Number and Algebra (Australian Curriculum): Year 5 - Ages 10-11 by Teacher Superstore

RIC-6089 11/625

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5) Published by R.I.C. Publications® 2012 Copyright© R.I.C. Publications® 2012 Revised edition 2013 ISBN 978-1-92175-073-1 RIC–6089 Titles available in this series:

A number of pages in this book are worksheets. The publisher licenses the individual teacher who purchased this book to photocopy these pages to hand out to students in their own classes. Except as allowed under the Copyright Act 1968, any other use (including digital and online uses and the creation of overhead transparencies or posters) or any use by or for other people (including by or for other teachers, students or institutions) is prohibited. If you want a licence to do anything outside the scope of the BLM licence above, please contact the Publisher.

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All material identified by is material subject to copyright under the Copyright Act 1968 (Cth) and is owned by the Australian Curriculum, Assessment and Reporting Authority 2013. For all Australian Curriculum material except elaborations: This is an extract from the Australian Curriculum. Elaborations: This may be a modified extract from the Australian Curriculum and may include the work of other authors. Disclaimer: ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that: • The content descriptions are solely for a particular year and subject; • All the content descriptions for that year and subject have been used; and • The author’s material aligns with the Australian Curriculum content descriptions for the relevant year and subject. You can find the unaltered and most up to date version of this material at http://www.australiancurriculum.edu.au/ This material is reproduced with the permission of ACARA.

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Australian Curriculum Mathematics resource book: Number and Algebra (Foundation) Australian Curriculum Mathematics resource book: Number and Algebra (Year 1) Australian Curriculum Mathematics resource book: Number and Algebra (Year 2) Australian Curriculum Mathematics resource book: Number and Algebra (Year 3) Australian Curriculum Mathematics resource book: Number and Algebra (Year 4) Australian Curriculum Mathematics resource book: Number and Algebra (Year 5) Australian Curriculum Mathematics resource book: Number and Algebra (Year 6)

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Internet websites In some cases, websites or specific URLs may be recommended. While these are checked and rechecked at the time of publication, the publisher has no control over any subsequent changes which may be made to webpages. It is strongly recommended that the class teacher checks all URLs before allowing students to access them.

View all pages online PO Box 332 Greenwood Western Australia 6924

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AUSTRALIAN CURRICULUM MATHEMATICS RESOURCE BOOK: NUMBER AND ALGEBRA (YEAR 5) Foreword Australian Curriculum Mathematics resource book: Number and Algebra (Year 5) is one in a series of seven teacher resource books that support teaching and learning activities in Australian Curriculum Mathematics. The books focus on the number and algebra content strands of the national maths curriculum. The resource books include theoretical background information, resource sheets, hands-on activities and assessment activities, along with links to other curriculum areas.

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

Format of this book............................................................ iv – v

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Identify and describe factors and multiples of whole numbers and use them to solve problems (ACMNA098)

– Teacher information ......................... 6 – Hands-on activities .......................... 7 – Links to other curriculum areas ........ 8

– Resource sheets ...............9–16 – Assessment ...................17–18 – Checklist ...............................19

• N&PV – 2

Use estimation and rounding to check the reasonableness of answers to calculations (ACMNA099)

– Teacher information ....................... 20 – Hands-on activities ........................ 21 – Links to other curriculum areas ...... 22

– Resource sheets ........... 97–110 – Assessment ............... 111–112 – Checklist .............................113

• F&D – 3

Recognise that the place value system can be extended beyond hundredths (ACMNA104)

– Teacher information ..................... 114 – Hands-on activities ...................... 115 – Links to other curriculum areas .... 116

• F&D – 4

– Resource sheets ......... 117–130 – Assessment ............... 131–132 – Checklist .............................133

Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies (ACMNA100)

– Teacher information ....................... 32 – Hands-on activities ........................ 33 – Links to other curriculum areas ...... 34

– Resource sheets .............35–46 – Assessment ...................47–48 – Checklist ...............................49

• N&PV – 4

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Solve problems involving division by a one-digit number, including those that result in a remainder (ACMNA101)

– Teacher information ....................... 50 – Hands-on activities ........................ 51 – Links to other curriculum areas ...... 51

• N&PV – 5

– Teacher information ....................... 94 – Hands-on activities ........................ 95 – Links to other curriculum areas ...... 96

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– Resource sheets .............52–60 – Assessment ...................61–62 – Checklist ...............................63

Compare, order and represent decimals (ACMNA105)

– Teacher information ..................... 134 – Hands-on activities ...................... 135 – Links to other curriculum areas .... 136

– Resource sheets ......... 137–151 – Assessment ............... 152–153 – Checklist .............................154

Answers ..................................................................................155

Money and Financial Mathematics ................................156–175 • M&FM – 1 Create simple financial plans (ACMNA106)

– Teacher information ..................... 156 – Hands-on activities ...................... 157 – Links to other curriculum areas .... 158

– Resource sheets ......... 159–171 – Assessment ............... 172–173 – Checklist .............................174

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• N&PV – 3

– Resource sheets .............23–28 – Assessment ...................29–30 – Checklist ...............................31

Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator (ACMNA103)

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Number and Place Value .................................................... 6–75 • N&PV – 1

• F&D – 2

Answers ..................................................................................175 Patterns and Algebra ....................................................176–221 • P&A – 1

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Use efficient mental and written strategies and apply appropriate digital technologies to solve problems (ACMNA291)

– Teacher information ....................... 64 – Hands-on activities ........................ 65 – Links to other curriculum areas ...... 66

– Resource sheets .............67–71 – Assessment ...................72–73 – Checklist ...............................74

Answers ....................................................................................75

Fractions and Decimals ...................................................76–155 • F&D – 1 Compare and order common unit fractions and locate and represent them on a number line (ACMNA102)

– Teacher information ....................... 76 – Hands-on activities ........................ 77 – Links to other curriculum areas ...... 78

– Resource sheets .............79–90 – Assessment ...................91–92 – Checklist ...............................93

Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction (ACMNA107)

– Teacher information ..................... 176 – Hands-on activities ...................... 177 – Links to other curriculum areas .... 178

– Resource sheets ......... 179–196 – Assessment ............... 197–198 – Checklist .............................199

• P&A – 2

Use equivalent number sentences involving multiplication and division to find unknown quantities (ACMNA121)

– Teacher information ..................... 200 – Hands-on activities ...................... 201 – Links to other curriculum areas .... 202

– Resource sheets ......... 203–217 – Assessment ............... 218–219 – Checklist .............................220

Answers ..................................................................................221 New wave Number and Algebra (Year 5) student workbook answers ........................................................222–231

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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FORMAT OF THIS BOOK This teacher resource book includes supporting materials for teaching and learning in all sections of the Number and Algebra content strand of Australian Curriculum Mathematics. It includes activities relating to all sub-strands: Number and Place Value, Fractions and Decimals, Money and Financial Mathematics, and Patterns and Algebra. All content descriptions have been included, as well as teaching points based on the Curriculum’s elaborations. Links to the Proficiency Strands have also been included. Each section supports a specific content description and follows a consistent format, containing the following information over several pages: • teacher information with related terms, student vocabulary, what the content description means, teaching points and problems to watch for • hands-on activities • links to other curriculum areas

• resource sheets • assessment sheets.

• a checklist

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Answers relating to the assessment pages are included on the final page of the section for each sub-strand (Number and Place Value, Fractions and Decimals, Money and Financial Mathematics, and Patterns and Algebra). (NOTE: The Foundation level includes only Number and Place Value, and Patterns and Algebra.)

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The length of each content description section varies.

Teacher information includes background information relating to the content description, as well as related terms and desirable student vocabulary and other useful details which may assist the teacher.

Related terms includes vocabulary associated with the content description. Many of these relate to the glossary in the back of the official Australian Curriculum Mathematics document; additional related terms may also have been added.

What this means provides a general explanation of the content description.

the teacher would use—and expect the students to learn, understand and use—during mathematics lessons.

points relating to the content description.

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The proficiency strand(s) (Understanding, Fluency, Problem Solving or Reasoning) relevant to each content CONTENTdescription DESCRIPTION are are listed. listed.

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What to to look watchforforsuggests suggestsany any What diffi culties and misconceptions difficulties and misconceptions the the students students might might encounter encounter or or develop. develop.

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Reference to relevant pages in New wave Number and Algebra (Year 5) student workbook.

Hands-on activities includes descriptions or instructions for games or activities relating to the CONTENTdescriptions content DESCRIPTIONs or elaborations. or elaborations. Some Some of the of the hands-on hands-on activities activities are supported are supported by resource by resource sheets. Where applicable, these will be stated for easy reference.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

FORMAT OF THIS BOOK Links to other curriculum areas includes activities in other curriculum areas which support the content description. These are English (literacy), Information and Communication Technology (ICT), Health and Physical Education (ethical behaviour, personal and social competence) and Intercultural Understanding (History and Geography, the Arts, and Languages). This section may list many links or only a few. It may also provide links to relevant interactive websites appropriate for the age group.

r o e t s Bo r e p ok u S Resource sheets are provided to support teaching and learning activities for each content description. The resource sheets could be cards for games, charts, additional worksheets for class use, or other materials which the teacher might find useful to use or display in the classroom. For each resource sheet, the content description to which it relates is given.

Assessment pages are included. These support activities included in the corresponding workbook. For each assessment activity, the elaboration to which it relates is given. Many of the questions on the assessment pages are in a format similar to that of the NAPLAN tests to familiarise students with the instructions and design of these tests.

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Cross-curricular links reinforce the knowledge that mathematics can be found within, and relate to, many other aspects of student learning and everyday life.

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o c . che e r o t r s super Each section has a checklist which teachers may find useful as a place to keep a record of the results of assessment activities, or their observations of hands-on activities.

Answers for assessment pages are provided on the final page of each sub-strand section.

Answers are also provided for New wave Number and Algebra (Year 5) student workbook. Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Number and Place Value—N&PV – 1

Identify and describe factors and multiples of whole numbers and use them to solve problems (ACMNA098)

RELATED TERMS

TEACHER INFORMATION

Factor

What this means

• A factor is a number that divides a larger number without leaving a remainder. • Every number has at least two factors: the number itself and the number 1. • Prime numbers have only two factors. Composite numbers have three or more factors.

• Students need to know the meaning of the terms ‘factors’ and ‘multiples’ and be able to identify them and apply them to solve problems.

• Common factors are factors that are exact divisors of two or more numbers. • The highest common factor of two or more numbers is the largest number that divides exactly into each number. Multiple

• A multiple is what results from multiplying a whole number by another whole number. Common multiples

Teaching points

• Practise basic multiplication facts as students need a good level of proficiency with division to determine the factors of a number. • Divisibility rules can be applied to determine if a number is divisible by another number without leaving a remainder. These rules may be found on many internet websites; e.g. <http://www. mathsisfun.com/divisibility-rules.html> and <http:// math.about.com/library/bldivide.htm>.

What to look for

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Common factors

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• Students who struggle with single-digit division because they lack fluency in multiplication facts and do not appreciate the connection between multiplication and division. Basic multiplication facts must be practised and the links with division made explicit.

• When two numbers are multiplied together, the answer is a common multiple; however‚ the two numbers may also have other common multiples which are achieved when each is multiplied by other whole numbers. • The lowest common multiple of two or more numbers is the smallest number that is common to each number.

See also New wave Number and Algebra (Year 5) student workbook (pages 2–8)

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Divisibility tests

• Learning multiplication facts helps the recall of factors of numbers up to 10. Applying divisibility tests helps to determine the factors of larger numbers. • If a number is divisible by two factors, it is also divisible by their product.

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Whole numbers

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• Whole numbers are the positive integers from zero to infinity that are used for counting. Number sequences

• When a set of numbers follows a regular pattern or sequence, the next, previous or missing numbers can be determined by analysing the pattern. Student vocabulary multiply

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

divide common factor common multiple

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Number and Place Value—N&PV – 1

HANDS-ON ACTIVITIES Multiplication table patterns Investigate the symmetry of a one-to-ten multiplication fact table (page 9) and see how because of the commutative property of multiplication, only 50% of the facts have to be learned.

Multiple skip counting Investigate the patterns created on a one-to-one hundred grid (page 9) by skip counting and colouring the multiples of different numbers.

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Factors and multiples

Times table shapes

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Use the blank factors and multiples times tables sheet (page 10). For any times table with students are not fluent, they write the 10 groups of four number sentences that relate the basic facts for that times table; e.g. 1 x 8 = 8, 8 x 1 = 8, 8 ÷ 8 = 1, 8÷ 1 = 8. In pairs, students use their sheets to test each other

For each times table, record the answers (multiples) on a ‘Times tables shape’ sheet (page 11). Highlight the ones place value digit of each answer. In order, draw lines among these numbers on the circle to create the times table shape. Compare the shapes made for different times tables.

Multiplication criss-cross Shuffle all the one-to-nine number cards from a deck of playing cards. Randomly select four cards to place in the centre sections of the multiplication square template (page 12). Record and answer the six multiplication facts that can be made by multiplying the two numbers in each horizontal, vertical and diagonal row. Write each fact in reverse on the left side of the sheet.

7 x 3 = 21 4x2=8 4 x 3 = 12 4 x 7 = 28 2x3=6 2 x 7 = 14

3 x 7 = 21

2x4=8

3x2=6

7 x 4 = 28

3 x 4=12

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Shuffle all the one-to-nine number cards from a deck of playing cards. Randomly select cards to make large numbers. Use the divisibility rules for numbers between two and ten to find factors for the random numbers made. See <http://www. mathsisfun.com/divisibility-rules.html> and <http://math.about.com/library/bldivide.htm>.

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Factors bingo

Prepare bingo cards (page 14) for the multiplication facts students need to practise. The caller rolls a 10-sided dice and calls out the number. Players cover any number on their card that is a multiple of that number and record the multiplication fact on the sheet (page 15) until a player has covered a horizontal‚ vertical or diagonal line of four numbers.

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Fibonacci sequence

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Join six sheets of 1 cm-squared paper together to create an approximate 60 cm x 60 cm square. Colour one single square in the centre of the large square. Mark a pencil dot in the centre of the square. In a different colour, colour the square immediately above the first and mark a pencil dot in the centre. To the left of this coloured rectangle and adjoining it, colour a square (2 cm x 2 cm) and mark the centre. Below this rectangle, colour another square (3 cm x 3 cm) and mark its centre. Join the centre dots in a circular, anti-clockwise sweep in the order that the squares were added. Continue adding new squares to the long side of the developing multicoloured rectangle, following an anticlockwise direction (to the right, above, to the left and below) until there is no room to add another square. Mark a dot in the centre of each square added and join as described. What have the joined pencil dots created? (a spiral) In a line, record the number of 1 cm squares for each square made from the centre. This is the Fibonacci sequence: (0) 1,1, 2, 3, 5, 8, 13, 21 … etc. What is the pattern of the sequence? (Each new number in the sequence is the sum of the previous two numbers.) Can you continue the sequence into three digits? (34, 55, 89, 144)

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Number and Place Value—N&PV – 1

LINKS TO OTHER CURRICULUM AREAS English • Read, write and spell all vocabulary associated with the descriptor. • Make as many small words as possible from the letters in words in the vocabulary list. • Write the vocabulary list in alphabetical order.

Information and Communication Technology • Find factor and multiple interactive games on the internet; for example: – <http://www.math-play.com/Factors-and-Multiples-Jeopardy/Factors-and-Multiples-Jeopardy.html> – <http://jamit.com.au/htmlFolder/FRAC1004.html#primeAndComposite>.

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Health And Physical Education

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History

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• Place four mats in the corners of a running area and one in the middle. Number the mats ‘2’, ‘3’, ‘4’, ‘5’ and ‘7’. Students move in ever changing directions using different actions; e.g. running, hopping, skipping. Teacher calls out a number from one to 100 (inclusive) and students move to a mat that is a factor of that number. Anyone running to an incorrect mat is out. • Each student writes a large number from one to 12 inclusive on a sheet of paper and pins it to his or her chest. They move as directed in a running area. The teacher calls out a number from one to 144. Students whose chest number is a factor of the number called out quickly search for a student with the matching second factor. • After playing this game a few times, students will recognise which numbers are factors for more numbers than others. • Working in pairs, students throw a ball to each other in different ways, while following the multiples of a given number; e.g. multiples of 3: six throws underarm right hand, nine throws underarm left hand, 12 throws overarm right hand, 15 throws overarm left hand, 18 bounces right hand, 21 bounces left hand, 24 underarm both hands, 27 overarm both hands, 30 bounces both hands.

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• Investigate ancient Egyptian multiplication.

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• Investigate the occurrence of the Fibonacci sequence in nature; for example, Fibonacci’s original studies investigated rabbits, the number of petals on flowers or the leaf arrangement on stems: <http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/fibnat.html>.

The Arts • Create shape patterns with the digits of common multiples.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Number and Place Value—N&PV – 1

RESOURCE SHEET One to ten multiplication fact table

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CONTENT DESCRIPTION: Identify and describe factors and multiples of whole numbers and use them to solve problems

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Number and Place Value—N&PV – 1

RESOURCE SHEET Factors and multiples

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Identify and describe factors and multiples of whole numbers and use them to solve problems

Sub-strand: Number and Place Value—N&PV – 1

RESOURCE SHEET Times tables shape

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CONTENT DESCRIPTION: Identify and describe factors and multiples of whole numbers and use them to solve problems

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Number and Place Value—N&PV – 1

RESOURCE SHEET

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Identify and describe factors and multiples of whole numbers and use them to solve problems

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Multiplication crisscross

Sub-strand: Number and Place Value—N&PV – 1

RESOURCE SHEET Divisibility rules

A number is divisible by two if … … the last digit is an even number; e.g. 342, 974, 786 A number is divisible by three if … … the sum of the digits is divisible by three;

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e.g. 724 24÷ 4 = 6 A number is divisible by five if … … the last digit is either 0 or 5; e.g. 390, 725, 560, 185

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A number is divisible by four if … … the last two digits are divisible by four;

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5 + 9 + 4 = 18 18 ÷ 3 = 6 A number is divisible by seven if … … the last digit when doubled and subtracted from the remaining digits, gives a difference that is divisible by seven;

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CONTENT DESCRIPTION: Identify and describe factors and multiples of whole numbers and use them to solve problems

e.g. 654 6 + 5 + 4 =15 15 ÷ 3 = 5

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o c . che e r o t r s su r A number is divisible by eight if … e p … the last three digits are divisible by eight; e.g. 847 7 x 2 = 14 84 - 14 = 70 70 ÷ 7 = 10

e.g. 463 896 896 ÷ 8 = 112 A number is divisible by nine if … … the sum of the digits is divisible by nine; e.g. 657 6 + 5 + 7 = 18 18 ÷ 9 = 2 Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Number and Place Value—N&PV – 1

RESOURCE SHEET Multiplication facts bingo cards

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Identify and describe factors and multiples of whole numbers and use them to solve problems

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Sub-strand: Number and Place Value—N&PV – 1

RESOURCE SHEET Multiplication facts bingo record cards

Number fact

Number called

Number covered

Number fact

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Number covered

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CONTENT DESCRIPTION: Identify and describe factors and multiples of whole numbers and use them to solve problems

Number called

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Number and Place Value—N&PV – 1

RESOURCE SHEET Solve the problems

Ned arranged 15 shirts equally on each strand of the washing line. If there are 5 strands, how many shirts are on each?

In Year 5, there are nine students in each of the four factions. How many students are in Year 5?

At school camp, the students are divided equally into six groups. If six students do caving first, how many students are on camp?

How many buses are needed to transport 250 students if each bus takes 50 passengers?

There are 24 children in the kindy class. How many student helpers are needed if there is to be one student for every four children?

On a driving trip, Nathan’s family drove 300 km each day. How many days did it take them to reach their destination, 3600 km away?

Jason trains four days a week with the swimming club. How often does he swim in a 13-week cycle?

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. te o Amira receives $6 for helping her c . It takes Achim four minutes to run ch father in the garden on the first e one kilometre. Ifr he runs at the same er o Saturday of each month. How much t slong will he take to run a sp speed, how r e does she earn from this source in one u 10 km race? year?

Jose and Elim walk the 3 km home from school each day. How far does each boy walk in one week?

One hundred and twenty people each bought a raffle ticket and raised $960 for the school fund. How much did each ticket cost?

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Identify and describe factors and multiples of whole numbers and use them to solve problems

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In how many ways could 24 students be seated so there were equal numbers of students in each row?

Assessment 1

Sub-strand: Number and Place Value—N&PV – 1

NAME:

DATE:

1. Fill in the missing factors (two outer circles) or multiples (centre circle). 6

36 8

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2. Which is the missing number? Shade the bubble.

(a) (i) 14‚ 21‚

(b) (i)

‚ 35‚ 42

‚ 64‚ 56‚ 48‚ 40

(ii)

(ii) (ii) (ii)

(e) (i)

(f)

‚ 36

‚ 20, 24, 28, 32

(ii)

3. Shade the bubble.

Use ‘divisibility rules’ to find the answer. Show any working out. Yes

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(d) Is the number 6322 a multiple of 8?

Yes

(e) Is the number 9 a factor of 8887?

Yes

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(a) Is the number 3 a factor of 3384?

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CONTENT DESCRIPTION: Identify and describe factors and multiples of whole numbers and use them to solve problems

(d) (i) 72, 63, 54,

Multiples of ...

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Each sequence shows multiples of which number?

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(b) Is the number 1242 a multiple of 6?

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

Yes

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Assessment 2

Sub-strand: Number and Place Value—N&PV – 1

NAME:

DATE:

1. Use your knowledge of factors and multiples to solve the problems.

9 Scoring

Outer ring – double points Middle ring – single points

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Inner ring – triple points

(a) What is the highest score that can happen with three darts? (b) What is the lowest score that can happen with three darts?

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Bullseye – 50

Record the multiplication facts to show how they are made. Multiplication facts

Score

Multiplication facts

Score

Multiplication facts

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Score

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(d) Using three darts each, four players each score 29 points. All do this with different combinations of scores. Investigate how they could have done this. Player 1

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Identify and describe factors and multiples of whole numbers and use them to solve problems

Checklist

Sub-strand: Number and Place Value—N&PV – 1

Identify and describe factors and multiples of whole numbers and use them to

Solves problems using factors and multiples

Explores divisibility rules to determine factors of larger whole numbers

Explores factors and multiples in number sequences

Recalls multiplication facts quickly

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STUDENT NAME

Recognises the link between factors and multiples

solve problems (ACMNA098)

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 2

Use estimation and rounding to check the reasonableness of answers to calculations (ACMNA099)

TEACHER INFORMATION

RELATED TERMS

What this means

Estimation

• An approximate calculation rather than a guess, using numbers that have been ‘rounded’.

• Students need to determine when using estimation is appropriate and be able to use rounding and mental computation skills to perform the estimation.

Rounding

Teaching points

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Rounding up

• When rounding a number to a certain place value‚ if the digit prior (to the right) to that being changed has a value of 5 or more‚ the number is most often rounded up (for example‚ to the nearest 10). When a rough estimation is required with which it is better to overestimate the quantity required (for example‚ making sure one has more than enough nails to build a cupboard)‚ rounding up is most often used.

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• When rounding a number to a certain place value‚ if the digit prior (to the right) to that being changed has a value of 4 or less‚ the number is most often rounded down (for example‚ to the nearest 10). When a rough estimation is required with which it is better to underestimate the quantity required (for example‚ the amount of cash on hand prior to buying petrol)‚ rounding down is most often used. Front-end estimation

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• The rounding of a number to the value of the left-most digit only. Context

• The meaningful (real-life) situation for which an estimate is being calculated. Student vocabulary approximate

context

estimate

rounding

rounding down rounding up

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Rounding down

• Rounding can be used: – to give an approximate answer when an exact answer is not required. This is generally to a round number (a number ending in zero‚ often to the nearest 10)‚ but students with good fluency with basic number facts may see that rounding to the nearest five (or midpoint of a given place value) gives a more accurate estimate – as a means to check the magnitude of an exact answer. • Depending on the context and student’s fluency with basic number facts, rounding can occur at any time in a calculation. • It is preferable to teach rounding and estimation in the context of a real problem. Students can determine the level of accuracy required, either an over- or underestimate, and apply that type of rounding. However, it is important to note that sometimes overestimating is preferred; for example, materials required for a job, cash required for goods bought, time taken to perform a task. Some students may recognise that overestimating with inaccurate rounding can lead to wastage; for example‚ of time or materials. • To round numbers, students need to be fluent with basic number facts and have developed a good understanding of place value; e.g. when estimating 72 x 83, a student might round the numbers down to 70 and 80 and then mentally calculate 7 x 8, understanding that they are multiplying 7 tens by 8 tens—which will give 56 hundreds or 5600. Reasoning in this way is preferable to subtracting and adding zeros (which often causes confusion). • Front-end estimation is the simplest method of rounding for division. The more left-hand digits that are included prior to the rounding, the more accurate the estimate is.

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• Adjusting the exact value of a number to make a calculation (an estimation) more straightforward. • Rounding can mean changing a number’s value to the nearest whole number, 10 or 100. With division, it can mean changing a number to an approximate multiple; for example, in 64 ÷ 9, the 64 could be changed to 63‚ to make 63 ÷ 9 = 7.

What to look for

• Students who lack fluency with basic number facts and are unsure about the mechanical procedure of rounding up or down when there is no context. • Students who have a poor understanding of place value and who may be able to mentally calculate a number fact but cannot determine by estimation what magnitude an answer should be. • Students who are unsure if a number should be rounded up or down when a context is given. • Students who do not appreciate that rounding can lead to over- and underestimation. See also New wave Number and Algebra (Year 5) student workbook (pages 9–14)

Proficiency strand(s): Understanding

Fluency

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

Problem solving

Reasoning

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

Sub-strand: Number and Place Value—N&PV – 2

HANDS-ON ACTIVITIES Running a house Collect multiple bills for the different services required for running a house; for example: council and water rates‚ and gas, electricity and telephone bills. Round the debit amounts of each to estimate the cost of each service for a year.

After-school activities Students collect information about the cost to their parents of after-school activities‚ such as swimming, dancing‚ music lessons or football. Round the numbers to estimate the cost of each activity over the year. Then determine the total cost of multiple activities each for several children.

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Collect a number of supermarket advertising brochures for students to use.

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Food shopping

Activity One – Round prices of individual items to estimate the total cost of a ‘wish’ shopping list of favourite foods. Activity Two – Estimate the cost of the food for a class party. Activity Three – What could be purchased with a $50 budget? Activity Four – Cut out and laminate various food items‚ labelled with their cost per unit weight or volume. By rounding, calculate the approximate cost of purchasing multiple units of any item. Activity Five – Consider the different types of consumable supermarket items often used by households weekly‚ excluding food; for example; laundry and bathroom products, pet food. Estimate the weekly total cost.

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Comparing rounding methods

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Use single-digit playing cards to create random large numbers (up to five digits). Compare answers to addition, subtraction and multiplication calculations solved using exact numbers, front-end estimated numbers‚ and numbers rounded to ten, 100 and beyond (page 25). In each case‚ most follow the rules for rounding (page 23).

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Think of and record real-life situations where each method may be appropriate for use; e.g. comparing cost price of items of different magnitude – MP3 player, entertainment system, car or house. Discuss problems that may arise with underand overestimates; e.g. running out of materials, ordering too much materials‚ not having enough money.

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Front-end estimation with division

Use single-digit playing cards to create random large numbers (up to five digits) as dividends. Randomly select a single card as the divisor. Think of and record a real-life situation in which the magnitude these numbers might be relevant. To estimate the outcome of division, focus on the divisor and the left-hand digits of the dividend. (Use the table on page 26.) Decide on the accuracy of estimate required and hence the number of left-hand digits to estimate from: Which number, that can be linked by a multiplication fact to the divisor, is closest to the first (or first two, three etc.) digit(s) of the dividend? Record this number and the related number fact. Round the dividend to this linked number. Divide the dividend by the divisor to arrive at the division estimate. Calculate the exact answer (quotient) and compare. Divisor

Dividend

Closest linked number

Related number fact

Rounded dividend

Quotient estimate

Exact answer

5928

2 x 30 = 60

6000

3000

2964

36 821

7 x 5 = 35

35 000

5000

5260

358 642

9 x 4 = 36

360 000

40 000

39 849

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 2

LINKS TO OTHER CURRICULUM AREAS English • Write a travelogue of places visited on an imaginary trip around the world. Use rounded numbers to describe the distances covered.

Science • Create a time line of the three ages of dinosaurs using ‘millions of years’ as the units of time. Round to the nearest tens of million of years. Collect pictures of dinosaurs to place on the time line in the correct period. Under the time line, complete the three sentences:

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– The Jurassic period lasted for about

million years.

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– The Triassic period lasted for about

million years.

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– The Cretaceous period lasted for about

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Information and Communication Technology

• <http://pbskids.org/cyberchase/math-games/glowlas-estimation-contraption/>

Geography • Research the population size of several countries; for example, those in South America. Record the information‚ then round the numbers to the nearest ten thousand, hundred thousand, million and ten million.

The Arts • Create a display of cloud types found across the different layers of the atmosphere.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

… to the nearest ten

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260

8770

264

8772

• The number has been rounded up.

• The number has been rounded down.

www.ricpublications.com.au

800 8130

795 8126

m 27.30 u 359 360

• The tens digit has increased and so the number is more.

6437

2349

128

6400

2300

100

• The number has been rounded down.

• The tens and ones digits have been decreased and so the number is less.

15 752

15 800

6300

1000

984 6295

800 751

• The number has been rounded up.

• The hundreds digit has increased and so the number is more.

– the tens and ones digits are changed to 0.

– the ones digit is changed to 0.

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

• The ones digits has been decreased and so the number is less.

– the hundreds digit is increased by one

• When the tens digit is any number from 5 to 9:

– the hundreds digit remains the same

• When the tens digit is any number from 0 to 4:

– the tens digit is increased by one

• When the ones digit is any number from 5 to 9:

How many tens does the number have?

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– the tens digit remains the same

• When the ones digit is any number from 0 to 4:

How many ones does the number have?

… to the nearest hundred

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What you do when rounding …

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CONTENT DESCRIPTION: Use estimation and rounding to check the reasonableness of answers to calculations Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEET Rounding

Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEET Front-end estimation and links with rounding to the nearest 10, 100 and beyond

In front-end estimation, focus on the left-hand digits of a number. The more digits looked at, the more accurate the estimate. 53 972 ÷ 4 = 13 493

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

Answer

50 000 ÷ 4

12 500

53 000 ÷ 4

13 250

53 900 ÷ 4

13 475

53 970 ÷ 4

13 492.5

3 4

2563 + 4796 = 7359 2000 + 4000

6000

2500 + 4700 7200 © R. I . C .Publ i cat i ons 3 4790 7350 •f orr evi e2560 w +p ur poseson l y• 2

7382 x 6 = 44 292

7300 x 6

7380 x 6

42 000

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7000 x 6

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Number of left-hand digits looked at

Number used in estimation

Rounded to nearest

70 000

10 000

75 000

1000

75 300

100

75 390

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Use estimation and rounding to check the reasonableness of answers to calculations

Teac he r

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Number of left-hand digits looked at

Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEET Comparing rounding methods: addition, subtraction and multiplication

Addition Exact calculation

… to nearest thousand

Subtraction

Exact calculation

Front-end estimation

… to nearest hundred

… to nearest thousand

… to nearest ten thousand

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… to nearest ten

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… to nearest hundred

… to nearest ten

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… to nearest ten thousand

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CONTENT DESCRIPTION: Use estimation and rounding to check the reasonableness of answers to calculations

Front-end estimation

o c . c e Exact calculation h Front-end estimation r … to nearest ten er o st super

… to nearest hundred

Multiplication

… to nearest thousand

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

… to nearest ten thousand

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEET

Rounded dividend

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Divisor

Dividend

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Related number fact Closest linked number

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Number of left-hand digits

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Use estimation and rounding to check the reasonableness of answers to calculations

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Quotient estimate

Exact answer

Front-end estimation with division

Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEET Rounding problems

Mrs Patel is driving to a town 143 km away. The speed limit varies between 50 and 60 km/hr. There are some road works along the way.

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The cost of Mr Hick’s last four weekly visits to the supermarket were $439, $397, $498 and $532. This week, he is trying out a new supermarket which he knows is a bit more expensive but it has some foreign foods that he would like to buy. He has mislaid his credit card and will have to pay in cash.

Do you think he is wise? Give reasons for your answer and show any calculations. Miss Angel wants to re-fence her garden. The measurements for the fencing are 123 metres for the two sides and 72 metres for the back. Fencing panels come in 5 metre widths.

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How much time do you think she should give herself to make the journey? Give reasons for your answer and show any calculations.

Each panel costs $93 for cash only.

How many panels will Miss Angel have to purchase and how much cash will she need to make sure she has enough to pay the bill? Give reasons for your answer and show any calculations.

Tess walks 1.3 km to school every day. Without a backpack weighing her down, she can walk at a speed of 5 km/hr.

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Jim wants to make cupcakes to celebrate his birthday at school. There are 32 students in his class and 11 staff to make cakes for.

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How much money do you think he should take with him so that he has enough to pay his bill? Give reasons for your answer and show any calculations.

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CONTENT DESCRIPTION: Use estimation and rounding to check the reasonableness of answers to calculations

Jordan’s new car can travel 100 km on 4.5 litres of fuel. The fuel tank capacity is 40 litres. He needs to make five trips of 136 km, 177 km, 206 km, 242 km and 128 km. He plans to do the five trips on one tank of fuel.

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How long should she give herself to walk to school on a day when her bag is heavy? Give reasons for your answer and show any calculations.

How many cakes should he bring in to make sure there is one for everyone? Give reasons for your answer and show any calculations.

Aden invites 12 friends to his party but he is not sure if they can all come. His mum has bought four 1.25 litre bottles of soft drink.

Zoe has saved $227 to spend during her six week summer holiday.

To make sure everyone has two drinks each, should he measure the drinks out in the 200 or 150 mL glasses? Give reasons for your answer and show any calculations.

