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Year 6

Primary

MATHEMATICS CURRICULUM Word Problems

What is the distance...?

For how long...?

What is the percentage

of...?

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Maths Word Problems

Lizzie Marsland

Year 6

Primary

MATHEMATICS CURRICULUM

Primary

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Acknowledgements:Author: Lizzie MarslandSeries Editor: Peter SumnerCover and Page Design: Kathryn Webster

Primary

Published by HeadStart Primary Ltd 2016 © HeadStart Primary Ltd 2016

The right of Lizzie Marsland to be identified as the author of this publication has been asserted by her in accordance with the Copyright, Designs and Patents Act 1998.

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, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the publisher.

A record for this book is available from the British Library - ISBN: 978-1-908767-31-8

T. 01200 423405E. [email protected]

HeadStart Primary LtdElker LaneClitheroeBB7 9HZ

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© Copyright HeadStart Primary Ltd 2016

INTRODUCTION

Year 6: NUMBER - Number and place value

Page ObjectivesPage 1 - Read and write numbers to at least 10,000,000Page 2 - Read and write numbers to at least 10,000,000Page 3 - Order and compare numbers to at least 10,000,000 - Order and compare numbers to at least 10,000,000Page 5 - Determine the value of each digit in numbers up to 10,000,000 Page 6 - Determine the value of each digit in numbers up to 10,000,000 Page 7 - Round any whole number to a required degree of accuracyPage 8 - Round any whole number to a required degree of accuracyPage 9 - Use negative numbers in context and calculate intervals across zeroPage 10 - Use negative numbers in context and calculate intervals across zero Page 11 - Solve problems involving number and place valuePage 12 - Solve problems involving number and place valuePage 13 - Solve problems involving number and place value

Year 6: NUMBER – Addition, subtraction, multiplication and division

Page 14 - Multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplicationPage 15 - Multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplicationPage 16 - Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long divisionPage 17 - Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long divisionPage 18 - Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division and interpret the remainder as a whole numberPage 19 - Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division and interpret the remainder as a fractionPage 20 - Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, rounding the remainder as appropriate for the context

CONTENTS Year 6

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CONTENTS Year 6

Page 21 - Interpret remainders appropriately for the context Page 22 - Divide numbers up to 4 digits by a two-digit whole number using the formal written method of short division, interpreting remainders according to the contextPage 23 - Perform mental calculations, including with mixed operations and large numbers Page 24 - Identify common factors, common multiples and prime numbersPage 25 - Identify common factors, common multiples and prime numbersPage 26 - Use knowledge of the order of operations to carry out calculations involving the four operationsPage 27 - Solve addition and subtraction multi-step problems in contextPage 28 - Solve problems involving addition, subtraction, multiplication and divisionPage 29 - Solve problems involving addition, subtraction, multiplication and divisionPage 30 - Solve problems involving addition, subtraction, multiplication and division

Year 6: NUMBER - Fractions (including decimals and percentages)

Page 31 - Use common factors to simplify fractionsPage 32 - Use common multiples to express fractions in the same denominationPage 33 - Compare and order fractions, including fractions greater than 1Page 34 - Compare and order fractions, including fractions greater than 1Page 35 - Add fractions with different denominators, using the concept of equivalent fractionsPage 36 - Subtract fractions with different denominators, using the concept of equivalent fractions Page 37 - Add or subtract fractions with different denominators, using the concept of equivalent fractionsPage 38 - Add or subtract mixed numbers, using the concept of equivalent fractions Page 39 - Multiply simple pairs of proper fractions, writing the answer in its simplest formPage 40 - Divide proper fractions by whole numbersPage 41 - Associate a fraction with division and calculate decimal fraction equivalentsPage 42 - Identify the value of each digit in numbers to three decimal placesPage 43 - Identify the value of each digit in numbers to three decimal placesPage 44 - Divide numbers by 10 giving answers upto 3 decimal placesPage 45 - Divide numbers by 100 giving answers upto 3 decimal placesPage 46 - Divide numbers by 1000 giving answers upto 3 decimal placesPage 47 - Multiply one-digit numbers with up to two decimal places by whole numbersPage 48 - Multiply one-digit numbers with up to two decimal places by whole numbersPage 49 - Use written divison methods in cases where the answer has up to two decimal placesSA

