Oxford mathematics 5 - Copy

Student Book PY P O x ford Ma thema tics Pr imar y Year s Programme Br ia n Mur r a y

1 Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trademark of Oxford University Press in the UK and in certain other countries. Published in Australia by Oxford University Press Level 8, 737 Bourke Street, Docklands, Victoria 3008, Australia. © Oxford University Press 2019 The moral rights of the author have been asserted First published 2019 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence, or under terms agreed with the reprographics rights organisation. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above. You must not circulate this work in any other form and you must impose this same condition on any acquirer. ISBN 978 0 19 031224 4 Edited by Philip Bryan Illustrated by Barbara Bakos Typeset by Newgen KnowledgeWorks Pvt. Ltd., Chennai, India Proofread by Rebecca Hill Printed in China by Leo Paper Products Ltd Acknowledgements Cover: Getty/Walker and Walker. Internal: Shutterstock.

To the teacher Ox ford Mathemat ics PY P prov ides st udent s w it h g u ided a nd i ndependent work to suppor t mat hemat ica l sk i l ls a nd u nder st a nd i ngs, a s wel l a s oppor t u n it ies for problem - solv i ng i n rea l-world contex t s. Teacher s w i l l f i nd t he suppor t i ng mater ia ls clea r, comprehen sive a nd ea sy to u se. W h i le t he ser ies of fer s complete coverage of t he PY P mat hemat ics scope a nd sequence, teacher s ca n a lso u se t he topics t hat f it wel l w it h ot her a rea s of work to suppor t st udent lea r n i ng across t he PY P c u r r ic u lu m. Student Books Each topic feat u res: • Gu ided prac t ice – a worked exa mple of t he concept, fol lowed by t he oppor t u n it y for st udent s to pract ise, suppor ted by ca ref u l sca f fold i ng • Independent prac t ice – f u r t her oppor t u n it ies for st udent s to con sol idate t hei r u nder st a nd i ng of t he concept i n d i f ferent ways, w it h a decrea si ng a mou nt of sca f fold i ng • E x tended prac t ice – t he oppor t u n it y for st udent s to apply t hei r lea r n i ng a nd ex tend t hei r u nder st a nd i ng i n new contex t s. Differentiation D i f ferent iat ion is key to en su r i ng t hat ever y st udent ca n access t he c u r r ic u lu m at t hei r poi nt of need. In add it ion to t he g radu a l relea se approach of t he St udent Book s, t he Teacher Book s help teacher s to choose appropr iate pat hways for st udent s, a nd prov ide act iv it ies for st udent s who requ i re ex t ra suppor t or ex ten sion.

O x ford Ma thema tics Pr imar y Year s Pro gramme 5 C ontents N U M B E R , PAT T E R N A N D F U NCT ION M E A SU R E M E N T, S H A P E A N D S PACE Unit 1 Number and place value Unit 5 Using units of measurement 1. Place value 2 6 10 1. Length and perimeter 72 14 76 2. Addition mental strategies 18 80 22 84 26 2. A rea 88 32 3. Addition written strategies 36 40 3. Volume and capacity 4. Subtraction mental strategies 4. Mass 5. Subtraction written strategies 5. Time 6. Multiplication mental strategies Unit 6 Shape 7. Multiplication written strategies 1. 2D shapes 92 96 8. Factors and multiples 2. 3D shapes 9. Divisibility 10. Division written strategies Unit 7 Geometric reasoning 1. A ngles 10 0 Unit 2 Fractions and decimals 1. Comparing and ordering fractions 44 48 52 Unit 8 Location and transformation 56 2. Adding and subtracting fractions 1. Transformations 104 10 8 3. Decimal fractions 112 116 2. Symmetr y 12 0 4. Percentages 3. Enlargements and reductions 4. Grid references Unit 3 Money and nancial mathematics 5. Giving directions 1. Financial plans 60 Unit 4 Patterns and algebra DATA H A N DL I NG 1. Number patterns 64 68 2. Number operations and properties Unit 9 Data representation and interpretation 1. Collecting and representing data 124 2. Representing and interpreting data 12 8 Unit 10 Chance 1. Chance 13 2 13 6 2. Chance experiments Glossar y 14 0 Answers 150

UNIT 1: TOPIC 1 Place value 5 6 In a number, the value of each digit depends 9 2 8 on its position, or place. 923856 is easier to read if we write it as 923 856. It also makes it easier to say the number: , Guided practice 1 Look at this number: 725 38 4. The 7 is wor th 70 0 0 0 0. Show the value of the other digits on the place value grid. senO Write the number, using gaps if necessar y sneT s d erdnuH sdnasuohT sdnasuoht neT sdnasuoht d erdnuH e.g. 7 0 0 0 0 0 700 000 a b Remember to c use a zero as d e a space-filler. 2 If we write thir t y-two thousand, ve hundred and nine in numerals, we use a zero to show there are no tens: 32 509 Write as digits: a nine thousand, three hundred and seven b twent y-ve thousand and for t y- six c one hundred and two thousand, seven hundred and one 3 Write in words: a 28 6 0 b 13 4 65 c 28 705 2 OX FOR D U N I V E RSI T Y PR E S S

Independent practice 1 What is the value of the red digit in each number? e.g. 8 5 306: 80 000 c 2 9 425: a 5 3 207: d 135 28 4: b 4 8 0 05: e 39 9 517: 2 Write each number from question 1 in words. e.g. 8 5 306: eight y-ve thousand, three hundred and six a b c d e 3 Write these numbers as numerals. a eight y- six thousand, two hundred and thir t y- one b one hundred and for t y-two thousand c six hundred and ft y- six thousand, three hundred and eight d one hundred and ve thousand, nine hundred and twent y- one 4 Circle the number that is one more than 25 78 9. 25 800 25 780 25 799 25 790 OX FOR D U N I V E RSI T Y PR E S S 3

5 Expand these numbers. The rst one has been done for you. 14 217: 10 000 + 4000 + 200 + 10 + 7 Remember to use spaces bet ween the digits where necessar y. a 25 123: 20 0 0 0 + b 6 3 382: c 6 0 0 4: d 125 381: e 8 6 0 0 9 4: 6 Use the digits on the cards to make: 6 1 5 3 9 7 a the largest number using all the cards. b the smallest number if “5” is in the ones place. c the largest number if the “7” is in the hundreds of thousands place. d the smallest number if the “1” is in the thousands place. 7 Write the number shown on each spike abacus as numerals and in words. a b Hth Tth Th H T O Hth Tth Th H T O numeral: numeral: words: words: 4 OX FOR D U N I V E RSI T Y PR E S S