To make sure she doesn’t run out of money before the end of the holiday, what is the most she should spend each day? Give reasons for your answer and show any calculations.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

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Sub-strand: Number and Place Value—N&PV – 2

RESOURCE SHEET Rounding game

Materials • one 10-sided dice • one hexagonal spinner with 10, 100 and 1000 each marked twice Rules

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• Roll the dice five times and use the numbers rolled to make a ten-of-thousands number. • Rearrange the five numbers to make five more five-digit numbers. • For each of the six numbers in turn, spin the spinner to give the place value to which the number is to be rounded. • Round the numbers and record them in the table. This game can be played by yourself with your teacher checking the work‚ or in small groups where each of you can check each other’s answers. Place value to round to

Rounded numbers

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Use estimation and rounding to check the reasonableness of answers to calculations

Numbers made

Assessment 1

Sub-strand: Number and Place Value—N&PV – 2

NAME:

DATE:

1. Round numbers to the nearest tens place value. (a) 423

(b) 82

(d) 164

(e) 55

(f) 798

(g) 11

(h) 76

2. Write the tens place value that each group rounds to. (a) 74, 69, 65, 71

(b) 26, 33, 28, 32

(d) 41, 35, 43, 36

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3. Use front-end estimation to calculate each answer. Show your working out. (b) 623 x 8

(d) 512 x 9

(e) 531 ÷ 6

(f) 118 x 6

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(a) 374 ÷ 4

Monday

$3.85, $9.45. $7.60, $12.25, $7.95

Tuesday

$3.55, 75c, $9.95, 85c, $7.95

Wednesday

$4.25, $12.95, 95c, $10.95, $8.25

Thursday

$2.75, $1.60, $5.30, $2.35, 95c

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CONTENT DESCRIPTION: Use estimation and rounding to check the reasonableness of answers to calculations

4. Dad went to the supermarket each day after work and always paid in cash notes. Estimate the total he spent each day and the minimum amount of money he would need in his wallet to be certain that he had enough to pay.

o c . c e Friday $8.95, $7.65, $5.45, $3.25, 45c her r o t s s r u e p 5. Find the number most closely linked by a multiplication fact to the bold number. Round the dividend to estimate the answer. Use a calculator to check. Divisor Dividend (a) 5

46 532

(b) 4

31 389

53 721

Closest-linked Related Rounded number number fact dividend

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Quotient estimate

Exact answer

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Assessment 2

Sub-strand: Number and Place Value—N&PV – 2

NAME:

DATE:

1. The new owner of a property wants to replace its old fence. To work out if she can afford the job, she must first work out a rough estimate of the cost. • The fencing she wants costs $27.85 per metre. • She needs 43 m of fencing down one side of the property, 124 m across the back and 62 m down the other side. She doesn’t need a fence across the front. Shade the bubble to show your answer.

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(a) To calculate the estimate, how should she round the figures? Down

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(b) Show the working for the estimate.

200 km

300 km

400 km

(ii) Day two:

367 km

200 km

300 km

400 km

(iii) Day three:

244 km

200 km

300 km

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(i) Day one:

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Shade the bubble to show your answer.

400 km

(b) When calculating fuel consumption, how should the numbers be rounded? up to 80 L down to 70 L . (ii) Fuel effit ciency to 10 L/100 km e 14 L/100 km: to 20 L/100 km o c . (iii) I made these decisions because che e r o r st super (i) Tank capacity 73 L:

350 km

400 km

(d) Using the rounded numbers from (a) and (b), about how much fuel will be needed for the whole trip? 180 L 30

60 L

6300 L

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Use estimation and rounding to check the reasonableness of answers to calculations

2. Using satellite navigation, a family has planned a three-day driving tour. The car’s fuel tank holds 73 L and the engine uses 14 L of fuel every 100 km.

Checklist

Sub-strand: Number and Place Value—N&PV – 2

Follows the correct steps for front-end estimation and calculates accurately

Follows the correct steps to round up and down and calculates accurately

Recognises the place value to which a number has been rounded and calculates accurately

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STUDENT NAME

Recognises the appropriate method of rounding for a given context.

Use estimation and rounding to check the reasonableness of answers to calculations (ACMNA099)

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 3

Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies (ACMNA100)

RELATED TERMS

TEACHER INFORMATION

Array

What this means

• An array is an arrangement of objects or numbers. Rectangular arrays, organised in columns and rows, are used to illustrate multiplication facts. Arrays are also used to separate the partitioned components of a multi-digit number and to illustrate multiplication strategies.

• As numbers become too large for students to calculate mentally, they will need to make notes on paper, apply a written method of calculation or use a calculator. • The point at which this occurs depends on a student’s working memory, fluency with basic facts, understanding of place value and ability to partition numbers. • Students often invent their own methods of written calculation‚ but these must be: – effective – giving the correct answer – efficient – performed simply and quickly – general – applicable to all cases.

Commutative law

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• The order in which numbers two are multiplied (or added) does not affect the outcome. Associative law

• When multiplying (or adding) three or more numbers, the order in which groups of numbers are multiplied (or added) does not affect the outcome. Distributive law

• When a group of numbers is added together‚ then multiplied by a number n, the same outcome is achieved as when each of the numbers in the group is first multiplied by the number n and the products are added together.

Teaching points

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• There are several algorithms (step-by-step procedures) that can be used in calculations: – The ‘area model’ (pages 39 and 40) involves partitioning numbers according to place value. It combines a diagram and interim calculations to reach the final answer. When working out a two-digit-by-two-digit multiplication calculation, the two numbers are split along place value lines and a diagram is drawn. The area of each part is then calculated. These are then added to find the total. This method is similar to the standard written algorithm‚ where the calculation is broken up into a series of manageable calculations and the results combined. – The ‘lattice method’ (pages 41 and 42) illustrates the partitioning of numbers and the application of the distributive law; e.g. 57 x 6 = (50 x 6) + (7 x 6) = 300 + 42 = 342.

Partitioning

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• Partitioning occurs when a number is separated into its place value parts (expanded notation) or into nonstandard place value parts. Product and partial product

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• The answer to a multiplication calculation is known as the product. When there are several multiplication calculations that form part of a multiplication algorithm‚ the answer to each calculation is known as a partial product. The final product is the sum of the individual partial products. Italian lattice method

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• A multiplication technique that applies the concept of partitioning and is illustrated within a lattice array.

What to look for

Student vocabulary array swapping separating product lattice

See also New wave Number and Algebra (Year 5) student workbook (pages 15–19)

Proficiency strand(s): Understanding Fluency Reasoning

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

Sub-strand: Number and Place Value—N&PV – 3

HANDS-ON ACTIVITIES Arrays • Roll two six-sided dice. Use the two numbers as the factors in a multiplication algorithm. Draw the two rectangular arrays that can be made and their corresponding addition and multiplication sentences. • Roll four ten-sided dice. Add the four numbers together and use this as the total number of objects in an array. How many arrays (with their corresponding addition and multiplication sentences) can be drawn? • Write number problems to go with these arrays.

Multiplying by single-digit numbers

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Use dice or number cards to generate two- and three-digit numbers. Use the resource sheet on page 37 to practise expanding and multiplying the numbers by single-digit numbers‚ determined by the throw of a ten-sided dice.

Multiplying lists of numbers by the same single digit

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Use the resource sheet on page 38 to write three multiplication problems in which up to six numbers need to be multiplied by the same single digit. Use the frameworks to show the calculations for each problem.

Area models

Practise calculating multiplication problems using area models. Split up the numbers to be multiplied and then multiply the numbers in each bracket. Show each part of the algorithm on grid paper. The grid on page 40 allows for problems up to 19 x 19. Colour to highlight each part. When confident, students can calculate problems with larger numbers. For less able students‚ use base 10 MABs.

Lattice multiplication

Use the resource sheet on page 42 to write two multiplication problems in which a large number (two or three digits) needs to be multiplied by a one- or two-digit number. Record each step of the calculation in the lattice.

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Write multiplication problems in which up to four-digit numbers need to be multiplied by one- or two-digit numbers. Use the resource sheet on page 44 to practise expanding and multiplying the numbers and determining the partial products before calculating the final product.

Vedic (Indian) maths

Using the website <http://www.jainmathemagics.com/page/5/default.asp>, write an explanation with examples of one of these ‘sutras’:

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• By the excess – The multiplication of numbers over a base • By the deficiency – The squaring of numbers under a base • By one more – The squaring of numbers ending in 5 • Vertically and crosswise • Digital sums: For multiplication by eleven

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 3

LINKS TO OTHER CURRICULUM AREAS English • Read, write and spell all vocabulary associated with the descriptor. • Make as many maths-related small words as possible from the letters in words in the vocabulary list. • Write vocabulary list in alphabetical order.

Information and Communication Technology • Lattice multiplication: <http://www.coolmath4kids.com/times-tables/times-tables-lesson-lattice-multiplication-1.html> • Napier rods: <http://threesixty360.wordpress.com/2009/06/11/the-second-bunch-of-ways-to-multiply/> • Vedic maths: <http://www.jainmathemagics.com/page/5/default.asp>

Design Technology

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• Napier’s bones – Make a set of Napier rods as described on the website: <http://threesixty360.wordpress.com/2009/06/11/the-secondbunch-of-ways-to-multiply/>. – Use them to calculate the products of large numbers by one- and two-digit numbers.

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• Explore Robinsunne’s multiplication clock and how to make one at: <http://www.robinsunne.com/robinsunnes_ multiplication_clock>.

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History

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• Investigate how people who used Roman numerals performed arithmetic calculations before the introduction of Arabic numerals. • Investigate the origin of the Italian lattice multiplication method. • Investigate ancient multiplication methods at <http://www.pballew.net/old_mult.htm>. • On a world map outline, shade in the areas of land where ancient and medieval methods of arithmetic calculations were used. • Research the invention of Napier’s bones by John Napier in the 16th century. • Research the invention of Genaille-Lucas rulers by Henri Genaille in the 19th century. • Research the ancient Indian Vedic methods for multiplication at <http://www.jainmathemagics.com/page/5/default. asp>.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET Which number first?

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This array shows 12 stars arranged in 3 rows‚ with 4 stars in each row. Three groups of four make 12. 4 + 4 + 4 = 12

3 x 4 = 12

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This array shows 12 stars arranged in 4 rows‚ with 3 stars in each row. Four groups of three make 12. 3 + 3 + 3 + 3 = 12

4 x 3 = 12

m . u

w ww

CONTENT DESCRIPTION: Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

The order in which two or more numbers are multiplied makes no difference to the answer.

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o c . che e r o 24 stars arrangedr t s 24 stars arranged super in 2 sets of 3 rows‚ in 4 sets of 2 rows‚

with 4 stars in each row.

with 3 stars in each row.

2 x (3 x 4) = 24

(2 x 3) x 4 = 24 2 x (3 x 4) = (2 x 3) x 4

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET

The same answer is obtained when; • a group of numbers is added and then multiplied by a number (n)

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

2 x (5 + 3)

ew i ev Pr

Teac he r

• or‚ each number is multiplied by the number (n) and their products are added.

(2 x 5) + (2 x 3)

©R . I . C.Publ i cat i ons 2 x (5 + 3) = (2 x 5) + (2 x 3) •f orr evi e r posesonl y• 2w x 8 =p 10u +6

2 groups of (5 + 3) is the same as 2 groups of 5 plus 2 groups of 3

w ww

m . u

16 = 16

This works for:

• multiplying large numbers by single-digit numbers 9 x 237

. t 9 x (200 + 30 + 7) = (9 x 200) + (9 x 30) + (9 x 7) e o c . che = 1800 + 270r e + 63 o t r s s r u e p = 2133

• multiplying lists of numbers by the same single-digit number. 9x6

9x3

9x5

9x4

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

9 x (6 + 3 + 5 + 4)

9 x 18

162

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CONTENT DESCRIPTION: Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

Multiplication and addition as partners

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET

+ +

Teac he r

ew i ev Pr =

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Three-digit numbers

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

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Two-digit numbers

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

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CONTENT DESCRIPTION: Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

Multiplying large numbers by single-digit numbers

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET

m + + . u

Number problem

+ +

Number problem

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

+ +

o c . che e r o t r s super +

+ + +

. te

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

CONTENT DESCRIPTION: Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

= + +

Number problem

w ww 38

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

ew i ev Pr

Teac he r

Multiplying lists of numbers by the same single-digit number

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET

Multiplication can be illustrated by using an area model which makes use of partitioning or ‘splitting up’ the numbers before multiplying out. 14 x 12 Splitting up: (10 + 4) x (10 + 2) Multiplying out: (10 x 10) + (10 x 2) + (4 x 10) + (4 x 2)

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

ew i ev Pr

Teac he r

m . u

w ww

CONTENT DESCRIPTION: Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

Using an area model

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o c . che H T O r e o t r super 1s 4 x

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

Partial products

Product www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET Multiplication can be illustrated by using an area model which makes use of partitioning (or ‘splitting up’) the numbers before multiplying out. Problem Splitting up

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

ew i ev Pr

Teac he r

Multiplying out

w ww

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m . u

o c . che H T O r e o r st super x

x x

Partial products

x x

+ Product

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

Using an area model

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET

• Write the digits of the larger number in the correct place value columns across the grid. • Write the digits of the one- or two-digit number in the correct place value row(s) to the right of the grid. • Multiply the number in each place value column by the number in each place value row(s). Write the answers in the squares of the appropriate rows and columns: the tens digit above the diagonal and the ones digit below. If there is no tens digit, write a zero.

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

ew i ev Pr

Teac he r

• Add the digits in the diagonals, starting with the ones column. 57 x 8 = 456 Hundreds

Tens

Ones

4 5 ©4 R. I . C .P u b l i ca t i on 8s 0 6 •f orr evi ew pur posesonl y•

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Thousands

591 x 36 = 21 276 Hundreds

Tens

Ones

m . u

w ww

o c . che e r o 1 2 0 t r s s r u e p 3 7 3 5 +1

5 0 2

4 7

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

Ones

3+1

Tens

CONTENT DESCRIPTION: Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

Italian lattice multiplication grids

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Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET

Number problem

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

Thousands

Hundreds

Tens

Ones

Tens Ones

ew i ev Pr

Teac he r

w ww

. te

Thousands

m . u

Number problem

o c . = chex e r o t r s su per Hundreds Tens Ones

Tens Ones

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

Italian lattice multiplication grids

Teac he r

• Multiply the number in each place value column by the number in each place value row.

• Write the numbers in the correct place value columns in the table.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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4000

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(40 x 2 =)

1200

(40 x 30 =)

Tens

2346 x 32 = 75 072

m .600 u

(300 x 2 =)

9000

60 000

(2000 x 2 =)

(300 x 30 =)

300

2000

(2000 x 30 =)

Hundreds

o c . che e r o t r s super

Thousands

Example: 2346 x 32

• List the answers (partial product) to all algorithms and add them to find the product of the original multiplication algorithm.

(6 x 2 =)

180

(6 x 30 =)

Ones

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

ew i ev Pr

. te

Tens

Ones

w ww 12 75 072

180

600

1200

4000

9000

60 000

Sum of partial products

CONTENT DESCRIPTION: Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET

Multiplying large numbers by one- and two-digit numbers

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

Ones

Tens

Thousands

Hundreds

o c . che e r o t r s super x

Tens

m . u

Hundreds

Tens

Ones

= Sum of partial products

Teac he r

Ones

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

Number problem

CONTENT DESCRIPTION: Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

Thousands

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

ew i ev Pr

w ww

. te

Ones

Sum of partial products

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET

Multiplying large numbers by one- and two-digit numbers

Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET

The ‘diagonal and crosswise’ sutra of Vedic mathematics lets students calculate multiplication problems without the need of knowing basic number facts beyond 5 x 5. This method does not explain how multiplication works but it is a quick and easy method for students to use to check answers when they cannot recall certain basic facts. It should not take the place of learning the basic facts‚ but its fun ‘trick’ nature may help and encourage fluency.

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

There are three steps to follow to calculate the product of two numbers. Subtract each number from 10 and record the difference beneath each number. This is the ‘lower pair’.

ew i ev Pr

Teac he r

Step 1

Step 2

Using either diagonal pair, subtract one from the other to give the tens digit of the answer.

Step 3

Multiply the lower pair to give the ones digit of the answer.

Examples © R. I . C.P ubl i cat i ons 4f xr 9e 8e 7l • o r v i ew pur pos sx on y •56 = 36 = 6

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10 – 8 = 2 10 – 7 = 3 8 – 3 = 5 or 7 – 2 = 5 2x3=6

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10 – 4 = 6 10 – 9 = 1 4 – 1 = 3 or 9 – 6 = 3 6x1=6

w ww

CONTENT DESCRIPTION: Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

Diagonal and crosswise

o c . che e r o t r s s r u e p 6 x 7 6 = = 36

As single-digit numbers are within the first power of 10, the digits of the answer are either tens or ones. If Step 3 produces a two-digit answer, the tens digit is added to the tens digit already determined. 6

10 – 6 = 4 10 – 6 = 4 6–4=2 4 x 4 = 16

10 – 6 = 4 10 – 7 = 3 6 – 3 = 3 or 7 – 4 = 3 4 x 3 = 12

The tens digit 2 is added to the tens digit of 16, giving the answer 36.

The tens digit 3 is added to the tens digit of 12, giving the answer 42.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Number and Place Value—N&PV – 3

RESOURCE SHEET Multiplication by line drawing

The digits of each number are represented as lines drawn, in a north-east direction for the first number and a south-east direction for the second number. The value of each digit in each place is represented by that number of lines.

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

Place value positions are separated by larger gaps. For example: 423 x 43 = 18 189

3 x 4 = 12

3x2= 6

ew i ev Pr

Teac he r

w ww

Ten thousands

m . u

. t e o Thousands Hundreds Tens Ones c . che 12 e r 6 o t r s sup r e + 12 8 2

4 x 3 = 12

1 The intersection points where the lines cross in each place value position, are counted, added up and ‘carried over’ if necessary. 46

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

Another lattice method for multiplication is line drawing.

Assessment 1

Sub-strand: Number and Place Value—N&PV – 3

NAME:

DATE:

1. Complete the lattice multiplication calculations. 4

(b)

(c)

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

2. Solve each problem by following the area model steps. Split up

Multiply out

Partial products

ew i ev Pr

Teac he r

Problem

(a) 29 x 12

Product

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Product

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CONTENT DESCRIPTION: Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

(a)

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o c . che e r o t r sit. su r pe 3. Write each algorithm in a different way‚ then solve (c) 31 x 19

Algorithm

Can also be written as ...

Product

Solution

(a) (4 x 6) + (3 x 6) (b) (5 x 8) + (5 x 7) 4. (a) If 3 x 7 = 21, 30 x 7 = (c) If 4 x 6 = 24, 40 x 60 =

(b) If 5 x 8 = 40, 5 x 80 = (d) If 2 x 9 = 18, 20 x 90 =

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Assessment 2

Sub-strand: Number and Place Value—N&PV – 3

NAME:

DATE:

Calculate how much it would cost for you and four friends to have dinner at the cafe. You all have a drink, a starter, a main meal and a dessert. Children’s

Teac he r

Drinks $2 Starters $4 Main meals $6 Desserts $3

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

(a)

–

(b) –

– = = R © . I . C.Publ i cat i on s – = or – = = or – = •f or=r evi ew pur pose so nl y• x = x –

–

ew i ev Pr

2. Use the ‘diagonal and crosswise’ method to find the answer to each problem. Show your working.

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How many prizes were given out altogether? Use the ‘splitting up’ chart to calculate the answer.

o c . che e r o t r s super

Hundreds

Tens

Ones

Sum of partial products

Ones

Tens

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m . u

3. A supermarket chain owned 32 stores across the country. In one year‚ it offered 787 prizes for each store to hand out to its customers.

Number sentence:

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

1. Show how you would work out the answer to the problem.

Checklist

Sub-strand: Number and Place Value—N&PV – 3

Solve problems involving multiplication of large numbers by one- or two-digit numbers using

Illustrates multiplication using an area model

Multiplies lists of numbers by the same number

Multiplies by one- and two-digit numbers using the lattice method

Expands large numbers to find partial products to multiply by one- and two-digit numbers

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

ew i ev Pr

Teac he r

STUDENT NAME

Draws and explains arrays to show multiplication algorithms

efficient mental, written strategies and appropriate digital technologies (ACMNA100)

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m . u

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 4

Solve problems involving division by a one-digit number, including those that result in a remainder (ACMNA101)

RELATED TERMS

TEACHER INFORMATION

Division

What this means

• The mathematical number operation in which objects and numbers are grouped into equal parts

• Students can divide using a single-digit divisor; e.g. 4 72 (no remainder), 6 97 (with a remainder).

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

Dividend

• The number being divided; e.g. in 54 ÷9 = 6, 54 is the dividend

• The number which is divided into the dividend; e.g. in 54 ÷ 9 =6, 9 is the divisor Quotient

• The answer to a division calculation; e.g. in 54 ÷ 9 = 6, 6 is the quotient

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• The amount left over when a number has been divided into equal parts. The value of the remainder is always less than that of the divisor; e.g. in 57 ÷ 9 = 6 r. 3, 3 is the remainder. • The remainder can be left as a whole number or expressed as a fraction or decimal of the divisor; e.g. in 57 ÷ 9 = 6 r. 3, the quotient can also be expressed as 6 1⁄3 or 6.33. Equivalent

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• Having the same value but expressed in different multiples or units; for example, ½ is the same as 2⁄4; 180 minutes is the same as three hours. Student vocabulary divisor dividend quotient multiply divide remainder equally share fraction decimal

What to look for

• Students may confuse the two representations of a division calculation and read them incorrectly; e.g. saying 48÷ 6 as ‘How many 48s in six’‚ or 6 48 as ‘six divided by 48’. • Students who lack fluency in basic multiplication facts may have problems with finding the associated multiplication fact to use when dividing without remainders‚ and the closest fact when dividing with remainders. • Students might think a remainder is a decimal; e.g reading 38 ÷ 5 = 7r.3 as 38 ÷ 5 = 7.3, when it is 7.6 or 73⁄5.

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Remainder

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Teac he r

Divisor

• Initially, relate the division calculation to its associated inverse multiplication fact; e.g. 48 ÷ 6 = 8 is the inverse of 6 x 8 = 48. • Use the two standard divisions signs; i.e. 48 ÷ 6 = 8 and 6 48. Teach students how to ‘read’ each; i.e. – 48 ÷ 6 is read as ‘48 divided by 6’ – 6 48 is read as ‘How many times does six go into 48?’ • Later, teach students the link between fractions and division; i.e. 1⁄6 of 48 is the same as 48 ÷ 6. • When confident with division calculations without a remainder, introduce ‘near basic facts’ (e.g. 49 ÷ 6) where the calculation is close to a known basic fact. Gradually increase the complexity of the calculation; e.g. 89 ÷ 8.

See also New wave Number and Algebra (Year 5) student workbook (pages 20–26)

o c . che e r o t r s super Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Number and Place Value—N&PV – 4

HANDS-ON ACTIVITIES Write number stories to go with division problems; e.g. 32 ÷ 6 can be said as; ‘How many groups of six students can be made from a class of 32? How many students, if any, cannot be placed in a group?’ Have students use concrete materials to demonstrate division by grouping; e.g. for 29 ÷ 3‚ distribute 29 marbles equally among three boxes. Two marbles will be remaining. In pairs, students create ‘missing number’ problems for each other, based on the ‘multiplying and dividing by a common factor’ rule (page 52). In pairs, students create division number stories which they then solve using any one or all of the methods explored on pages 53 to 55.

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

Play ‘Division challenge’: one student volunteers to stand in front of the class and solve on the whiteboard any division problem the class gives to him or her. (The teacher may need to set parameters.)

LINKS TO OTHER CURRICULUM AREAS

English • • • •

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Teac he r

In pairs, students use playing cards and take turns to generate numbers for division calculations. In turn‚ a student writes a problem using the numbers‚ then both try to solve it. Whoever is best of three (or five) is the division champion for the day.

Read, write and spell all vocabulary associated with the descriptor. Make as many small words as possible from the letters in words in the vocabulary list. Write the vocabulary list in alphabetical order. Research the number of players required for different team sports. Based on the number of students in the year group or combined year groups, determine how many teams for each sport your school could field at an international festival of all sports.

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• Write an explanation of how the ancient Egyptians performed division calculations. Use: <http://homepage.mac.com/shelleywalsh/MathArt/EgyptDivide.html> <http://www.gap-system.org/~history/HistTopics/Egyptian_papyri.html>.

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History

• Research to find out what the Rhind papyrus is and how it got its name. (See above.)

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o c . che e r o t r s super

• Research to find out the symbols used by the ancient Egyptians to count in base ten. • Calculate the average length of office of Australian Prime Ministers in the 20th century. Determine how much above or below the average each PM’s term was.

Geography • Research the population and area of different countries and‚ by use of division, their population densities.

Technology and Enterprise • Make a model to illustrate how time as we know it is divided into seconds, minutes, hours, days and weeks.

Science • Describe how the lunar month is divided by the phases of the moon. (Round the lunar month to 30 days.) • Research simple favourite recipes. Determine how much of each ingredient would be required for making a single serve. For recipes that require only one egg, the capacity of one beaten egg could be determined and divided. Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Number and Place Value—N&PV – 4

RESOURCE SHEET Multiplying and dividing by a common factor

If numbers in a number sentence are multiplied or divided by a common factor, the multiplied or divided numbers are equivalent to the original numbers; for example, 12 ÷ 4 = 24 ÷ 8 Both numbers on the left-hand side have been multiplied by 2. Because both numbers have been treated in the same way, the answers to both sides of the equation are equal; i.e. 12 ÷ 4 = 3 24 ÷ 8 = 3

r o e t s Bo r e p ok u S x2

÷2

ew i ev Pr

21 ÷ 3 = 42 ÷ 6

32 ÷ 8 = 16 ÷ 4

(21 x 2) ÷ (3 x 2) = 42 ÷ 6

(32 ÷ 2) ÷ (8 ÷ 2) = 16 ÷ 4

42 ÷ 6 = 42 ÷ 6 = 7

16 ÷ 4 = 16 ÷ 4 = 4

÷3

72 ÷ 9 = 24 ÷ 3

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• Recognising the relationship between the divisor and dividend; e.g.

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•

o c . che ? = 2 x 27 = 54 r e o r st super Applying number families; e.g. ? ÷ 3 = 54 ÷ 6 ? ÷3 =9

27 ÷ 3 = ? ÷ 6

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When students understand this rule, they can solve equivalent ‘missing number’ problems in a variety of ways.

6 = 2 x 3; so‚

or 27 ÷ ? = 54 ÷ 6 27 ÷ ? = 9

? =9x3

27 ÷ 9 = ?

27 = 9 x 3

27 ÷ 9 = 3

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Solve problems involving division by a one-digit number, including those that result in a remainder

Teac he r

Students can prove this rule for themselves.

Sub-strand: Number and Place Value—N&PV – 4

RESOURCE SHEET Using known basic facts for division – 1

If you do not know all the basic multiplication and division facts, you can still use those you do know to find the answer to a division problem. Example: 558 ÷ 6 Step 1 Think of the highest related x6 fact you know with a product less than 558; e.g. 5 x 6 = 30‚ so 50 x 6 = 300.

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

Teac he r

Step 2 Subtract that product from the dividend of the original question, 558. 558 – 300 = 258. Continue the process until the difference is either zero or less than the divisor (6). Subtraction facts

50 x 6 =

300

558 – 300 = 258

30 x 6 =

180

258 – 180 = 78

12 x 6 =

78 – 72 = 6

1x6=

6–6=0

©93R . I . C. Pub l i c at i o n sixes are needed to go from 558 to 0‚s so ÷ 6 = 93. •f orr evi ew 558 pu r posesonl y• Example: 693 ÷ 8

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Step 1 Think of the highest related x7 fact you know with a product less than 693; e.g. 5 x 8 = 40, so 50 x 8 = 400.

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CONTENT DESCRIPTION: Solve problems involving division by a one-digit number, including those that result in a remainder

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Multiplication facts

. te o c Continue until the difference is either zero or less than. the divisor (8). che e facts r o Multiplication facts Subtraction r st super

Step 2 Subtract that product from the dividend, which is 693. 693 – 400 = 29

50 x 8 =

400

693 – 400 = 293

20 x 8 =

160

293 – 160 = 133

10 x 8 =

133 – 80 = 53

5x8=

53 – 40 = 13

1x8=

13 – 8 = 5

86 eights are needed to go from 693 to less than 8, so 693 ÷ 8 = 86 with remainder 5. Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

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Sub-strand: Number and Place Value—N&PV – 4

RESOURCE SHEET Using known basic facts for division – 2

If you do not know all the basic multiplication and division facts, you can still use those you do know to find the answer to a division problem. Example: 637 ÷ 7 Think of the highest related divisor fact you know with a product less than the dividend; e.g. 5 x 7 = 35‚ so 50 x 7 = 350.

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

Subtract that product from the dividend, which is 637. 637 – 350 = 287. Ones

– –

Check your answer by backtracking with multiplication

(50 x 7)

(50 x 7) + (30 x 7) + (11 x 7) = (50 + 30 + 11) x 7

(30 x 7)

91 x 7 (Use calculator) = 637

–

(11 x 7)

637 – 637 = 0

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Example: 786 ÷ 9 Think of the highest related divisor fact you know with a product less than the dividend; e.g. 5 x 9 = 45‚ so 50 x 9 = 450.

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Subtract that product from the dividend, which is 786. 786 – 450 = 287.

Continue until the difference is either zero or less than the divisor.

Hundreds

7 –

– – –

. te8

Tens

Ones

o c . 6 che e (50 r o t 0r (50 x 9) s x 9) ++(20(7xx9)9)+ (10 x 9) super 7

Check your answer by backtracking with multiplication

= (50 + 20 + 10 + 7) x 9

(7 x 9 )

(87 x 9)

(20 x 9)

87 x 9 (Use calculator) = 783

(10 x 9) 786 – 783 = 3

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

remainder 3 R.I.C. Publications® www.ricpublications.com.au

CONTENT DESCRIPTION: Solve problems involving division by a one-digit number, including those that result in a remainder

Tens

ew i ev Pr

Hundreds

Teac he r

Continue until the difference is either zero or less than the divisor.

Sub-strand: Number and Place Value—N&PV – 4

RESOURCE SHEET Using known basic facts for division – 3

If you do not know all the multiplication and division facts, you can still use those you do know to find the answer to a long division problem. Example: 207 ÷ 9 Step 1

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

Think of the highest related x9 fact you know with a product less than 207; e.g. 9 x 2 = 18‚ so 9 x 20 = 180. Step 2

Teac he r

Subtract that product from 207. 207 – 180 = 27.

3 3

–

207 ÷ 9 = 23

Example: 693 ÷ 8 Step 1

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2P 7 b © R. I . C. u l i cat i ons – 2 7 •f orr evi ew p r posesonl y• 0u

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CONTENT DESCRIPTION: Solve problems involving division by a one-digit number, including those that result in a remainder

ew i ev Pr

Continue until the difference is either zero or less than the divisor (9).

Think of the highest related x8 fact you know with a product less than 495; e.g. 8 x 6 = 48‚ so 8 x 60 = 480.

. te o Subtract that product from 495. 495 – 480 = 15. c . c e Continue untilh thee difference is either zero or o lessr than the divisor (9). t r s s r up e 6 1 Step 2

1 6

–

495 ÷ 8 = 61 remainder 7

8 7

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Number and Place Value—N&PV – 4

RESOURCE SHEET Using simple basic facts for division

Even if you know only your x10, x5 and x2 multiplication and division facts, you can still use them to find the answer to a long division problem. Example: 950 ÷ 7 • You know that 10 x 7 = 70‚ so 100 x 7 = 700.

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• You know that 2 x 7 = 14‚ so 20 x 7 = 140. • You know that 5 x 7 = 35.

0 5

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Solve problems involving division by a one-digit number, including those that result in a remainder

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Teac he r

Continue until the difference is either zero or less than the divisor (9).

Sub-strand: Number and Place Value—N&PV – 4

RESOURCE SHEET Division problems

There are two types of division problems: Sharing a given number of items (the dividend) among a given number of people or things (the divisor). There are 24 students (dividend) in a class but only 8 computers (divisor). How many students have to share each computer?

r o e t s Bo r e p ok u S 24 ÷ 8 = 3

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There are 32 apples (dividend) to be distributed among 4 baskets (divisor). How many apples for each basket? 32 ÷ 4 = 8

56 ÷ 7 = 8

How many of a given quantity (the divisor) can be made from (or are needed for) a given amount (the dividend)?

How many vases are needed? 9 8 72

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CONTENT DESCRIPTION: Solve problems involving division by a one-digit number, including those that result in a remainder

There are 56 L of milk (dividend) to be distributed among 7 customers (divisor). How much milk will each customer receive?

There are 6 beds (divisor) in each ward at the hospital. There are 24 new patients (dividend) needing beds for the night. How many vacant wards are needed?

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o c . che e r o t r s It takes the whites of s 5 eggs (divisor) to make a pavlova dessert. r up e How many desserts can be made with 45 eggs (dividend)? 4 6 24

9 5 45 It takes 7 L of milk (divisor) to make 1 kg of cheese. How much cheese can be made from 35 L of milk (dividend)? 5 7 35

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 4

RESOURCE SHEET Division problems with remainders

The remainder of a division problem can be expressed as a whole number or as a fraction or decimal of the divisor. How it is expressed depends on the type of problem and its context. Sharing a given number of items (the dividend) among a given number of people or things (the divisor).

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There are 38 kg potatoes (dividend) to be distributed among 8 sacks (divisor). 38 ÷ 8 = 4 remainder 6 4 kg will be put in each sack‚ plus 6⁄8 = ¾ = 0.75 kg.

There are 73 books (dividend) to be distributed on 8 library shelves (divisor). 73 ÷ 8 = 9 remainder 1 9 books will be placed on each shelf and the remaining book placed randomly on one of the shelves.

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Each player will receive one whole muffin‚ plus 3⁄9 = 1⁄3.

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How many of a given quantity (the divisor) can be made from (or are needed for) a given amount (the dividend)?

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38 m of material (dividend) is available to make shirts. It takes 3 m (divisor) to make one shirt.

o c . ch12e shirts can be made. e r The remaining 2 m of material can be used fort something else. o r s super 45 students (dividend) on school camp are sleeping in 6-person (divisor) tents. 38 ÷ 3 = 12 remainder 2

45 ÷ 6 = 7 remainder 3 One extra tent will be needed to accommodate the three remaining students‚ so 8 tents will be required altogether. 17 cats (dividend) are housed in pens that each accommodate 2 cats (divisor). 17 ÷ 2 = 8 remainder 1 One extra pen will be needed to accommodate the remaining cat‚ so 9 pens will be required altogether. 58

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Solve problems involving division by a one-digit number, including those that result in a remainder

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There is $19 (dividend) to be distributed among 4 prize winners (divisor). 19 ÷ 4 = 4 remainder 3 Each winner takes home $4¾ = $4.75.

