

2 km

7 Looking for links

1 25

2 24

3 35

4 543.7

5 18

6 7

7 12.5 or 1212

8 24

8 Calendar

1 25 days

2 59 days

3 5 months

4 4 leap years

5 16th January

9 Poles and spaces

1 7770 m or 7 km 770 m

2 The green lights are 50 cm apart.

3 101 posts

UNIT 17: Statistics / sets / integers1 Making a graph

Check students have drawn the graph correctly. The title of the graph should be ‘Heights of students in Mr Yen’s class’.

Each column of the horizontal axis should be marked to show 134 cm, 138 cm, 142 cm, 146 cm, 150 cm and 154 cm, and the axis should be labelled ‘Height’.

The scale of the vertical axis should be numbered from 1 to 6. The vertical axis should be labelled ‘Number of students’. The bars should be shaded correctly to show:

two students at 134 cm

four students at about 138 cm

five students were about 142 cm tall

seven were 146 cm


five were about 150 cm

four were about 154 cm.

2 Tally chart, pie chart and line graph

a Adventure //// ////

Mystery //// //// ////

Fantasy //// //// /

Science //// //// //// ///

Science fiction //// ///

b Check students’ pie charts have filled in proportionately from a total of 60, i.e. Adventure 9 parts, Mystery 14 parts, Fantasy 11 parts, Science 18 parts and Science fiction 8 parts.

c Check students’ line graphs. The horizontal axis should be labelled with the subjects, and the vertical axis numbered in 2s to 20. Put a dot in each column to represent the numbers for each topic. Connect the dots with straight lines.

3 Understanding pie charts

1 14

2 45 degrees

3 food stall

4 Safari swing

5 90 degrees

6 x = 40 degrees

4 Comparing charts

1 The bar graph best shows progress over time. Reasons might include that the order is more obvious, the comparison of totals is more noticeable, or the range of scores low to high can more easily be compared.

2 The pie chart would be good for comparing several different categories or topics that are part of one whole or one event, as the proportions could be quickly seen.

5 Probability

1 Students’ responses will vary.

2 Check individual student’s responses compared to question 1.



5 0

6 1

Enrichment: 6 Venn diagrams

1 Students may label ovals in any order.

2 a–f Results within each labelled oval are shown here (Note: students may not use a matching layout, but will be correct once numbers match the labels.)

In the overlap between camping and hiking, 8

In the open part of Hiking not overlapping , 4

In the open part of Camping not overlapping, 1

In the overlap between climbing and camping, 9

In the overlap between hiking and climbing, 3

In the open part of Climbing not overlapping, 7

There should be no number in the very centre where all three ovals overlap.

g In the box, but not in the ovals, should be ‘1’.

3 12 students like all three activities.

Enrichment: 7 Negative numbers

1 $6

2 a 10 metres deep

b −3 + −10 = −13

The total depth is −13

3 4

4 −8

5 3

6 −10

7 13

9 13

10 5

11 3

12 −10

13 −6

14 7


Enrichment: 8 Integers on the number line

1 −17, −15, −11, −1, 4

2 a 5

b −8

c 0

d −8

e 0

f 25

g 20

h −89

i −2

j 2

Enrichment: 9 Comparing integers

1 <

2 >

3 <

4 >

5 <

6 >

7 <

8 >

9 <

10 >

11 −1

12 1

13 10

14 −41

15 91

Enrichment: 10 Calculating with integers

1 −155

2 8

3 −36

4 74

5 28

6 30

7 −98

8 31

9 159, 62, −7, −103, −160

10 23, 3, −13, −33, −34


1 20

2 −2

3 −98

4 −62

5 −1

6 −33

7 −292

8 16

9 0

10 0


1 100

2 −25

3 5

4 45

5 94

6 −196

7 −11

8 −6

9 7

10 −225

Enrichment: 13 Coordinates

1–5 Check students’ placement of drawings.

6 (−4, 3)

7 (−5, −3)

8 Check top-right quadrant for students’ response.

9 Check students’ answer on the vertical axis at +5.

10 Check top-left quadrant for students’ response.

UNIT 18: Consumer / Business Mathematics1 Profit or loss

1 $3.49 profit

$3.85 loss

$114.49 selling price

2 $22.50 – $14.25 = $8.25 profit

3 $2790 profit

2 Profit or loss percent

1 $100 loss 20% loss

2 $81.90 selling price $3.90 profit

3 $500 profit ~8%

4 $220 15%

5 $624 selling price $26 loss

6 She wants to make $40 (160% × $25), so will sell the oranges at 3 for $4, or $1.33 each.

3 Problem-solving profit and loss

1 $57 375

2 a $1.02

b $0.82 is less than the original $0.85, so he would not make a profit.