Extended practice 1 This table shows unusual record - breaking activities. Plac e Ac tivit y Number USA Number of dogs on a dog walk together Spain People salsa dancing together Poland People ringing bells together Hong Kong People playing percussion instruments together Singapore People line dancing together Por tugal People making a human adver tising sign Mexico People doing aerobics at the same time India Trees planted by a group in one day USA People in a conga line England The longest scar f ever knitted (in centimetres) Complete the number column in the table by rewriting the numbers below in order, from the lowest to the highest number. The events are in order from low to high. Record numbers 80 241 10 021 119 986 38 633 322 0 0 0 3117 34 309 3868 11 967 10 102 2 The following numbers are from the list in question 1. They have been rounded in various ways. Write the actual number for each. a 80 000 f 10 0 0 0 b 40 000 g 10 0 0 0 0 c 3000 h 12 0 0 0 d 300 000 i 4000 j 35 000 e 10 10 0 3 Rounded to the nearest ten thousand, the 20 0 6 population of Noosa in Queensland was 50 0 0 0 people. The actual number can be made by using each of these digits once: 1 2 5 6 9 List as many of the 12 numbers that could be the actual population as you can. OX FOR D U N I V E RSI T Y PR E S S 5

UNIT 1: TOPIC 2 Addition mental strategies For $100: What is 250 + 252? Imagine you were on a T V quiz show and had 4 seconds to answer the question. There are several strategies you could use to come up with the right answer. However, in only 4 seconds you would probably have to use a mental strategy. Guided practice 1 You could use the near- doubles strategy for 252 + 250: Double 250 is 50 0. Then add 2 = 502. Fill in the gaps. Problem Find a near- double Now I need to: Answer 250 + 250 = 500 add 2 more 502 e.g. 252 + 250 add 10 more a 150 + 16 0 150 + 150 = b 126 + 126 125 + c 14 0 0 + 14 50 2 You could split the numbers. For example, 250 + 252 is the same as: 20 0 + 50 + 2 + 20 0 + 50. Fill in the gaps. Problem Expand the numbers Join the par tners Answer e.g. 252 + 250 200 + + 2 + 200 + 200 + 200 + + + 2 = 500 + 2 502 a 66 + 34 60 + 6 + 30 + 4 60 + 30 + 6 + 4 = 9 0 + 10 b 14 0 + 230 10 0 + 4 0 + 20 0 + 30 10 0 + 20 0 + 4 0 + 30 = 30 0 + 70 c 1250 + 23 47 3 You could use the jump strategy on an empt y number line: e.g. What is 50 + 52? a What is 105 + 8 4? + 80 +4 + 50 +2 10 5 50 10 0 10 2 Answer: 50 + 52 = 102 Answer: 105 + 8 4 = b What is 1158 + 130? c What is 2424 + 505? + 10 0 + + + 115 8 2424 Answer: 1158 + 130 = Answer: 2424 + 505 = 6 OX FOR D U N I V E RSI T Y PR E S S

Independent practice 1 Another mental strategy for adding is the compensation strategy. It uses rounding For 74 + 19, we can round 19 to 20 and say 74 + 20. Use the compensation strategy tosolve these. Problem Using rounding it Now I need to: Answer b e c ome s: 93 e.g. 74 + 19 74 + 20 = 94 take away 1 56 + 40 = 96 add 1 a 5 6 + 41 25 + 70 = 9 5 take away 1 125 + 6 0 = 18 5 add b 25 + 6 9 c 125 + 62 d 136 + 19 8 136 + e 19 5 + 24 9 f 1238 + 501 g 16 4 5 + 19 9 8 2 Use the compensation strategy to solve these. a 35 + 99 b 24 + 101 c 173 + 19 8 d 14 07 + 10 02 e 1451 + 14 9 9 f 1562 + 10 0 4 3 Use the jump strategy to solve these. a 125 + 38 = b 16 4 + 47 = c 1193 + 8 42 = d 2585 + 1321 = OX FOR D U N I V E RSI T Y PR E S S 7

4 Practise the split strategy with these addition problems. Problem Expand the numbers Join the par tners Answer 257 e.g. 125 + 132 10 0 + + 5 + 10 0 + +2 10 0 + 10 0 + + +5+2 a 173 + 125 b 124 0 + 2130 c 5125 + 123 4 d 7114 + 2365 e 25 6 4 + 4236 5 Use your choice of strategy to nd the answer. Be ready to explain the strategy you used. a 713 + 19 0 = b 14 9 0 + 14 9 0 = c 20 0 9 + 20 0 9 + 20 0 9 = d 18 6 4 + 313 4 = e 24 9 9 + 10 02 = f 1236 + 247 = g 24 9 9 + 24 9 9 = h 3130 + 236 0 = 8 OX FOR D U N I V E RSI T Y PR E S S

Extended practice Improving your estimating and rounding skills can help you save time with mental calculations. 1 Look at these facts and gures. Show how you would round the numbers by underlining or highlighting one of the numbers. World fac t Metres Rounded number a Krubera: the deepest cave in the world 2191 m 210 0 or 220 0? b c Cehi: the tenth - deepest cave in the world 1502 m 150 0 or 16 0 0? d e Mont Blanc: the highest mountain in Europe 4 8 07 m 4 8 0 0 or 4 9 0 0? f g Mont Maudit: the tenth - highest mountain in Europe 4466 m 4 4 0 0 or 4 50 0? h Mt Everest: the highest mountain in the world 8850 m 8 8 0 0 or 8 9 0 0? Mt Kosciusko: the highest mountain in Australia 2228 m 220 0 or 230 0? Mammoth Cave: the longest cave in the world. 590 600 m 50 0 0 0 0 or 6 0 0 0 0 0? Wind Cave: the four th - longest cave in the world 212 500 m 20 0 0 0 0 or 30 0 0 0 0? 2 Circle the number that will make the information correct. a The total of the depths of Krubera and Cehi caves is about 3500 m, 3700 m, 3600 m, 3400 m b Mont Blanc is about 20 m, 200 m, 30 m, 300 m taller than Mont Maudit. c If you walked the lengths of the Mammoth Cave and the Wind Cave you would have travelled about 700 km, 70 km, 80 km, 800 km 3 Sarah goes shopping in a bargain shop. She has $11 to spend. She goes to the checkout with these items: Paint set: $1.9 9 Ball: 9 9c Calculator: $1.9 9 Cuddly toy: $1.9 9 Pen set: $1.25 Notebook: 4 9c Geometr y set: $1.9 9 Stickers: $1.29 a To the nearest dollar, how much more than $11 is the total? b Which item should Sarah put back to be closest to a total of $11? OX FOR D U N I V E RSI T Y PR E S S 9

UNIT 1: TOPIC 3 Addition written strategies T O T O 8 3 4 One of the most common written strategies for addition is 1 to set the numbers out ver tically. You star t with the ones and 3 add each column in turn. + 2 5 Sometimes you need to trade from one column to the next. + 2 5 5 9 6 3 Guided practice 1 Complete the following. T O H T O H T O Th H T O 2 6 1 3 3 3 7 5 3 6 4 1 + 2 3 + 1 4 1 + 1 2 3 + 1 2 2 5 2 Complete the following. T O H T O H T O H T O You need 7 8 9 to trade 1 5 with these. 1 1 1 1 6 6 8 2 3 + 2 9 + 1 5 6 + 2 8 6 + 3 Star t with the ones and add each column in turn. H T O Th H T O a b c 2 4 9 3 2 4 6 3 2 2 8 6 + 1 3 7 + 1 3 7 7 + 1 5 5 3 7 Tth Th H T O Hth Tth Th H T O d e 4 2 7 4 2 4 3 4 5 3 6 + 3 2 3 7 8 + 2 6 5 5 9 5 10 OX FOR D U N I V E RSI T Y PR E S S