Sub-strand: Number and Place Value—N&PV – 4

RESOURCE SHEET The link between divisions and fractions

In a familiar context, students are subconsciously aware of the link between division and fractions‚ such as halves, quarters and thirds. They know that a bag containing 12 lollies can be shared equally among 4 children and each will receive 3 lollies. Instinctively, they will see that 4 muffins divided among 3 will result in 11⁄3 each. This awareness can be highlighted by teaching students this link and referring to all basic multiplication and division facts. Initially, familiar facts should be used so that students can self-check.

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Just as there are two types of division problems, there are two ways to teach the link between division and fractions.

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Sharing a given number of items (the dividend) among a given number of people or things (the divisor).

Repeat the process‚ using 3 tubs instead of 4.

Students will see that ¼ of 12 is 3 and 1⁄3 of 12 is 4. Can they see the link? (12 ÷ 4 = 3 and 12 ÷ 3 = 4)

Practise with other facts to improve students’ fluency.

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parts the whole is being divided into (4). By sharing, we discover how much each part (quarter) is worth (3) in relation to the original amount (12).

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CONTENT DESCRIPTION: Solve problems involving division by a one-digit number, including those that result in a remainder

Count out 12 small objects. Divide them equally among 4 tubs. How many objects in each tub? The contents of each tub represent ¼ of the original amount; 1 because it is one tub and 4 because there are four tubs altogether.

. tobjects, how many groups of 3 can be made? What o Using 12 smalle fraction of 12 is 3? c . 4 groups of 3 can be made (there are 4 threes in 12). The contents of each group c e h r represent ¼ of the original amount; 1 because it is oneo group and 4 because there are e t r s s r upe four groups altogether. How many of a given quantity (the divisor) can be made from (or are needed for) a given amount (the dividend)?

Repeat the process making 4 groups instead of 3. Again, students will see that ¼ of 12 is 3 and 1⁄3 of 12 is 4. Can they see the link? (12 ÷ 4 = 3 and 12 ÷ 3 = 4) Using this method, we discover what fraction of the original amount (12) a given number (3) is. Practise with other facts to improve students’ fluency.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 4

RESOURCE SHEET The link between divisions and fractions

because

6÷6=1

1⁄7 of 7 = 1

because

7÷7=1

1⁄6 of 12 = 2

because

12 ÷ 6 = 2

1⁄7 of 14 = 2

because

14 ÷ 7 = 2

1⁄6 of 18 = 3

because

18 ÷ 6 = 3

1⁄7 of 21 = 3

because

21 ÷ 7 = 3

1⁄6 of 24 = 4

because

24 ÷ 6 = 4

1⁄7 of 28 = 4

because

28 ÷ 7 = 4

1⁄6 of 30 = 5

because

30 ÷ 6 = 5

1⁄7 of 35 = 5

because

35 ÷ 7 = 5

1⁄6 of 36 = 6

because

36 ÷ 6 = 6

1⁄7 of 42 = 6

because

42 ÷ 7 = 6

1⁄6 of 42 = 7

because

42 ÷ 6 = 7

1⁄7 of 49 = 7

because

49 ÷ 7 = 7

1⁄6 of 48 = 8

because

48 ÷ 6 = 8

1⁄7 of 56 = 8

because

56 ÷ 7 = 8

1⁄6 of 54 = 9

because

54 ÷ 6 = 9

1⁄7 of 63 = 9

because

63 ÷ 7 = 9

1⁄6 of 60 = 10

because

60 ÷ 6 = 10

1⁄7 of 70 = 10

because

70 ÷ 7 = 10

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1⁄6 of 6 = 1

When you understand the link between single fractions and division, the next step is to calculate multiple fractions. 5⁄7 of 42 = 5 x (1⁄7 of 42)

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= 30

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=5x6

If 1⁄8 of 8 = 1‚

3⁄8 of 8 = 3

because

3x1=3

If 1⁄8 of 48 = 6‚

7⁄8 of 48 = 42

because

7 x 6 = 42

If 1⁄8 of 56 = 7‚

3⁄8 of 56 = 21

because

3 x 7 = 21

If 1⁄8 of 64 = 8‚

5⁄8 of 64 = 40

because

5 x 8 = 40

If 1⁄8 of 72 = 9‚

7⁄8 of 72 = 63

because

7 x 9 = 63

If 1⁄8 of 80 = 10‚

3⁄8 of 80 = 30

because

3 x 10 = 30

If 1⁄8 of 16 = 2‚ 5⁄8 of 16 = 10 because 5 x 2 = 10 . tof 24 = 3‚ 7⁄8 of 24 = 21 because 7 x 3 = 21 o If 1⁄8e c . If 1⁄8 of 32c = 4‚ 3⁄8 of 32 = 12 because 3 x 4 = 12 e her r o st s If 1⁄8 of 40 = 5‚ 5⁄8 of 40 =u 25p because 5 x 5 = 25 er

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Solve problems involving division by a one-digit number, including those that result in a remainder

These tables show the link between division and fractions.

Assessment 1

Sub-strand: Number and Place Value—N&PV – 4

NAME:

DATE:

Solve each problem using the table given. (a) 492 ÷ 6 =

(b) 738 ÷ 8 =

Multiplication facts Subtraction facts

–

Hundreds

Tens

Ones

How many 9s?

–

(

) (

)+(

(

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–

)

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=( + + )x © R. I . C.Publ i cat i o ns ( x ) x •f orr evi ew pur posesonl y• =

–

(d)

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eights

(c)

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CONTENT DESCRIPTION: Solve problems involving division by a one-digit number, including those that result in a remainder

sixes

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(e)

+ o c . = ch=e e r o t r s supe r 4 8 7 3

–

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

+ +

www.ricpublications.com.au

Assessment 2

Sub-strand: Number and Place Value—N&PV – 4

NAME:

DATE:

1. Show your working to solve each problem. (b) Five strawberries are used to make one milkshake. (i) How many can be made with 237? (ii) Would there be any strawberries left over?

(d) Marker pens come in packs of eight. How many packs must the school buy if 372 students are to have one pen each?

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helicopter have to make to rescue 45 people stranded at high tide?

Shade the bubble to show your answer.

o c . c e (a) ¼ of 36 6 3 r 5 h 15 27 e o r st super (b) 1⁄7 of 42 3 9

3. Find the fraction of each number.

2. Use doubling or halving, or tripling or ‘thirding’ to solve the equations. (a) 45 ÷ 9 = (b) 32 ÷

÷3

= 16 ÷ 4 8

(d)

÷ 7 = 28 ÷ 14 3

(e)

÷ 3 = 36 ÷ 6

÷ 12 8

(f) 12 ÷ 4 = 62

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(f) A helicopter carries © R. I . C.Pu b l i c at i ons seven passengers. How many trips •f orr evi ew pu r po sesonl y• would a single

(e) A surf club insists on a minimum of one adult in the water for every three swimmers. How many adults are needed if there are 137 swimmers in the water?

9 6

(d) 1⁄8 of 56

(e) 1⁄5 of 45

(f) 1⁄6 of 54

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Solve problems involving division by a one-digit number, including those that result in a remainder

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Teac he r

(a) 357 potato seedlings are planted in rows of nine. (i) How many rows have been planted? (ii) Are there any seedlings left over?

STUDENT NAME

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

Use the context of a problem to determine how to treat a remainder

Uses the link between division and fractions to solve problems

Uses known multiplication and division facts to solve problems

Checks answers to division algorithms by backtracking with multiplication

Uses a range of strategies to show the process of a division algorithm

Recognises the equivalence of numbers multiplied or divided by a common factor

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Sub-strand: Number and Place Value—N&PV – 4

Checklist

Solve problems involving division by a one-digit number, including those that result in a remainder (ACMNA101)

Sub-strand: Number and Place Value—N&PV – 5

Use efficient mental and written strategies and apply appropriate digital technologies to solve problems (ACMNA291)

RELATED TERMS

TEACHER INFORMATION

Mental calculation

What this means

• A calculation that is performed in the head using a knowledge and understanding of number facts and place value.

• Students will need to make sensible choices as to when it is appropriate to solve a calculation mentally, with pen and paper or with the aid of a calculator. Students should be encouraged to try mental methods first. Written calculations are effectively a series of mental calculations performed in a logical sequence. As the answer to each step is recorded, the memory space is ‘cleared’, ready for the next mental calculation to be performed. Mental and written calculations can be checked for accuracy using a calculator. If there is a discrepancy, students should check that they have entered the information on the calculator correctly first before checking their written calculations as this would be more time consuming.

Written strategy

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Algorithm

• The layout of a number operation so that a calculation can be performed and steps identified. Digital technology

• An electronic device (such as a pocket calculator) that is programmed to calculate answers to number problems after information has been input by the user.

Teaching points

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• A calculation process recorded on paper with individual steps that may be performed mentally. This can be useful‚ as errors can be identified and rectified.

• Students need to be taught to make appropriate calculation choices and then execute them. Interactive games on the computer such as ‘Beat the calculator’ (where students are given a series of calculations to complete either mentally or with a calculator)‚ help students to decide when to abandon mental calculation. Different students have a greater or lesser capacity for mental calculation and so will resort to using a calculator at different times.

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• • When multiplying (or adding) three or more numbers, the order in which groups of numbers are multiplied (or added) does not affect the outcome.

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Distributive law

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Student vocabulary multiple factor

What to look for • Be aware of students who perform mental calculations slowly or inaccurately as a result of a lack of fluency with basic number and facts, and understanding of place value‚ and an inability to follow the steps of algorithms accurately. • Look for students who arrive at an incorrect result because they have recorded the answers to steps of a procedure inaccurately; e.g. reversal (as in 43 instead of 34) in incorrect place value columns if they are working with plain paper rather than squared. • Students who arrive at an incorrect result when using a calculator because they have keyed in the information incorrectly or have not pressed the keys hard enough to register.

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Associative law

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See also New wave Number and Algebra (Year 5) student workbook (pages 27–32)

partitioning mentally written strategy algorithm procedure calculator

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

Sub-strand: Number and Place Value—N&PV – 5

HANDS-ON ACTIVITIES Tables snap Two students deal a deck of cards between them. Player One lays down a card. Player Two lays down a card. The first player to call out the correct product of the two numbers takes the cards. This player lays down the next card. Play continues until one player (the winner) has taken all of the cards.

Teacher–student role reversal Students volunteer to play the role of teacher and explain the procedure of different strategies to solve a multiplication or division problem. Students can be filmed in their role as teacher and the videos used as resources for explaining the strategy to others.

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Problem–solution match-up

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Prepare a selection of number stories on cards that require more than one operation to solve. Write the solutions to the problems on separate cards, showing all the calculations. In pairs, students read and discuss a problem then look through the solution cards to find the matching answer. Store the cards in packs of four for students to complete in one session. They will be able to self-correct as all cards should be matched.

Guess-check-improve

Working in pairs and taking turns, one students uses a calculator to find the answer to a division problem involving a three-digit dividend and a two-digit divisor. Then he or she records the answer secretly. The second student follows the ‘Guess-check-improve’ strategy (page 67) to find the answer. Students challenge each other to take the least number of guesses to reach the answer.

Which works best for me?

Students generate numbers for multiplication and division problems using dice, spinners or playing cards. They work through each problem using at least three different methods. If different answers are obtained, they can check through their calculations to find their errors. Finally, they can check their answers using a calculator.

First to 5000!

Best fit

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Students generate numbers for multiplication and division problems using dice, spinners or playing cards. Alternate one multiplication problem with one division problem. Students record the answer to each problem and, using a calculator, keep a running total. The first student to reach 5000 is the winner for the week. If no-one achieves this in one week, decide to rollover to the next week or reduce the total to make it achievable.

o c . che e r o t r s super

Write multiplication and division problems and consider which strategies are the most appropriate to use to solve them.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 5

LINKS TO OTHER CURRICULUM AREAS English • • • •

Write procedures to explain how to solve a multiplication or division problem using a given strategy. Give an oral report on a preferred method of multiplication or division. The report must include reasons for preference. Write an oral explanation of a given method for solving multiplication or division problems. Write a short report on a famous mathematician.

Information and Communication Technology • History of number representations <http://www.teachingideas.co.uk/maths/numbersys.htm> • Number bases <http://motivate.maths.org/conferences/conf67/c67_Number_bases.shtml> • An online maths game that times how many multiplication‚ division‚ subtraction and addition problems can be solved in two minutes: <http://arithmetic.zetamac.com>.

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History

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• Visit <http://www.teachingideas.co.uk/maths/numbersys.htm> for information on the development of number representation since ancient times.

The Arts

• Create a display showing the various representations of number since ancient times. • Visit <http://www.robinsunne.com/robinsunnes_multiplication_clock> for instructions to make a multiplication clock.

Technology

• Create a board game in which players have to answer multiplication and division questions to proceed. Include rewards for correct answers (such as an extra turn) and penalties for incorrect ones (such as having to backtrack two spaces).

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o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

Sub-strand: Number and Place Value—N&PV – 5

RESOURCE SHEET Guess-check-improve strategy for finding answers

By using this strategy to answer number problems, students use the knowledge they gain about their original guess to inform their choice for an improved ‘guess’. This is a mathematically sound process‚ as opposed to randomly choosing numbers in the hope that one might eventually be the correct answer. Start by providing a calculation that is beyond the students’ ability to calculate mentally; for example: 783 ÷ 27 (= 29). Students offer a possible quotient which may be random or chosen by rounding 783 and 27 (for example, 24).

Teac he r

Guess

Using this information, students make a second possible guess knowing that it must be above 24.

© R. I . C.Publ i cat i ons •f orr evi ew pur posesonl y• Also, 27 is close to 25 and students will know that there are four lots of 25 in At this stage, some may use reasoning to recognise that 135 is half of 270 and that 270 ÷ 27 = 10, so 135 ÷ 27 must be five.

Improve

100 with one more for 35, to make five.

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It is easier to estimate using rounding when the dividend has been reduced after the first guess. If the numbers have been rounded down, the guess will need to be raised by one, if they have been rounded up, the guess will be reduced by one.

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CONTENT DESCRIPTION: Use efficient mental and written strategies and apply appropriate digital technologies to solve problems

Using a calculator, 24 is checked as a possible quotient by multiplying it by 27 and comparing the product with 783. 24 x 27 = 648 783 – 648 = 135 24 was a reasonable guess‚ but is too low by 135.

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Check

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o c . che 644 ÷ 28 (= 23). r e o (with calculator) Check (with calculator) Improve t r s super 28 x 20 = 560 Too small: 644 – 560 = 84

Organising guess-check working in a table will help students to work systematically and make realistic improved guesses; for example:

Guess 20 22

28 x 22 = 616

Almost there‚ but still too small: 644 – 616 = 28

28 x 23 = 644

Spot on! 644 – 644 = 0

For the guess-check-improve strategy to work efficiently, it is important to remember: • record all guesses • organise working in a table • guess-check-improve is not random • the ‘improve’ step is vital. Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 5

RESOURCE SHEET Time-saving strategies for multiplication and division

Division by 5 Division by 5 can often be made easier to calculate mentally if the number is first doubled and then divided by 10. If the ones digit of the dividend is zero, it is simpler to divide by 10 and then multiply by two.

4235 ÷ 5

= 8470 10 = 847

Example 3

Example 4

7935 ÷ 5

3240 ÷ 5

8640 ÷ 5

= 7935 x 2 10

= 3240 x 2 10

= 8640 x 2 10

= 15 870 10

= 324 x 2

= 864 x 2

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Teac he r

= 4235 x 2 10

r o e t s Bo r e p ok u S Example 2

= 648

= 1728

= 1587

Multiplication by 5

Multiplication by 5 can be facilitated by multiplying by 10 and then dividing by two.

Example 1. Example 2i © R. I C.Publ i cat ons 343 x 5 276 x 5 •f or r e v i e w p u r p osesonl y• = 343 x 10 = 276 x 10

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= 3430 2

2 = 138 x 10

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= 1380

= 1715

. tesimple task to perform but it can be made even o Dividing by 2 is a relatively simpler by c making use of the distributive law and partitioning the numbers to suit. the calculation. che e r o t r Example 1 Example s 2 super Dividing by 2

= 3430 2

= 2760 2

= (2000 + 1400 + 20 + 10) 2

= (2000 + 600 + 100 + 60) 2

= 1000 + 700 + 10 + 5

= 1000 + 300 + 50 + 30 = 1380

= 1715

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Use efficient mental and written strategies and apply appropriate digital technologies to solve problems

Example 1

Sub-strand: Number and Place Value—N&PV – 5

RESOURCE SHEET Multiplication and division by 25

Students need to know and understand the relationship between 25 and 100, 1000 etc. 100 ÷ 25 = 4 1000 ÷ 25 = 40

Teac he r

Example 1

Example 2

32 x 25

57 x 25

= 32 x 100 4

= 57 x 100 4

= 32 x 100 4

= 57 x 100 4

= 8 x 100

= 5700 4

= 800

= 1425

Example 3

Example 4

= 124 x 100 4

= 131 x 100 4

= 31 x 100

= 13 100 4

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Division

. te

= 3100

= 3275

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x 25 131 x 25 © R124 . I . C .Publ i ca t i o ns = 124 x 100 = 131 x 100 4 p •f orr evi ew ur poses4onl y•

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CONTENT DESCRIPTION: Use efficient mental and written strategies and apply appropriate digital technologies to solve problems

Multiplication

o c . che e r o Example 2 s Example 3 t r su r e p 8250 ÷ 25 1475 ÷ 25

This method uses the relationship between 25 and powers of 10, and the partitioning of the dividend. Example 1 7325 ÷ 25 = (7000 + 300 + 25) 25

= (8000 + 200 + 50) 25

= (1000 + 400 + 75) 25

= (7 x 1000) + (3 x 100) + 25 25 25 25

= (8 x 1000) + (8 x 100) + 50 25 25 25

= (1 x 1000) + (4 x 100) + 75 25 25 25

= (7 x 40) + (3 x 4) + 1

= (8 x 40) + (2 x 4) + 2

= (1 x 40) + (4 x 4) + 3

= 280 + 12 + 1

= 320 + 8 + 2

= 40 + 16 + 3

= 293

= 330

= 59

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Number and Place Value—N&PV – 5

RESOURCE SHEET Multiplication and division using factors

Multiplication and division by larger, two-digit numbers can be made simpler by using a factor of the multiplier or divisor and performing the calculation in two or more steps. Multiplication Partitioning the numbers and using the larger factor first will help to make the calculation even simpler.

57 x 12

132 x 21

= (132 x 7) x 3

= (50 x 4) + (7 x 4) x 3

= (100 x 7) + (30 x 7) + (2 x 7) x 3

= (200 + 28) x 3

= (700 + 210 + 14) x 3

= (200 + 20 + 8) x 3

= (700 + 210 + 10 + 4) x 3

= (600 + 60 + 24)

= (2100 + 630 + 30 + 12)

= 684

= 2772

Teac he r

= (57 x 4) x 3

Example 2

176 ÷ 8

540 ÷ 36

= (176 ÷ 4) ÷ 2

= (540 ÷ 4) ÷ 9

= 44 ÷ 2

= 135 ÷ 9

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Example 1

. te = 22

= 15

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o c . cheExample 3 e r o t r s s r u 888 ÷p 24 e = (888 ÷ 4) ÷ 2 ÷ 3 = (222 ÷ 2) ÷ 3 = 111 ÷ 3 = 37

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Use efficient mental and written strategies and apply appropriate digital technologies to solve problems

Example 2

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Division

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Example 1

Sub-strand: Number and Place Value—N&PV – 5

RESOURCE SHEET Multiplication algorithms

Some students will confidently use the algorithm. Have students verbally explain the procedure to ensure they have full comprehension of the concept of place value. Can they explain that the algorithm is a formal way to illustrate the partitioning method and why the partial products must be added to give the product?

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Large numbers by a single-digit number H

9 x 7 ones

9 x 3 tens (or 30)

5 0 9 x 6 hundreds (or 600) © R4. I . C0.Pu bl i ca t i ons 5r 7i 3 p 3 r •f or ev ew u posesonl y• +

Ten Th

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o c 7 x1 = 7 . che e r o 1 4 s r s rt0 7 x 20 = 140 upe 3

7 7

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Large number by a two-digit number

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CONTENT DESCRIPTION: Use efficient mental and written strategies and apply appropriate digital technologies to solve problems

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Teac he r

7 x 500 = 3500

30 x 1 = 30

30 x 20 = 600

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

30 x 500 = 15 000

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Assessment 1

Sub-strand: Number and Place Value—N&PV – 5

NAME:

DATE:

1. In two guesses, find the answer to each problem. Use a calculator to check each guess. Show all your working. (a) 594 ÷ 9

(b) 812 ÷ 7

Guess 1

Check

Check 2

Too large/ small by: Guess 2

Check 2

592 ÷ 37 =

817 ÷ 43 =

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Guess 2

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2. Use the ‘multiply by 2/divide by 10’ strategy and ‘multiply by 10/divide by 2’ strategy to solve the problems.

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Show your working.

. te o c 3. Use partitioning and the relationship between 25 and 100, 1000 etc. to solve the . c e problems. her r o t s÷ 25 (d) 3975 ÷ 25 sup(c)e8350 r (b) 49 x 25 (a) 64 x 25

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Use efficient mental and written strategies and apply appropriate digital technologies to solve problems

Too large/ small by:

Assessment 2

Sub-strand: Number and Place Value—N&PV – 5

NAME:

DATE:

1. Use factors to solve the problems. (a) 43 x 9

(b) 62 x 9

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2. Use the algorithm to solve each problem.

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(c)

H R TI © . . CO.Publ i cat i ons 3 7 9 orr evi ew ur posesonl y• x •f 7 p

(b)

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CONTENT DESCRIPTION: Use efficient mental and written strategies and apply appropriate digital technologies to solve problems

T 8

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(a)

3. (a) Write a multiplication problem using the number sentence 31 x 7 =

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(b) Work out the answer, showing all calculations.

4. (a) Write a division problem using the number sentence 365 ÷ 12 =

(b) Work out the answer, showing all calculations.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Checklist

Sub-strand: Number and Place Value—N&PV – 5

Use efficient mental and written strategies and apply appropriate digital

Performs division calculations using factors of the or divisor

Performs multiplication calculations using factors of the multiplier

Uses relationships between multiples and their factors to solve calculations

(Guess-CheckImprove)

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STUDENT NAME

Uses knowledge gained to assist predictions of problem answers

technologies to solve problems (ACMNA291)

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Answers

Sub-strand: Number and Place Value

N&PV – 1 Page 17 4 3

(a) (c) (e) (a) (d)

(i) 28 (i) 42 (i) 16 yes no

Page 18

3 8

4 12

6 12

4 8

9 6

12 8

Page 48

1. 2.

(ii) 7 (b) (i) 72 (ii) 8 (ii) 6 (d) (i) 45 (ii) 9 (ii) 4 (f ) (i) 15 (ii) 3 (b) yes (c) no (e) no

Divisor

Dividend

46 532

5 x 9 = 45

45 000

9000

9306.4

31 389

4 x 8 = 32

32 000

8000

7847.25

53 721

9 x 6 = 54

54 000

6000

5969

Related number fact

Rounded dividend

Quotient estimate

Exact answer

Assessment 2

(a) Up; to make sure he has enough materials for the job and money to pay for them. (b) Teacher check – depends on method of rounding used. Possibilities include

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50 + 130 + 70 = 250 m fencing @ $30/m = $7500

10 – 7 = 3 10 – 8 = 2 7 – 2 = 5 or 8 – 3 = 5

2 x 1 = 1

3 x 2 = 6

Hundreds

Tens

Ones

Sum of partial products

7 8 7 (700 x (80 x 30 =) (7 x 30 =) 30 =) 3 2400 210 21 000 (700 x 2 =) (80 x 2 =) (7 x 2 =) 2 1400 160 14

21 000 2400 1400 210 160 14 25 184

Assessment 1

Teacher check for number facts used. (a) 82 (b) 92 remainder 1 (d) 62 (e) 218 remainder 1

Page 62 1.

2. 3.

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

Assessment 2

(i) 39 rows (i) 47 milk shakes 4 ½ hours 46 adults 15 (b) 8 14 (e) 18 9 (b) 6 7 (e) 9

(ii) (ii) (d) (f ) (c) (f ) (c) (f )

yes – 6 seedlings yes – 2 strawberries 47 packs 7 trips 3 36 9 9

N&PV – 5 Page 72 1. 2. 3.

Assessment 1

(a) 66 (a) 285 (a) 1600

(b) 116 (b) 463 (b) 1225

Page 73

1. 2.

45 + 125 + 65 = 235 m (240 m) fencing @ $30/m = $7200

(a) 387 (a)

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4 4 8

1 8 0 9

H 3

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4 11 6

6 9 0 5

(a) (i) 300 k (ii) 400 k (iii) 200 k (b) (i) down to 70 L (ii) to 20 L (iii) to make sure the car does not run out of fuel (c) 350 k (d) 180 L

2 2

(b)

Assessment 1

(a) 1645 (b) 2976 (c) 2407 (a) 29 x 12 = (20 + 9) x (10 + 2) = (20 x 10) + (20 x 2) + (9 x 10) + (9 x 2) = 200 + 40 + 90 + 18 = 348 (b) 23 x 17 = (20 + 3) x (10 + 7) = (20 x 10) + (20 x 7) + (3 x 10) + (3 x 7) = 200 + 140 + 30 + 21 = 391

3. 4.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

2 2

(d) 4310 (d) 159

Assessment 2

o c . che e r o t r s super Rounding up to five, then rounding the total to ten

N&PV – 3 1. 2.

8 – 1 = 7 or 9 – 2 = 7

Page 61

Closest linked number

Page 47

Rounding up to ten

10 – 9 = 1

N&PV – 4

(a) 420 (b) 80 (c) 40 (d) 160 (e) 60 (f ) 800 (g) 10 (h) 80 (a) 70 (b) 30 (c) 60 (d) 40 Context always requires rounding up. (a) 360 ÷ 4 = 90 (b) 600 x 8 = 4800 (c) 240 ÷ 3 = 80 (d) 500 x 9 = 4500 (e) 420 ÷ 6 = 70 (f ) 120 x 6 = 720 Monday‚ $43‚ $45; Tuesday‚ $24‚ $25; Wednesday‚ $39‚ $40; Thursday‚ $15‚ $15; Friday‚ $28‚ $30

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Number sentence: 787 x 32

Assessment 1

Page 30

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2. 3.

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N&PV – 2 1.

10 – 8 = 2

(a) 150 (b) 6 (c) 2 – 1 x 2; 3 – 1 x 3; 4 – 1 x 4, 2 x 2; 5 – 1 x 5; 6 – 1 x 6, 2 x 3, 3 x 2; 7 – 1 x 7; 8 – 1 x 8, 2 x 4; 9 – 1 x 9, 3 x 3; 10 – 2 x 5; 12 – 2 x 6, 3 x 4; 14 – 2 x 7; 15 – 3 x 5; 16 – 2 x 8; 18 – 2 x 9, 3 x 6; 21 – 3 x 7; 24 – 3 x 8; 27 – 3 x 9; 50 – 1 x 50 (d) Examples could include: Player 1 – triple 3, double 6, single 8 (9 + 12 + 8 = 29) Player 2 – single 9, triple 4, double 4 (9 + 12 + 8 = 29) Player 3 – single 3, triple 6, single 8 (3 + 18 + 8 = 29) Player 4 – triple 8,single 3, single 2 (24 + 3 + 2 = 29)

Page 29

Assessment 2

(5 x 2) + (5 x 4) + (5 x 6) + (5 x 3) = 5 x (2 + 4 + 6 + 3) = 5 x 15 = $75 (a) (b) 8 x 9 7 x 8

Assessment 2

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Tens

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Ones

Assessment 1

(c) 37 x 19 = (30 + 7) x (10 + 9) = (30 x 10) + (3 x 9) + (7 x 10) + (7 x 9) = 300 + 270 + 70 + 63 = 703 (a) (4 x 6) + (3 x 6) = 6 x (3 + 4) = 6 x 7 = 42 (b) (5 x 8) + (5 x 7) = 5 x (8 + 7) = 5 x 15 = 75 (a) 210 (b) 400 (c) 2400 (d) 1800

(a) 217 (a) 30 remainder 5

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O 3 6 8 0 0 8 O 9 7 3 0 0 3

(b) 558 (c)

(b) Teacher check (b) Teacher check

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Sub-strand: Fractions and Decimals—F&D – 1

Compare and order common unit fractions and locate and represent them on a number line (ACMNA102)

RELATED TERMS

TEACHER INFORMATION

Fraction

What this means

• An expression showing one number being divided by another

Students need to know that the denominator of a fraction expresses the number of unit parts a whole is divided into and, as the denominator increases, the size or value of each unit part decreases.

Denominator

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Numerator

• The dividend of a fraction, sitting above the ‘divided by’ line • The number of equal parts of a fraction being considered Unit fraction

• A fraction with a numerator of one; a single equal part of a whole

Teaching points

Explore the relationships among fractions. An easy way to model this is to use standard-sized number lines to compare the relative size of unit fractions with different denominators. For example, a half is twice the size of a quarter because the whole has been divided into half as many unit parts.

What to look for

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• The divisor of a fraction, sitting below the ‘divided by’ line • The number of equal parts into which a whole (number) is divided

• Students who do not recognise that the size of the denominator is inversely proportional to its value; e.g. 1⁄7 is less than 1⁄5. • Students who think fractions are the same as whole numbers and therefore think 1⁄8 is larger than 1⁄5.

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See also New wave Number and Algebra (Year 5) student workbook (pages 33–38)

o c . che e r o t r s super Proficiency strand(s):

Student vocabulary Fraction Numerator Denominator

Understanding Fluency Problem solving Reasoning

Unit fraction Fraction families

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Fractions and Decimals—F&D – 1

HANDS-ON ACTIVITIES How big is this fraction? Give each student four squares of paper (about 20 cm x 20 cm). Fold the first in half horizontally and label each half as ½; the second square in half and then into quarters and label each quarter as ¼. Next‚ fold the third square in half, into quarters and then into eighths and label each eighth as 1⁄8. Finally‚ fold the fourth square in half, into quarters and eighths and then into sixteenths and label each sixteenth as 1⁄16. Students will see that as the denominator of the unit fraction increases, the size it denotes decreases. The whole is divided into more parts so each equal part is smaller. Repeat the exercise with triangles and circles.

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Fractions mix and match

Make several copies of pages 83 to 85. Cut out and laminate all the fraction shapes. To make the master bases, make sufficient copies of any of the pages so that each player has one each. Cut around the outside of the rectangle. In the centre of the reverse side write ‘One whole’‚ then laminate.

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The object of the game is to be the first to cover one’s own master base with fraction pieces. Any combination of fractions is allowed‚ but there must be no overlapping. Players take turns to throw a 10-sided dice. The number thrown indicates the denominator of the unit fraction that a player can pick up and place on his or her master board. If that unit fraction does not fit, the player has to wait for his or her next turn. A player can choose to miss a turn if he or she wants a particular fraction. The zero on the dice indicates 1⁄10 while the nine on the dice indicates 1⁄12. If a player throws a seven, he or she misses a turn.

Unit fraction bingo card challenge

Label a resealable plastic bag for each group of fractions: ½, ¼, 1⁄8; 1⁄3, 1⁄6, 1⁄12; 1⁄5, 1⁄10. Use the templates on page 86 to prepare several laminated unit fractions to place in each bag. One-half cards can be made separately.

Before making the bingo cards, use the templates on page 87 to record different positions and combinations of unit fractions for each bingo card. Mark and label the cards appropriately and laminate them. When sufficient different cards for each category have been made, the game can be played.

Fraction rummy

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Make up and laminate 56 playing cards (see page 87): eight halves, 16 quarters and 32 eighths. The aim of the game is to collect as many ‘wholes’ as possible. Shuffle cards and deal seven to each of between two to four players. Place the remaining cards in a pile, facedown on the table. Each player examines his or her cards and groups together all similar unit fractions. Players take turns to pick up and discard cards. The discarded cards must be fanned so that all cards are visible. A player can pick up one card from the facedown pile or as many cards as he or she chooses from the discarded pile. If a player wants a discarded card, he or she must also take any cards that have been placed on top of it. Players must discard one card to complete their turn. As each whole is made, it is placed face up on the table. Play ends with the first player to use up all of his or her cards.

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Ordering fractions/Missing fractions

Create fraction cards (see page 89) and use them to place in ascending or descending order.

Arrange some in order‚ with spaces between. Students place the remainder in the spaces in the correct ascending or descending order.

Less than or greater than? Make several copies of the fraction cards on page 90. Cut and laminate. Working in pairs or small groups, one student shuffles the cards and deals 20 to each player. Without looking at their cards, players set them out in 10 pairs. On paper, players record their pairs and, reading from left to right, insert the correct sign: less than or greater than. Players hand their papers to the player to their left for checking.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Fractions and Decimals—F&D – 1

LINKS TO OTHER CURRICULUM AREAS English Write an explanation to help a fellow student understand what an Egyptian fraction is: The Egyptians always used unit fractions, referring to any others as ‘vulgar’ fractions. They expressed fractions as the sum of a set of unit fractions; for example, ¾ as ¼ + ¼+ ¼, 5⁄6 as 1⁄6 + 1⁄6 + 1⁄6 + 1⁄6 + 1⁄6. The same fraction was never repeated in any set. Some students may see that some unit fractions can be added to make new unit fractions; e.g. ¼ + ¼ = ½‚ 1⁄6 + 1⁄6 = 1⁄3.

Information and Communication Technology

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History and Geography

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• The history of fractions can be explored at <http://scienceray.com/mathematics/the-history-of-fractions/> and <http:// www.basic-mathematics.com/history-of-fractions.html> • An explanation of Egyptian fractions can be viewed at <http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/ egyptian.html>‚ <http://www.mathcats.com/explore/oldegyptianfractions.html> and <www.mathexpression.com/ understanding-fractions.html>

• Research the ancient Egyptian hieroglyphic numeral system and write all the powers of 10 from one to one million. • On an outline of a world map, colour the regions where ancient civilisations are known to have lived; for example: Egyptians, Romans, Greeks, Chinese, Mayans, Aztecs, Babylonians.