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CONTENTS Year 6

Page 50 - Solve problems which require answers to be rounded to specified degrees of accuracyPage 51 - Recall and use equivalences between simple fractions, decimals and percentages, including in different contextsPage 52 - Solve problems involving percentagePage 53 - Solve problems involving percentage Page 54 - Solve problems involving fractionsPage 55 - Solve fraction problemsPage 56 - Solve problems involving fractions, decimals and percentagesPage 57 - Solve problems involving fractions, decimals and percentages

Year 6: RATIO AND PROPORTION

Page 58 - Solve problems involving the relative size of quantities using division and multiplicationPage 59 - Solve problems involving the relative size of quantities using division and multiplication0 - Solve problems involving the calculation of percentagesPage 61 - Solve problems involving the comparison of percentagesPage 62 - Solve problems linking percentages, angles and pie chartsPage 63 - Solve problems involving scaling by multiplicationPage 64 - Solve problems involving scaling by divisionPage 65 - Solve problems involving scaling by multiplication and divisionPage 66 - Solve problems involving scaling of shapesPage 67 - Solve problems involving unequal groupings using knowledge of fractions and multiplesPage 68 - Solve problems involving unequal quantitiesPage 69 - Solve problems involving unequal quantities

Year 6: ALGEBRA

Page 70 - Solve problems involving finding missing numbers using simple formulaePage 71 - Solve problems with linear number sequencesPage 72 - Express missing number problems algebraicallyPage 73 - Express missing number problems algebraicallyPage 74 - Solve problems involving equations with two unknown numbersPage 75 - Enumerate possibilites of combinations of two variblesSA

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CONTENTS Year 6

Year 6: MEASUREMENT

6 - Solve problems involving the calculation and conversion of units of measure, using decimal notation up to three decimal placesPage 77 - Solve problems involving the calculation and conversion of units of measure, using decimal notation up to three decimal placesPage 78 - Solve problems involving the calculation and conversion of units of measure, using decimal notation up to three decimal placesPage 79 - Solve problems involving converting measurements of lengthPage 80 - Solve problems involving converting measurements of massPage 81 - Solve problems involving converting measurements of volume Page 82 - Solve problems involving converting measurements of time Page 83 - Solve problems converting between miles and kilometresPage 84 - Solve problems, involving perimeter and areaPage 85 - Solve problems, recognising that shapes with the same areas can have different perimeters and vice versaPage 86 - Solve problems using formula for areaPage 87 - Solve problems using formula for volumePage 88 - Solve problems by calculating the area of parallelogramsPage 89 - Solve problems by calculating the area of triangles Page 90 - Solve problems by calculating the area of compound and mixed shapesPage 91 - Solve problems by calculating and comparing the volume of cubes and cuboids using cubic centimetresPage 92 - Solve problems by calculating and comparing the volume of cubes and cuboids using cubic metresPage 93 - Solve problems by calculating and comparing the volume of cubes and cuboids extending to other units (mm3 and km3)

Year 6: GEOMETRY - Properties of shapes / Position and direction

Page 94 - Solve problems involving 2D shapes using dimensions and anglesPage 95 - Solve problems involving 2D shapes using dimensions and anglesPage 96 - Solve problems involving the properties of 3D shapesPage 97 - Solve problems involving nets of 3D shapes Page 98 - Solve problems involving the properties of 2D and 3D shapesPage 99 - Solve problems involving angles in trianglesPage 100 - Solve problems involving angles in quadrilaterals

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CONTENTS Year 6

Page 101 - Solve problems involving angles in regular polygonsPage 102 - Solve problems involving the parts of a circlePage 103 - Solve problems involving anglesPage 104 - Solve problems involving position and directionPage 105 - Solve problems involving position and direction