Independent practice 1 Look for a pattern in the answers for each row. a 8 5 b 5 3 8 c 7 0 6 6 + 3 8 + 6 9 6 + 5 2 7 9 d 8 7 2 3 9 e 6 2 f 1 5 8 9 + 3 6 2 1 7 + 5 9 + 7 4 3 g 1 5 0 7 8 h 2 4 8 9 3 6 i 7 2 + 1 9 4 6 5 + 2 0 7 7 1 8 + 3 9 j 9 2 4 k 1 8 6 5 1 l 1 8 6 1 2 8 + 1 2 9 8 + 1 4 6 8 2 + 2 5 8 3 1 6 2 Look for linking numbers to save time in written addition. 2 5 a b c d e 2 7 2 1 4 1 8 4 4 7 5 5 9 3 1 4 Link 2 2 1 3 1 2 3 5 1 0 1 2 1 8 Link 2 5 2 3 1 9 6 2 2 6 1 3 5 8 9 8 + 2 6 + 1 8 + 2 7 9 + 1 7 0 + 6 0 9 + 5 9 8 9 0 3 On a holiday, Jack spent $29 5 on food, $207 on travel, $ 9 85 for his hotel, $ 92 on presents and $213 on enter tainment. He wanted to know how much he had spent and used a calculator and found that the total was $1612. a If you round the numbers, is Jack ’s answer reasonable? b How much did Jack spend altogether? OX FOR D U N I V E RSI T Y PR E S S 11

T O T O When you write an addition problem ver tically, it is 4 5 1 5 impor tant to keep the digits in the correct columns. + 3 4 If you don’t, you will get the wrong answer. 7 + 3 7 4 8 7 8 2 4 Rewrite these problems ver tically, then solve them. a 114 + 137 b 927 + 138 c 739 + 278 H T O H T O Th H T O + + + d 173 + 33 + 38 e 55 4 + 5 37 + 4 9 f 6 37 + 77 + 829 H T O Th H T O Th H T O + + + g 1452 + 257 + 2318 h 35 174 + 257 + 2318 + 624 Th H T O Tth Th H T O + + i 61 28 6 + 4 35 + 24 + 325 j 579 + 4529 + 33 + 65 8 9 + 527 Tth Th H T O Tth Th H T O + + 12 OX FOR D U N I V E RSI T Y PR E S S

Extended practice 1 Find four different solutions to make this addition correct. a b c d + + 6 + 6 + 6 6 3 3 3 3 2 A football team can have more than Game Possible number 20 0 0 0 0 spectators at their home 1 games in a season. 2 Here is some information about one 3 famous football team. 4 5 • Number of home games: 12. 6 • Total number of spectators: 7 • 212 052. 8 • Average attendance per home 9 • game: 17 671. 10 Ever y game had more than 11 10 0 0 0 spectators. 12 No games had exactly the same Total number of spectators. List the possible number of spectators for each game. Make sure the total is 212 052. Use the grid to help you keep the numbers in columns. 3 Find the total of 30 521 + 85 365 + 7570 and you will see that the digits in the answer make a pattern. Make three other three - line addition problems with the same answer. Working- out spac e OX FOR D U N I V E RSI T Y PR E S S 13

UNIT 1: TOPIC 4 Subtraction mental strategies Round numbers are easier to work with. Can you work out the answer to 76 – 19 in your head? We could say 76 – instead of 76 – 19. 76 – = 56. We took away 1 too many, so we add 1 back to the answer. So, 76 – 19 = 57 Guided practice 1 Use the compensation strategy (rounding) to solve these. Fill in the gaps. Problem Using rounding, it Now I need to: Answer b e c ome s: e.g. 76 – 19 76 – 20 = 56 add 1 back a 5 3 – 21 5 3 – 20 = 3 3 take away 1 more b c 85 – 28 85 – 30 = 55 add 2 back d e 167 – 22 167 – 20 = 147 take away more f g 14 6 – 19 8 346 – 1787 – 39 0 5 8 4 0 – 310 0 6178 – 39 9 5 Splitting numbers can make subtraction easier. For example, 479 – 135 = ? • Split (expand) the number you are taking away: 135 becomes + and • • First take away : 479 – = 379 • • Next take away : 379 – = 349 Then take away : 349 – = 344 So, 479 – 135 = 3 4 4 2 Use the split strategy. Fill in the gaps. Problem Expand the number Take away the Take away Take away Answer 1st par t the 2nd par t the 3rd par t e.g. 479 – 1 5 1 5 = 10 0 + +5 479 – 10 0 = 379 379 – = 349 349 – 5 = 344 344 a 257 – 126 126 = 10 0 + 20 + 6 257 – 10 0 = b 5 4 8 – 224 224 = c 765 – 4 42 d 878 – 236 e 9 9 9 – 753 14 OX FOR D U N I V E RSI T Y PR E S S

Independent practice 1 Use the compensation strategy to solve these — or nd your own sensible shor t cut. a 47 – 22 b 18 4 – 29 c 5 47 – 231 d 2455 – 1219 e 56 67 – 2421 2 Use the split strategy to solve these — or nd another shor t cut. a 45 – 24 b 464 – 343 c 676 – 25 4 d 5727 – 3325 e 8958 – 5635 3 The split strategy can be used on an open number line. Fill in the gaps. e.g. What is 900 – 350? a What is 776 – 423? –3 – 20 – 400 – 50 – 300 776 550 600 900 Answer: 900 – 350 = 550 Answer: 776 – 423 = b What is 4 87 – 26 4? c What is 1659 – 5 36? – 200 487 16 5 9 Answer: 4 87 – 26 4 = Answer: 1659 – 5 36 = OX FOR D U N I V E RSI T Y PR E S S 15

$3.75 up to $3.80 is 5c Another strategy for subtraction $3.80 up to $4 The difference is to count up. is another 20c between $ 3.75 and $5 is $1.25. Tina buys a sandwich for $ 3.75. $4 up to $5 That is another She gives a $5 note. To work out is $1 way of saying the change, the shopkeeper star ts $5 – $ 3.75 = $1.25. at $ 3.75 and counts up to $5. The change is 5c + 20c + $1 = $1.25 4 Use the counting-up strategy to nd the change if you paid for each item with a $10. a a toy at $7.50 b a book at $ 8.75 c a melon at $ 3.50 d a calculator at $ 4.45 e a game at $5.35 f a pencil set at $2.15 You can also use the counting - up strategy to nd the difference between ordinar y numbers. For example, what is the difference between 20 0 and 155? • 155 up to 16 0 is • 16 0 up to 20 0 is Altogether I counted up 45, so the difference between 20 0 and 155 is 45. 5 Use the counting - up strategy to work out the difference between these numbers. a 10 0 – 57 = b 150 – 128 = c 20 0 – 135 = d 151 – 118 = e 10 05 – 8 9 0 = f 250 0 – 239 0 = 6 Use a mental strategy of your choice to nd the answers to these problems. Be ready to explain the strategies you use. a 8 9 – 19 = b 65 – 14 = c 78 – 21 = d 150 – 75 = e 1515 – 1220 = f 20 0 0 – 1450 = 16 OX FOR D U N I V E RSI T Y PR E S S