The Arts

• Students use a range of art techniques to decorate the squares, triangles and circles from the ‘How big is this fraction’ hands-on activity. Glue each shape on to thin card before decorating to make each piece more sturdy. • Make fraction shape mobiles using the decorated fraction shapes from the above activity. Tie a knot in a length of string. Thread the string through the underside of the ‘whole’ shape. Tie another knot before threading through the first half shape. Repeat until all shapes, from largest to smallest, have been attached. Make a loop in the string for hanging. The length of string between one piece and the next will depend on the height available for hanging. There will be 31 pieces in total, including the one whole. At 1 cm between each shape, the height of the mobile will be 30 cm. • Dramatise a whole being divided into its unit fractions. For example, have six students face each other in a circle with arms held high and hands touching. To music, they slowly and symmetrically separate into their ‘equal’ parts. Fraction groups could perform together so the increase in number can be seen; i.e. halves, quarters and eighths, thirds, sixths and twelfths and fifths and tenths. • Design a poster showing fractions of the ancient Egyptian hieroglyphic numeral system. • Use different craft materials to create a fraction wall showing the relative value of some fractions; e.g. ½, ¼ and 1⁄8 or 1⁄3, 1⁄6 and 1⁄12.

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET Compare ½, ¼ and 1⁄8 unit fractions – 1

When a whole, one ‘thing’ or the number one is divided into equal parts, each part is a ‘unit fraction’.

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

Name of each part (unit fraction)

two (2)

one half (½)

four (4)

one quarter (¼)

eight (8)

one eighth (1⁄8)

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Teac he r

The ‘denominator’ (or name) of a unit fraction depends on how many parts the whole has been divided into.

½+½=1

¼P +¼ +b ¼ +l ¼c =a 1 t © R. I . C. u i i ons 1⁄8 + 1⁄8+ 1⁄8 + 1⁄8 + 1⁄8 + 1⁄8 + 1⁄8 + 1⁄8 = 1 •f orr evi ew pur posesonl y•

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½ is two times greater than ¼‚ and four times greater than 1⁄8.

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The greater the number of equal parts a whole is divided into, the smaller each unit fraction is.

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CONTENT DESCRIPTION: Compare and order common unit fractions and locate and represent them on a number line

When all the unit fractions are added, they form the whole again:

o c . 1⁄8 is half as small as ¼‚ che e and a quarter of the size of ½r . o t r s super As the number of unit fractions increases‚ ¼ is half as small as ½‚ but two times greater than 1⁄8.

the size of each unit fraction decreases.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET Compare ½, ¼ and 1⁄8 unit fractions – 2

Fractions can be compared using: • number lines

one whole

r o e t s Bo r ½ e p ok u S ¼ 2⁄4 ¾

1⁄8

2⁄8

3⁄8

4⁄8

5⁄8

6⁄8

• shapes

2⁄2 4⁄4

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0 0

7⁄8

8⁄8

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One whole can fit

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1⁄8 o c . che e r o r st super 1⁄8

One half can fit

One quarter can fit

one whole (1⁄1) one half (½) one quarter (¼)

two halves (2⁄2) two quarters (2⁄4)

two eighths (2⁄8)

four quarters (4⁄4) four eighths (4⁄8) eight eighths (8⁄8) 80

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Compare and order common unit fractions and locate and represent them on a number line

Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET Compare 1⁄3, 1⁄6 and 1⁄12 unit fractions – 1

When a whole, ‘one thing’ or the number one is divided into equal parts, each part is a ‘unit fraction’.

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

Name of each part (unit fraction)

three (3)

one third (1⁄3)

six (6)

one sixth (1⁄6)

twelve (12)

one twelfth (1⁄12)

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Teac he r

The ‘denominator’ or (or name) of a unit fraction depends on how many parts the whole has been divided into.

1⁄3 + 1⁄3 + 1⁄3 = 1

1⁄6. + 1⁄6P + 1⁄6 +b 1⁄6 +l 1⁄6 + 1⁄6 =t 1i © R. I . C u i c a ons 1⁄12 + 1⁄12+ 1⁄12 + 1⁄12 + 1⁄12 + 1⁄12+ 1⁄12 + 1⁄12 + 1⁄12 + 1⁄12+ 1⁄12 + 1⁄12 = 1 •f or r evi ew pur posesonl y•

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1⁄3 is two times greater than 1⁄6, and four times greater than 1⁄12.

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The greater the number of equal parts a whole is divided into, the smaller each unit fraction is.

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CONTENT DESCRIPTION: Compare and order common unit fractions and locate and represent them on a number line

When all the unit fractions are added, they form the whole again:

o c . che1⁄12 is half as small as 1⁄6, r e o and a quarter of the size of 1⁄3. t r s s uper 1⁄6 is half as small as 1⁄3, but two times greater than 1⁄12.

As the number of unit fractions increases, the size of each unit fraction decreases.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET Compare 1⁄3, 1⁄6 and 1⁄12 unit fractions – 2

Fractions can be compared using: • number lines

one whole

1⁄6

1⁄12

• shapes

2⁄12

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3⁄12

2⁄6

4⁄12

3⁄6

5⁄12

6⁄12

4⁄6

7⁄12

8⁄12

3⁄3

5⁄6

6⁄6

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9⁄12

10⁄12 11⁄12 12⁄12

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1⁄3

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One whole can fit

1⁄6

1⁄12

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1⁄12

1⁄12 co 1⁄12

. che e r o t r s super One third can fit

One sixth can fit

one whole (1⁄1) one third (1⁄3) one sixth (1⁄6)

three thirds (3⁄3) two sixths (2⁄6)

two twelfths (2⁄12)

six sixths (6⁄6) four twelfths (4⁄12) twelve twelfths (12⁄12) 82

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Compare and order common unit fractions and locate and represent them on a number line

Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET

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r o e t s Bo r e p ok u S ½

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CONTENT DESCRIPTION: Compare and order common unit fractions and locate and represent them on a number line

Fraction mix and match – ½, ¼, 1⁄8

1⁄8

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1⁄8 Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET

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1⁄12

1⁄6

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1⁄12

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

1⁄12

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

CONTENT DESCRIPTION: Compare and order common unit fractions and locate and represent them on a number line

r o e t s Bo r 1⁄3 e p ok u S

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Teac he r

Fraction mix and match – 1⁄3, 1⁄6, 1⁄12

Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET

Teac he r

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1⁄5 r o e t s Bo r e p ok u S 1⁄5

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1⁄5

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CONTENT DESCRIPTION: Compare and order common unit fractions and locate and represent them on a number line

Fraction mix and match – 1⁄5, 1⁄10

. te o c 1⁄10 1⁄10 . che e r o t r s super

1⁄10 Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

1⁄10 R.I.C. Publications®

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET Unit fraction bingo card challenge – 1

1⁄8

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1⁄5

1⁄12 1⁄12

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1⁄10

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1⁄10 1⁄5 © R. I . C.Publ i cat i o ns 1⁄10 •f orr evi ew pur posesonl y• 1⁄10 1⁄5

o c . che e r o t r s su 1⁄6per 1⁄3 1⁄3

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Compare and order common unit fractions and locate and represent them on a number line

Teac he r

1⁄8

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r o e t s Bo r e p ok 1⁄8 u ¼S

Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET

Teac he r

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

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m . u

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CONTENT DESCRIPTION: Compare and order common unit fractions and locate and represent them on a number line

Unit fraction bingo card challenge – 2

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET Fraction rummy

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

1⁄8

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

CONTENT DESCRIPTION: Compare and order common unit fractions and locate and represent them on a number line

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Teac he r

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

Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET Ordering fractions/Missing fractions

1⁄8

1⁄16

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1⁄7

1⁄10

½0 m . u

1⁄5 w ww

CONTENT DESCRIPTION: Compare and order common unit fractions and locate and represent them on a number line

r o e t s Bo r e p ok u S 1⁄3 1⁄6 1⁄12

o c . che e r o t r s super

1⁄14

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

1⁄9

R.I.C. Publications®

1⁄18

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Sub-strand: Fractions and Decimals—F&D – 1

RESOURCE SHEET Less than or greater than?

1⁄6

1⁄10

1⁄8

1⁄9

1⁄16 90

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1⁄12

o c . che e r o t r s super Greater than

Less than

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Compare and order common unit fractions and locate and represent them on a number line

1⁄5

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1⁄3 r ¼ o e t s Bo r e p ok u S

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Assessment 1

Sub-strand: Fractions and Decimals—F&D – 1

NAME:

DATE:

Shade a bubble to show your answer. 1. The bigger fraction is: (a)

1⁄5 or

(b)

1⁄9 or

1⁄12

(c)

¼ or

1⁄3

2. The missing fraction is: (a) 1⁄3‚

1⁄6‚

(b) ?‚

1⁄8‚

?‚

½4

1⁄9

1⁄12

r o e t s Bo r e p ok u S 1⁄16‚

1⁄32

1⁄10‚

1⁄5

1⁄15

½0

(a) 0

(c)

1⁄3 or

1⁄10 or

1⁄16

¼ or

1⁄6

4. Shade the bubble to show each fraction on the number line.

m . u

(a) 1⁄3

w ww

CONTENT DESCRIPTION: Compare and order common unit fractions and locate and represent them on a number line

(b) 0

ew i ev Pr

Teac he r

3. The point on the number line is:

(b) 2¼

. t1e

(a) 1⁄6

(b) 1⁄3

o c . c e 5. Use a ruler as a guideh to mark the fraction on the number line. r er o t s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

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Assessment 2

Sub-strand: Fractions and Decimals—F&D – 1

NAME:

DATE:

Shade a bubble to show your answer. 1. Eight students each had an equal share of a pizza. Which shape shows how the pizza would have been divided?

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

3. Colour the shape to show the fraction.

1⁄6 l © R. I . C.P(b) ub i cat i ons •f orr evi ew pur posesonl y•

w ww

. te

(d) ¼

m . u

(a) 1⁄3

o c . che e r o t r s super

4. (a) Arrange the fractions from smallest to largest. ½, 1⁄6, 1⁄10, 1⁄9, 1⁄5

(b) Arrange the fractions from largest to smallest. ¼, 1⁄8, 1⁄12, 1⁄7, 1⁄3

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Compare and order common unit fractions and locate and represent them on a number line

ew i ev Pr

Teac he r

2. Which shape has not been divided into four equal parts?

Checklist

Sub-strand: Fractions and Decimals—F&D – 1

Compare and order common unit fractions and locate and represent them

Represents unit fractions on a number line

Locates unit fractions on a number line

Arranges unit fractions in ascending order of size

Arranges unit fractions in descending order of size

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

ew i ev Pr

Teac he r

STUDENT NAME

Compares the value of unit fractions

on a number line (ACMNA102)

w ww

. te

m . u

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Fractions and Decimals—F&D – 2

Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator (ACMNA103)

RELATED TERMS

TEACHER INFORMATION

Fraction

What this means

• An expression showing one number being divided by another

Students need to be taught different strategies to add and subtract fractions with the same denominator.

Denominator

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

• The divisor of a fraction, sitting below the ‘divided by’ line • The number of equal parts into which a whole (number) is divided

• The dividend of a fraction, sitting above the ‘divided by’ line • The number of equal parts of a fraction being considered Unit fraction

• A fraction with a numerator of one • A single equal part of a whole Proper fraction

• A fraction with a value less than 1 • The numerator is less than the denominator

• With addition and subtraction of fractions with like denominators, the denominator remains the same and only the numerators are added/subtracted. • Convert mixed fractions into improper fractions before adding/subtracting. • A whole is made up of the number of unit fractions that match the denominator of each unit fraction. • To simplify an improper fraction, the number of unit fractions required to make a whole is subtracted from the numerator (the number of unit fractions of the improper fraction). • To simplify an equivalent fraction‚ you determine the highest common factor of the numerator and denominator.

ew i ev Pr

Teac he r

Numerator

Teaching points

Improper fraction

w ww

• A fraction with a value greater than one and in which the numerator is greater than the denominator Mixed fraction

. te

• A combination of a whole number and a proper fraction Highest common factor

o c . che e r o t r s super

• The highest number that will divide into (with no remainder) both the numerator and the denominator of a fraction Simplify

• Students who mix up the position and meaning of the numerator and denominator. • Students who struggle to understand the relationship between the numerator and denominator. • Students who cannot add/subtract samedenominator fractions unaided. • Students who cannot simplify equivalent and mixed fractions.

m . u

fraction but whose numerator and denominator are different

See also New wave Number and Algebra (Year 5) student workbook (pages 39–43)

• To reduce a fraction to the point where there is no common factor of the numerator and denominator

Proficiency strand(s): Student vocabulary

Fraction

Numerator

Denominator

Unit fraction

Proper fraction

Equivalent fraction

Improper fraction

Mixed fraction

Highest common factor

Simplify

Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Fractions and Decimals—F&D – 2

HANDS-ON ACTIVITIES Number line addition and subtraction Use the number lines (page 98) and spinners (page 100). In pairs, students take turns to spin a spinner twice to give the two numbers which represent the numerator of a fraction, to work with for addition and subtraction. While one counts, the other checks.

Fraction wheels Make several copies of the fraction wheels (page 99) and the spinners (page 100). Students spin the appropriate spinner twice to give the number of unit fractions to colour in the outer (higher number) and inner (lower number) wheels. On the back of the wheels, students record the fraction sentences for the addition and subtraction of the fractions.

What’s the fraction?

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

Talk about fractions

ew i ev Pr

Teac he r

Make several copies of the shapes on pages 101 and 102. In pairs, students each select three shapes and use two colours to colour them in. They swap shapes and record on the back of each the fraction that is coloured by each colour.

Give pairs of students five large (about 20 cm diameter) paper circles between them. Keep one circle whole. Fold and cut one circle in half. Make statements about the two pieces and compare them with the whole. Students can determine‚ for example, that they are equal. Each is one-half. Two halves join together to make one whole. A whole is bigger than a half‚ while a half is smaller than a whole. Fold and cut the next circle into quarters. Make statements about the four pieces and compare them with the whole and the halves. Repeat the exercise with the other two circles, cutting them into eighths and sixteenths.

Addition and subtraction of fractions

Use the fraction pieces from ‘Talk about fractions’ to model addition and subtraction of fractions with the same denominator, simplifying answers where necessary. More whole circles may be required.

Fraction wall

. te

m . u

w ww

Divide the class into mixed-sized groups; for example: two, three, four, five, six and eight. Provide one paper pizza for each group. ‘Pizzas’ must be of identical size. Each group divides their pizza so that each student in the group receives an equal portion. When all pizzas have been shared, the whole class discusses the size of each student’s portion. Work out which group has the largest/smallest sized pieces. Why? If the pizzas were real and each student could eat his/her piece, would it be fair? Why not? What have they learned from the exercise? (The greater the number of equal pieces something is divided into, the smaller each piece is.)

o c . che e r o t r s super

Using strips of paper, make fraction walls showing the relationship/equivalence of connected fractions; e.g. ½, ¼, 1⁄8, 1⁄16 or 1⁄3, 1⁄6, 1⁄12. Use two colours to make an alternate pattern on each row of the wall.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Fractions and Decimals—F&D – 2

LINKS TO OTHER CURRICULUM AREAS English • Design a flip book of fractions‚ with an explanation of each unit fraction and how it relates in size to a whole and other fractions. • Write the text for a poster explaining how to add and subtract fractions. • Give a written or oral report to explain how you made the different pieces of artwork from The Arts section below.

Information and Communication Technology • All about fractions: <http://www.coolmath4kids.com/fractions/> • Fraction addition: <http://www.coolmath4kids.com/fractions/fractions-10-adding-with-like-denominators-01.html> • Fraction subtraction: <http://www.coolmath4kids.com/fractions/fractions-11-subtracting-with-like-denominators-01.html>

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

Health And Physical Education

ew i ev Pr

Teac he r

Make a salad using slices of fruits or vegetables that have been cut into equal pieces. Take a portion of salad and determine the amount of each fruit or vegetable.

The Arts

• Draw around different regular shapes on coloured paper. Fold and cut each shape into equal parts. Use the parts to make a picture by arranging them on a sheet of paper of contrasting colour and gluing them in place. • Design a ‘How to add and subtract fractions’ poster. Include text from the English activity above.

w ww

. te

m . u

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEET Addition and subtraction of fractions on a number line

To add or subtract fractions, start on the first fraction and count on (or back) the value of the second fraction—count on for addition and count back for subtraction. Examples:

r o e t s Bo r e p ok ½ u S

• ½ + ½ = 2⁄2 = 1

• ¼ + 2⁄4 = ¾

• ¾ – ¼ = 2⁄4 = ½

2⁄4

4⁄4

Examples:

2⁄8

. te

1⁄16 2⁄16

4⁄8

5⁄8

6⁄8

7⁄8

8⁄8

o c . che e r o t r s10⁄16 11⁄16 12⁄16 13⁄16 14⁄16 15⁄16 16⁄16 su er 3⁄16 4⁄16 5⁄16 6⁄16 7⁄16p 8⁄16 9⁄16

• 7⁄16 + 5⁄16 = 12⁄16 = ¾

3⁄8

m . u

1⁄8

w ww

CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

Examples:

2⁄2

ew i ev Pr

Teac he r

• 1–½=½

• 7⁄16 – 3⁄16 = 4⁄16 = ¼

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

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1⁄8

2⁄16

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

1⁄10

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

2⁄12

3⁄12

7⁄16

4⁄12

5⁄12

2⁄6 m . u

4⁄10

2⁄5

6⁄16

6⁄12

3⁄6

5⁄10

8⁄16

4⁄8

9⁄16

6⁄10

3⁄5

10⁄16

4⁄6

7⁄10

11⁄16

Teac he r 5⁄8 6⁄8

7⁄12

8⁄12

CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

1⁄12

3⁄10

5⁄16

3⁄8

2⁄4

4⁄16

o c . che e r o t r s super

2⁄10

1⁄5

3⁄16

1⁄6

Showing twelfths – 1⁄12

Showing sixths – 1⁄6

Showing tenths – 1⁄10

Showing fifths – 1⁄5

1⁄16

2⁄8

. te

Showing sixteenths – 1⁄16

Showing eighths – 1⁄8

13⁄16

7⁄8

12⁄16

9⁄12

5⁄6

10⁄12

8⁄10

4⁄5

15⁄16

11⁄12

9⁄10

14⁄16

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

ew i ev Pr

w ww

Showing quarters – ¼

12⁄12

6⁄6

10⁄10

5⁄5

16⁄16

8⁄8

4⁄4

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEET

Addition and subtraction of fractions on a number line

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEET Fraction wheels for the addition and subtraction of fractions

r o e t s Bo 1⁄10 r e p ok u S

Teac he r

ew i ev Pr

. te

m . u

w ww

CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

1⁄5

o c . che e r o t r s super 1⁄8

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

1⁄16

R.I.C. Publications®

www.ricpublications.com.au

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEET Spinners for fraction wheel and number line activities

Teac he r

ew i ev Pr

11 10

o c . che e r o t r s super

13 14

. te

w ww 1

m . 3 u

6 2

8 100

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

5 3

7 1

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

CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

r o e t s Bo r e p ok u S 10 2

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEET

Teac he r

ew i ev Pr

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

. te

m . u

w ww

CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

What’s the fraction? – 1

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

101

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEET

w ww

. te

102

m . u

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

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

ew i ev Pr

Teac he r

What’s the fraction? – 2

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEET Proper, improper and mixed fractions

A proper fraction has a value of less than one. The numerator is less than the denominator. There are two types of proper fraction.

Teac he r

Equivalent

A proper fraction with no common factors of the numerator and denominator.

A proper fraction with the same value as a simple fraction‚ but with common factors of the numerator and denominator.

r o e t s Bo r e p ok u S 2⁄3 ¾ 4⁄5 5⁄6

ew i ev Pr

Improper

½ = 2⁄4 = 3⁄6 = 4⁄8 = 5⁄10 = 6⁄12

A fraction with a value of more than one. The numerator is greater than the denominator. 9⁄5 7⁄4 12⁄8 20⁄16 13⁄7 8⁄3

combination of a whole number and a proper fraction. © AR . I . C.Publ i cat i ons 13⁄5 36⁄7 2½ 12⁄3 4¾ •f orr evi ew pur posesonl y•

Mixed

Proper

Improper

A proper fraction has a value of less than one. The numerator is less than the denominator. There are two types of proper fraction.

m . u

w ww

CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

Proper

Simple

Equivalent

A proper fraction with the A proper fraction with . same value as a simple te no common factors fraction‚ c buto with common of the numerator and . factors of the numerator cdenominator. e her r and denominator. o t s s r pe ½ = 2⁄4 = 3⁄6 = 4⁄8 = 5⁄10 = 6⁄12 2⁄3 ¾ 4⁄5 5⁄6 u A fraction with a value of more than one. The numerator is greater than the denominator. 9⁄5 7⁄4 12⁄8 20⁄16 13⁄7 8⁄3

Mixed

A combination of a whole number and a proper fraction. 13⁄5 36⁄7 2½ 12⁄3 4¾

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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103

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEET Proper fractions – 1

There are two types of proper fractions: simple and equivalent. Every simple fraction has many equivalent fractions. To find a simple fraction’s equivalent fractions, the numerator and denominator are multiplied by the same number.

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

ew i ev Pr

Teac he r

Simple fraction

1x2 2 = 2x2 4 1x3 3 = 2x3 6

1x5 5 = 2 x 5 10

w ww

. te

1x6 6 = 2 x 6 12

m . u

o c . c e Equivalent of other unit fractions hfractions r e o t r s s r u e p Unit Equivalent fractions

fraction

104

1⁄3

2⁄6, 3⁄9, 4⁄12, 5⁄15, 6⁄18

2⁄8, 3⁄12, 4⁄16, 5⁄20

1⁄5

2⁄10, 3⁄15. 4⁄20

1⁄6

2⁄12, 3⁄18

1⁄7

2⁄14, 3⁄21

1⁄8

2⁄16, 3⁄24

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

For example:

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEET Proper fractions – 2

To simplify an equivalent fraction, the numerator and denominator are divided by their highest common factor. For example:

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

Teac he r

4÷2 2 = 6÷2 3

6÷3 2 = 15 ÷ 3 5

12 ÷ 4 3 = 20 ÷ 4 5

12⁄20

. te

m . u

6⁄15

Simple fractions

ew i ev Pr

4⁄6

w ww

CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

Equivalent fractions

Highest common factor

o c . che e r o t r s super 5

10 ÷ 5 2 = 15 ÷ 5 3

18⁄24

18 ÷ 6 3 = 24 ÷ 6 4

14⁄21

14 ÷ 7 2 = 21 ÷ 7 3

10⁄15

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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105

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEET Improper and mixed fractions

Improper fractions are simplified to mixed fractions. Number of equal parts Number of equal to make one whole parts remaining

Mixed fraction

4⁄3

four-thirds

3⁄3

4⁄3 – 3⁄3 = 1⁄3

11⁄3

5⁄4

five-quarters

4⁄4

5⁄4 – 4⁄4 = ¼

1¼

8⁄5

eight-fifths

5⁄5

8⁄5 – 5⁄5 = 3⁄5

13⁄5

7⁄6

seven-sixths

6⁄6

7⁄6 – 6⁄6 = 1⁄6

10⁄9

ten-ninths

9⁄9

10⁄9 – 9⁄9 = 1⁄9

ew i ev Pr

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

11⁄6 11⁄9

Mixed fractions are often converted to improper fractions before addition and subtraction.

12⁄3 1¼ 13⁄5

106

as a fraction

one whole and one-half

1+½

one whole and two-thirds

1 + 2⁄3

w ww

1½

in words

equal parts 2⁄2 + ½

fraction

m . u

Mixed fraction

. twhole and onee o 1+¼ 4⁄4 + ¼ c one-quarter . che e r o one whole and t r s5⁄5 + 3⁄5 su r 1 +p 3⁄5 e three-fifths 3⁄3 + 2⁄3

3⁄2 5⁄3 5⁄4

8⁄5

15⁄7

one whole and five-sevenths

1 + 5⁄7

7⁄7 + 5⁄7

12⁄7

17⁄9

one whole and seven-ninths

1 + 7⁄9

9⁄9 + 7⁄9

16⁄9

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

What it means

Teac he r

Improper fraction

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEET Addition of proper fractions

To add proper fractions with the same denominator, the numerators are added while the denominator remains the same. The answer may be:

Examples

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

2⁄5 + 2⁄5 =

Teac he r

3⁄8 + 1⁄8 =

3+1 8

= 4⁄8 =½

2⁄5 + 3⁄5 =

2+3 5

©IfR I . C.P l i cat i ons = 5⁄5 the. numerator isu theb same as the denominator, there are the same =1 •f orr evi ew pu posesonl y • enough unit fractions tor make as the

denominator

one whole and the answer is written as 1 (a whole).

3⁄8 + 5⁄8 =

3+5 8

. te

an improper fraction

m . u

= 8⁄8

w ww

CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

= 4⁄5

ew i ev Pr

a proper fraction

If the answer is a proper fraction, it may need to be simplified by dividing the numerator and denominator by the highest common factor.

2+2 5

3⁄5 + 4⁄5 =

3+4 5

5⁄8 + 7⁄8 =

5+7 8

o c = 7⁄5 . che e = 5⁄5 + 2⁄5 r o r st s r e If the answer isu anp improper = 12⁄5 fraction, it needs to be changed

to a mixed fraction. The fraction part may then need to be simplified.

= 12⁄8 = 8⁄8 + 4⁄8 = 14⁄8 = 1½

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

107

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEET Addition of mixed fractions

Here are two different ways to add mixed fractions. The three-step method Example

Step

r o e t s Bo r e p ok u S 15⁄7 + 14⁄7

12⁄7 + 11⁄7 =

2. Add the improper fractions.

12 + 11 7

= 23⁄7

How many wholes, each divided into seven equal parts‚ can be made from 23⁄7?

3. Change the answer to a mixed fraction.

= 32⁄7

w ww

m . u

The four-step method

Example

Step

. te

25⁄9 + 37⁄9

o c . 1. Add the fraction parts. c e her r o t s = 12⁄9 super 12⁄9 = 9⁄9 + 3⁄9 2. If necessary, convert the answer to a mixed fraction and simplify the fraction part. 3. Add all the whole numbers. 4. Combine the whole number and fraction part. 108

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

5⁄9 + 7⁄9 =

5+7 9

= 13⁄9 = 11⁄3 2 + 3 + 1= 6 61⁄3

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

CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

15⁄7 + 14⁄7 = (7⁄7 + 5⁄7) + (7⁄7 + 4⁄7) = 12⁄7 + 11⁄7

ew i ev Pr

Teac he r

1. Change the mixed fractions to improper fractions.

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEET Subtraction of proper fractions

r o e t s Bo r e p ok u S 4⁄5 – 1⁄5 =

4–1 = 3⁄5 5

7⁄9 – 2⁄9 =

7–2 = 5⁄9 9

5⁄7 – 2⁄7 =

5–2 = 3⁄7 7

ew i ev Pr

Teac he r

To subtract proper fractions with the same denominator, the smaller numerator is subtracted from the larger numerator while the denominator remains the same.

17⁄18 – 2⁄18 =

17 – 2 18

19⁄24 – 1½4 =

19 – 11 24

= 15⁄18 © R. I . C.Publ i cat i on s 5⁄6 •f orr evi ew pur poses=o nl y•

= 1⁄3

Sometimes, the answer needs to be simplified.

. te

m . u

= 8⁄24

w ww

CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

Examples

11 – 5 12

o c . che e =½ r o t r s 13 – 9 super 13⁄16 – 9⁄16 = 11⁄12 – 5⁄12 =

= 6⁄12

= 4⁄16 =¼

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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109

Sub-strand: Fractions and Decimals—F&D – 2

RESOURCE SHEET Subtraction of mixed fractions

Here are two different ways to subtract mixed fractions. The three-step method Example

Step

r o e t s Bo r e p ok u S 43⁄8 – 15⁄8

2. Subtract the improper fractions.

35⁄8 – 13⁄8 =

35 – 13 = 22⁄8 24

How many wholes, each divided into eight equal parts‚ can be made from 22⁄8?

3. If necessary, change the answer to a mixed fraction and simplify the fraction part.

m . u

= 2¾

w ww

The four-step method

Example

. te

67⁄10 – 43⁄10

2. If necessary, simplify the fraction part.

4⁄10 = 2⁄5

Step

o c . che 1. Subtract the fraction parts. 7⁄10r –e 3⁄10 = 4⁄10 o r st super 3. Subtract the whole numbers.

4. Combine the whole number and fraction parts. 110

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

6–4=2

22⁄5

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CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

43⁄8 – 15⁄8 = (8⁄8 + 8⁄8 + 8⁄8 + 8⁄8 + 3⁄8) – (8⁄8 + 5⁄8)

ew i ev Pr

Teac he r

1. Change the mixed fractions to improper fractions.

Assessment 1

Sub-strand: Fractions and Decimals—F&D – 2

NAME:

DATE:

1. Show the fraction sentence on the number line and circle the answer. (a) 5⁄6 + 5⁄6 = 0

(b) 7⁄8 – 3⁄8 = 0

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

Teac he r

ew i ev Pr

(d) 13⁄16 – 9⁄16 = 0

2. Colour the shapes to show the fraction sentences. Write each answer in the box.

+ 5⁄6

. te+

= 1⁄6

(b)

+ 7⁄12

m . u

(a)

w ww

CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

1⁄12

o c . c e he=r 8⁄9 + 5⁄9 7⁄8 r+ 3⁄8 o t s super 3. Write the fraction sentences shown by the fraction wheels. (c)

(a)

(d)

(b)

= =

(c)

1⁄12

1⁄8

–

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

1⁄9

– R.I.C. Publications®

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111

Assessment 2

Sub-strand: Fractions and Decimals—F&D – 2

NAME:

DATE:

Shade a bubble to show your answer. 1. Which is an equivalent fraction of each fraction? (a) ½ =

4⁄6

3⁄6

10⁄12

(b) ¾ =

6⁄8

4⁄5

9⁄12

6⁄9

8⁄10

3⁄9

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

(a) 3⁄6 – ½ (b) 8⁄12 – 2⁄3

3. Which is the correct mixed fraction for each improper fraction? (a) 5⁄3

51⁄3

32⁄3

12⁄3

(b) 7⁄4

1¾

14⁄7

4¾

11⁄6

12⁄3

15⁄12

Show how you calculate the answer.

ew i ev Pr

Teac he r

w ww

. te

m . u

4. Jude cut his pizza into eight portions. He gave a piece each to his three friends and ate ¼ himself. How much pizza was left?

5. Antek made up some cordial drink by adding 5⁄8 litres of cordial syrup to 17⁄8 litres of water. He needed 3 litres of cordial altogether. How much more did he need to make?

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6. The canteen manager ordered 45⁄12 kg of wholemeal flour and 211⁄12 kg of self-raising flour. How much plain flour does she need to have a total of 10 kg of flour?

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CONTENT DESCRIPTION: Investigates strategies to solve problems involving addition and subtraction of fractions with the same denominator

2. What is the highest factor used to simplify each fraction?

Checklist

Sub-strand: Fractions and Decimals—F&D – 2

Investigates strategies to solve problems involving addition and subtraction of fractions with

Solves problems involving fractions of the same denominator

Converts improper fractions to mixed fractions and vice versa

Uses highest common factor to simplify fractions

Uses handson activities to subtract fractions

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STUDENT NAME

Uses hands-on activities to add fractions

the same denominator (ACMNA103)

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Sub-strand: Fractions and Decimals—F&D – 3

Recognise that the place value system can be extended beyond hundredths (ACMNA104)

RELATED TERMS

TEACHER INFORMATION

Decimal number system

What this means

• The base ten number system that is the most widely used number system in the world today.

• Students need to extend their understanding of the place value system to the thousandths‚ using division by 1000. • Students need to use their knowledge of place value and division by 10, 100 and 1000 to recognise equivalences of units of measure; for example‚ ½ metre is also 50⁄100 centimetres and 500⁄1000 millimetres.

Decimal number

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Decimal fraction

• Any fraction whose denominator is any power of ten. All fractions can be converted to decimal fractions. Decimal

• The term ‘decimal’ is used to describe a decimal numeral that include a decimal point. Decimals are a way of writing fractions without using numerator on a denominator. A decimal can be created from a decimal fraction by dividing its numerator by its denominator.

Teaching points

• All numbers in the base ten system are decimal numbers. We only write the zero digit where it is required. That is why whole numbers are not usually written as 3.0, 47.00 etc. • Although the denominator of a decimal fraction is a power of 10, any fraction can be converted to a decimal. • The place value of decimals extends to the right of the ones column, decreasing by a power of 10 with each place—in the same way as whole numbers extend to the left, increasing by a power of 10 each time.

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• Every number in the base ten system is a decimal number. The value of each ‘place’ is ten times greater than the one to its right or ten times smaller than the one to its left.

What to look for © R. I . C .Publ i cat i ons •f orr evi ew pur posesonl y•

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• Represented by a full stop (or sometimes a comma), this separates the whole number from the decimal. Decimal place

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• The ‘place value’ position to which a decimal extends; e.g. one decimal place – 3.7, two decimal places – 5.43, three decimal places – 9.421 etc.

Student vocabulary decimal point decimal place decimal fraction

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Decimal point

• Students who think that the largest decimal number is the one with the most digits because they are thinking of all numbers as whole numbers; for example: 1⁄1000 is bigger than 1⁄100 or 0.125 is bigger than ½. • Students who think, for example‚ $4.125 is the same as $5.25 because they see the .125 as a whole number of cents which equals $1.25. They add this to the $4, giving $5.25. See also New wave Number and Algebra (Year 5) student workbook (pages 44–48)

o c . che e r o t r s super Proficiency strand(s): Understanding Fluency Problem solving Reasoning

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Fractions and Decimals—F&D – 3

HANDS-ON ACTIVITIES Closest to one Use a spinner, dice or playing cards to select digits and create five numbers to three decimal places. Record the numbers in the table (page 122). Look at the five numbers and put them in order (1 to 5) from the largest to the smallest.

Words, decimal fractions and decimals

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Equivalent decimal fractions

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Use the table on page 123 to show how decimal fractions can be written with words, as fractions and as decimals. Fill in one section on each line and give to students to complete the other two sections.

Demonstrate the relationship in size between the whole number of one (1) and the tenths, hundredths and thousandths place value (See pages 124–128.) For example: one is equivalent to ten tenths, one hundred hundredths and one thousand thousandths; one tenth is equivalent to ten hundredths and one hundred thousandths. Students choose a fraction between 1⁄10 and 9⁄10 and‚ using the same colour for each, colour the fraction as tenths, hundredths and thousandths.

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Prepare the decimal fraction calling lists (page 129) and sufficient bingo cards (page 128) for all students. As the equivalent decimal fractions are called out, students must work out if they have the corresponding decimal on their card and if so, they cross it out in pencil. Determine how many crosses in a row are required for a win.

Place value games

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Throw a ten-sided dice (from 0 to 9) to generate numerators for tenths, hundredths and thousandths decimal fractions. Record each fraction made and write it as a decimal (page 130). Working in pairs, students can check and correct each other’s work.