3 a $378

b 45%

4 Simple interest

1 $2000 × 2% × 6 years = $240

2 The first year he received $15 interest.

5 VAT

1 $69 2 $170 3 $97.75

6 Hire purchase

1 $774 2 $625

7 Exchanging currency

1 Che paid more (equivalent of TT $44.80)

2 US $60 US

3 BB $16

4 BZ $144

Review 11 C

2 B

3 A

4 D

5 A

6 B

7 A

8 A

9 D

10 C

Review 21 D

2 D

3 D

4 A

5 A

6 A

7 A

8 D

9 B

10 A

Review 31 B

2 A

3 C

4 B

5 D

6 B

7 D

8 C

9 A

10 B

Review 4 1 5 267 392.041

2 16 575

3 108

4 16, 32, 48, 64, 80

5 (81 + 3) ÷ 7 = 12

6 13 (Note: pattern is +0, +1, +2, +3, +4)

7 64 cm2

8 n = 15

9 97.4

10 796

Review 5 1 482 637 591

2 5996

3 225

4 36 students

5 ×, ÷, ×

6 300.6, 30.653, 30.65, 30.603

7 145 000 mℓ

8 87.5% or 8712%

9 1.4944

10 112

Review 61 $6000

2 4787

3 144 boxes

4 2 × 3 × 17

5 28 (Note: order of operations applies.)

6 20 ribbons

7 n = 5

8 1.06 m

9 568

10 4 156

196 CD-ROMactivitydirectory•BrightSparksTeacher’sBook6

CD-ROM activity directory (list of topics) for level 6

C GeometryLines and angles

Lines, segments and rays: revision

Angles

Measuring and drawing angles

Polygons and Polyhedra

Polygons

Congruence and tessellations

Polyhedra

Nets and solids

Geometry problems

Parts of a circle

Rotational symmetry

Movement geometry

D Algebra and measuresAlgebra

Substitution in algebra

Simplifying algebraic expressions

Equations

Patterns and sequences

Functions and relationships

Area

Area of compound shapes

Area of any triangle

Area of parallelograms

Area of trapeziums

Surface area and volume

Surface area of cuboids

Volume of cuboids

A Understanding numbers Integers and decimals

Integers

Rounding and approximation

Large numbers

Decimal numbers

Number properties

Rules of divisibility

Multiples

Factors

Squares, roots and powers

Fractions

Equivalent fractions

Comparing and ordering fractions

Improper fractions and mixed numbers

Fractions and percents

B Numerical operationsMixed operations

Addition and subtraction of whole numbers

Order of operations: +, –, ×, ÷

Brackets

Multiplication

Multiplying by a two-digit number

Multiplying by a three-digit number

Multiplying decimals

Division

Dividing by a single-digit number

Remainders as decimals

197BrightSparksTeacher’sBook6•CD-ROMactivitydirectory

E Ratio, proportion and percentages Ratio

Ratio: revision

Ratio and measures

Rate and speed

Proportion

Proportion: revision

Direct proportion

Scale drawings

Percentages, profit and loss

Expressing percentages

Percentages of quantities

Percentage increase

Percentage decrease

F Handling dataCoordinates

Coordinates: revision

Moving shapes on grids

Functions and coordinates

Coordinates in four quadrants

Graph and charts

Straight-line graphs

Time graphs

Pie charts

Probability and statistics

Probability: revision

Mode and mean

198 LinktoResources•BrightSparksTeacher’sBook6

Link to Caribbean Teacher Resources websiteThere are free worksheets for Level 3 on Macmillan Caribbean’s Teacher Resources website. (copy this URL into your browser):

http://www.macmillan-caribbean.com/pages.aspx/primary/Bright_Sparks_Resources/

Some sample topics included for this year level are:

Level 6 / Standard 5 / 10–11+ yrs

Numeracy Substituting percent, fraction, decimal Arithmetic Putting in signs to make equations true Money Savings and interest Challenges Tessellations Games / Activities Products in a queue Enrichment Simple algebra

For Advanced Students, or for additional practice, Teachers may also find the following resource pages useful :

Standard 6 (Belize) / 12+ yrs / Transition to secondary / Lower 1st Form

Some sample topics included for this year level are: Numeracy Working with rates Arithmetic Using formulae in problem solving Money Personal budget Challenges Scale and proportion Games / Activities Probability Enrichment Working with number puzzles

199Bright Sparks Teacher’s Book 6

Notes

200 Bright Sparks Teacher’s Book 6

Notes

second edition teacher’s book 6 - macmillan caribbean · 2017-06-29 · general introduction 4...

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