Extended practice 1 A football game star ts at 1:30 pm and ends at 3:05 pm. How long does it last? 2 The difference between two 3 - digit numbers is 57. What might the numbers be? 3 Iva receives $2.45 change after paying with a note. Which banknote might have been used and how much was spent? 4 What is 4235 – 397? Explain how you got the answer. 5 Bob, Bill and Ben buy the same model of car from different dealers. Bob pays $74 6 4 for his car. Bill pays $193 more than Bob, but Bill pays $19 3 less than Ben. How much do Bill and Ben pay for their cars? 6 Fill in the gaps to show three more ways to make the subtractions correct. e.g. 6 1 3 – 5 3 5 = 7 8 a 6 3 – 5 = 7 8 b 6 3 – 5 = 7 8 c 6 3 – 5 = 7 8 OX FOR D U N I V E RSI T Y PR E S S 17

UNIT 1: TOPIC 5 Subtraction written strategies Some written subtractions involve trading. Here is a When you write the reminder of how it works, using MAB and small numbers, algorithm, you trade such as 5 4 – 25. in the same way. 4 There aren’t enough ones. Take away Trade a ten 5 ones. Take away for 10 ones. Trade a ten. 2 tens. That leaves 4 tens. T O 4 1 Now there are 10 + 4 ones = 14 – 2 5 3 7 There are s till 5 4 Star t with 5 4 . (4 tens and 14 ones). That leaves 4 9. The answer (You c annot take away5on e s .) 2 9 is 29. Guided practice 1 You could use MAB to help with the trading as you complete these algorithms. a b c d T O H T O H T O H T O 3 1 7 3 7 2 5 – 2 4 – 1 2 7 – 2 3 5 – 3 1 8 e f g h Th H T O Th H T O Th H T O Th H T O 7 2 7 3 4 3 6 1 5 2 5 3 6 7 7 1 – 1 1 4 7 – 1 2 6 7 – 3 7 4 7 – 2 7 7 3 i j k Tth Th H T O Tth Th H T O Tth Th H T O 8 3 4 1 9 4 3 7 2 4 7 0 7 3 5 – 6 1 2 3 2 – 2 5 4 6 5 – 3 7 4 8 8 Subtraction l with larger Hth Tth Th H T O 8 1 3 5 1 8 – 2 4 5 8 7 9 18 OX FOR D U N I V E RSI T Y PR E S S

Independent practice 1 Practise trading with subtraction. Look for patterns in the answers. a b c d e H T O H T O H T O H T O H T O 4 1 0 5 0 8 8 1 2 8 7 2 9 5 3 – 8 9 – 7 6 – 2 6 9 – 2 1 8 – 1 8 8 2 Once you know how to subtract with trading, it doesn’t matter how large the numbers are. Complete the following. There is a pattern in the answers. a b c d Th H T O Th H T O Th H T O Th H T O 1 8 2 1 3 7 1 4 5 6 4 3 6 1 5 5 – 5 8 7 – 1 3 6 9 – 2 1 8 7 – 1 5 8 8 e f h Th H T O Th H T O Th H T O g Th H T O 8 3 2 6 8 4 1 3 9 7 5 3 9 9 3 4 – 2 6 4 8 – 1 6 2 4 – 9 8 8 – 5 8 3 There is also a pattern in the answers to these 5 - digit subtractions. a b c Tth Th H T O Tth Th H T O Tth Th H T O 1 3 4 6 5 2 6 0 8 1 3 8 9 8 1 – 2 3 5 4 – 3 8 5 9 – 5 6 4 8 d e f Tth Th H T O Tth Th H T O Tth Th H T O 6 6 1 3 3 7 7 2 4 1 9 1 2 3 5 – 2 1 6 8 9 – 2 1 6 8 6 – 2 4 5 6 9 4 Use the digits 1, 3, 2, 6, 4 and 7. Make the largest number using all the digits and the smallest number using all the digits. Find the difference between the two numbers. OX FOR D U N I V E RSI T Y PR E S S 19

5 Rounding and estimating can help 6 One algorithm in each pair is wrong. you avoid making careless mistakes. Estimate the answer, then circle Imagine you subtract 18 9 from 913 the correct algorithm. and get an answer of 824. If you round and estimate you know the a answer is wrong. 9 0 0 – 20 0 = 70 0, so the answer must be around 70 0. 6 1 2 6 1 2 Write an algorithm and nd the exact answer. – 4 8 8 OR – 4 8 8 1 2 4 2 2 4 b 9 1 5 2 9 1 5 2 – 2 9 5 8 OR – 2 9 5 8 Working- out spac e 7 1 9 4 6 1 9 4 c 1 4 2 0 5 1 4 2 0 5 – 6 9 4 7 OR – 6 9 4 7 7 2 5 8 8 2 5 8 Sometimes when you trade, there is nothing in the next column. Here’s what to do: More ones are needed … Trade a hundred. Trade a ten. … but there are no tens That leaves 2 hundreds. That leaves 9 tens. 2 1 2 9 1 1 5 – 1 4 7 – – … so trade FROM the Now there are 10 tens. Now there are 15 ones. hundreds TO the tens r s t . 7 Practise trading across two columns with these subtractions. a b c d e H T O H T O H T O H T O H T O 6 0 2 4 0 6 9 0 3 – 1 3 4 – 2 4 8 – 1 7 7 – 2 5 8 – 5 3 4 f g h Th H T O Tth Th H T O Hth Tth Th H T O 3 4 0 7 2 6 0 5 9 5 3 0 7 7 2 – 2 5 8 9 – 1 2 3 8 2 – 1 4 4 8 4 6 20 OX FOR D U N I V E RSI T Y PR E S S

Extended practice 1 Follow the rules to write three subtraction algorithms. Each algorithm must: • be 5 digits take away • 5 digits • be different from the other two have the answer 9 9 9. 2 This table shows the size of the crowd at some spor ting events around the world. Use the information to answer the questions. Spor t Size of crowd Year Plac e Working- out spac e Gaelic Football 90 556 19 61 Dublin, Ireland Hurling 84 865 19 5 4 Dublin, Ireland Australian football 121 6 9 6 1970 Melbourne, Australia Rugby Union 10 9 874 20 0 0 Sydney, Australia NFL 102 36 8 19 57 Los Angeles, USA a What is the difference between the biggest and smallest crowds in the table? b By how many was the American Football (NFL) crowd bigger than the Gaelic Football crowd? c What is the difference between the total of the two Irish games and the total of the two Australian games? d Use rounding strategies to circle the correct response. The difference between the size of the crowds at the Rugby Union and Hurling games was about: 22 000 23 000 24 000 25 000 3 The world’s smallest dog is a Yorkshire terrier. It is only 76 mm from the ground to its shoulder. The tallest dog is a great dane, which measures 105 4 mm high from the ground to its shoulder. If they were side by side, what would 21 be the difference in their heights? OX FOR D U N I V E RSI T Y PR E S S