Students can use pages 124 to 127 to double-check answers. For example‚ consider 347⁄1000. By looking at the thousandths sheet (page 127), 347 is represented by three large columns (each equivalent to one tenth) four large rectangles (each equivalent to one-hundredth)‚ and seven small rectangles (each equivalent to one thousandth). Also available for use are the Middle school maths games‚ created by Richard Korbosky and published by R.I.C. Publications®‚ which includes activities on decimal place value. (These are also available as an interactive version.)

Decimal problems Using the units of currency, weights and measure, students write problems to test one another, for example; ‘Who collected the greatest weight of rubbish: Achim with 2 kg and 237 g, Kyle with 2½ kg or Gavin with 1826⁄1000 kg?’ Answer: Kyle‚ because 2½ kg is greater than 2 kg and 237 g‚ which is greater than 1826⁄1000 kg.

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Sub-strand: Fractions and Decimals—F&D – 3

LINKS TO OTHER CURRICULUM AREAS English • Discuss the benefits of global decimalisation of money, weights and measure. Make notes of the key points. • Research and write a report on the currency used by Mauritania and Madagascar.

Information and Communication Technology • All about decimals: <http://www.coolmath.com/prealgebra/02-decimals/index.html> • Place value game: <http://www.mrnussbaum.com/placevaluepirates1.htm>

History

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• Create a time line showing the first countries to use decimal currency. Give the name of each country’s currency. • Research the history of units of weights and measures.

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Geography

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• Find out which two countries do not use decimal currency. Where are they located? What currency do they use? • On a blank world map, locate the countries that use the dollar, renminbi‚ pound‚ yen and euro. Use five colours to colour the map accordingly.

Health and Physical Education

• Find different recipes for the same healthy snack that list the ingredients by weight and capacity. Compare the amount of ingredients in each recipe.

The Arts

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• Show students an image of ‘Colours for a large wall’ by American artist Ellsworth Kelly. Count the number of colours used and how many squares there are of each colour. Discuss randomly and geometrically coloured squares. Use a range of art techniques to decorate the individual squares of a 100-square copied onto A3 paper. Calculate the proportion of squares decorated using each technique and express as a decimal of the whole. • Design a poster to illustrate the information found about weights and measures through history.

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET What are decimals?

A decimal is the part of a decimal number that has a value of less than one. The decimal point separates the whole number from the decimal. Decimal places are the place value positions of the digits that come after the decimal point.

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The value of a digit in the tenths column is ten times less than it would be in the ones column. Decimal

Tens

Ones

Tenths

1 1 10

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Whole number

Tens

Ones

Decimal

Tenths

Hundredths

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

The value of a digit in the hundredths column is one hundred times less than it would be in the ones column and tens times less than it would be in the tenths column.

8 10

3 100

. te o c The value of a digit in the thousandths column is one thousand times less than . c e it would be in the ones column, one hundred times less than it would be in the h r e o tenths column and tens times less than it would be in the hundredths column. r st s uper Whole number

Tens

Ones

Decimal

Tenths

Hundredths Thousandths

6 10

7 100

4 1000

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

1 1000

0.001

1 100

0.1

m .10 u 10 000

0.001

10 1000

0.01

10 100

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

Third decimal place

Thousandths

Second decimal place

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0.01

1 10

0.1

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Hundredths

First decimal place

Tenths

0.001

100 10 000

0.01

100 1000

0.1

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Divide numerator and denominator by 100

100 100 000

1000 10 000

0.1

Divide numerator and denominator by 1000

1000 1 000 000

0.001

1000 100 000

0.01

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Divide numerator and denominator by 10

Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET

Equivalent decimal fractions

Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET Using decimals

Decimals are used in the units of money, length, weight and capacity. Money All but two countries across the world use decimal currency (for example: the dollar, the euro and the pound). Each unit of currency has its decimal part (for example: cents and pence).

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There are 100 cents in one dollar. $1¼

$31⁄5

$1.25

$1 and 25 cents

125 cents

$4.50

$4 and 50 cents

450 cents

$3.20

$3 and 20 cents

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$4½

A cent is 1⁄100 of one dollar.

320 cents

Length The main unit of length is the metre (m) but for smaller lengths, centimetres (cm) and millimetres (mm) are used. There are 100 cm in one metre.

A centimetre is 1⁄100 of one metre.

3 m 25 cm

325 cm

3250 mm

5½ m

5.5 m

5 m 50 cm

550 cm

5500 mm

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3¼ m

Weight The main unit of weight is the kilogram (kg) but for lighter weights, grams (g) are used.

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

There are 1000 mm in one metre.

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There are 1000 g in one kilogram.

2¾ kg 35⁄8 kg

A gram is 1⁄1000 of one kilogram.

7500 g o c . ch 2.75 kg 2 kg 750 g 2750 g e r ekgsupe o st 3.625r 3 kg 625 g 3625 g r 7.5 kg

7 kg 500 g

Capacity The main unit of capacity is the litre (L) but for smaller capacities, millilitres (mL) are used. There are 1000 mL in one litre.

A millilitre is 1⁄1000 of one litre.

37⁄8 L

3.875 L

3 L 875 mL

3875 mL

93⁄8 L

9.375 L

9 L 375 mL

9375 mL

6½ L

6.5 L

6 L 500 mL

6500 mL

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Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET Converting decimal fractions to decimals

Decimal

1⁄10

0.1

2⁄10

0.2

3⁄10

0.3

4⁄10

0.4

Decimal fraction

Decimal

1⁄100

0.01

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

If the numerator of the fraction is less than 10 it is placed in the hundredths column and a zero is placed in the tenths column.

6⁄100 9⁄100

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The numerator of the fraction is placed in the tenths column.

Decimal fraction

0.06 0.09

27⁄100 © R. I . C.Publ i cat i o53⁄100 ns 0.27 0.53 •f orr evi ew pur poses89⁄100 onl y0.89 • If the numerator is 10 or more, the digits are placed in the tenths and hundredths columns.

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If the numerator of the fraction is less than 10 it is placed in the thousandths column and a zero is placed in the tenths and hundredths columns.

Decimal fraction

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Decimal

o 0.009 c . che e r o t r s s r u e p If the numerator is between 10 and 99, the

120

1⁄1000

0.001

9⁄1000

digits are placed in the hundredths and thousandths columns and a zero is placed in the tenths column.

34⁄1000

0.034

72⁄1000

0.072

If the numerator is one hundred or more, the digits are placed in the tenths, hundredths and thousandths columns.

469⁄1000

0.469

652⁄1000

0.652

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

Tenths

Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET The zero digit

The zero digit is only used when it is needed.

When is the zero digit needed?

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When it acts as a place holder within a number; for example: Ones

Teac he r

Tens

Tenths

Hundredths Thousandths

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The zeros in the ones and the hundredths columns are essential. They show that there are no units of value in those columns.

Ones 4

Tenths

Hundredths

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When is the zero digit not needed?

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

If the zeros are removed, the value of the number is altered.

When it falls outside the place value columns of a number; for example:

Hundreds 0

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Ones

Tenths Hundredths Thousandths

Ten thousandths

o 0 c . che and ten thousandths columns eare unnecessary. r The zeros in the hundreds o r stof the number. serence r uptoe They make no diff the value 3

If the zeros are removed, the value of the number remains the same. Tens

Ones

Tenths .

Hundredths Thousandths

073.6920 does equal 73.692

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Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET Closest to/Furthest from one

0 0 0

Ones 0

0 0 0 0

0 0 0

. . . . .

Ones 0 0 0 0 0 122

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Tenths

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Order

Hundredths

Thousandths

Order

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Ones 0

Thousandths

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. . . . .

Hundredths

Thousandths

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Order

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Tenths

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. . . . .

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

Ones

Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET Words, decimal fractions and decimals

Decimals

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

Decimal fractions

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Words

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Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

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One

Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

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

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Teac he r

Tenths

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Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

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Teac he r

Hundredths

Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

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Thousandths

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Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET Decimal bingo cards

0.09

0.03

0.4

128

0.005

0.01

0.06

0.009

0.006

0.9

0.003

0.7. 0.04 0.08 0.002 ©R I . C. Publ i cat i on s 0.05 •f orr evi ew pur posesonl y• 0.2

0.2

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0.02

r o e t s B r 0.5 e 0.3 0.007 o0.001 p ok u S

0.8

0.004

0.008

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0.6

0.07

. t 0.7 e

0.08

0.04

0.9

0.003

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0.006

0.8

0.001

0.5

0.01

0.05

0.009

0.03

0.007

0.6

0.002

0.07

0.3

0.02

0.09

0.004

0.1

0.008

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

0.4

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0.1

Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET Decimal bingo – Calling list

1⁄10

10⁄100

10⁄1000

r o e t s 200⁄1000 2⁄100 Bo 20⁄1000 r e p ok u S

1⁄1000

2⁄1000

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3⁄10

30⁄100

300⁄1000

3⁄100

30⁄1000

3⁄1000

4⁄10

40⁄100

400⁄1000

4⁄100

40⁄1000

4⁄1000

6⁄10

60⁄100

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600⁄1000

6⁄100

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1⁄100

20⁄100

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2⁄10

100⁄1000

60⁄1000

5⁄1000

6⁄1000

o c . che700⁄1000 70⁄100 7⁄100 70⁄1000 e r o r st super

7⁄1000

8⁄10

80⁄100

800⁄1000

8⁄100

80⁄1000

8⁄1000

9⁄10

90⁄100

900⁄1000

9⁄100

90⁄1000

9⁄1000

7⁄10

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Sub-strand: Fractions and Decimals—F&D – 3

RESOURCE SHEET Place value check

Decimal fraction

Ones

Tenths

Hundredths

Thousandths

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

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

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Assessment 1

Sub-strand: Fractions and Decimals—F&D – 3

NAME:

DATE:

1. Write each number as a decimal fraction. (a) 0.347

(b) 0.62

(d) 0.6

(e) 0.561

(f)

0.19

2. (a) Write the numbers in order from smallest to greatest. 0.176

r o e t s Bo r e p ok u S 0.53

0.3

0.09

0.7

0.274

0.7

0.271

0.096

0.53

0.4

0.83

3. Circle the three decimal fractions with the same value. (a) 1⁄10

(b) 700⁄1000

1⁄100

10⁄100 70⁄1000

10⁄1000 70⁄100

100⁄1000

7⁄100

7⁄10

4. Write > (greater than) or < (less than).

(a)

2589

(b)

1467

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5. What are the largest and smallest three decimal-place numbers you can make using the numbers in each box?

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

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(b) Write the numbers in order from greatest to smallest.

largest:

smallest:

largest:

smallest:

largest:

smallest:

o 0. c . chein words and as a decimal.r e 6. Write each decimal fraction o t r s s r u e p Decimal Words (c)

fraction

Decimal

(a) 3⁄10 (b) 7⁄100 (c) 63⁄1000

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Assessment 2

Sub-strand: Fractions and Decimals—F&D – 3

NAME:

DATE:

1. A group of students measured different parts of their body and recorded the information in the table. Feet

Hand span

Cubit

Achim

180⁄1000 m

13⁄100 m

237⁄1000 m

Joan

16⁄100 m

105⁄1000 m

20⁄100 m

Salma

175⁄1000 m

Eyal

21⁄100 m

r o e t s Bo r e p ok u S 12⁄100 m

210⁄1000 m

135⁄1000 m

24⁄100 m

(c) Who has the shortest cubit? (d) Who has the widest hand span? 2. Which drink bottle holds almost half a litre? Shade the bubble.

Ln 95i m 0.2. L u 15 P © R. I . C bl i ca4t o s 300 mL •f orr evi ew pur posesonl y•0.52 L

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Tim 4.75 kg

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Tam 4374 g

Tem 4½ kg

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3. Write the names of the cats in weight order from lightest to heaviest.

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Tom 4189 g

4. In a swimming competition, the following times were achieved for the 25-metre freestyle.

132

Zoe

13.71 seconds

Wanda

13.48 seconds

Jamilla

13.89 seconds

Peta

13.62 seconds

Emma

…… seconds

Whose time must Emma beat in order to win the competition?

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Recognise that the place value system can be extended beyond hundredths

(b) Who has the biggest feet?

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(a) In the table, write each measurement as a decimal.

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Relates knowledge of decimals to practical applications

Writes decimal fractions in decimal and word form

Recognises the equivalence of decimal fractions, e.g. 10⁄100 has the same value as 1⁄10

Orders decimals by increasing and decreasing size

Understands the relationship in size between 1⁄10, 1⁄100 and 1⁄1000

Understands place value of digits after the decimal point

STUDENT NAME

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Sub-strand: Fractions and Decimals—F&D – 3

Checklist

Recognise that the number system can be extended beyond hundredths (ACMNA104)

133

Sub-strand: Fractions and Decimals—F&D – 4

Compare, order and represent decimals (ACMNA105)

RELATED TERMS

TEACHER INFORMATION

Decimal number system

What this means

• A base ten number system that is the most widely used number system in the world today.

• Students should be able to locate decimals such as 1⁄10, 1⁄100 and 1⁄1000 on a number line.

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

Decimal number

Teac he r

Decimal fraction

• Any fraction whose denominator is any power of ten. All fractions can be converted to decimal fractions. Decimal

• Decimals have a fractional equivalence. Provide opportunities to link a picture or model of decimals to their fractional equivalences. • Although the denominator of a decimal fraction is a power of 10, any fraction can be converted to a decimal. • The place value of decimals extends to the right of the ones column, decreasing by a power of 10 with each place—in the same way as whole numbers extend to the left, increasing by a power of 10 each time.

What to look for

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• Every number in the base ten system is a decimal number. The value of each ‘place’ is ten times greater than the one to its right or smaller than the one to its left.

• Students who think that the largest decimal number is the one with the most digits; for example, 0.125 is greater than 0.5 because they think of each number as whole numbers. • Students who think that 1⁄1000 is greater than 1⁄100. • Students who do not understand decimal place value; for example: that 0.50 and 0.5 are the same.

See also New wave Number and Algebra (Year 5) student workbook (pages 49–55)

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Decimal point

• Represented by a full stop or a comma, this separates the whole number from the decimal. Decimal place

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• The ‘place value’ position to which a decimal extends; e.g. one decimal place (3.7), two decimal places (5.43), three decimal places (9.421) etc.

Student vocabulary decimal point decimal place decimal fraction

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Fractions and Decimals—F&D – 4

HANDS-ON ACTIVITIES Decimal breakdown Use the number cards from a deck of playing cards to generate four-digit numbers up to 9.999. Follow the decimal breakdown procedure on page 137 to break down the numbers made. Use the tables on page 138. Use the decimal models on pages 143 and 144 to represent each number made. Colour and cut out the appropriate number of tenths, hundredths and thousandths and glue within a ‘ones’ box. Then glue that box on to a large sheet of coloured paper to the right of the required number of ones boxes to represent the whole part of the number. Separate with a decimal point. Write the number under the boxes. That all the decimal parts are contained within a ones box demonstrates that the decimal part is less than one.

Decimal money

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

Use the number cards from a deck of playing cards to generate three-digit numbers up to 9.99. Follow the decimal breakdown procedure on page 137 to break down the numbers made and record then in the tables on page 139.

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Take that amount of play dollars and cents from the resource centre. Use the decimal models on pages 143 and 144 to represent the money. Colour and cut out the appropriate number of tenths and hundredths and glue within a ‘ones’ box. Place this ones box on to a large sheet of coloured paper to the right of the required number of ones boxes to represent the number of whole dollars. Place the money next to the ones, tenths and hundredths boxes that represent it. Write the amount of money underneath the boxes and money. Take a photograph of the representation.

Decimal length, mass and volume

Measure the length, mass and volume of different items around the classroom. Use the tables on pages 140–142 to record the measurements and break down the decimal component of the measurement.

Comparing decimals

Randomly select a tenth number card (page 146) and use counters to represent it on one of the tables on page 145. Notice that the number of boxes in each table determines how many need to be covered. For example‚ 0.4 can be represented on the first table by simply covering four boxes. But with the fourth table (a 4-by-10 grid)‚ 0.4 is represented by covering 16 boxes.

Ordering decimals

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Randomly select five decimal number cards (pages 146–148). Place in order of size from largest to smallest and smallest to largest. Say what each means and how their values compare with each other.

Number lines

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Randomly select 5 tenths number cards (page 146) and locate each decimal amount on the ones number line (page 150).

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Repeat with the hundredths cards (page 147) on the tenths number line‚ and the thousands cards (page 148) on the hundredths number line.

Games with number lines

Make number lines (See page 151). Cut out and join the two pieces for each number line.

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Sub-strand: Fractions and Decimals—F&D – 4

LINKS TO OTHER CURRICULUM AREAS English • Research how the Dewey decimal classification system‚ used by libraries all over the world, is organised. Choose one class to illustrate how the number changes as the subject of the books alters. • Give an oral presentation, including working through examples on a board, to explain how decimal fractions can be written as decimals in place value columns and how they compare in size.

Information and Communication Technology • • • • •

All about decimals: <http://www.coolmath.com/prealgebra/02-decimals/index.html> Place value game: <http://www.mrnussbaum.com/placevaluepirates1.htm> Chinese number rods (PDF download): <orion.math.iastate.edu/mathnight/activities/modules/.../countright.pdf> Ancient Egyptian numeral system (powers of ten): <http://www.recoveredscience.com/const102egynumerals1.htm> Metric clock: <http://www.minkukel.com/en/time/metric_clock.htm>

• Investigate the Chinese counting rod system.

Science

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History

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• Find out about the metric clock and how it compares with the 12-hour clock we are most familiar with. When, where and by whom was it introduced?

The Arts

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• Research to find the hieroglyphics that were used to denote the powers of 10 in the Ancient Egyptian number system. Use a range of art techniques to create a display illustrating the powers of 10.

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Fractions and Decimals—F&D – 4

RESOURCE SHEET Decimal breakdown – 1

All decimal numbers are made up of combinations of the digits from zero to nine. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 The position of each digit within a number tells us its value.

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

Some decimal numbers include a part with a value less than one.

In the number 4.257, 4 is a whole number and 0.257 is less than one.

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This part is separated from the whole part of the number by a decimal point.

Look at the number in its place value columns.

Each column gives a digit a value that is tens times greater than it would be in the next column to the right. Ones

Tenths

Hundredths

Thousandths

4 . 2 5 7 © R. I . C. Publ i cat i ons Separate • the f number into its i diff erent orr ev e wparts. pur posesonl y• 4

2⁄10

5⁄100

7⁄1000

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To write the decimal fractions as decimals, use a calculator to divide the numerator by the denominator.

. tparts in the correct place value columns and add them Put the different e to return to the o c complete number. . che e r o O Tths Hths Thths t r s super 4

0.2

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0.007

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RESOURCE SHEET Decimal breakdown – 2

Number: Whole number: O

Decimal part: Tths

Hths

Thths

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Convert decimal fractions to decimals. .

Add the whole number and the decimal parts. O

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Hths

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Thths

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Number:

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Add the whole number and the decimal parts. O

Tths

Hths

Thths

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CONTENT DESCRIPTION: Compare, order and represent decimals

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Decimal part:

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Sub-strand: Fractions and Decimals—F&D – 4

RESOURCE SHEET Decimal money

Total amount: Dollars:

Cents:

Tths

Hths

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Convert decimal fractions to decimals. .

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Add the whole number and the decimal parts. O

Tths

Hths

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Cents:

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CONTENT DESCRIPTION: Compare, order and represent decimals

Dollars:

Hths

Separate parts.

o c decimal fractions to decimals. . chConvert e r . +o er t s s r u e p Add the whole number and the decimal parts. .

Tths

Hths

. . +

. .

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Sub-strand: Fractions and Decimals—F&D – 4

RESOURCE SHEET Length

Total amount: Metres:

Centimetres:

Tths

Hths

r o e t s Bo r e p ok u S Separate parts.

Convert decimal fractions to decimals. .

Add the whole number and the decimal parts. O

Tths

Hths

. .

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Tths

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. . + te o c Convert decimal fractions to decimals. . c e her r . + t o s s r u e p Add the whole number and the decimal parts. Separate parts.

Tths

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CONTENT DESCRIPTION: Compare, order and represent decimals

Centimetres:

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Metres:

Sub-strand: Fractions and Decimals—F&D – 4

RESOURCE SHEET Decimal weight

Total weight: Kilograms:

Grams:

Tths

Hths

Thths

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Convert decimal fractions to decimals. .

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Add the whole number and the decimal parts. O

Tths

Hths

Thths

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CONTENT DESCRIPTION: Compare, order and represent decimals

Kilograms:

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Grams:

Tths

Hths

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Total weight: Thths

o c . chConvert e decimal fractions to decimals. r er o t s + upe .s +r Separate parts.

Add the whole number and the decimal parts. O

Tths

Hths

Thths

. . . +

. .

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Sub-strand: Fractions and Decimals—F&D – 4

RESOURCE SHEET Decimal volume

Total volume: Litres:

Millilitres:

Tths

Hths

Thths

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Convert decimal fractions to decimals. .

Add the whole number and the decimal parts. O

Tths

Hths

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Thths

. . .

Total volume: Millilitres:

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Tths

Hths

Thths

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Add the whole number and the decimal parts. O

Tths

Hths

Thths

. . . +

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CONTENT DESCRIPTION: Compare, order and represent decimals

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Litres:

Sub-strand: Fractions and Decimals—F&D – 4

RESOURCE SHEET Decimal models – 1

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One

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CONTENT DESCRIPTION: Compare, order and represent decimals

Tenths

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RESOURCE SHEET Decimal models – 2

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Hundredths

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CONTENT DESCRIPTION: Compare, order and represent decimals

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Thousandths

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RESOURCE SHEET

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Decimal models – 3

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CONTENT DESCRIPTION: Compare, order and represent decimals

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RESOURCE SHEET

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Decimal number cards – Tenths

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CONTENT DESCRIPTION: Compare, order and represent decimals

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Decimal number cards – Hundredths

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RESOURCE SHEET

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Decimal number cards – Thousandths

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CONTENT DESCRIPTION: Compare, order and represent decimals

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RESOURCE SHEET

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CONTENT DESCRIPTION: Compare, order and represent decimals

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Decimal number lines – 1

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RESOURCE SHEET

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Decimal number lines – 2

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CONTENT DESCRIPTION: Compare, order and represent decimals

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RESOURCE SHEET

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Assessment 1

Sub-strand: Fractions and Decimals—F&D – 4

NAME:

DATE:

1. In each pair, which decimal has the greater value? Shade a bubble to show your answer. (a)

0.05

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(b)

0.3

0.09

(c)

0.07

0.007

(d)

0.15

0.1

(e)

0.27

0.275

(f)

0.064

0.64

2. Write the decimal fraction for each decimal. (a) 0.14 (d) 0.03 (g) 0.98

(e) 0.582

(f) 0.025

(h) 0.048

(i) 0.5

(k) 0.07

(l) 0.002

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(j) 0.004

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3. Write the decimal for each decimal fraction. (a) 4⁄10

(b) 579⁄1000

(d) 61⁄1000

(e) 2⁄100

(f) 39⁄100

4. Write your answers from Question 3 in order from the smallest to the greatest.

5. What value is the arrow pointing to on the number line?

(a)

(b)

(c)

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6. Draw an arrow to show each value on the number line.

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o c . c e he 7. Which arrows on the number line show the numbers? r o t r s s uper x w v z u y 3.2

(a) 3.27

8.71

3.4

(b) 3.35

8.72

(a) 8.716

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3.3

3.5

8.73

(b) 8.724

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8.74 (c) 8.738

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CONTENT DESCRIPTION: Compare, order and represent decimals

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Assessment 2

Sub-strand: Fractions and Decimals—F&D – 4

NAME:

DATE:

1. In a class of 30 students, 0.6 are girls. How many boys are in the class?

2. Some of the students measured their height. Put them in order from the tallest to the shortest. 156.1 cm

Cooper

154⁄100 m

Franz

15⁄10 m

Jan

1517⁄1000 m

Paolo

155.4 cm

Sandra

1519 mm

Tallest

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Ajay

3. At the local show, several gardeners entered the ‘heaviest pumpkin competition’. Which pumpkin won? Shade a bubble to show your answer.

Sunday

0.184 L

Monday

0.525 L

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CONTENT DESCRIPTION: Compare, order and represent decimals

4. For one week during the winter, students measured how much rain fell each day by collecting it in a measuring jug. 0.759 L . t Wednesdaye0.337 L o c . Thursday 0.180 L ch e r o Friday 0.760 L e t r s super Saturday 0.330 L Tuesday

(a) On which day did it rain the most? (b) On which day did it rain the least? (c) Which day was wetter, Saturday or Wednesday? (d) On which day was the rainfall about the same as on Sunday? (e) On which day was about 500 mL of rainwater collected? (f) On which day was about three-quarters of a litre collected? Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Checklist

Sub-strand: Fractions and Decimals—F&D – 4

Recognises decimals in units of measure and money

Represents decimals on a number line

Orders decimals from largest to smallest and vice versa

Compares the size of decimals

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STUDENT NAME

Converts unit fractions to decimals and vice versa

Compare, order and represent decimals (ACMNA105)

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Answers

Sub-strand: Fractions and Decimals

F&D – 1 Page 91 1. 2. 3. 4.

Assessment 1

(a) ½ (b) 1⁄9 (a) 1⁄12 (b) ¼ (a) 1⁄3 (b) 1⁄16 (a) second bubble (b) second bubble (c) first bubble Teacher check

Page 92 1. 2. 3. 4.

Page 132

1. 2.

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1. 2. 3. 4. 5. 6.

Assessment 1

Teacher check (a) 10⁄6 = 5⁄3 = 12⁄3 (c) 16⁄12 = 4⁄3 = 11⁄3 Teacher check (a) 6⁄6 = 1 (c) 13⁄9 = 14⁄9 (a) 7⁄8 – 3⁄8 = 4⁄8 (b) 11⁄12 – 5⁄12 = 6⁄12 = ½ (c) 8⁄9 – 4⁄9 = 4⁄9

Page 112

(a) 3⁄6 (a) 3 (a) 12⁄3 3⁄8 pizza ¼ litre 22⁄3 kg flour

(b) 4⁄8 = ½ (d) 4⁄16 = ¼ (b) 8⁄12 = 2⁄3 (d) 10⁄8 = 12⁄8 = 1¼

4. 5. 6. 7.

(a) 0.1 (b) 0.3 (c) (d) 0.15 (e) 0.275 (f ) (a) 14⁄100 (b) 376⁄1000 (c) (e) 582⁄1000 (f ) 25⁄1000 (g) (i) 5⁄10 (j) 4⁄1000 (k) (a) 0.4 (b) 0.579 (c) (d) 0.061 (e) 0.02 (f ) 0.008, 0.02, 0.061, 0.39, 0.4, 0.579 (a) 4.3 (b) 5.1 (c) Teacher check (a) w (b) z (c)

Page 153 1. 2. 3. 4.

Assessment 1

3. 4. 5.

12 Ajay, Paolo, Cooper, Sandra, Jan, Franz Pumpkin D, 7374 g (a) Friday (b) Thursday (c) (d) Thursday (e) Monday (f )

Wednesday Friday

(b) 9⁄12 (b) 4 (b) 1¾

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5.6

Assessment 1

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(d) 3⁄100 (h) 48⁄1000 (l) 2⁄1000

Assessment 2

F&D – 3 Page 131

0.07 0.64 6⁄10 98⁄100 7⁄100 0.008 0.39

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F&D – 2 1.

(a) Achim – feet: 0.18, hand span: 0.13, cubit: 0.237; Joan – feet: 0.16, hand span: 0.105, cubit: 0.2; Salma – feet: 0.175, hand span: 0.12, cubit: 0.21; Eyal – feet: 0.21, hand span: 0.135, cubit: 0.24; (b) Eyal (c) Joan (d) Eyal 495⁄1000 L Tom, Tam, Tem, Tim Wanda’s

F&D – 4

Assessment 2

first bubble third bubble Teacher check (a) 1⁄10, 1⁄9, 1⁄6, 1⁄5, ½ (b) 1⁄3, ¼, 1⁄7, 1⁄8, 1⁄12

Page 111

Assessment 2

347⁄1000 (b) 62⁄100 (c) 8⁄10 6⁄10 (e) 561⁄1000 (f ) 19⁄100 0.09, 0.176, 0.274, 0.3, 0.53, 0.7 0.83, 0.7, 0.53, 0.4, 0.271, 0.096 1⁄10, 10⁄100, 100⁄1000 700⁄1000, 70⁄100, 7⁄10 > (b) < > (d) >,< largest 0.985, smallest 0.258 largest 0.764, smallest 0.146 largest 0.974, smallest 0.347 three tenths, 0.3 seven hundredths, 0.07 sixty-three thousandths, 0.063

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Sub-strand: Money and Financial Mathematics—M&FM – 1

Create simple financial plans (ACMNA106)

RELATED TERMS

TEACHER INFORMATION

Account

What this means

• A record of income and expenditure

• Students need to know how to: – record income and expenditure and calculate savings over a set period of time – calculate the length of time it will take to save a set amount by saving regularly – calculate the cost of entry to a venue for different numbers of adults and children.

Income

• Money that is earned Expenditure

• Money that is spent

• The difference between money earned and money spent Credit

• A positive balance–money earned is greater than money spent Debt

Teaching points

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Balance

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• Use real-life financial goals and realistic costs. • When solving word problems‚ identify: – the different pieces of information – the question(s) being asked – and disregard unnecessary information – the different number operations required to solve the problem

• A negative balance–money earned is less than money spent

to look for © R. I . C.PWhat ub l i cat i ons • The result when the• income from r a financial f o r evi ew pur posesonl y• venture is greater than the expenditure Loss

Earn

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• The result when the income from a financial venture is less than the expenditure

• To make money Spend

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• To use money to buy goods or services Save

• Students who have difficulties working out the different parts of a problem. • Students who do not recognise that amounts of money are set out as decimal amounts and therefore follow the place value pattern of all decimal numbers. • Students who do not line up decimal points correctly when calculating monetary amounts, and place digits in incorrect columns.

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Profit

See also New wave Number and Algebra (Year 5) student workbook (pages 56–63)

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• To reduce spending and keep money for a specific purpose

Proficiency strand(s): Student vocabulary account

income

expenditure

balance

credit

debt

profit

loss

earn

spend

Understanding Fluency Problem solving Reasoning

save

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Sub-strand: Money and Financial Mathematics—M&FM – 1

HANDS-ON ACTIVITIES Saving for something special • Students browse through department store catalogues to find an item they would like to save up for (up to the value of $50). The item and its cost is recorded on the savings chart (page 159). Cut out and laminate the ‘Income’ and ‘Expenditure’ cards (page 160) and place them in a container. For each week with the chart, students choose two cards and record the greater amount in the income column and the lesser amount in the expenditure column. For each week, students calculate the amount they have saved‚ which is added to the previous week’s balance to give the new balance. Students stop ‘saving’ when they have enough money to buy their chosen item. After ‘purchasing’ the item, they calculate how much money they have left over.

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Fundraising cake sales

• Students roll two six-sided dice to determine the amount of each cake type in their stalls, and record the number on the sheet (pages 161 and 162). They use the information on the sheets to calculate the profit they would make from selling all of their cakes.

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Calculating the cost of a class excursion

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• Students roll two 10-sided dice to determine the number of students going on the excursion and record the number on the sheet (page 163). They use the information on the sheets to calculate the total cost of the excursion for each student and adult.

After-school activities

• One student is elected banker‚ who gives each other student a $300 float of play money (available from two-dollar shops). Students roll a six-sided dice to move from space to space around the board from top to bottom. When they land on a square marked with an activity, they mark that activity on their score card under ‘Monday’. As they land on a new activity, they mark the other four days of the week. With each successive landing, the players pay the required amount until they have paid for each of their five activities. They do this for a total of four times‚ which may require travelling around the board a number of times. – A player has a second throw of the dice if he or she throws a six – A player misses a turn if he or she lands on an activity that has already been filled; that is‚ an activity that has already been paid for four times – As each payment is made, players must record the balance of their account on the balance sheet. – Play ends with the first player to pay for each of his or her activities for four weeks.

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Money to spend

• Use the budget planner (page 168) to record a proposed budget to spend on clothing, sports or electronic equipment or other items of your choice

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Sub-strand: Money and Financial Mathematics—M&FM – 1

LINKS TO OTHER CURRICULUM AREAS English • Write a report comparing the cost of basic household items that have increased in price since your parents and grandparents were young. • Use supermarket and department store catalogues to prepare a budget for: – furnishing a house as you would like – buying a week’s supply of groceries – giving yourself a complete new wardrobe.

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Information and Communication Technology

• Comparing today’s prices with those of yesteryear: <http://zsuzsybee.hubpages.com/hub/Life-75-Years-Ago-ComparedTo-Now> and <http://www.talkfinance.net/f32/cost-living-today-vs-1960-a-3941/>

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History

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• Research the average wage and cost of basic household items from 10, 20, 50 or 100 years ago. Make up a weekly household budget for each era.

Geography

• Compile a shopping list of foods originating from different countries; for example: Swiss cheese, Italian pizza, Turkish bread, Greek yoghurt, Spanish chorizo, New Zealand lamb, Danish pastry, Mexican tacos. Compare the cost of each item in different major supermarkets. Make a display of the food wrappers against the national flags.

The Arts

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• Make a montage of pictures of currency from different nations against a backdrop of famous natural features or structures from each country.

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Money and Financial Mathematics—M&FM – 1

RESOURCE SHEET Saving for something special

Item: Week

Income (I)

Expenditure (E)

Savings (S) (S = I – E)

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Balance (B) (new B = S + old B)

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Cost:

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Sub-strand: Money and Financial Mathematics—M&FM – 1

RESOURCE SHEET Income and expenditure cards

$1.05

$2.10

$2.35

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$1.25

$3.25

$4.25

$5.25

$3.45

$4.50

$5.40

$2.65 $3.50 $4.55 $5.50 . te o c . che e r o $2.70 $3.65 $4.75 $5.60 t r s super

$1.75

$2.75

$3.75

$4.85

$5.75

$1.80

$2.90

$3.95

$4.95

$5.90

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Create simple financial plans

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$2.60

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$1.60

$5.20

$1.50

$1.55

$5.10

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$1.20

$3.15 $4.05 r o e t s Bo r e p ok u S $2.25 $3.20 $4.15

$1.30

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RESOURCE SHEETS Fundraising cake sale – Cupcakes

Income Number of cupcakes

Income from cupcakes

cupcakes @ 60c each

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packs of 2 cupcakes @ $1 a pack

Total number of cupcakes

Total income from cupcakes

Expenditure

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packs of 5 cupcakes @ $2 a pack

The cakes are prepared individually or in batches. The more cakes there are in a batch, the less it costs to prepare them.

$2.80

$2.40

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Number of cakes in a batch

$2.70

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CONTENT DESCRIPTION: Create simple financial plans

Calculate the cost of providing the cakes for your stall.

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Profit Calculate the profit made from your cake stall.

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Sub-strand: Money and Financial Mathematics—M&FM – 1

RESOURCE SHEET Fundraising cake sale – Family cakes

Income Number of Income from family cakes family cakes small family cakes @ $4 each

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medium family cakes @ $6 each

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large family cakes @ $10 each

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Total number of cakes

Total income from family cakes

Expenditure

Size of cake

Cost of ingredients

$1.75 © R. I . C.Publ i cat i ons medium $2.50 •f orr evi ew pur posesonl y• large $3.00 small

Profit

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Calculate the profit made from your cake stall.

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CONTENT DESCRIPTION: Create simple financial plans

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Calculate the cost of providing the cakes for your stall.

Sub-strand: Money and Financial Mathematics—M&FM – 1

RESOURCE SHEET Calculate the cost of a class excursion

Number of people going on the excursion Students Adults (minimum of 1 for every 6 students)

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Number of seats on bus

Cost

$120

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Cost of transport

$150

$200

$270

Cost of entry to venue

©R . I . C .Pub l i ca on s of 10 Single ticket Groups oft 5i Groups Student $7.50 $33.75 $67.50 •f orr evi ew pur p oseson l y• $12.25

$55.10

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Calculate

CONTENT DESCRIPTION: Create simple financial plans

Total number of people on excursion Total cost of trip for students

Total cost of transport for each student Cost of entry for students

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$110.25

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Adult

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Sub-strand: Money and Financial Mathematics—M&FM – 1

RESOURCE SHEET Creating a budget

When creating a budget, all the necessary information needs to be collected before the final costs are known. Recording this information in a simple graphic organiser makes it easier to see what is required. In this example, a sports teacher wants to know how much it would cost each term to run a school swimming squad at the local pool and how much each student would have to pay.

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

Number of students who want to join the squad

32 swimmers

Maximum number of students to swim in each lane

8 swimmers

Number of days to train

5 days

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Some decisions are made by the person creating the budget.

1b hour © R. I . C.Pu l i cat i ons 2e groups •f orr evi ew pu r p o s sonl y• 16 swimmers in each group:

How long to train each day

– Group A trains 3 times a week – Group B trains 2 times a week

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Number of groups within the squad

Number of lanes required each day

16 swimmers ÷ 8 in a lane = 2 lanes

. thire Cost of lanee

$9 per lane per hour

Pool entry

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How much will it cost to run the swimming squad each term? How much will students need to pay?

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Lane hire each term

$9 x 2 lanes x 5 days x 10 weeks = $900

Pool entry Group A Group B

$2.50 x 3 sessions x 10 weeks = $75 $2.50 x 2 sessions x 10 weeks = $50

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CONTENT DESCRIPTION: Create simple financial plans

Some decisions are determined by others and cannot be altered.

Sub-strand: Money and Financial Mathematics—M&FM – 1

RESOURCE SHEET Goods and services tax (GST)

Having established the basic cost of running the swimming squad, the sports teacher may then want to purchase some equipment. Such equipment incurs a goods and services tax (GST). The GST is an additional 10% charge on the cost of an item which the purchaser has to pay.

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‘Per cent’ means ‘per 100’ or ‘every hundred’. For example, for every $100 an additional $10 is charged. Ten per cent has the same value as 1⁄10 or one-tenth. • To calculate GST, multiply the cost price of an item by 1⁄10 (one-tenth).

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• To calculate the actual price of an item, add the GST to the cost price. Cost price

GST

Actual price

$100

$10

$110

$73

$7.30

$80.30

$275.10

$3026.10

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$2751

Equipment required for swimming squad

Quantity Cost per item Total cost per item . te o kickboard 8 $18 + GST = $18 + $1.80 = $19.80 c $19.80 x 8 = $158.40 . che e $17.60 x 8 = $140.80 r fins 8 pairs $16 + GST = $16 + $1.60 = $17.60 o r st super

CONTENT DESCRIPTION: Create simple financial plans

Item

pull buoys

$17 + GST = $17 + $1.70 = $18.70

$18.70 x 8 = $149.60

hand paddles

8 pairs

$15 + GST = $15 + $1.50 = $16.50

$16.50 x 8 = $132.00

Total cost

$580.80

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RESOURCE SHEET

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After-school activities – Play board

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CONTENT DESCRIPTION: Create simple financial plans

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Sub-strand: Money and Financial Mathematics—M&FM – 1

RESOURCE SHEET After-school activities – Score card

Week/ Activity

Monday

Tuesday

Wednesday

Thursday

Friday

1 2 3

Activity

Balance

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$300

1 2 3 4 5 6

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8 9

10 11 CONTENT DESCRIPTION: Create simple financial plans

12 13 14

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RESOURCE SHEET Budget proposal

Type of equipment Cost per item

Quantity

Total cost per item

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Individual items

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Total cost of all items 168

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Create simple financial plans

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Sub-strand: Money and Financial Mathematics—M&FM – 1

RESOURCE SHEET GST and purchase receipts

Most goods are subject to GST. GST is always included in the advertised price of goods. The cost of services also have GST added to their sub-total.

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The receipt for any purchase includes the GST component‚ which the service provider has to pay to the government. A receipt is given for the purchase of all goods and services.

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The receipt is proof that the goods or services have been sold by the service provider and bought by the customer. The service provider’s receipt has the same number as the customer’s receipt.

Department store

Items

GST calcu 10% or 1⁄10

lation

f $138.00 25.40 © R. I . C. Publ i ca t i o n s 1⁄10 x $138.0 0 = $13.80 toaster 30.60 In cos meo •f orr evi ew15.10 pur pos e • fon r sel lly er clock

towel

55.20

candles

11.70 $138.00

Total includes GST

$13.80

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Total

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Total – GS T

$138.00 –

$13.80 = $

124.20 Cost to cu stomer Total = $1 38

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cushions

o c GST c. alculatio Services c $ n e h r er o10% or 1⁄10 of $280 t Lopping trees $175 s s r u e p 1⁄10 x $28 Mowing lawn $35 Gardening services

CONTENT DESCRIPTION: Create simple financial plans

Pruning shrubs

$30

Pulling weeds

$25

Clearing rubbish

$115

Sub-total

$280

GST

$28

Total

$308

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

Income fo

0 = $28

r service

Sub-total

provider

= $280

Cost to cu stomer Sub-total + GST $280 + $2 8 = $308

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• decide the steps that need to be taken to find the answer to the question.

How much profit did the class make? How many packs of sausages were needed? 96 ÷ 24 = 4

How much did it cost to make 96 hot dogs?

The cost

Step 1

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

m . Total cost u $34 + $44 = $78

How much did 16 packs of rolls cost? $2.75 x 16 = $44

How many packs of rolls were needed? 96 ÷ 6 = 16

Step 3

Income = $144

$144 – $78 = $66

Profit is the income minus the cost

The profit is the amount of money remaining after expenses have been paid.

How many hot dogs were sold and at what price?

96 hot dogs at $1.50 each = 96 x 1.5 = $144

The profit

Step 2

The income

How much did four packs of sausages cost? $8.50 x 4 = $34

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CONTENT DESCRIPTION: Create simple financial plans

• Ninety-six hot dogs were sold at $1.50 each.

• A pack of six hot dog rolls costs $2.75.

• A pack of 24 sausages costs $8.50.

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The question

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The information

For example, if a pack of 24 sausages costs $8.50 and a pack of six hot dog rolls costs $2.75, how much profit did a class make when it sold 96 hot dogs at $1.50 each?

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• separate the information from the question,

To solve word problems, it is necessary to:

Sub-strand: Money and Financial Mathematics—M&FM – 1

RESOURCE SHEET

Solving word problems

Sub-strand: Money and Financial Mathematics—M&FM – 1

RESOURCE SHEET Money problems

(For these problems‚ GST has been included in the price.) Petrol costs $1.45 per litre. Zara’s dad’s car holds 70 litres.

• Would there be any change from $50 if someone bought three packs?

• How much would half a tank of petrol cost him?

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A 500-g family-size pack of lollies sells at $5.45 and a standard 375-g pack sells at $4.50. • How much cheaper per kg is the better value pack?

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Bottled water is sold in packs of 24 at $14.40 a pack.

A theatre has a 300-seat capacity and costs $4500 to hire for one day. • How much would a drama group have to charge for each ticket if it played to a full audience at a matinee and an evening performance and made a profit of $3000.

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• How much more does he need to save to buy a $60 leisure centre pass?

CONTENT DESCRIPTION: Create simple financial plans

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A 500-g packet of cereal costs $3.75 and a 750-g packet costs $6.30.

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Ahmed has saved $2.75 a week for 13 weeks.

• Which packet is the better value?

o c . che e r o t r s uper At a show, fair ride tickets can bes bought for $5 each, $30 for a strip of

Eve has saved $67.70 and wants to buy a USB thumb drive for $17.45 and a computer game for $34.95.

seven or $50 for a strip of 12. • What saving is made by buying: (a) a seven strip of tickets?

• Does she have enough money to also buy a bracelet for $13.25?

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Assessment 1

Sub-strand: Money and Financial Mathematics—M&FM – 1

NAME:

DATE:

1. Complete the table to find out how much Sam saved in four weeks. Week

Income

Expenditure

$5.35

Savings

Balance

Sam saved $7.60

$6.25

$10.80

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$2.50

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in four weeks.

$3.35

(a) On average, how many faces were painted each hour? 18

(b) On average, how long did each painting take?

(c)

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2. The total income from a face painting stall was $180. The cost of materials used was $25. The face painter worked for three hours and painted 36 faces.

12 minutes

18 minutes

$12

$18

(e) How much profit did the painter make?

$137 $162 . te 3. Tickets for the zoo are sold in different ways. o c . c e Two adults plus hertwo children s r Single child Single adult Four children Four adults o t s r u pe $30 $8 $14 $50 $40 $180

(a) What is the cheapest option for a group of three adults and six children?

(b) How much cheaper is it to buy a ‘four adults’ ticket than four ‘single adult’ tickets?

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CONTENT DESCRIPTION: Create simple financial plans

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(d) How much GST at 10% does the painter have to pay altogether?

Assessment 2

Sub-strand: Money and Financial Mathematics—M&FM – 1

NAME:

DATE:

1. For each example, calculate: (i) the cost to the customer (ii) the GST (iii) the income for the seller/service provider.

Hardware store

(a)

Items nails

(b)

Plumbing services

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3.50

Items

call out fee

17.10

taps

295

garden rake

11.50

pipes

180

pool salt

7.30

washing line

washers

19.60

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hammer

Sub-total

Total

GST

Total includes GST

Total

Income for seller/ service provider

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o c . (a) $50 if Jill buysc a book for $19.95 and a pencil case fore $7.80? her r o t s super

CONTENT DESCRIPTION: Create simple financial plans

2. How much change is there from:

(b) $10 if Anton buys a drink for $2.35, a biscuit for $1.70 and a cake for $3.85? 3. Will there be enough money to buy: (a) a guitar for $275 and a soccer ball for $45 if Jack has saved $310? (b) nail varnish for $3.95, a ring for $7.85 and a chain for $12.35 if Sandra has $25?

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STUDENT NAME

174

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

Identifies and calculates GST component of invoice/ receipt

Organises information to create budget for financial venture

Calculates total cost of an excursion, given data on numbers and individual costs involved

Calculates income, expenditure and profit of a financial venture

Keeps record of income, expenditure and saving over time

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Checklist Sub-strand: Money and Financial Mathematics—M&FM – 1

Create simple financial plans (ACMNA106)

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Answers

Sub-strand: Money and Financial Mathematics

M&FM – 1 Page 172 1. 2.

$18.40 (a) 12 faces (c) $5 (e) $137 (a) $84

Page 173 1.

(b) 5 minutes (d) $18 (b) $6

Assessment 2

(a) (i) $59 (ii) $5.90 (iii) $53.10 (b) (i) $542 (ii) $54.20 (iii) $596.20 (a) $22.25 (b) $2.10 (a) No (b) Yes

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Assessment 1

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Sub-strand: Patterns and Algebra—P&A – 1

Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction (ACMNA107)

RELATED TERMS

TEACHER INFORMATION

Number pattern/sequence

What this means

• A list of numbers in sequence that follow a specific rule

Students should be able to:

Repeating number pattern

• describe patterns and number sequences involving fractions, decimals and whole numbers resulting from addition and subtraction by using a number line, diagrams or tables • continue patterns accurately • write rules for patterns • recognise and describe the reverse rules for patterns • identify the rules for patterns where the operation is not clear • create their own patterns and number sequences involving fractions, decimals and whole numbers resulting from addition and subtraction by using a number line, diagrams or tables.

Growing number pattern

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• A list of numbers that increase or decrease in size as the rule is followed Routine growing pattern

• A list of numbers that increase or decrease in size with a constant difference Non-routine growing pattern

• A list of numbers that increase or decrease in size with a non-constant but predictable difference.

Teaching points

Term

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• A set of random numbers that form a unit that is repeated

Variable

• A symbol for an unknown number Constant

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• A number on its own Coefficient

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• The number by which a variable is multiplied Operator

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• Show students number patterns in tables, on number lines and in number sequences. • Show patterns as a range of addition and subtraction operations involving fractions, decimals and whole numbers. • Ask the questions: ‘What is happening in the pattern? What comes next in the pattern? What is the rule for the pattern?’ • The relationship between a number and its position within a sequence is the link between the rule that creates the pattern and the generalisation or algebraic expression.

• A number within a pattern/sequence

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• An arithmetic symbol for addition, subtraction, multiplication and division

What to look for

• Students who think of decimals as whole numbers and continue patterns beyond 1.0 as 0.10, 0.12, 0.14 instead of 1.0, 1.2, 1.4. This shows a lack of understanding of decimals and place value. See also New wave Number and Algebra (Year 5) student workbook (pages 64–70)

Student vocabulary Number pattern Number sequence Growing pattern Routine pattern Non-routine pattern

176

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

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Sub-strand: Patterns and Algebra—P&A – 1

HANDS-ON ACTIVITIES Number patterns • Photocopy onto card, cut out and laminate the number cards from pages 186 to 196. Separate the cards to create patterns. • Create patterns using fractions, decimals and whole numbers. Use addition (plus) and subtraction (minus) moves in your pattern. Include single-step and multi-step patterns with addition or subtraction, and multi-step patterns with a mixture of addition and subtraction.

Partner patterns

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• In pairs students create number patterns. Swap patterns for partner to solve. Write the solution on a record sheet (page 182). Record patterns on a number line (pages 184–185) for others to continue by using the rule. Reverse the numbers in the pattern and its record the reverse rule and generalisation.

Number patterns concentration

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• Lay the laminated number cards required for a specific number line facedown on a table. With a partner, take turns picking a card to place on the number line. Cards must be placed in sequence. If the card chosen is not the next in the pattern, it must be replaced facedown on the table. The game ends when all the cards have been placed. The person to have picked the greater number of cards to complete the sequence is the winner.

Dice patterns

• The first throw of a dice determines the operation of the pattern: an even number for addition and an odd number for subtraction. The second throw determines the number to be added or subtracted. Use more than one dice (or a dice with many sides) to make larger numbers. For multi-step patterns, repeat the two throws of the dice. Record the difference(s) on the record sheet (page 182) and then work out the pattern. Complete the rest of the record sheet.

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• Use the rules of the card game rummy to play ‘number pattern rummy’‚ in which a sequence of seven numbers must be made.

Semi-regular tessellation patterns • Create semi-regular (two shapes) tessellation patterns by drawing around regular shapes. Take note on a record sheet (page 182) the number of each shape required to create a repeating unit. Have students consider how many of a shape would be used if x number of a second shape two is used? For example‚ for every regular hexagon, two equilateral triangles are required.

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Matchstick shapes

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• Use school-safe matchsticks to make a series of regular 2-D shapes starting with a side length of one matchstick, then two matchsticks, then three etc. For each shape, record how the number of matchsticks per length alters the number of matches of the perimeter.

Triangular patterns • Draw around one equilateral triangle to give the first term in a pattern. Create the second term by drawing around two triangles side by side and inverting a third triangle to fit in the space between. The third term has three triangles at its base with two inverted, two in the next row with one inverted‚ and one at the apex. Continue the pattern. Record the total number of smaller triangles in each triangle against the length of the base. Describe the pattern, then write the rule and the generalisation.

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Sub-strand: Patterns and Algebra—P&A – 1

LINKS TO OTHER CURRICULUM AREAS English • Students research to find and read the story of the grain of rice and present the mathematical explanation orally and in a table. Use: <http://www.watersfoundation.org/index.cfm?fuseaction=content.display&id=93> • Research the construction of the Fibonacci sequence and Pascal’s and Sierpinski’s triangles. Write a procedure to describe how to construct one of them. Use step-by-step illustrations to clarify the text. • Write a brief report on either Fibonacci, Pascal or Sierpinski. • Look for and describe any patterns you see in Pascal’s triangle. • Look for and describe any connection you see between the Fibonacci sequence and Pascal’s and Siepinski’s triangles. (Colouring all the even numbers in the Pascal triangle will reveal the Sierpinski pattern. By going up one row and along one square of Pascal’s triangle and adding the values, the Fibonacci sequence is revealed.)

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Information and Communication Technology

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• Interactive number pattern game: <http://www.bbc.co.uk/schools/ks2bitesize/maths/number/number_patterns/play. shtml> • Fibonacci sequence: <http://www.mathsisgoodforyou.com/topicsPages/number/famousequences.htm> • Pascal’s triangle: <http://www.mathsisfun.com/pascals-triangle.html> • Sierpinski’s triangle: <http://www.mathsisfun.com/sierpinski-triangle.html>

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The Arts • Create semi-regular (two shapes) tessellation patterns. Have students work out how many of each shape creates a repeating unit? How many of a shape would be used if x number of another shape is used. Use a range of art techniques to decorate the patterns. • Create display borders of repeating patterns of shapes and other decorative objects. • Use contrasting material scraps to create a Sierpinski-style carpet wall hanging.

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RESOURCE SHEET Solving number patterns – 1

Number patterns can be shown in different ways. • In tables: Position

Pattern

1⁄8

r o e t s Bo r e p ok u S 3⁄8

5⁄8

7⁄8

11⁄8

13⁄8

15⁄8

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–1

• As number sequences: 6

5.25

4.5

3.75

2.25

1.5

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• On number lines: The arrow shows the direction of the pattern—addition to the right, subtraction to the left. 10

0.75

In whichever way number patterns are presented, © R . I . C . P u b l i c a t i o ns there are five steps to follow to solve them. •What f or r evi ew pur posesonl y• is the difference between consecutive numbers?

If the numbers in the pattern are increasing, record the difference as a plus number; e.g. + 3. If the numbers are decreasing, record as a minus number; e.g. – 3.

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Step 1

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

Difference

Step 2

Step 3

In your own words, describe what is happening to the numbers in the pattern. Are they increasing or decreasing in value? Are they going up or down by the same or different amounts?

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o c . c e her r Write the description as a rule. o t sto start at (the first number of the su Always begin with stating which number r e p sequence) and say which mathematical operation (+, –, x, ÷) to perform to give the next number in the pattern. This rule must apply to all the numbers in the pattern.

Step 4

Look for a link among each number in the pattern and how to work out any number’s position within the pattern.

Step 5

Write a general rule (algebraic expression—using letters in place of actual numbers) that will describe how to calculate a number anywhere in the pattern without having to write out the whole pattern.

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RESOURCE SHEET Solving number patterns – 2

Pattern (Sequence)

Difference

Teac he r

Position (Element)

r o e t s Bo r e p ok u S 6

Rule

Start at 3 and add three each time.

Link between position and pattern

The pattern number is three times the position number.

in own words

in the pattern at that position.

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Description

The difference among consecutive numbers in the sequence is three. Three is added to each number to obtain the next number in the pattern.

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Position Pattern

Difference Description

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2 5 8 11 14 17 20 23 26 . t e o c . +3 +3 +3 +3 +3 +3 +3 +3 c e her r o t er Each numbers in u the p pattern iss three more than the last.

Rule

To get the next number in the pattern, add three.

Link between position and pattern

A position number multiplied by three and with two added gives the pattern number for the next position.

Generalisation in own words

Multiply the previous position number (n-1) by 3 and add 2.

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29 +3

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

In a routine growing pattern, the same operation is applied to each number in the sequence; for example:

Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET Solving number patterns – 3

Position

Pattern

Difference

Teac he r

Each number in the pattern is four more than the last. To find the next number in the pattern, add four.

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Rule

Link between position and pattern

The difference between each pattern number and its position number increases by three each time.

Generalisation in own words

To get the pattern number, multiply the position number by four and subtract three.

Pattern (Sequence)

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Position (Element)

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

Description

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

. te +1 +2 +3 +4 +5 +6 o +7 +8 +9 c . c e by one each time. Description The between numbers increases hdiff r eerence o r st super Difference

Rule

Start at 1 and add one to get the second number, then add two for the third number, then three for the fourth and so on.

Link between position and pattern

Adding the previous position number and its pattern number gives the next pattern number.

Generalisation in own words

Find the sum of the previous position number and its corresponding pattern number.

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Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET

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Generalisation in your own words

Link between position and pattern

Rule

Description

Difference

Pattern (Sequence)

Position (Element)

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

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

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Teac he r

Number patterns in tables

Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET Creating number patterns

Working with number patterns: • consolidates students’ understanding of number concepts and extends their thinking • highlights specific difficulties that students have in performing number operations; for example‚ with place value.

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• Create number patterns with numbers that students can confidently work with and vary the complexity of the patterns so they are appropriate to the students’ needs.

• Give students the opportunity to work out the reverse operation so that patterns can be extended backwards.

Reverse operation

Pattern

–17

1.3

+0.3

–17

1.6

+0.3

–17

1.9

+0.3

–17

2.2

+0.3

–17

2.5

+0.3

2.8

+0.3

3.1 …

+0.3

–0.3

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–17

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

• challenges students to make generalisations about patterns and then represent them using symbols.

. te 3 3.25 5 5.25 7 7.25 9 … o–1.75, –0.25 …1 1.25 c . che e r o … 10 9¼ 8¼ 7½r 6½ 5¾ 4¾s 4 3… +1, +¾ s upert

Multi-step

+0.25

+1.75 +0.25

–¾

–1

+1.75 +0.25

–¾

–1

+1.75 +0.25

–¾

–1

–¾

+1.75

–1

Multi-step with mixed operations …1

14⁄5

+4⁄5

13⁄5

–1⁄5

… 100

+4⁄5

200

+100

22⁄5 –1⁄5

150

–50

21⁄5

24⁄5

+4⁄5 –1⁄5

225

+75

325

+100

33⁄5

32⁄5 …

+4⁄5

–1⁄5

275

350 …

–50

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

+1⁄5, –4⁄5 –75, +50, –100

+75

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2⁄4

1⁄3

2⁄3

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1⁄6

1⁄3

2⁄5

3⁄5

4⁄5

2⁄3

1⁄3

5⁄8

2⁄4

5⁄6

2⁄3

1⁄5

2⁄5

1⁄6

1⁄3

7⁄8

1⁄8

2⁄3

1⁄3

2⁄4

5⁄6

2⁄3

3⁄8

m . 3⁄5 4⁄5 2 1⁄5 u

1⁄3

2⁄3

2⁄5

3⁄5

1⁄6

1⁄3

5⁄8

4⁄5

1⁄3

2⁄3

2⁄4

1⁄3

3⁄8

2⁄3

1⁄5

2⁄5

3⁄5

4⁄5

5⁄6

2⁄3

1⁄8

1⁄3

2⁄3

1⁄3

1⁄6

7⁄8

Teac he r

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

CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

General rule

1⁄5

3⁄8

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General rule

1⁄8

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General rule

2⁄4

1⁄5

5⁄8

2⁄5

2⁄3

1⁄3

3⁄5

5⁄6

2⁄3

7⁄8

4⁄5

1⁄6

1⁄3

1⁄8

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184

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RESOURCE SHEET

Fraction patterns on number lines

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1 0.2 0.3 0.4 0.5

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

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

0.5

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General rule

0.5

0.25 0.5 0.75

General rule

0.5

0.25 0.5 0.75

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0.25 0.5 0.75

0.5

0.25 0.5 0.75

0.5

0.25 0.5 0.75

0.25

0.5 10 0.5 11 0.5 12 0.5

0.25 0.5 0.75

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General rule

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 0.125 0.25 0.375 0.5 0.625 0.75 0.875 2 0.125 0.25 0.375 0.5 0.625 0.75 0.875 3 0.125

General rule

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET

Decimal patterns on number lines

185

Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET Patterns with fractions – 1

1⁄3

2⁄3

¾ 1⁄5 r o e t s B r e oo p u k S 2⁄5 3⁄5 4⁄5 3⁄6

5⁄6

4⁄7

5⁄7

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3⁄8

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1⁄7

3⁄9

4⁄9

5⁄9

6⁄9

7⁄9

8⁄9

1⁄8 . t

7⁄8

186

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

1⁄6

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Teac he r

Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET Patterns with fractions – 2

1½

11⁄3

12⁄3

1¾ 11⁄5 r o e t s B r e oo p u k S 12⁄5 13⁄5 14⁄5

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11⁄6

13⁄6

15⁄6

I . C.Pu bl i cat i ons 13⁄7 11⁄7© R. 12⁄7

•f orr evi ew pur posesonl y•

15⁄7

16⁄7

11⁄8 . t

13⁄8

m . u

14⁄7

13⁄9

14⁄9

15⁄9

16⁄9

17⁄9

18⁄9

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

1¼

15⁄8

o c . che e r o t r 17⁄8 11⁄9 12⁄9 s sup er

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Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET Patterns with fractions – 3

2½

21⁄3

22⁄3

2¾ 21⁄5 r o e t s B r e oo p u k S 22⁄5 23⁄5 24⁄5 23⁄6

25⁄6

I . C. Publ i cat i on23⁄7 s 21⁄7 © R. 22⁄7

•f orr evi ew pur posesonl y•

25⁄7

26⁄7

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23⁄8

25⁄8

23⁄9

24⁄9

25⁄9

26⁄9

27⁄9

28⁄9

21⁄8. t

27⁄8

188

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

21⁄6

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Teac he r

2¼

Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET Patterns with fractions – 4

3½

31⁄3

32⁄3

3¾ 31⁄5 r o e t s B r e oo p u k S 32⁄5 33⁄5 34⁄5

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31⁄6

33⁄6

35⁄6

I . C.Pu bl i cat i ons 33⁄7 31⁄7© R. 32⁄7

•f orr evi ew pur posesonl y•

35⁄7

36⁄7

31⁄8 . t

33⁄8

m . u

34⁄7

33⁄9

34⁄9

35⁄9

36⁄9

37⁄9

38⁄9

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

3¼

35⁄8

o c . che e r o t r 37⁄8 31⁄9 32⁄9 s sup er

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET Patterns with fractions – 5

4½

41⁄3

42⁄3

4¾ 41⁄5 r o e t s B r e oo p u k S 42⁄5 43⁄5 44⁄5 43⁄6

45⁄6

I . C. Publ i cat i on43⁄7 s 41⁄7 © R. 42⁄7

•f orr evi ew pur posesonl y•

45⁄7

46⁄7

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43⁄8

45⁄8

43⁄9

44⁄9

45⁄9

46⁄9

47⁄9

48⁄9

41⁄8. t

47⁄8

190

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

41⁄6

ew i ev Pr

Teac he r

4¼

Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET Patterns with decimals – 1

0.1

0.2

0.3

0.5 0.6 r o e t s B r e oo p u k S 0.7 0.8 0.9

Teac he r

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1.0

1.1

1.2

I . C.Pu bl i cat i ons 1.5 1.3© R. 1.4

•f orr evi ew pur posesonl y•

1.7

1.8

1.9 . t

2.0

m . u

1.6

2.5

2.6

2.7

2.8

2.9

3.0

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

0.4

2.1

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Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET Patterns with decimals – 2

3.1

3.2

3.3

3.5 3.6 r o e t s B r e oo p u k S 3.7 3.8 3.9 4.1

4.2

I . C. Publ i cat i on s 4.3 © R. 4.4 4.5

•f orr evi ew pur posesonl y•

4.7

4.8

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5.0

5.1

5.5

5.6

5.7

5.8

5.9

6.0

4.9. t

5.2

192

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

4.0

ew i ev Pr

Teac he r

3.4

Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET Patterns with decimals – 3

0.25

0.75

1.25

2.25 2.75 r o e t s B r e oo p u k S 3.25 3.75 4.25

Teac he r

0.125

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4.75

0.375

1.625

1.875

m . u

1.375

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

1.75

2.125 . t

2.375

3.625

3.875

4.125

4.375

4.625

4.875

2.625

o c . che e r o t r 2.875 3.125 3.375 s su per

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Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET Patterns with whole numbers – 1

Teac he r

5or 6 e t s B r e oo p u k S 8 9

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19 . t

194

18 21

o c . che e r o t r 23 24 s s uper

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET Patterns with whole numbers – 2

35 36 r o e t s B r e oo p u k S 37 38 39

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I . C.Pu bl i cat i ons 45 43© R. 44

•f orr evi ew pur posesonl y•

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

49 . t

o c . che e r o t r 52 53 54 s sup er

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Sub-strand: Patterns and Algebra—P&A – 1

RESOURCE SHEET Patterns with whole numbers – 3

65 66 r o e t s B r e oo p u k S 67 68 69 71

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

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Teac he r

Assessment 1

Sub-strand: Patterns and Algebra—P&A – 1

NAME:

DATE:

1. Use the numbers in the top row of the table to make the different patterns. 1, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23 Rule

Start at

(a) + 4

(b) +3

Pattern

r o e t s Bo r e p ok u S 23

Description

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Teac he r

Pattern

(a) 4.4, 3.7, 3.0, 2.3, 1.6, 0.9 (b) 19, 20, 22, 25, 29, 34 (c) 1⁄8, ¼, 3⁄8, ½, 5⁄8, ¾

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

2. Describe each pattern.

3. Complete the table. Number of matchsticks per side

Number of matchsticks in perimeter

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Description of pattern

General rule

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Assessment 2

Sub-strand: Patterns and Algebra—P&A – 1

NAME:

DATE: A

3 0.25 0.5 0.75 4 0.25 0.5 0.75 5 0.25 0.5 0.75 6 0.25 0.5 0.75 7 Shade the bubble.

(a) 3.75

(b) 6.25

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

Position

5 1

7 2

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Teac he r

2. Create a pattern on the number line and complete the table.

10 11 12 13 14 15 16 17 3

Pattern

Difference

Description

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Link

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3. Colour fractions of each circle to show a pattern and complete the table.

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Position

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Pattern Difference Description Rule Link 198

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction

1. Which point on the number line represents

Checklist

Sub-strand: Patterns and Algebra—P&A – 1

Describe, continue and create patterns with fractions, decimals and whole numbers

Creates own patterns with rules and generalisations

Writes a general rule for the pattern

Identifies a link between the each number in a pattern and its position in the sequence

Writes a rule explaining how to continue a pattern

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

STUDENT NAME

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Teac he r

Identifies and describes the pattern in a number sequence

resulting from addition and subtraction (ACMNA107)

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Sub-strand: Patterns and Algebra—P&A – 2

Use equivalent number sentences involving multiplication and division to find unknown quantities (ACMNA121)

RELATED TERMS

TEACHER INFORMATION

Balance

What this means

• The state which must exist between the two expressions of an equation

Students should be able to:

Equivalent

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

• Having the same value but expressed in different terms; e.g. 3 x 4 = 60 ÷ 5 Commutative law

Associative law

• When multiplying (or adding) three or more numbers, the order in which groups of numbers are multiplied (or added) does not affect the outcome. Distributive law

Teaching points

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• The order in which numbers two are multiplied (or added) does not affect the outcome.

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• write number sentences to represent and answer questions • balance both sides of an equation involving multiplication or division • employ a range of strategies to solve different multiplication and division problems.

• Give students the opportunity to: – learn strategies for developing number sentences based on the semantic structure of a written question – balance both sides of an equation – use manipulative materials‚ such as MABs, play money, number balances and calculators. • Students need to write the number sentence before using the materials or technology.

• The representation of a number problem expressed using numbers and operation symbols

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Equation

• A mathematical statement which asserts the equality of two expressions Unknown term

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• The variable whose value is the solution to the equation Expression

• are unable to write the number sentence that reflects the semantic structure in the word problem • guess the operation required to solve a problem • do not work out both sides of the equation to find out which part of the problem is missing • do not understand the inverse relationship of multiplication and division and try to solve multiplication problems using repeated addition only

m . u

Number sentence

Students who:

• The group of terms on each side of the equals sign Student vocabulary balance equivalent number sentence equation

See also New wave Number and Algebra (Year 5) student workbook (pages 71–78)

Proficiency strand(s): Understanding Fluency Problem solving Reasoning

expression term greater than‚ less than

200

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Patterns and Algebra—P&A – 2

HANDS-ON ACTIVITIES Number balance Photocopy on to card, then cut and laminate several copies of the numbers and signs on page 216. Working in pairs, students each create balanced number sentences which they then swap with their partner to check. When checked, students record sentences on the sheet on page 215.

Missing term Using the cards and record sheet from pages 215 and 216, students create number sentences with a missing term for a partner to solve.

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Balanced number problems

Teac he r

Using units of currency and weight, students practise the process described on page 211 and write number problems with a missing term to be solved.

Restoring the balance

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Students use the cards and record sheets from pages 215 and 216 to create inequalities using the ‘greater than’ and ‘less than’ signs. They discuss what can be done to restore balance. For example, 6 x 7 < 9 x 7 could become either (6 x 7) + (3 x 7) = 9 x 7 or 6 x 7 = (9 x 7) – (3 x 7)

Algebra spy

Select one student in the class to be the spy. Do not reveal the student’s name to anyone! Create the same amount of number sentences as there are letters in the ‘spy’s’ name. Assign a letter to each number sentence. Underneath the number sentences‚ draw dashes equal to the number of letters in the spy’s name. Write the value of the missing term under the dash of each letter.

Algebra bingo

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m . u

w ww

Use the number sentences created in the ‘Missing term’ activity as call-out questions for ‘Algebra bingo’. Prepare bingo cards that reflect the answers to the missing terms. Allow students time to record the number sentences which will help them work out the solution before reading the next question.