UNIT 1: TOPIC 6 Multiplication mental strategies 5 50 7 12 120 Multiplying by ten is 1.3 m 13 m easy— but you don’t just add a zero. The digits 70 move one place bigger. 3 × 10 30 15 15 0 (If you added a zero to multiply 1.3 m by 10, the answer would be 1.30 m and that’s the same length. It is clearly not the product of 1.3 m and 10.) Guided practice 1 Complete the grid. e.g. a b c d e f T O T O T O T O T O H T O H T O 4 7 8 6 9 1 4 1 9 0 × 10 4 2 Multiply each of these by 10. Remember, mo ve the digits one place a 1.5 m 1.4 m x 10 = ? to the left when 1.4 you multiply by ten. 14 b 2.2 L c 4.5 t d $1.70 e 3.8 cm f 3.6 m g $2.75 When you multiply by 3 Multiply by 10 0. 10 0, the digits move t wo places to the left. a 14 b 17 For example, what is 11 × 100? c 13 d 27 Th H T O e 23 f 45 1 1 1 1 0 0 g 64 h 3.7 m i $1.25 22 OX FOR D U N I V E RSI T Y PR E S S

Independent practice Once you know the ten trick, you can use it to multiply by multiples of 10. In 5 × 30, 30 is the same as 3 tens, so change it to 5 × 3 tens. 5 × 3 = 15 so 5 × 3 tens = 15 tens, or 150. 1 Fill in the gaps. × 20 × 30 Rewrite the problem and solve Rewrite the problem and solve 6 × 30 = 6 × 3 tens 6 × 20 = 6 × 2 tens 6 × 3 tens = 6 × 2 tens = 12 tens a 6 12 tens = 120 So, 6 × 20 = 120 b 9 c 8 d 7 2 Fill in the gaps. To multiply by 4, × 8 a b c d e Strategy you can double a 5 12 15 50 40 number and then double it again. 2 16 Double To multiply by 8, you can double a 4 32 Double again number three times. 8 64 Double again OX FOR D U N I V E RSI T Y PR E S S 23

If you double one number and halve the other, it can make multiplication easier. It works like this: Imagine you didn’t know that 5 × 6 = 30. You could double 5 and halve 6. This would give the same answer: 10 × 3 = 30. 3 Fill in the gaps. Problem and Produc t 10 × 3 30 e.g. 5× 6×7 10 × 9 a 3 × 14 b 5 × 18 c 3 × 16 d 5 × 22 e 6 × 16 f 4 × 18 4 Here is a mental strategy for multiplying by 5. ×5 First multiply Then halve it Multiplication fac t by 10 e.g. 14 14 0 70 14 × 5 = 70 a 16 b 18 c 24 d 32 e 48 5 Use your choice of strategy to nd the product. Be ready to explain how you got the answer. a 18 × 10 b 14 × 10 0 c 2.5 m × 10 d 3 4 × 10 e 14 × 20 f 150 × 5 g 13 × 8 h 9 × 40 i $1.75 × 10 j 8 × 60 24 OX FOR D U N I V E RSI T Y PR E S S

Extended practice 1 Use the split strategy to multiply by 15. × 15 × 10 Halve it Add the t wo Multiplication fac t 120 to nd × 5 answe r s 12 × 15 = 180 e.g. 12 60 120 + 60 = 180 a 16 b 14 c 20 d 30 e 25 2 At the beginning of the year, Dee’s mum gave her two spending - money choices. Working- out spac e • Choice 1: “Would you like $10 a week • this year?” Choice 2: “Would you prefer 10c for the • rst four weeks, then double it for the next four weeks, then double it for the next four weeks, and so on for the rest of the year?” Dee remembered the ten trick and said, “52 weeks × $10 is $520. I’ll take Choice 1 thanks, Mum.” Was this the better choice? How much would Dee have got if she’d taken Choice 2? 3 Tran is reading books for his school’s read - a -thon. He writes down how many pages he reads each day for a week. a Use mental strategies to nd the total number of pages Tran reads during the week. Monday 48 Tuesday 48 b Explain the way you found the answer. Wednesday 48 Thursday 48 Friday 48 Saturday 45 Sunday 45 OX FOR D U N I V E RSI T Y PR E S S 25

UNIT 1: TOPIC 7 Multiplication written strategies You can work out multiplication problems by breaking the numbers down by place value and marking them off on grid paper. This is called an area model, because as you calculate the total number of squares marked off, you are nding the area of the rectangle. 30 6 8 36 × 8 8 × 36 = 8 × 30 + 8 × 6 = 24 0 + 4 8 = 288 Guided practice 1 7 × 34 = 7 × +7× Would the product be the same if I multiplied = + by the ones first? = 4 30 7 7 × 30 = 7×4= 2 5 × 28 = 5 × +5× = + = 20 8 5×8= 5 5 × 20 = 26 OX FOR D U N I V E RSI T Y PR E S S

Independent practice Shade the model and ll in the blanks to nd the product. 1 6 × 32 = × + × = + = 30 2 27 6 2 5 × 35 = × + × = + = 3 7 × 48 = × + × = + = OX FOR D U N I V E RSI T Y PR E S S

Guided practice 4 2 42 × 4 is the same as 2 × 4 and 4 tens × 4, so the 4 × 4 4 answer is 8 plus 16 tens (16 0). You can make written multiplication shor t by writing a contracted algorithm. 1 6 8 You star t with the ones and then multiply each column in turn to nd the product. If you need to trade, 1 4 you can do it like this. 3 × 4 1 3 6 1 Complete the algorithms. Trade if necessar y. a 1 b c d e 4 3 6 5 2 9 9 2 3 8 × 4 × 3 × 2 × 7 × 4 2 Solve these problems in the same way. It works the same with larger numbers. Start at the ones, a b c 1 2 5 1 4 2 2 5 3 × 2 × 4 × 3 d e f g 3 2 5 4 1 5 3 4 8 4 7 5 × 3 × 6 × 2 × 3 h i j k 1 6 2 3 1 2 7 2 2 1 7 3 1 2 3 2 × 4 × 5 × 4 × 8 28 OX FOR D U N I V E RSI T Y PR E S S

Independent practice 1 Once you understand the shor t form of multiplication, it doesn’t matter how big the number is that you are multiplying. Star t at the ones column and complete each column in turn. a b c d 1 6 2 3 1 7 3 4 2 5 1 6 4 2 3 0 × 4 × 4 × 3 × 5 e f g 1 2 3 2 6 2 1 5 3 8 3 3 6 4 0 × 3 × 2 × 7 h i j 2 3 8 5 2 3 6 3 7 4 2 4 7 3 7 × 5 × 5 × 9 2 Find the product. Look for a pattern in the answers. a b c 3 7 0 3 7 3 7 0 3 7 3 7 0 3 7 × 3 × 6 × 9 d e f 7 4 0 7 4 7 9 3 6 5 7 4 0 7 4 × 6 × 7 × 9 g h i 2 5 9 2 5 9 1 2 6 9 8 4 1 4 2 8 5 7 × 3 × 7 × 7 3 To multiply an amount of money, star t with the column of least value. Complete these in the same way as the example. e.g. 1 2 a b c $ $ $ $ × 4 × 3 × 5 × 6 $ $ $ $ 29 OX FOR D U N I V E RSI T Y PR E S S

The ten trick In the ten trick , ever y thing When you are multiplying by a multiple moves over one plac e. of ten, youneed to remember the “tentrick”. 2 3 × 2 0 Then jus t multiply × 2. 4 6 0 4 Use the ten trick to complete these algorithms. a 1 7 b 1 4 c 1 6 d 1 6 e 2 7 × 2 0 × 2 0 × 3 0 × 4 0 × 3 0 0 0 ( 17 × 3 ) + ( 17 × 2 tens ) Multiplying by a 2- digit number 2 7 1 7 is like doing two multiplications 1 1 in one. You split the number you are multiplying by. × 3 × 2 0 What is 17 × 23? 5 1 Add the t wo answer s 3 4 0 There are t wo multiplic ations. to nd the total. 5 1 + 3 4 0 3 9 1 17 × 23 = 3 91 5 Split the numbers to multiply. Use separate paper to work out the answers to questions d, e and f. a What is 15 × 24? b What is 16 × 23? c What is 19 × 25? Make t wo multiplic ations. Make t wo multiplic ations. Make t wo multiplic ations. 1 5 1 5 1 6 1 6 1 9 1 9 × 4 × 2 0 × 3 × 2 0 × 5 × 2 0 0 0 0 Add the answer s. Add the answer s. Add the answer s. + + + 15 × 24 = 16 × 23 = 19 × 25 = f What is 19 × 45? d What is 16 × 39? e What is 15 × 37? Make t wo multiplic ations. Make t wo multiplic ations. Make t wo multiplic ations. OX FOR D U N I V E RSI T Y PR E S S 30

Extended practice Distance from Melbourne to: An aeroplane ies millions of kilometres in Adelaide, Australia 651 km its lifetime. This table shows the distances Bangkok, Thailand 736 3 km from Melbourne airport, in Australia, to Chicago, USA 11 559 km other airports around the world. Dar win, Australia 314 3 km Edmonton, Canada 13 9 9 3 km Frankfur t, Germany 16 30 8 km Glasgow, Scotland 16 9 62 km Honolulu, USA 8 870 km Istanbul, Turkey 14 619 km Johannesburg, South Africa 10 326 km Kuala Lumpur, Malaysia 6 36 0 km Los Angeles, USA 12 76 4 km 1 Choose an ef cient multiplication method to nd the answers. You may need extra paper for your calculations. a What is the distance of a return trip to Istanbul? b How far does a plane y if it makes three return trips to Bangkok? c If a plane ies to and from Chicago eight times, how far does it y? d A plane ies from Melbourne to Dar win and back twice a day for two weeks. What distance does it cover? e If a plane travelled to and from Johannesburg 50 times, would it have own a million kilometres? 2 People can earn points for the distances they travel on cer tain airlines. ABC Airlines offers one point for ever y kilometre that its passengers y. a Olivia ies from Melbourne to Adelaide on business each day and back home again from Monday to Friday. How many points does she earn in two weeks? b Tran travels from Kuala Lumpur to Melbourne once a month to visit his family. How many points does he earn in a year? c How many points does a family of four people earn by going on a holiday to Frankfur t? 3 A plane ies from Melbourne to Adelaide and back twice a day. How many kilometres does it y in one week? OX FOR D U N I V E RSI T Y PR E S S 31

UNIT 1: TOPIC 8 1 is a factor Factors and multiples of every whole A fac tor is a number that will divide evenly into another number: 2 is a factor of 4. A multiple is the result of multiplying a number by a whole number: 6 is a multiple of 3 (3 × 2 = 6). Guided practice 1 Circle the factors of each number. a The factors of 8 are: 1 2 3 4 5 6 7 8 b The factors of 5 are: 1 2 3 4 5 c The factors of 9 are: 1 2 3 4 5 6 7 8 9 d The factors of 6 are: 1 2 3 4 5 6 e The factors of 2 are: 1 2 f The factors of 4 are: 1 2 3 4 g The factors of 7 are: 1 2 3 4 5 6 7 h The factors of 3 are: 1 2 3 2 Write the rst ten multiples of each number. e.g. 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 a 3: b 6: c 9: d 2: e 4: f 8: g 7: h 5: 32 OX FOR D U N I V E RSI T Y PR E S S

Independent practice 1 Write the factors of each number. a 15 b 16 c 20 d 13 e 14 f 18 2 Which numbers between 21 and 30 have exactly: a two factors? b four factors? c three factors? d six factors? 3 a List all eight factors of 24. b Which number between 30 and 4 0 has even more factors than 24? List its factors. 4 The number 2 is a factor of ever y even number. Factors that are the same The fac tors of 16 are: for more than one number are called common factors The fac tors of 20 are: The common fac tors 1 2 4 of 10 and 20 are: a The factors of 4 are: b The factors of 6 are: The factors of 8 are: The factors of 8 are: The common factors of 4 and 8 are: The common factors of 6 and 8 are: c The factors of 14 are: d The factors of 12 are: The factors of 21 are: The factors of 18 are: The common factors of 14 and 21 The common factors of 12 and 18 are: are: OX FOR D U N I V E RSI T Y PR E S S 33

5 For each row, circle the numbers that are multiples of the red number. e.g. a 5 15 21 25 40 50 57 60 65 69 75 85 10 0 8 12 22 24 26 28 30 34 36 40 42 48 b 4 8 12 16 20 24 30 32 36 44 48 56 60 20 21 27 28 35 37 42 47 49 56 60 c 8 14 12 18 21 24 27 36 39 45 55 63 72 9 d 7 e 9 6 How do you know that: a 74 is a multiple of 2? b 4 8 is a multiple of 3? c 10 01 is not a multiple of 10? d 5551 is a not a multiple of 5? 7 When numbers share the same multiples, we call them common multiples. List the multiples of 2 and 3 as far as 30. Circle the common multiples. 2 4 6 Multiples of 2: 3 6 Multiples of 3: 8 Find a common multiple of 4 and 5 between 1 and 30. 9 Find a common multiple of 2 and 3 between 31 and 4 0. 10 What is the lowest common multiple of: a 6 and 9? b 3 and 4? c 5 and 7? d 3 and 5? e 5 and 9? f 4 and 7? 34 OX FOR D U N I V E RSI T Y PR E S S