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Sub-strand: Patterns and Algebra—P&A – 2

LINKS TO OTHER CURRICULUM AREAS English • Give an oral presentation to describe the concept of balance. Use materials or pictures to enhance your presentation, for example; silhouettes of children (page 221) on a see saw.

Information and Communication Technology • A interactive game for balancing equations: <http://illuminations.nctm.org/activitydetail.aspx?id=26> • A brief explanation of how algebra was used in ancient times and how it is used today: <www.youtube.com/ watch?v=BJNhKz1DVIA>

Design and Technology

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

• Design and make a seesaw for use in the English activity.

History

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• Draw a time line to show the development of algebra from Egypt, Babylon, Greece, India, Arabia and Europe. Colour the regions on an outline of a world map to show the location of each place.

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Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEET Three signs of the number sentence

Three types of number sentence can be made using different signs to separate the left and the right sides of the sentence.

7+8>4+5 (15 > 9)

8x6>4x6 (48 > 24)

r o e t s Bo r e p ok u S Anything subtracted or divided on the left must be less than the value of the right.

The value of the left side of the sentence is less than the value of the right side.

12 ÷ 4 < 24 ÷ 2 (3 < 12)

. te

7+5=8+4 (12 = 12) 6–4=5–3 (2 = 2) 4x6=8x3 (24 = 24) 12 ÷ 4 = 12 ÷ 2 (6 = 6)

must be less than the value of the right. Anything subtracted or divided on the left must be greater than the value of the right.

m . u

5+4<8+7 (9 < 15)

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CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

12 ÷ 2 > 12 ÷ 4 (12 > 3)

Anything added or multiplied on the left must be greater than the value of the right.

ew i ev Pr

Teac he r

6–3>6–4 (3 > 2)

The value of the left side of the sentence is greater than the value of the right side.

o c . The value of the left side of the sentence is equal to the ch e r value of the right side. er o t s s r u e p Anything added or multiplied on the left must equal that the value of the right. Anything subtracted or divided on the left must equal that on the right.

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RESOURCE SHEET The ‘equals’ sign: What does it mean? – 1

In algebra, the ‘equals’ sign separates the two sides (expressions) of a number sentence and indicates that the value of each side is equal. That is‚ the number sentence is balanced. Fact

Number sentence

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

17 = 17

Carly saved $39 for the trip and Kym saved $39.

$39 = $39

Ely and Cooper each had five model cars and they each bought four more.

5+4=5+4

ew i ev Pr

Sita made 12 cupcakes and then made 24 more.

Mena made 18 cupcakes and then made 18 more.

12 + 24 = 18 + 18

They both made the same number of cupcakes. Amy ate two biscuits at school each day.

Tia ate five biscuits each day of the weekend.

2x5=5x2

© R. I . C.Publ i cat i ons Antek scored 12 points in each of two matches. •f orr evi ew pur poseson l y• Bruno played three matches and scored eight points in each. 12 x 2 = 3 x 8 They both ate the same number of biscuits.

Chinh had 18 lollies and shared them among six friends.

w ww

Danh had 15 lollies and shared them among five friends. Their friends all received the same number of lollies.

m . u

They both scored the same number of points.

18 ÷ 6 = 15 ÷ 5

. tethree weeks to save $27. o Hoa took $36 ÷ 4 = $27 ÷ 3 c . csame They both saved the amount each week. e h r e o t r s Hui took 37 minutes to walk to school onp Monday‚ but s r u e was nine minutes faster by Friday. Giang took four weeks to save $36.

Lian took 33 minutes on Monday and was five minutes faster by Friday.

37 – 9 = 33 – 5

They both took the same time to walk to school on Friday. Bai’s best javelin throw is 25 metres‚ but in the windy conditions he only managed 23 metres. Eric’s best throw is 19 metres and he only reached 17 metres.

25 - 23 = 19 - 17

They both threw two metres fewer on the windy day. 204

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CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

Teac he r

Jack and Jacob each have 17 marbles.

Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEET Type and order of operations in number sentences

The type of computation operation does not have to be the same on both sides of a number sentence. 6 + 18 = 30 – 6 There can be more than one operation on one or both sides of a number sentence. 8 + 4 x 3 = 96 ÷ 4 – 4

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

Teac he r

ew i ev Pr

When there is more than one type of operation, there is a convention which must be followed relating to the order in which the operations are calculated. This order can be remembered using mnemonics. Two examples are: BIMDAS – brackets, indices, multiplication, division, addition, subtraction PEMDAS – parentheses, exponentials, multiplication, division, addition, subtraction

© R. I . C.Publ i cat i ons Any numbers in brackets (parentheses) (4 + 8) x (12 – 6) othe r(12 r e ecalculated w pu othe se s onl y• The (4 +• 8) f and –v 6) i are fir rstp and two answers multiplied Order of calculation

Any numbers with indices (exponentials) 32 + 8 – 2

m . u

together.

The 32 is calculated first and the answer added to 8 before subtracting 2.

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CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

The key fact is that the result of the operation (or combined operations) is the same for both sides. This is necessary for the number sentence (equation) to be balanced.

. te

Multiplication and division

o c . che e r o t r s super

Neither operation has priority over the other. Reading from the left, whichever comes first is calculated first; for example: • 36 ÷ 12 x 8 The 36 ÷ 12 is calculated first and the answer multiplied by 8 • 8x9÷6 The 8 x 9 is calculated first and the answer divided by 6. Addition and subtraction Neither operation has priority over the other. Reading from the left, whichever comes first is calculated first; for example: • 45 – 9 + 4 The 45 – 9 is calculated first and the answer added to 4. • 56 + 8 – 36 The 56 + 8 is calculated first and 36 is subtracted from the answer. Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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RESOURCE SHEET Equivalence in number sentences

Students are familiar with finding the missing number by using arithmetic computation and their knowledge of inverse relationships. Example 1 – addition/subtraction + 20 = 35 + 15

– 40 = 60 – 10

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

The right side expression is calculated first giving an answer which the left side expression must equal. + 20 = 50

– 40 = 50

= 50 + 40

= 30

= 90

Example 2 – multiplication/division x 6 = 12 x 3

÷ 8 = 54 ÷ 6

The right side expression is calculated first‚ giving an answer which the left side expression must equal.

w ww

. te

= 36 ÷ 6

=9x8

= 72

m . u

The inverse relationship between multiplication and division is used to calculate the unknown.

o c . che e r o t r s super 12 x 3 = 6 x

The concept of equivalence in number sentences as related to algebra is often difficult for students to grasp. Rather than solving a problem by computation, they need to focus on the relationship between the two sides of the number sentence. Always ask: What must be done to balance the equation?

42 ÷ 6 =

÷9

• 12 is 2 x 6 (or 6 is 12 ÷ 2 ) To keep the balance‚

42 is 7 x 6 To keep the balance‚ must be 7 x 9

must be 2 x 3 • 3 is 6 ÷ 2 (or 6 is 3 x 2 ) To keep the balance‚ must be 12 ÷ 2

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CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

Teac he r

= 50 – 20

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The inverse relationship between addition and subtraction is used to calculate the unknown.

Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEET Equivalent number sentences

6x9=

Looking at this number sentence, it is easy to see the relationship between the two expressions.

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

The unknown must be 9 as one of the known factors on the right side is the same as the known factor on the left‚ so the unknown factor on the right must be the same as the other known factor on the left.

Teac he r

3x9=

5x7=

8x9=

6x7=

7x8=

ew i ev Pr

4x8=

= 4 x 12

What is the relationship between the left and right expressions?

© R. I . .P ub l i ca t i ns 8C is two times greater than 4.o For both sides to be equal, if 8 is two times greater than 4, •f o rr ev i ewmust pu ptimes os es o12:nl y• the unknown be r two less than 12 ÷ 2 = 6

=9x7

m . u

What is the unknown factor in these number sentences?

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CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

What is the unknown factor in these number sentences?

x 14 = 7 x 8

. te

9 x 4 = 12 x

8x5=

=4x6

x 20

x 6 = 10 x 2

o c . 12 x 4 = (7 x 4) + (3 x 4) + ( x 4)e che r o t r What is the relationship between the left and right expressions? s su r pe

) lots of 4. The left side has 12 lots of 4, the right side has (7 + 3 + For both sides to be equal, the right side must have 12 lots of 4. As 7 + 3 = 10, +2 is needed to make 12. Therefore‚ the unknown factor is 2. What is the unknown factor in these number sentences? 8 x 7 = (4 x 7) + (3 x 7) + ( 24 x 9 = (10 x 9) + (

x 7)

x 9) + (5 x 9)

16 x 3 = (8 x 3) + (5 x 3) + ( 18 x 2 = (

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

x 3)

x 2) + (4 x 2) + (2 x 2)

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Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEET Equivalent multiplication fact number sentences – 1

8 x 6 = (1 x 6) + (7 x 6)

9 x 6 = (2 x 6) + (7 x 6)

8 x 6 = (2 x 6) + (6 x 6)

9 x 6 = (3 x 6) + (6 x 6)

8 x 6 = (3 x 6) + (5 x 6)

9 x 6 = (4 x 6) + (5 x 6)

8 x 6 = (4 x 6) + (4 x 6)

7 x 6 = (1 x 6) + (6 x 6)

6 x 6 = (1 x 6) + (5 x 6)

7 x 6 = (2 x 6) + (5 x 6)

6 x 6 = (2 x 6) + (4 x 6)

7 x 6 = (3 x 6) + (4 x 6)

6 x 6 = (3 x 6) + (3 x 6)

5 x 6 = (1 x 6) + (4 x 6)

4 x 6 = (1 x 6) + (3 x 6)

8 x 7 = (1 x 7) + (7 x 7)

9 x 7 = (2 x 7) + (7 x 7)

8 x 7 = (2 x 7) + (6 x 7)

9 x 7 = (3 x 7) + (6 x 7)

8 x 7 = (3 x 7) + (5 x 7)

9 x 7 = (4 x 7) + (5 x 7)

8 x 7 = (4 x 7) + (4 x 7)

7 x 7 = (3 x 7) + (4 x 7)

6 x 7 = (3 x 7) + (3 x 7)

5 x 7 = (1 x 7) + (4 x 7)

4 x 7 = (1 x 7) + (3 x 7)

5 x 7 = (2 x 7) + (3 x 7)

4 x 7 = (2 x 7) + (2 x 7)

3 x 7 = (1 x 7) + (2 x 7)

2 x 7 = (1 x 7) + (1 x 7)

. te

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4 x 6 = (2 x 6) + (2 x 6)

w ww

5 x 6 = (2 x 6) + (3 x 6)

o c . c e 7 x 7 = (1 x 7) + (6 x 7) 6 x 7 = (1 x 7) + (5 x 7) h r er o t s super 7 x 7 = (2 x 7) + (5 x 7) 6 x 7 = (2 x 7) + (4 x 7)

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CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

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

ew i ev Pr

Teac he r

9 x 6 = (1 x 6) + (8 x 6)

Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEET

8 x 8 = (1 x 8) + (7 x 8)

9 x 8 = (2 x 8) + (7 x 8)

8 x 8 = (2 x 8) + (6 x 8)

9 x 8 = (3 x 8) + (6 x 8)

8 x 8 = (3 x 8) + (5 x 8)

9 x 8 = (4 x 8) + (5 x 8)

8 x 8 = (4 x 8) + (4 x 8)

7 x 8 = (1 x 8) + (6 x 8)

6 x 8 = (1 x 8) + (5 x 8)

7 x 8 = (2 x 8) + (5 x 8)

6 x 8 = (2 x 8) + (4 x 8)

7 x 8 = (3 x 8) + (4 x 8)

6 x 8 = (3 x 8) + (3 x 8)

5 x 8 = (1 x 8) + (4 x 8)

4 x 8 = (1 x 8) + (3 x 8)

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

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Teac he r

9 x 8 = (1 x 8) + (8 x 8)

9 x 9 = (1 x 9) + (8 x 9)

8 x 9 = (1 x 9) + (7 x 9)

9 x 9 = (2 x 9) + (7 x 9)

8 x 9 = (2 x 9) + (6 x 9)

9 x 9 = (3 x 9) + (6 x 9)

8 x 9 = (3 x 9) + (5 x 9)

7 x 9 = (3 x 9) + (4 x 9)

6 x 9 = (3 x 9) + (3 x 9)

5 x 9 = (1 x 9) + (4 x 9)

4 x 9 = (1 x 9) + (3 x 9)

5 x 9 = (2 x 9) + (3 x 9)

4 x 9 = (2 x 9) + (2 x 9)

3 x 9 = (1 x 9) + (2 x 9)

2 x 9 = (1 x 9) + (1 x 9)

m . u

5 x 8 = (2 x 8) + (3 x 8)

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CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

Equivalent multiplication fact number sentences – 2

. te 9 x 9 = (4 x 9) + (5 x 9) 8 x 9 = (4 x 9) +o (4 x 9) c . c e 7 x 9 = (1 xh 9) + (6 x 9) 6 x 9 =r (1 x 9) + (5 x 9) er o t s super 7 x 9 = (2 x 9) + (5 x 9) 6 x 9 = (2 x 9) + (4 x 9)

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Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEET

6x2=4x3

2x6=3x4

8x2=4x4

2x8=4x4

9x2=6x3

2x9=3x6

10 x 2 = 5 x 4

2 x 10 = 4 x 5

8x3=6x4

3x8=4x6

10 x 3 = 5 x 6

3 x 10 = 6 x 5

9x4=6x6

4x9=6x6

10 x 4 = 5 x 8

4 x 10 = 8 x 5

2x5=5x2

2x7=7x2

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8x7=7x8

9x7=7x9

4x8=8x4

6x8=8x6

7x8=8x7

3x9=9x3

m . u

5x9=9x5 6x9=9x6 . te7 x 9 = 9 x 7 o 8x9=9x8 c . che e r o t 2 x (3 x 4) = 3 x (4 x r 2)s 2 x (3 x 4) = 4 x s (2 x 3) uper

210

3 x (4 x 5) = 4 x (5 x 3)

3 x (4 x 5) = 5 x (3 x 4)

4 x (5 x 6) = 5 x (6 x 4)

4 x (5 x 6) = 6 x (4 x 5)

5 x (6 x 7) = 6 x (7 x 5)

5 x (6 x 7) = 7 x (5 x 6)

6 x (7 x 8) = 7 x (8 x 6)

6 x (7 x 8) = 8 x (6 x 7)

7 x (8 x 9) = 8 x (9 x 7)

7 x (8 x 9) = 9 x (7 x 8)

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

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

ew i ev Pr

Teac he r

Equivalent multiplication fact number sentences – 3

Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEET Balanced number problems

Give students numerous opportunities to determine how balance can be restored to an unbalanced equation. Example 1

Write the illustration as a number sentence:

ew i ev Pr

Teac he r

r o e t s Bo r e p o 3 x 100 gu weight 6k x weights S 3 x 100 = 6 x

Example 2

DOLLAR

6 x 50c

. te

= 6 x 50

m . u

Write the illustration as a number sentence:

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CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

Use a weighing scale and weights of 25 g, 50 g, 100 g, 250 g, 500 g to determine the correct denomination of weight that will restore balance to the scales.

Use 5c, 10c, 20c, 50c, $1 and $2 coins to determine the correct denomination of money that will restore balance to the equation.

o c . che Example 3 r e o r st super

2 x tubes of tennis balls

x tube of tennis balls

Write the illustration as a number sentence: 2x6=

Students draw the number of tubes of tennis balls to balance the equation. Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEET Equivalent multiplication fact number word problems

Write and solve equivalent number sentences to match each question. When six is multiplied by eight it equals the product of four and which number?

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

Nine is multiplied by which number to give an answer that equals the product of 12 and six?

w ww

. te

m . u

What number must be multiplied by six to make the answer equal to the product of 15 and two?

Three times which number is equal to the product of nine and two?

By how much must six be multiplied for the product to equal that of eight and three?

When six is multiplied by itself, it gives an answer that equal how many times four?

Three and four give a product equal to twice which number?

212

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Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

The product of seven and nine is equal to how many times 21?

ew i ev Pr

Teac he r

What number when multiplied by three gives an answer that equals 12 multiplied by two?

Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEET Equivalent division fact number word problems

Write and solve equivalent number sentences to match each question.

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

What number has to be divided by 12 to equal 18 divided by three?

Ninety-six divided by which number gives an answer equal to 64 divided by eight?

. te

m . u

The quotient of 12 divided by two equals the quotient of 18 divided by which number?

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CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

When 28 is divided by seven it gives an answer equal to which number divided by six?

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Teac he r

The quotient of eight divided by four equals the quotient of 12 divided by which number?

What number has to be divided by eight to equal 36 divided by four?

Forty-nine divided by which number gives an answer equal to 35 divided by five?

The quotient of 27 divided by nine equals the quotient of 24 divided by which number?

When 30 is divided by five it gives an answer equal to which number divided by nine?

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Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEET Problems – Making tables

When information from a problem is put into a table, it is easier for the students to recognise the arithmetic relationship among expressions.

Jill

The table shows that Jack’s time is always 2 x Jill’s and Jill’s time is always Jack’s ÷ 2.

ew i ev Pr

Teac he r

Jack

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

Bill earns three times as much money for jobs he does at home than Ben does. If Bill earns $27, how much does Ben earn? If Ben earns $6, how much does Bill earn? 3

Ben

w ww

The table shows that Bill’s earnings are always 3 x Ben’s and Ben’s earnings are always Bill’s ÷ 3.

. te

m . u

Bill

o c . che For every point Brad scores in basketball, Jim always scores fie ve times more. r o If Brad scores seven r points, how many does Jim score? t s s r u e p If Jim scores 45 points, how many does Brad score? Jim

Brad

The table shows that Jim’s score is always 5 x Brad’s and Brad’s score is always Jim’s ÷ 5

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CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

Jack takes twice as long to get ready in the morning as Jill does. If Jill takes 15 minutes, how long does Jack take? If Jack takes 20 minutes, how long does Jill take?

Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEET Creating equivalent number sentences

First expression

Equivalent expression

Teac he r

ew i ev Pr

r o t = eB s r e oo p u k S = = =

. te

m . u

w ww

CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

o c . = che e r o t r s super = = =

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Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEET Creating equivalent number sentences

2 3 or eB st 5

oo k

w ww

0 x

216

. te

m . u

–

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

r e p u S

ew i ev Pr

Teac he r

Sub-strand: Patterns and Algebra—P&A – 2

RESOURCE SHEET

Teac he r

ew i ev Pr

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

. te

m . u

w ww

CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

Balancing silhouettes

o c . che e r o t r s super

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Assessment 1

Sub-strand: Patterns and Algebra—P&A – 2

NAME:

DATE:

1. Write the unknown term. (a) 12 x 5 = (5 x 5) + ( (c) (3 x 4) x 5 = 3 x (4 x

x 5) + (3 x 5) )

(b) 36 = (

x 6) + (2 x 6)

(d) (7 x 8) x 9 = 7 x (

)

2. Write a number sentence to solve each word problem. (a) Three times which number equals the product of six and five?

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

Teac he r

(b) Which number divided by two is equal to 81 divided by nine?

(a)

3x8=

(b)

72 ÷ 9 =

÷7

w ww

m . u

. te o 4. Explain how you would solve the unknown term. c . c e h r (a) (b) er o x 12 6 x 10 = 56 t s ÷7= ÷6 super

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CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

3. Write a word problem to match the number sentence.

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Assessment 2

Sub-strand: Patterns and Algebra—P&A – 2

NAME:

DATE:

The table compares the time in minutes that Jack and Jill spend doing chores at home. Jack

Jill

1. Shade a bubble to show your answer.

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

Jack spends twice as much time doing chores than Jill does.

(a)

Jack spends half as much time doing chores as Jill does. (b)

When Jack spends half an hour doing chores, Jill spends one hour.

(c)

ew i ev Pr

Teac he r

When Jill spends half an hour doing chores, Jack spends one hour. Jack’s time = 2 x Jill’s time Jill’s time = 2 x Jack’s time

(b) How much time would Jill spend working if Jack worked for 50 minutes?

Jul.

Aug. Sep. Oct. Nov. Dec.

$10

$12

$14

$16

$18

$20

$22

$24

Ben

$12

$18

$24

$30

$36

$42

$48

$54

$60

$66

$72

. te (a) Bill saves

m . u

Bill

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CONTENT DESCRIPTION: Use equivalent number sentences involving multiplication and division to find unknown quantities

2. (a) How much time would Jack spend working if Jill worked for quarter of an hour?

o c . c e her r (b) Ben saves as much money as Bill. o t s swordpproblem. er 4. Write number sentences for eachu 3. Finish the word problems.

as much money as Ben.

(a) Bill: (b) Ben: 5. (a) How much will Ben have saved if Bill saves $13? (b) How much will Bill have saved if Ben saves $75?

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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219

Checklist

Sub-strand: Patterns and Algebra—P&A – 2

Use equivalent number sentences involving multiplication and division to find

Uses partitioning to solve multiplication and division problems

Balances both sides of multiplication and division number sentences to find an unknown factor

Writes word problems to represent equivalent number sentences

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

ew i ev Pr

Teac he r

STUDENT NAME

Writes equivalent number sentences to represent word problems

unknown quantities (ACMNA121)

w ww

. te

220

m . u

o c . che e r o t r s super

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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Answers

Sub-strand: Patterns and Algebra

Page 219

P&A – 1 Page 197

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

1, 5, 9, 13, 17, 21 5, 8, 11, 14, 17, 20 23, 19, 15, 11, 7, 3 The numbers decrease by 0.7 each time. The difference between numbers increase by one each time. The numbers increase by 1⁄8 each time.

2. 3. 4.

3. Number of matchsticks per side

Number of matchsticks in perimeter

Teac he r 1.

2. 3.

Three is added each time.

The perimeter number is three times the side number.

Assessment 2

(a) B (b) G (c) E (d) C Teacher check Teacher check

P&A – 2 Page 218 1.

(a) 4 (c) 5

(a) 3 x (b)

=6x5

÷ 2 = 81 ÷ 9

(a) 3 x 8 = 12 x 2; The product of three and eight equals the product of which number and two?

w ww

ew i ev Pr

Page 198

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

Description of pattern General rule

(a) Jack spends twice as much time doing chores than Jill does. (b) When Jill spends half an hour doing chores, Jack spends one hour. (c) Jack’s time = 2 x Jill’s time (a) Jack = 2 x 15 = 30 minutes (b) Jill = 50⁄2 = 25 minutes (a) one-third (b) three times (a) Bill = 1⁄3 Ben (b) Ben = 3 x Bill (a) $39 (b) $25

m . u

Assessment 1

Assessment 2

(b) 72 ÷ 9 = 56 ÷ 7; Seventy-two divided by nine equals which number divided by seven? 4.

. te

o c . che e r o t r s super

(a) 6 x 10 = 5 x 12; Because 6 x 10 = 60, the unknown must be whatever has to be multiplied by 12 to get 60.

(b) 56 ÷ 7 48 ÷ 6; Because 56 ÷ 7 = 8, the unknown must be whatever gives the answer eight when it is divided by six. Fifty six is 8 x 7 so the unknown must be 8 x 6.

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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221

NEW WAVE NUMBER AND ALGEBRA (YEAR 5) STUDENT WORKBOOK ANSWERS Page 8

N&PV – 1 Page 2 1. 2.

Handball target

Answers will vary. Teacher check

Complete the factor asteroids

Answers will vary. Teacher check

N&PV – 2 6

✘

✓

✘

Number

Rounded off to the nearest ten

Number

Rounded off to the nearest ten

✘

746

750

774

770

✘

✓

✘

789

790

268

270

✘

✓

✘

178

180

✘

727

730

672

670

✘

641

640

687

The number 1904 is a multiple of

✘

✓

✘

779

780

277

The number 4408 is a multiple of

✘

✓

✘

107

110

272

The number 3199 is a multiple of

✘

✓

✘

122

120

891

581

580

774

678

680

659

The number 1467 is a multiple of The number 9382 is a multiple of The number 1005 is a multiple of The number 3856 is a multiple of The number 9385 is a multiple of The number 7705 is a multiple of

Page 3

The laws of dividing – try 6

Divisible by 6 – 1032, 498, 564, 642, 834, 426

Page 4

The laws of dividing – this time try 7

Round ‘em up and round ‘em off

896

900

996

1233

1230

6573

6677

6680

7104

ew i ev Pr

Teac he r

The number 2296 is a multiple of

r o e t s Bo r e p ok u S Page 9

690

280

270 890 770

660

1000

6570

Divisible by 7 – 476, 413, 644, 406

Page 5

The laws of dividing – try 8

7100

Number

567

600

7714

7700

Divisible by 9 – none

6634

6600

2655

2700

Page 7

7802

7800

9223

9200

5693

5700

6872

6900

5822

5800

4457

4500

5801

5800

1277

1300

Page 6

The laws of dividing – try it with 9

The mystery of multiples Counting by multiples of ...

48 42

w ww

Missing number is ...

. te

63 36 48

9 6 4 5

4456

4500

2712

2700

2978

3000

8491

8500

1109

1100

7748

7700

1288

1300

6590

6600

o c . che e r o t r s super 3451

3500

8755

8800

4458

4500

6903

6900

1129

1100

4464

4500

Page 10

Pete the plumber rounds off

Jobs done

Actual cost

Rounded to nearest $10

Rounded to nearest $100

Jan #1

$1234

$1230

$1200

125

Jan #2

$1566

$1570

$1600

148

Jan #3

$2238

$2240

$2200

150

Jan #4

$1849

$1850

$1800

128

Jan #5

$2902

$2900

112

Jan #6

$1229

$1230

$1200

162

Jan #7

$1185

$1190

$1200

Total

$12 203

$12 210

$12 100

I am – 72, 56, 70, 56, 72, 42

222

Rounded off to the nearest hundred

m . u

Rounded off to the nearest hundred

Number

Divisible by 8 – 1032, 1376, 834

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

Jobs done

Actual cost

Rounded to nearest $10

Rounded to nearest $100

Feb #1

$2295

$2300

Feb #2

$2817

$2820

$2800

Feb #3

$2093

$2090

$2100

Feb #4

$2473

$2470

$2500

Feb #5

$2109

$2110

$2100

Feb #6

$2248

$2250

$2200

Feb #7

$2871

$2870

$2900

Total

$16 906

$16 910

Mar #1

$2119

$2120

Mar #2

$2321

Mar #3

$3067 $3107

Mar #5

$3902

Mar #6

$3188

Mar #7

$3029

Total

$20 733

3. 4.

The wide, big land for me

(a) Area – square km

Rounded off to the nearest million

Rounded off to the nearest hundred thousand

Rounded off to the nearest ten thousand

$16 900

2 526 786

3 000 000

2 500 000

2 530 000

$2100

QLD

1 723 936

2 000 000

1 700 000

1 720 000

$2320

$2300

1 335 742

1 000 000

1 300 000

1 340 000

$3070

$3100

978 810

1 000 000

980 000

$3110

$3100

NSW

800 628

1 000 000

800 000

$3900

VIC

227 010

200 000

230 000

$3190

$3200

TAS

64 519

100 000

60 000

$3030

$3000

ACT

2356

$20 740

$20 700

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

860 380 450 930 300 200 900 900 4900 30 340

(b) Teacher check Teacher check

Percentage Percentage % of % of State/ rounded off State/ rounded off Australia’s Australia’s to nearest Territory to nearest Territory area area 5% 5% WA 33 35% NSW 10.04 10% QLD 22.5 25% VIC 3.0 5% NT 17.5 20% TAS 0.9 0% SA 12.7 15% ACT 0.1 0%

The business of rounding off

(a) 40 (b) (e) 30 (f ) (i) 490 (j) (m) 330 (n) (a) 600 (b) (e) 200 (f ) (i) 300 (j) (m) 400 (n) (q) 2900 (r) (a) 70 (b) (e) 60 (f ) Teacher check

50 590 320 560 700 500 300 500 4700 90

(d) (h) (l) (p) (d) (h) (l) (p) (t) (d)

390 80 740 840 500 100 800 900 5700 110

Page 12

w ww

Lamb BBQ chops Whole milk

Pizza base 30 cm Pizza base 30 cm Pizza base 30 cm

10.52

. te 2.99

3.79 3.79 3.79

6.08

Bananas

3.06

Green apples

6.35

Raisins

1. 2. 3. 4. 5. 6. 7.

A super docket

Buddy’s Supa Market!

Sandwich ham

Page 14

11.29

Rounded to nearest $

Rounded to nearest 50c

$11

$10.50

My Australian holiday

No, his 3 weeks will cost $525. He can spend only $20 per day. Yes, she is correct and she could spend $26 a day on lunches. No, his 3 weeks of breakfast and lunch will cost him $39 a day. Yes, she has exactly the right amount of money to do this. Yes, he is correct and could afford $33 a day. Yes, she is correct and could afford $30 a day. Yes, she is correct and could afford $29 a day.

N&PV – 3 Page 15

m . u

Page 13

State/ Territory

Answers will vary. Teacher check Page 11

$80.05 $82 $80 Teacher check

ew i ev Pr

Teac he r

Mar #4

1. 2. 3.

Use the lattice method 5

o c . che e r o t r s super $4

$6.50

$11

$11.50

BBQ sausages

3.94

Tinned tuna

2.73

$2.50

Tinned tuna

2.73

$2.50

4 +1

2.73

$2.50

Tinned tuna

2.73

$2.50

Olive oil spray

3.99

Iceberg lettuce

1.75

Lamingtons

3.67

$3.50

Muesli

4.12

3 +1

0 5

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

3 3

1 +1

7 2 +1

R.I.C. Publications®

5 +1

6 2

2 2

2 0

8 2

6 1

2 +2

Tinned tuna

6 1 +1

3 5 4 3

8 6

1 2

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223

If and then

Page 19 Problem

Then

6 x 3 = 18

60 x 3 = 180

5 x 6 = 30

50 x 60 = 3000

4 x 9 = 36

40 x 9 = 360

7 x 7 = 49

70 x 70 = 4900

7 x 5 = 35

70 x 5 = 350

2 x 6 = 12

20 x 60 = 1200

6 x 8 = 48

60 x 8 = 480

9 x 9 = 81

90 x 90 = 8100

3 x 7 = 21

30 x 7 = 210

3 x 6 = 18

30 x 60 = 1800

9 x 8 = 72

90 x 8 = 720

4 x 7 = 28

40 x 70 = 2800

2 x 6 = 12

20 x 6 = 120

8 x 6 = 48

80 x 60 = 4800

5 x 5 = 25

50 x 5 = 250

5 x 9 = 45

50 x 90 = 4500

3 x 4 = 12

30 x 4 = 120

4 x 6 = 24

40 x 60 = 2400

5 x 3 = 15

50 x 3 = 150

6 x 7 = 42

60 x 72 = 4200

$159. Teacher check $95. Teacher check

Page 17

63 x 37

5 2

2 0 0

57 x 24

48 x 32

Page 20

2 4

1. 2.

0 3.

8 2

6 4

1 1

4 6

1. 2. 3. 4. 5.

4 +2 2

Page 18 Problem 33 x 18 27 x 19 32 x 24 45 x 23 62 x 34 59 x 16 37 x 25 49 x 27

224

w ww

2 +1

5 +1

. te

Use the area model – 1

Split out (30 + 3) x (10 + 8) (20 + 7) x (10 + 9) (30 + 2) x (20 + 4) (40 + 5) x (20 + 3) (60 + 2) x (30 + 4) (50 + 9) x (10 + 6) (30 + 7) x (20 + 5) (40 + 9) x (20 + 7)

300 + 90 + 60 + 18 = 468 800 + 200 + 80 + 20 = 1100 1500 + 450 + 60 + 18 = 2028 1800 + 420 + 90 + 21 = 2331 1000 + 200 + 140 + 28 = 1368 1200 + 80 + 240 + 16 = 1536 1000 + 400 + 120 + 48 = 1568

5 x 9 = 45, 91 7 x 4 = 28, 73 5 x 7 = 35, 73r4 4 x 3 = 12, 32r2 5 x 7 = 35, 73r4 4 x 8 = 32, 87r2 5 x 3 = 15, 33r1 4 x 10 = 40, 101r3

Multiply out (30 x 10) + (30 x 8) + (3 x 10) + (3 x 8) (20 x 10) + (20 x 9) + (7 x 10) + (7 x 9) (30 x 20) + (30 x 4) + (2 x 20) + (2 x 4) (40 x 20) + (40 x 3) + (5 x 20) + (5 x 3) (60 x 30) + (60 x 4) + (2 x 30) + (2 x 4) (50 x 10) + (50 x 6) + (9 x 10) + (9 x 6) (30 x 20) + (30 x 5) + (7 x 20) + (7 x 5) (40 x 20) + (40 x 7) + (9 x 20) + (9 x 7)

4 6

1 2

1. 2. 3. 4.

Soccer squads

6 teams, 3 sitting out 5 teams, 4 sitting out 8 teams, 6 sitting out 7 teams, 2 sitting out 8 teams, 5 sitting out

Page 22 4 +1

Make the connection

11 x 4 = 44, 112 8 x 3 = 24, 86 4 x 6 = 24, 64r1 3 x 8 = 24, 85r2 4 x 3 = 12, 36 3 x 7 = 21, 76r1 4 x 9 = 36, 92r3 3 x 7 = 21, 73

Page 21

600 + 90 + 160 + 24 = 874

6 x 1 = 6, 121 6 x 3 = 18, 33

6 x 7 = 42, 72r5

5 x 6 = 30, 51r4

Partial products

N&PV – 4

Multiply out (30 x 20) + (30 x 3) + (8 x 20) + (8 x 3) (30 x 10) + (30 x 3) + (6 x 10) + (6 x 3) (40 x 20) + (40 x 5) + (4 x 20) + (4 x 5) (50 x 30) + (50 x 9) + (2 x 30) + (2 x 9) (60 x 30) + (60 x 7) + (3 x 30) + (3 x 7) (50 x 20) + (50 x 4) + (7 x 20) + (7 x 4) (40 x 30) + (40 x 2) + (8 x 30) + (8 x 2) (50 x 20) + (50 x 8) + (6 x 20) + (6 x 8)

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

52 x 39

Split out (30 + 8) x (20 + 3) (30 + 6) x (10 + 3) (40 + 4) x (20 + 5) (50 + 2) x (30 + 9) (60 + 3) x (30 + 7) (50 + 7) x (20 + 4) (40 + 8) x (30 + 2) (50 + 6) x (20 + 8)

ew i ev Pr

44 x 25

Lattice method activity 2

36 x 13

56 x 28

Teac he r

1. 2.