Extended practice 1 In a biscuit factor y, they make a lot of the same item. When the biscuits are ready, they have to decide how many should be put in each packet. a If 50 biscuits were baked in an hour, they could put all 50 in one box. Find another ve factors of 50 that will show the other ways that 50 biscuits could be packed. b What would be a sensible number of biscuits to put in each packet? 2 A donut machine makes a batch of four donuts ever y minute. a Circle the numbers of donuts that it is possible for the machine to make: 16 24 30 36 50 52 90 96 b How many donuts does the machine make in one hour? c If the machine slowed down to three donuts a minute, which numbers in question 2a would it be possible for the machine to make? d Which of the numbers of donuts can be made from both the faster and slower speeds? 3 Pencils at the Pixie Pencil Company come off the conveyor belt in batches of 9 6. Find all the options for the number of pencils that could go in a packet. 4 The Bigfoot Sock Company makes 10 0 socks a day. Ever y sock is the same size and colour. a How many different ways could the socks be packed? b Bigfoot’s customers will not accept an odd number of socks. What are the options for the number of socks that could be put in a pack from one batch? OX FOR D U N I V E RSI T Y PR E S S 35

UNIT 1: TOPIC 9 Divisibility 2 is one of 2 is NOT one your fac tor s. of your fac tors. factor of half of all the whole numbers. Guided practice Every even number is divisible by 2. 1 Circle all the numbers that are exactly divisible by 2. 18 43 29 78 514 707 10 0 0 20 01 123 4 990 2223 118 2 4 is an even number. a Is ever y even number divisible by 4? b Test your answer by circling the even numbers that are exactly divisible by 4. 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 3 All three of these numbers are exactly divisible by 4: 3 20, 716 and 5812. What do you notice about the divisibilit y of the red par t (tens and ones) of each number? 4 Circle all the numbers that are exactly divisible by 4. 112 620 425 426 428 340 3 42 716 714 410 412 5 Without using numbers that appear on this page, write: a a 3 - digit number that is exactly divisible by 2. b a 3 - digit number that is exactly divisible by 4. c a 4 - digit number that is exactly divisible by 2. d a 4 - digit number that is exactly divisible by 4. 36 OX FOR D U N I V E RSI T Y PR E S S

Independent practice There is a way to test for divisibilit y. Test to see if a number can be It can if … E xample divided exac tly by: 2 the number is even In 135 792 the last digit is an even number so 135 972 is an even number. (135 792 ÷ 2 = 67 8 9 6) 3 the sum of the digits in In 24 the sum of the digits is 2 + 4 = 6. the number is divisible (6 ÷ 3 = 2) by 3 4 the last two digits can be In 132, the last 2 digits are 32. divided by 4 32 can be divided by 4 (132 ÷ 4 = 3 4) 5 the number ends in 5 or 9 5 ends in 5 (9 5 ÷ 5 = 19). a0 6 the number is even and it 78 is even and the sum of its digits is is divisible by 3 7 + 8 = 15. 15 is divisible by 3 (78 ÷ 6 = 13). the last three digits are In 10 4 8, the last 3 digits are 0 4 8. 4 8 is a number that can be 8 divided by 8 divisible by 8 (10 4 8 ÷ 8 = 131). the sum of the digits in the In 15 3, the sum of the digits is 1 + 5 + 3 = 9. 9 number is divisible by 9 15 3 is divisible by 9 (15 3 ÷ 9 = 17). 10 the number ends in a zero 5 4 3 210 ends in a zero so it is divisible by 10. (5 4 3 210 ÷ 10 = 5 4 321) 1 Use the divisibilit y tester for this activit y. Circle the numbers that are exactly divisible by: a 411 207 433 513 b 552 775 630 751 c 711 702 522 603 d 888 24 8 24 4 884 e 819 693 539 252 f 8 02 820 990 10 01 OX FOR D U N I V E RSI T Y PR E S S 37

2 A prime number has just two factors: 1 and itself. 37 is a prime number because it can only be divided by 1 and 37. Use the divisibilit y tester to help you circle the only other prime number on this par t of a 10 0 char t. 3 Numbers that have more than two factors are called composite numbers. 35 is a composite number. It has four factors: 1, 35, 5 and 7. Which numbers in question 2 are exactly divisible by: a 2 and 4? b 3 and 6? c 4 and 8? d 2 and 3? e 2, 4 and 8? f 3 and 11? g 2, 3, 4, 6 and 9? 4 39 is divisible by 3. What are its other factors? 5 How do you know that 5 31 is divisible by 3? Circle one answer. a It has a 3 in it. b It is an odd number. c The sum of the digits is divisible by 3. 6 Circle the number that is divisible by 4. 4446 9 324 24 42 123 4 7 Jack has 24 6 model cars. He wants to put them in groups of 4. a How could you explain to Jack that it is not possible to do that? b Would it be possible to put them in groups of 3? c Explain your answer to question 7b. d How many more cars would Jack need to be able to make groups of 4? 38 OX FOR D U N I V E RSI T Y PR E S S

Extended practice 1 When a number is divisible by another number, it is also divisible by the factors of that number. For example, 2 and 3 are factors of 6. So, if a number is divisible by 6, it can also be divided by 2 and by 3. Prove it for yourself with the following. Circle the numbers that are divisible by 6, 2 and 3. 24 54 72 96 48 78 2 Use the Venn diagram to show which of the following numbers are divisible by 3, by Divisible by 3 12 Divisible by 4 4 and by both 3 and 4. 6 8 15 20 44 45 48 72 76 81 92 96 3 a How do you know that 30 6 is divisible by 6? b Find all the single - digit numbers that will divide exactly into 30 6. 4 There is a number between 70 0 and 730 that is divisible by ever y single - digit number except 7. What is the special number? 5 Complete the Venn diagram to show some numbers that are divisible by 4, 5 and by Divisible by 4 Divisible by 5 both 4 and 5. OX FOR D U N I V E RSI T Y PR E S S 39

UNIT 1: TOPIC 10 Division written strategies One written way to Let ’s share ÷ 2 is the same as ÷ 2 and ÷2 86 marbles solve a division problem ÷2= is to split the number you ÷2=3 are dividing by to nd the quotient. So, ÷2= + = 3 Guided practice 1 Split these numbers to nd the quotient. a What is 6 8 ÷ 2? b What is 6 9 ÷ 3? 6 8 ÷ 2 is the same as 60 ÷ 2 and 8 ÷ 2 69 ÷ 3 is the same as 60 ÷ 3 and 9 ÷ 3 60 ÷ 2 = 60 ÷ 3 = 8÷2= 9÷3= So, 6 8 ÷ 2 = + = So, 69 ÷ 3 = + = c What is 8 4 ÷ 2? d What is 124 ÷ 4? 8 4 ÷ 2 is the same as 8 0 ÷ 2 and 4 ÷ 2 124 is the same as 10 0 ÷ 4 and 24 ÷ 4 ÷2= 10 0 ÷ 4 = ÷2= 24 ÷ 4 = So, 8 4 ÷ 2 = + = So, 124 ÷ 4 = + = e What is 122 ÷ 2? f What is 145 ÷ 5? 122 ÷ 2 is the same as ÷2 = 145 ÷ 5 is the same as ÷5 = and 22 ÷ 2 and ÷5 ÷2= ÷5= 22 ÷ 2 = ÷5= So, 122 ÷ 2 = + So, 145 ÷ 5 = + 40 OX FOR D U N I V E RSI T Y PR E S S