38 x 23

Use the area model – 2

6. 4 teams, 1 sitting out 7. 5 teams, 0 sitting out 8. 8 teams, 0 sitting out 9. 3 teams, 4 sitting out 10. 10 teams, 4 sitting out 11. 11 teams, 2 sitting out

Make the connection again!

4 x 1= 4, 195 3 x 1 = 3, 149r2 4 x 2 = 8, 202r2 3 x 2 = 6, 287r1 4 x 8 = 32, 85r2 3 x 1 = 3, 110r2 4 x 4 = 16, 43 3 x 1 = 3, 194r1 4 x 1 = 4, 167 3 x 2 = 6, 294r2

5 x 1 = 5, 187r3 4 x 2 = 8, 29 5 x 3 =15, 39r4 4 x 5 =20, 56 5 x 1 = 5, 103r1 4 x 7 = 28, 73r1 5 x 1 = 5, 110r2 4 x 1 = 4, 105r1 5 x 1 = 5, 169r2 4 x 2 = 8, 209r1

o c . che e r o t r s super Partial products

300 + 240 + 30 + 24 = 594

200 + 180 + 70 + 63 = 513

Page 23

6 x 3 = 18, 38r1

m . u

Page 16

6 x 5 = 30, 57r5 6 x 1 = 6, 118r1 6 x 1 = 6, 101r3 6 x 3 = 18, 38r1

Not all golfers get a group!

600 + 120 + 40 + 8 = 768

Date

Number of golfers

Number of groups (4)

Leftovers

800 + 120 + 100 + 15 = 1035

Jan 1

97 ÷ 4 = 24 groups

1 player

Jan 2

Jan 3

Jan 4

Jan 5

109

Jan 6

123

Jan 7

Jan 8

Jan 9

102

Jan 10

114

Jan 11

106

Jan 12

133

1800 + 240 + 60 + 8 = 2108 500 + 300 + 90 + 54 = 944 600 + 150 + 140 + 35 = 925 800 + 280 + 180 + 63 = 1323

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

Jan 13

Page 29

Jan 14

Jan 15

105

1. 58 2. 83 Teacher check

Jan 16

108

Page 30

Jan 17

116

6525 ÷ 25 = 261

8375 ÷ 25 = 335

3725 ÷ 25 = 149

9175 ÷ 25 = 367

2325 ÷ 25 = 93

5025 ÷ 25 = 201

219 x 25 = 5475

526 x 25 = 13 150

377 x 25 = 9425

619 x 25 = 15 475

785 x 25 = 19 625

971 x 25 = 24 275

Jan 18

Jan 19

111

Jan 20

139

Jan 21

145

Jan 22

Jan 23

122

Jan 24

Jan 25

118

Jan 26

37 chairs – 0 left over 18 tables – 0 left over 9 computers – 2 left over 32 lockers – 1 left over 22 calculators – 2 left over

Page 25

The relationship between 25 and 100

Page 31

Use the algorithm to solve!

r o e t s Bo r e p ok u S 19

639 x 7 = 4473

577 x 6 = 3462

483 x 4 = 1932

298 x 9 = 2682

Page 32

378 x 46 = 17 388

619 x 27 = 16 713

634 x 75 = 47 550

291 x 69 = 20 079

824 x 47 = 38 728

6. 7. 8. 9.

337 x 52 = 17 524

48 pens – 2 left over 44 cushions – 1 left over 69 scrap books – 2 left over 189 sheets of cover paper – 0 left over

A large number by a 2-digit number

F&D – 1 Page 33 1. 2.

You ate how much pizza?

357 x 8 = 2856

ew i ev Pr

1. 2. 3. 4. 5.

884 x 9 = 7956

Setting up a new school

Teac he r

Page 24

Some really good guesses

Zac – 2 pizzas, 1 piece left over

A fraction of a long jump

Teacher check Abbey 1.8 m Eddie 0.6 m Indy 3 m

Bianca 0.8 m Frankie 4 m Jessie 1.6 m

Yuri – 2 pizzas, 6 pieces left over Xavier – 1 pizza, 5 pieces left over

Page 34

Udom – 4 pizzas, 2 pieces left over

Charli 3.6 m Gina 1.2 m Ky 3.2 m

Donna 1.6 m Harry 3.6 m Liam 4.2 m

Which is bigger?

Tina – 4 pizzas, 4 pieces left over

Ronnie – 4 pizzas, 0 pieces left over

Quentin – 3 pizzas, 6 pieces left over Sean – 5 pizzas, 3 pieces left over

Page 35

Peta – 4 pizzas, 6 pieces left over

1. 2. 3. 4. 5.

N&PV – 5

w ww

Page 26

Multiplication using tables

Teacher check Page 27

256 ÷ 8 = 32

739 ÷ 15 = 49r4 532 ÷ 14 = 38 792 ÷ 12 = 66

. te

(b) 3⁄24 or 1⁄8 (b) 4⁄24 or 1⁄6 (b) pink (b) brown by 9 squares (b) blue by 6 squares

My favourite fraction flavours

o c . che e r o t r s super

1344 ÷ 6 = 224

765 ÷ 12 = 63

1062 ÷ 18 = 59 1360 ÷ 16 = 85

936 ÷ 24 = 39

928 ÷ 16 = 58

122 ÷ 16 = 7r10

1065 ÷ 15 = 71

702 ÷ 18 = 39

1416 ÷ 24 = 59

924 ÷ 28 = 33

1836 ÷ 36 = 51

1648 ÷ 24 = 68r16

1392 ÷ 48 = 29

644 ÷ 28 = 23

1044 ÷ 36 = 29

960 ÷ 24 = 40

444 ÷ 12 = 37

1. 2. 3. 4. 5.

No pineapple or raspberry 12⁄36 or 1⁄3 1⁄6 either pineapple or orange 12⁄36 or 1⁄3 = 12 cans 9 cans = 9⁄24 or ¼

Page 37

608 ÷32 = 19

Teacher check

1584 ÷ 24 = 66

Page 38

I’m marking the spot

Colour the shapes

Teacher check

Page 39

Multiplication using factors

57 x 15 = 855

49 x 12 = 588

73 x 24 = 1752

38 x 14 = 532

51 x 18 = 918

113 x 12 = 1356

67 x 24 = 1608

121 x 18 = 2178

83 x 24 = 1992

123 x 21 = 2583

127 x 32 = 4064 97 x 12 = 1164

Teacher check Teacher check Teacher check Teacher check Teacher check

Page 36

Division using factors

896 ÷ 8 = 112

Page 28

(a) (a) (a) (a) (a)

Colour in these fractions

m . u

Vivian – 5 pizzas, 5 pieces left over

The adding of fractions

Teacher check

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

add

1½

2¾

3¾

9¾

2⁄4

3¼

4½

1¼

4¼

10¼

2¼

3½

4¾

1½

4½

10½

1¾

4¼

R.I.C. Publications®

www.ricpublications.com.au

225

Page 44 add

1½

2¾

3¾

9¾

1¼

2¾

5¼

22⁄4

5¼

6½

3¼

6¼

12¼

3¾

5¼

6½

7¾

4½

7½

13½

Order – heavy to light

Wombat weight – kg

Wang

2.984

Wayne

2.663

Wade

2.629

Walter

2.628

25⁄12

1⁄12

7⁄12

9⁄12

42⁄12

Warren

2.608

5⁄6

33⁄12

11⁄12

35⁄6

15⁄12

17⁄12

Wally

2.419

7⁄12

8⁄12

37⁄12

12⁄12

14⁄12

49⁄12

3⁄12

28⁄12

4⁄12

33⁄12

10⁄12

47⁄12

Wendy

2.319

Equal 9

Wynne

2.138

Equal 9

Wozza

2.138

25⁄12

1½

10⁄12

33⁄12

24⁄12

310⁄12

11⁄12

33⁄12

27⁄12

4½

41⁄12

57⁄12

Page 41 8⁄3 19⁄6 24⁄5 12⁄7 16⁄6 1¾ 14⁄3 11⁄5

(a)

42⁄12

Willy

2.114

15⁄12

47⁄12

Wanda

2.109

18⁄12

410⁄12

53⁄12

Wynette

2.027

32⁄12

64⁄12

69⁄12

Walt

2.012

Page 45 1.

School swimming records

Lane 1 – 8th Lane 2 – 5th Lane 3 – 6th Lane 4 – 3rd (a) 2.01 sec (b) 0.05 sec (c) 0.03 sec Lane 1 – 8th Lane 2 – 4th Lane 3 – 7th Lane 4 – 5th (a) 2.33 sec (b) 0.04 sec (c) 1 second

Lane 5 – 1st Lane 6 – 7th Lane 7 – 2nd Lane 8 – 4th

Lane 5 – 1st Lane 6 – 3rd Lane 7 – 6th Lane 8 – 2nd

Page 46

Find that fraction

Teacher check

Page 43

39⁄12

24⁄7 51⁄3 1¾ 32⁄5 52⁄3 31⁄7 4¼ 6½

w ww

1.–2.

7⁄12

Improper and mixed fractions 9. 10. 11. 12. 13. 14. 15. 16.

Page 42

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

Mixed and improper fractions 9. 10. 11. 12. 13. 14. 15. 16.

2.407

. te

Adding and subtracting in colour (b)

⁄12

(c)

(d) 6

⁄12

(e) 7

(g)

⁄10

⁄12

Runner’s name

School

Annie

Southern PS

41.11

Zac

Eastern PS

41.27

Donna

Eastern PS

41.52

Emma

Western PS

41.59

Sienna

Western PS

42.23

(h) ⁄10

⁄12

Bridget

Southern PS

42.34

Angus

Western PS

42.41

⁄12

Peumike

Eastern PS

42.59

Aaron

Western PS

43.13

Jacqui

Eastern PS

43.28

Dylan

Western PS

43.47

Jeff

Western PS

43.56

Ronny

Southern PS

44.03

Amber

Western PS

44.11

Suzie

Western PS

44.39

22⁄10

226

Time minutes and seconds

Position

o c . che e r o t r s super 7

(f )

District cross-country

m . u

21⁄3 15⁄6 13⁄5 15⁄7 24⁄6 2¼ 31⁄3 23⁄5

Wallis

ew i ev Pr

Page 40

Teac he r

add

27⁄12

1. 2. 3. 4. 5. 6. 7. 8.

Wombat name

add

1. 2. 3. 4. 5. 6. 7. 8.

Wombat weigh-in

Carl

Eastern PS

44.51

Caleb

Western PS

45.12

Benjamin

Southern PS

45.28

Ryan

Eastern PS

45.47

Emily

Eastern PS

46.09

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

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

Page 47

Roll me a decimal

Page 52

Teacher check

Page 48

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

Look alike but not

0.024, 0.042, 0.044, 0.22, 0.23, 0.44 0.0038, 0.3, 0.33, 0.35, 0.38, 3.8 0.007, 0.066, 0.067, 0.607, 0.67, 0.7 0.05, 0.059, 0.095, 0.59, 0.9, 0.95 0.049, 0.094, 0.4, 0.49, 0.9, 0.94 1.3, 0.13, 0.03, 0.013, 0.01, 0.003 0.5, 0.45, 0.05, 0.045, 0.04, 0.005 0.76, 0.7, 0.67, 0.067, 0.06, 0.0067 0.9, 0.59, 0.095, 0.059, 0.05, 0.009

Page 49

1960

8.24 m

1961

3 cm

8.28 m

1961

4 cm

8.31 m

1962

3 cm

Ralph Boston

8.31 m

1964

Equalled

Ralph Boston

8.34 m

1964

3 cm

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

8.35 m

1965

1 cm

Igor Ter-Ovanesyan

8.35 m

1967

Equalled

Bob Beaman

8.90 m

1968

55 cm

Mike Powell

8.95 m

1991

5 cm

88.5 92.5

2FBI

94.5

2. 3. 4.

Smooth

95.3

Page 53

The Edge

96.1

Rain gauge – teacher check

Nova

96.9

Triple H

100.1

BFM

100.9

WOW FM

101.7

WSFM

101.9

Hope

103.2

2DAY

104.1

Triple M

104.9

Triple J

105.7

Mix

106.5

2SER

107.3

Sydney – June Sydney – March Sydney – April Sydney – May Sydney – February Sydney – January Melbourne – January or July Melbourne – July or January Melbourne – February Melbourne– June Melbourne – August Melbourne – March

Page 54 1. 2.

Radio dial – teacher check

Decimal apples

w ww

1.0, 1.01, 1.11, 10.00, 10.01, 10.1, 10.11, 11.0, 11.01, 11.11, 11.12, 12.0, 12.01, 12.1, 12.11, 12.12, 20.0, 20.01, 20.1, 20.11, 20.12, 21.0, 21.01, 21.1, 21.11, 21.12, 22.0, 22.01, 22.1, 22.11, Largest – 22.11, 22.1, 22.01 Smallest – 1.0, 1.01, 1.11

Page 51

Which is the wetter city?

Page 50

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

0.74 m 74 cm Teacher check

ew i ev Pr

Teac he r

2RRR

SBS Radio

Increase (m)

Ralph Boston

2. 3.

8.21 m

Ralph Boston

Frequency

2. 3.

Year

Station

Distance

Igor Ter-Ovanesyan

FM radio stations

Rank

Athlete

. te

Adelaide 92 hours, Brisbane 89.4 hours, Melbourne 71.9 hours Melbourne June

3.6

July

3.7

May

3.9

Aug.

4.7

Brisbane Mar.

6⁄10 1⁄100 774⁄1000 76⁄100 376⁄1000 984⁄1000 0.49 0.1 0.37 0.039 0.097 0.345

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

5.6

June

4.7

Feb.

6.6

July

June

6.8

May

5.6

May

6.9

Aug.

6.1

Sept.

7.1

Apr.

7.2

Sept.

5.7

Jan.

7.4

Apr.

7.3

Oct.

6.3

July

7.4

Oct.

8.4

Mar.

6.8

Aug.

7.9

Mar.

8.7

Nov.

Oct.

Nov.

9.3

Dec.

7.5

Dec.

8.1

Dec.

9.3

Feb.

8.1

Sept.

8.2

Feb.

Jan.

Nov.

8.4

Jan.

10.5

Total

71.9

Total

89.4

Total

Most sunshine: Adelaide Jan. 10.5, Adelaide Feb. 10, Adelaide Nov. 9.3, Adelaide Dec. 9.3, Melbourne Jan. 9 Least sunshine: Melbourne June 3.6, Melbourne July 3.7, Melbourne May 3.9, Melbourne Aug. 4.7, Adelaide June 4.7 Adelaide for summer, autumn and spring. Brisbane for winter.

Page 55 1. 5. 9. 13.

Adelaide

6.5

o c . che e r o t r s super Apr.

Decimals and fractions

(a) 25⁄100 (b) 9⁄100 (d) 7⁄1000 (e) 443⁄1000 (g) 22⁄1000 (h) 17⁄100 (j) 559⁄1000 (k) 88⁄100 (m) 34⁄100 (n) 53⁄100 (p) 16⁄100 (q) 8⁄10 (c), (e), (b), (d), (e), (b), (a) (a) 0.7 (b) 0.11 (d) 0.008 (e) 0.54 (g) 0.22 (h) 0.019 (j) 0.231 (k) 0.597 (m) 0.3 (n) 0.817 (p) 0.185 (q) 0.9

Here comes the sun!

m . u

Long jump world records

0.7 0.13 0.016 0.08

Decimals and decimal fractions 2. 6. 10. 14.

0.31 0.9 0.199 0.059

R.I.C. Publications®

3. 7. 11. 15.

0.008 0.101 0.234 0.1

www.ricpublications.com.au

4. 8. 12. 16.

0.121 0.88 0.44 0.18

227

Page 60

M&FM – 1 1. 2. 3. 4. 5. 6. 7. 8.

Showbags galore

2 x Candy bag, 0 x Giant chip, 2 x Big chocolate,1 x Dentist bag 1 x Candy bag, 1 x Giant chip, 1 x Big chocolate, 1 x Dentist bag 2 x Candy bag, 4 x Giant chip, 2 x Big chocolate, 0 x Dentist bag 0 x Candy bag, 0 x Giant chip, 3 x Big chocolate, 3 x Dentist bag 2 x Candy bag, 2 x Giant chip, 2 x Big chocolate, 2 x Dentist bag 0 x Candy bag, 2 x Giant chip, 2 x Big chocolate, 2 x Dentist bag 1 x Candy bag, 1 x Giant chip, 1 x Big chocolate, 2 x Dentist bag 3 x Candy bag, 3 x Giant chip, 3 x Big chocolate, 1 x Dentist bag

Page 57

Leaf busting spray

$37

Ryan Roof Clearing

$41

Garvey’s Gutters

$27

Gutters R Us

$36

Guts n Gutters

$28

Page 58 1.

Teac he r

That’s a Clean Roof

Total

$165

$181.50

$204

$224.40

$169

$185.90

$200

$220

$178

$195.80

$209

$229.90

$28.71

$7.74

$85.14

000919

$5.31

$58.39

000900

$2.30

$25.25

000901

$2.78

$30.53

000902

$2.19

$24.04

000903

$1.96

$21.56

000904

$2.22

$24.37

000905

$2.67

$29.32

000906

$2.83

$31.08

Page 61

$3.20 $2.90 $3.80 $2.70 $4 $3.60 $4 $3.80 $4.90

GST (10% of rounded price) $0.32 $0.29 $0.38 $0.27 $0.40 $0.36 $0.40 $0.38 $0.49

Product Rounded up/ price down

$140.80

$107.80

$2.45

$2.40/$2.50

$0.24/$0.25

$110

$11

$121

$6.11 $7.14 $6.86 $5.66 $7.03 $6.99 $8.11 $7.33 $8.21

$6.10 $7.10 $6.90 $5.70 $7 $7 $8.10 $7.30 $8.20

$0.61 $0.71 $0.69 $0.57 $0.70 $0.70 $0.81 $0.73 $0.82

$5.78

$5.80

$0.58

Just Swimming

$147

$14.70

$161.70

GST equation Product price + GST $3.18 + $0.32 $2.93 + $0.29 $3.77 + $0.38 $2.66 + $0.27 $4.04 + $0.40 $3.61 + $0.36 $4.00 + $0.40 $3.83 + $0.38 $4.91 + $0.49 $2.45 + $0.24 $2.45 + $0.25 $6.11 + $0.61 $7.14 + $0.71 $6.86 + $0.69 $5.66 + $0.57 $7.03 + $0.70 $6.99 + $0.70 $8.11 + $0.81 $7.33 + $0.73 $8.21 + $0.82

$3.50 $3.22/$3.20 $4.15 $2.93/$2.95 $4.44/$4.45 $3.97/$3.95 $4.40 $4.21/$4.20 $5.40 $2.68/$2.69/ $2.70 $6.72/$6.70 $7.85 $7.55 $6.23/$6.25 $7.73/$7.75 $7.69/$7.70 $8.92/$8.90 $8.06/$8.05 $9.03/$9.05

$5.78 + $0.58

$6.36/$6.40

You pay (inc GST)

w ww

Too Blue to Be True Summer Daze Cleaning No Fuss Poolworks Stand Up Pool Cleaning Krystal Klear Just Swimming

GST and the sun! + GST $0.98 $1.23

. te

Total cost

Change

$10.78

$9.20

$13.53

o c . che e r o t r s super $6.45

Page 62

$7.25

$11.60

$1.16

$10.40

$1.04

$19.50

$1.95

$21.45

$17.90

$1.79

$19.69

$18.20

$1.82

$20.02

$19.10

$1.91

$21.01

$11.95

$1.20

$13.15

$6.85

$7.70

$0.77

$8.47

$11.55

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

$12.10

$1.21

$13.31

$6.70

Page 63

$11.45

$1.15

$12.60

$7.40

1. 3. 5. 7.

228

Coffee anyone?

$9.80

$124.30

$12.30

$2.61

000918

$12.80

$11.30

$9.80

000917

$98

$113

Cost

$15.76

$128

Total

Page 59

$25.96

$1.43

$10.70

+ GST

No Fuss Poolworks

$2.36

000916

$107

Subtotal

Too Blue to Be True

000915

$3.18 $2.93 $3.77 $2.66 $4.04 $3.61 $4.00 $3.83 $4.91

Stand Up Pool Cleaning

Krystal Klear

Cost to customer

Call the poolman – but which one?

Summer Daze Cleaning

Total payable

ew i ev Pr

$25

GST included

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

The cost of guttering

Up the spout!

Order number

$12.76 $11.44

m . u

Page 56

Adding GST

$8.55

$28.55 $30.30 $30 $29

On the weekend

3 colas, 6 pies, 2 waters 6 colas, 4 pies, 1 water 4 colas, 4 pies, 3 waters 3 colas, 3 pies, 5 waters 4 colas, 4 pies, 4 waters 0 colas, 6 pies, 6 waters 4 colas, 5 pies, 3 waters 3 colas, 3 pies, 6 waters 6 colas, 3 pies, 3 waters 6 colas, 6 pies, 0 waters

Pocket money nightmare

17 weeks 19 weeks 11 weeks 13 weeks

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

2. 4. 6. 8.

17 weeks 15 weeks 14 weeks 17 weeks

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

P&A – 1 Page 64

1.2

Follow the pattern

3.5

5.8

Rule

Start at

Next 10 numbers

9, 15, 21, 27, 33, 39, 45, 51, 57, 63

–3

101

98, 95, 92, 89, 86, 83, 80, 77, 74, 71

+5, –1

21, 20, 25, 24, 29, 28, 33, 32, 37, 36

27, 31, 35, 39, 43, 47, 51, 55, 59, 63

–6

144

138, 132 ,126, 120, 114, 108, 102, 96, 90, 84

–3, +7

9, 16, 13, 20, 17, 24, 21, 28, 25, 32

+12

18, 30, 42, 54, 66, 78, 90, 102, 114, 126

13.8

17.2

–9, +2

114

105, 107, 98, 100, 91, 93, 84, 86, 77, 79

46.2

19, 27, 35, 43, 51, 59, 67, 75, 83, 91

61.3

+10, –2

210

220, 218, 228, 226, 236, 234, 244, 242, 252, 250

+3, –5

125

128, 123, 126, 121, 124, 119, 122, 117, 120, 115

–6, +3

117

111, 114, 108, 111, 105, 108, 102, 105, 99, 102

+12, –6

40, 34, 46, 40, 52, 46, 58, 52, 64, 58

+9, –7

43, 36, 45, 38, 47, 40, 49, 42, 51, 44

–4, +8

8, 16, 12, 20, 16, 24, 20, 28, 24, 32

10.4

12.7

13.8

16.1

18.4

20.7

25.3

27.6

46.2

48.5

50.8

53.1

55.4

57.7

61.3

63.6

65.9

68.2

70.5

72.8

75.1

1.2

4.6

11.4

14.8

18.2

21.6

Pattern is +3.4

r o e t s Bo r e p ok u S 20.6

27.4

30.8

34.2

49.6

56.4

59.8

63.2

66.6

64.7

68.1

71.5

74.9

78.3

81.7

Page 66

Number sequences to solve

Pattern

The next 8 in the sequence

+6, –2

+9, –3

–12, +14

ew i ev Pr

Teac he r

8.1 Pattern is +2.3

11 21.5 8.75 17.5 14.5

10.5 24.5 8.5 19.7 12.5

Pattern

Description of rule/pattern

+5, +7

11, 12, 15, 20, 27, 36

+1, +3, +5, +7, +9

–3, +5

39, 44, 43, 48, 47, 52, 51

+5, –1

17, 24, 29, 36, 41, 48

+7, +5

65, 55, 60, 50, 55, 45, 50

-10, +5

66, 68, 64, 66, 62, 64, 60

+2, –4

88, 85, 91, 88, 94, 91, 97

-3, +6

112, 121, 130, 139, 148

22, 30, 26, 34, 30, 38, 34

+8, –4

33, 40, 34, 41, 35

+7, –6

67, 76, 85, 84, 93, 102, 101

+9, +9, –1

+11, –3

14, 21, 18, 25, 22, 29

+7, –3

+5, +10

133, 144, 154, 165, 175

+11, +10

39, 44, 55, 60, 71, 76, 87

+5, +11

2. Pattern +1.5, –0.5 8 +1.5, +3 8 6.5 +0.75, –0.25 7.25 +1.1, +2.2 7.6 +3.5, –2 10

7.5 11 7 9.8 8

The next 8 in the sequence 9 8.5 10 9.5 12.5 15.5 17 20 7.75 7.5 8.25 8 10.9 13.1 14.2 16.4 11.5 9.5 13 11

1.2

13.8

Pattern

Identify and continue

2.4 15

. te 3.6

16.2

7.2

–3, +15

Pattern is +1.2 17.4

18.6

19.8

49.8

52.2

53.4

61.3

62.5

63.7

64.9

66.1

67.3

68.5

3.0

Pattern

8.4

48.6

2.1

o c . che e r o t r s super

4.8

47.4

1.2

–5, +9 +6, +9

46.2

The next 8 in the sequence

m . u

w ww

Page 65

3.9

4.8

5.7

6.6

Pattern is +0.9 13.8

14.7

15.6

16.5

17.4

18.3

19.2

46.2

47.1

48.9

49.8

50.7

51.6

61.3

62.2

63.1

64.9

65.8

66.7

1.2

2.7

4.2

5.7

7.2

8.7

10.2

Pattern is +1.5 13.8

15.3

16.8

18.3

19.8

21.3

22.8

46.2

47.7

49.2

50.7

52.2

53.7

55.2

61.3

62.8

64.3

65.8

67.3

68.8

70.3

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

+9, –11

–11, +15

+8, +2

+13, –6

Page 67 1. 2.

The next 8 in the sequence

+7, –8

Make your own pattern

Teacher check +2, +1, +3 –6, +10, +11 –3, +5, –4 +3, –5, +4 +3, +4, +5 –14, +6, +3 +11, +10, +9 +5, +5, +10 +6, +7, +8 –9, +3, –10 –4, +10, –5

R.I.C. Publications®

www.ricpublications.com.au

229

Page 68

Identify and continue – 2

⁄8

12⁄8

1. 1.4

2.8

4.2

5.6

8.4

9.9

11.3

12.7

14.1

15.5

12.3

13.7

15.1

16.5

17.9

19.3

20.7

18.9

20.3

21.7

23.1

24.5

25.9

27.3

31.3

30.4

29.5

28.6

15 ⁄8

16 ⁄8

175⁄8

182⁄8

187⁄8

26 ⁄8

27 ⁄8

28 ⁄8

29 ⁄8

301⁄8

435⁄8

442⁄8

447⁄8

454⁄8

461⁄8

466⁄8

473⁄8

⁄10

14⁄10

18⁄10

22⁄10

26⁄10

3 136⁄10

16.1

15.2

77.5

76.6

75.7

74.8

56.1

55.2

54.3

53.4

12.3

15.5

18.7

21.9

Pattern is + ⁄10 4

27.7

112⁄10

116⁄10

124⁄10

128⁄10

132⁄10

23 ⁄10

24 ⁄10

25 ⁄10

673⁄10

677⁄10

681⁄10

685⁄10

689⁄10

693⁄10

697⁄10

61⁄2

73⁄4

r o e t s Bo r e p ok u S 14.3

13.4

12.5

73.9

72.1

52.5

51.6

50.7

25.1

28.3

31.5

Pattern is –0.9 17.9

43⁄8

15 ⁄8

32.2

36⁄8

Pattern is + ⁄8

2. 33.1

31⁄8

Pattern is +1.4 8.5

24⁄8 5

9.8

7.1

17⁄8

Page 70

Fiddle with fractions – 2

⁄4

11⁄2

23⁄4

Pattern is +3.2

44.8 87.1 92.6 4. 12.9

51.2

54.4

57.6

60.8

90.3

93.5

96.7

99.9

103.1

106.3

95.8

102.2

105.4

108.6

111.8

12.2

11.5

10.8

10.1

9.4

10 ⁄4

11 ⁄2

123⁄4

161⁄2

173⁄4

201⁄4

27 ⁄4

30 ⁄4

31 ⁄2

8.7

21⁄3

22⁄3

31⁄3

7 ⁄3

192⁄3

201⁄3

202⁄3

661⁄3

662⁄3

671⁄3

⁄3

32⁄3

51⁄3

48.3

47.6

46.9

46.2

45.5

81.8

81.1

80.4

79.7

78.3

101.2

100.5

99.8

99.1

98.4

97.7

15.5

26.5

37.5

48.8

151⁄4

161⁄2

211⁄2

223⁄4

32 ⁄4

351⁄4

32⁄3

41⁄3

81⁄3

82⁄3

211⁄3

212⁄3

672⁄3

681⁄3

82⁄3

101⁄3

82.5

51⁄4

Pattern is +11⁄4

Pattern is –0.7

49.7

ew i ev Pr

Teac he r

Pattern is +1⁄3

Pattern is +5.5

Pattern is +1 ⁄3 2

69.3

74.8

80.3

85.8

91.3

96.8

122⁄3

141⁄3

172⁄3

191⁄3

1.3

6.8

12.3

17.8

23.3

28.8

34.3

28 ⁄3

30 ⁄3

33 ⁄3

35 ⁄3

382⁄3

112.7

118.2

123.7

129.2

134.7

140.2

145.7

551⁄3

582⁄3

601⁄3

632⁄3

651⁄3

25⁄8

34⁄8

43⁄8

52⁄8

61⁄8

77⁄8

14 ⁄4

182⁄8

191⁄8

306⁄8

315⁄8

324⁄8

767⁄8

776⁄8

56⁄10

63⁄10

Page 69 ⁄4

11⁄2

21⁄4

. te 3

133⁄4

19 ⁄2

20 ⁄4

343⁄4

351⁄2

361⁄4

Pattern is +7⁄8 33⁄4

41⁄2

51⁄4

15 ⁄8

16 ⁄8

173⁄8

271⁄4

281⁄8

297⁄8

721⁄2

733⁄8

742⁄8

751⁄8

28⁄10

31⁄2

42⁄10

49⁄10

o c . che e r o t r s super

Pattern is + ⁄4

121⁄4

Fiddle with fractions – 1

m . u

w ww

63.8

141⁄2

151⁄4

21 ⁄4

22 ⁄2

23 ⁄4

373⁄4

381⁄2

391⁄4

163⁄4

Pattern is + ⁄10 7

21⁄2

51⁄2

81⁄2

Pattern is +1 ⁄2 1

111⁄2

191⁄2

221⁄2

251⁄2

28 ⁄4

29 ⁄4

31 ⁄4

32 ⁄4

34 ⁄4

35 ⁄4

37 ⁄4

553⁄4

571⁄4

583⁄4

601⁄4

613⁄4

631⁄4

643⁄4

177⁄10

184⁄10

191⁄10

198⁄10

205⁄10

212⁄10

34 ⁄10

35 ⁄10

36 ⁄10

37 ⁄10

386⁄10

517⁄10

524⁄10

531⁄10

538⁄10

545⁄10

552⁄10

559⁄10

161⁄2 1

11⁄3

22⁄3

31⁄3

42⁄3

Pattern is +2⁄3 111⁄3

122⁄3

131⁄3

142⁄3

151⁄3

232⁄3

241⁄3

252⁄3

261⁄3

272⁄3

48 ⁄3

49 ⁄3

50 ⁄3

51 ⁄3

230

P&A – 2 Page 71

⁄3

3. 2

1. 2. 5. 8. 11.

How did you do that?

12 36 15 40 22

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

3. 6. 9. 12.

16 24 27 10

4. 6 7. 3 10. 26

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

Page 72 1.

Making equations

(a) (c) (e) (g) (i) (k)

8+5–4=9 9÷3+2=5 9 ÷ 3 + 8 = 11 8÷4+5=7 15 ÷ 5 + 12 = 15 6 x 5 – 9 = 21 or 6 x 5 – 21 = 9 (m) 18 ÷ 6 + 2 = 5

Page 73

(b) (d) (f ) (h) (j) (l)

Right column

48 62 30 64 176 96 28 72 121 32 79 84

Page 75

Teac he r 4.

28 72

The order of operations – BIMDAS (b) (f ) (j) (b) (f ) (j) (b) (f ) (j) (b) (f ) (j)

61 86 102 35 99 42 22 52 108 34 62 83

(d) 87 (h) 58

(d) 24 (h) 72

(d) 56 (h) 32

(d) 61 (h) 47

(a) (e) (i) (a) (e) (i) (a) (e) (i) (a) (e) (i)

6 9 7 6 3 4 5 23 56 7 6 3

Well-known tables sentence trees (b) (f ) (j) (b) (f ) (j) (b) (f ) (j) (b) (f ) (j)

5 10 4 2, 5 7 10 9 4 72 31, 2 27 80

(d) 1 (h) 3

(d) 3 (h) 11

(d) 6 (h) 6

(d) 11 (h) 4

. te

m . u

w ww

Sean works three times as often as Sally. Sally works a quarter as often as Simon. Simon works twice as often as Sienna. Sienna works twice as often as Sally. Sally: 12 Sienna: 24 Sean: 36 Sally: 15 Sienna: 30 Sean: 45 Judy earns a quarter as much money as Jenni. Jaxon earns eight times as much money as Jai. Jai earns half as much money as Jenni. Jenni earns four times as much money as Judy. Judy: $6.50 Jai: $13 Jenni: $26 Judy: $7.50 Jai: $15 Jenni: $30

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ew i ev Pr

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(a) (e) (i) (a) (e) (i) (a) (e) (i) (a) (e) (i)

Right column

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

Left column

More equivalent number sentences

Left column

Match the questions to the numbers in the middle

15 ÷ 5 + 7 = 10 5+6–4=7 9 x 3 + 3 = 30 2 x 6 + 5 = 17 4 x 3 + 9 = 21 7 x 7 + 8 = 57

(n) 9 x 4 + 5 = 41

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o c . che e r o t r s super

Solving problems with tables – 2

Rupert finds three times as many 20 cent coins as Riley. Remy finds three times as many 20 cent coins as Rachel. Rachel finds double as many 20 cent coins as Riley. Riley finds a sixth as many 20 cent coins as Remy. Riley: 13 Rachel: 26 Rupert: 39 Riley: 15 Rachel: 30 Rupert: 45 Annie eats twice as many pizzas as Audrey. Alan eats two thirds as many pizzas as Audrey. Angus eats five times as many pizzas as Alan. Audrey eats half as many pizzas as Annie. Alan: 12 Annie: 36 Angus: 60 Alan: 17 Annie: 51 Angus: 85

Australian Curriculum Mathematics resource book: Number and Algebra (Year 5)

R.I.C. Publications®

www.ricpublications.com.au

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