Independent practice Division can be set out in an Step 1 3 Step 2 algorithm. You put the number in a 4 tens split into Trade the ten “box” and split it up. This is called groups of three for 10 ones. shor t division. Imagine the problem makes 1 group of Now there are is 42 ÷ 3. This is how it works: three tens and 12 ones. 1 ten lef t over. 1 Find the quotient using the shor t division method. 1 1 4 5 6 2 36 5 85 6 78 3 72 7 84 3 5 95 6 84 3 87 8 96 7 91 2 These problems contain larger numbers but you can solve them in the same way. 11 2 4 4 6 8 5 560 3 651 2 850 6 696 3 954 5 585 7 798 2 674 6 690 3 645 4 896 3 378 7 791 2 898 6 684 OX FOR D U N I V E RSI T Y PR E S S 41

When the digit in the rst column cannot be divided, this is what you do. 1 There aren’tenough hundreds to make groups 2 Trade the ten for 10 ones. of 2. We star t with 11 tens. That makes 18 ones. 18 split into groups of 2 = 9 11 tens split into groups of 2 = 5r1 2 5 2 59 18 11 8 3 Find the quotient. 8 5 2 1 1 3 162 144 136 74 132 4 268 270 399 468 6 282 680 372 644 198 Remember to write the digits in the correct c olumns. 294 5 395 w r ong right H T O H T O 59 59 Sometimes the number you are dividing will not split equally. When this happens you have a remainder. This can be shown 2 11 8 2 11 8 using “r” for remainder. For example, 13 ÷ 3 = 4 r1. 4 Find the quotient. Use “r” to show the remainder. r 4 57 2 51 5 77 6 80 693 3 952 582 782 167 6 275 265 277 293 547 9 394 199 42 OX FOR D U N I V E RSI T Y PR E S S

Extended practice 1 Not ever y number can be divided equally by other numbers. Write algorithms to nd the quotient for these. Use remainders where necessar y in the answers. a 97 ÷ 5 b 72 ÷ 3 c 145 ÷ 6 d 386 ÷ 7 2 In real life, we have to work out what to do with remainders. You know that 7 ÷ 2 = 3 r1, 9 ÷ 2 = 4 r1 and 13 ÷ 2 = 6 r1. What is the best way to express the answers in these real - life situations? a Seven donuts shared between two people. b Two people are given nine marbles. How many can each person have? c Two sisters share $13. How much do they each get? 3 At a school, there are 161 children in the six senior classes. Class Number of a To nd the mean (average) number of students per s tudent s class, divide the total number of students by the 3W 25 number of classes. The mean is: 3/ 4D 26 4M b Complete the table to show the actual number that 5S could be in each class. Two classes have been lled 5/6H in. None of the other classes has the same number of 6T students as any other class. 4 Three people share a prize of $10 0. a Calculate how much money each should receive. b They ask a bank to change the money so that they can each have a fair share. List the coins and notes that each might have. 5 At a chicken farm, 30 0 0 eggs a day are packaged. They are put into boxes. Each box can hold 8 dozen eggs. How many boxes are needed for 30 0 0 eggs? OX FOR D U N I V E RSI T Y PR E S S 43

UNIT 2: TOPIC 1 Comparing and ordering fractions What does a fraction A fraction can be par t A fraction can be par t look like? of a whole thing. of a group of things. The number on the top 1 1 is the numerator. The 4 2 number on the bottom A quar ter of the Half of the is the denominator circle is blue. beads are red. Guided practice 1 Write the fractions in words and numbers. a What fraction b What fraction c What fraction d What fraction is red? is green? is blue? is black? one sixth one 1 1 2 Shade the shapes. a 3 b 3 c 2 d 3 e 5 Shade Shade Shade Shade Shade 8 4 3 5 6 3 What fraction of the group is shaded? 4 Shade each group to match the fraction. a 3 b 5 c 2 d 2 10 12 5 4 44 OX FOR D U N I V E RSI T Y PR E S S

Independent practice 1 Write the missing fractions on the number lines. a 1 3 5 5 b c 9 10 d 1 4 1 e 3 8 1 3 f 4 6 g 0 2 Which fraction in each pair is closer to 1? Use the number lines in question 1 to help. 1 3 1 1 1 1 a or ? b or ? c or ? 8 6 8 4 3 4 1 1 1 1 3 7 d or ? e or ? f or ? 10 3 5 2 4 8 7 4 5 1 7 7 g or ? h or ? i or ? 5 2 10 8 8 10 3 Which fractions are the same distance along the number line as 1 ? 2 OX FOR D U N I V E RSI T Y PR E S S 45

4 Use the number lines on page 4 9 to help you order each group from smallest to largest 4 1 3 2 a , , 1, , 5 5 5 5 7 3 9 2 6 b , , 1, , , 10 10 10 10 10 1 1 1 1 1 c , , , , 2 4 8 10 5 3 3 3 3 3 d , , , , 8 10 4 6 3 2 2 2 2 2 e , , , , 5 8 3 10 6 5 Use the symbols > (is bigger than), < (is smaller than) or = to complete these number sentences. a 3 7 b 1 1 c 3 1 d 2 2 4 8 4 8 6 2 3 6 e 3 1 2 5 g 9 4 h 3 6 8 2 8 10 5 5 10 f 4 5 2 3 i 6 6 0 1 a Circle the two fractions that describe the position of the triangle on the number line. 6 6 6 1 6 3 and and and 8 10 8 4 8 4 b Circle the fraction that describes how far from 1 the triangle is. 2 2 2 3 8 4 3 c Draw a diamond that is of the way from 0 to 1. 8 7 a Divide the rectangle into eighths. b 2 Shade 8 c What other fraction describes the fraction that you have shaded? 46 OX FOR D U N I V E RSI T Y PR E S S

Extended practice 1 a Divide the rectangle b into three equal par ts. Shade one par t. c Write two fractions that describe the shaded par t. 2 Write these fractions at the correct place on the number line. a 1 b 1 c 3 d 3 e 7 4 2 4 8 8 0 1 3 Write a fraction that is: a bigger than a quar ter but smaller than a half. b smaller than two -thirds but bigger than a half. c bigger than a third but smaller than a half. d bigger than ve - sixths but smaller than one. e smaller than an eighth but larger than a twelfth. 4 Some people say it is impossible to fold a square of paper in half, then half again, and so on, more than eight times. Is it true? How many times can you keep folding a piece of paper in half? When you can go no fur ther, write down the number of folds, then open the paper out and nd the fraction that the folds have split the paper into. • Number of folds: • Fraction: Was the result as you expected? Write a sentence to say how easy or how dif cult you found this task. OX FOR D U N I V E RSI T Y PR E S S 47