Year 6: STATISTICS

Page 106 - Interpret pie charts and use these to solve problemsPage 107 - Interpret pie charts and use these to solve problemsPage 108 - Interpret pie charts and use these to solve problems Page 109 - Interpret line graphs and use these to solve problems Page 110 - Interpret line graphs and use these to solve problems Page 111 - Interpret line graphs and use these to solve problems Page 112 - Calculate and interpret the mean as an averagePage 113 - Calculate and interpret the mean as an averagePage 114 - Calculate and interpret the mean as an average

Answers - Year 6

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INTRODUCTION

These problems have been written in line with the objectives from the Mathematics Curriculum. Questions have been written to match all appropriate objectives from each ‘content domain’ of the curriculum.

Solving problems and mathematical reasoning in context is one of the most difficult skills for children to master; a ‘real life’, written problem is an abstract concept and children need opportunities to practise and consolidate their problem-solving techniques.

As each content domain is taught, the skills learnt can be applied to the relevant problems. This means that a particular objective can be reinforced and problem-solving and reasoning skills further developed. The pages can be reproduced and used either within or outside the mathematics lesson at school. They are also very useful as a homework resource.

The questions are arranged, in general, so that the more difficult questions come towards the bottom of the page. This means that differentiation can be achieved with the lower ability children working through the earlier questions and the higher ability going on to complete the whole page. The CD-ROM contains editable copies of each page. These can be edited and saved, as required, to provide extra practice or additional differentiated problems. The electronic versions on the CD-ROM can also be used on an interactive whiteboard, facilitating class discussion and investigation.

An example of a step-by-step method to solve word problems can be found on the following page. This can be edited, enlarged or used as a poster for classroom display and/or copies given to each child to be used as a check for each question answered.

Important parts of each question have been highlighted in ‘bold’ font. Once children have become more proficient at problem solving, it may be appropriate to remove these prompts on the editable page. Children can then be encouraged to use a highlighter or underline the important parts themselves.

Since a structured approach to problem solving supports learning, developing a whole-school approach is very worthwhile. SA

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Read the problem carefully.

Find the question.

Identify the important parts.

Decide on the operation or operations.

Carry out the operation or operations.

Check your answer.

Feel very pleased with yourself.

TO SOLVE A WORD PROBLEM

Follow these steps:

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Number and place value

NUMBER

These are all about

number and place value!

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© Copyright HeadStart Primary Ltd 2016 3 Name .............................................

Here is a sequence of numbers. Put the numbers in order of size from smallest to largest.

8452 7482 7452 7432

Bishra is trying to decide which number is the largest: 35,494 or 53,949. Which one should he choose?

Lola puts these numbers in order of size from largest to smallest. What should her new list look like?

532,873 532,200 532,998

Mr Robinson asks his class to write down the number which is larger than 854,323 but smaller than 854,325. What number should they write?

Mrs Thorn asks her class which of these numbers is smaller: 1,263,892 or 1,326,982. What should their answer be?

Which number is larger: 2,999,998 or 2,999,989?

Which number is smaller: 5,842,978 or 5,841,989?

Christian wants to put these numbers in order from smallest to largest. What should his new list of numbers look like?

5,389,000 10,900,500 8,938,030 3,500,089

NUMBER - Number and place value Year 6

Order and compare numbers to at least 10,000,000

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At New Hall Primary there are 545 pupils. How many pupils attend the school to the nearest hundred?

There are 4389 Red Flyers fans at the rugby match. Round the number of fans to give an approximate attendance.

Miss Cranston buys a new sports car for £18,599. Is the cost of the car nearer to £18,500 or £18,600?

Ayesha rounded a football crowd of 84,762 to the nearest ten thousand. Was this the best way to estimate the attendance? Explain your answer.

Thomas writes down the number 349,400. Round this number to the nearest hundred thousand.

There are 2,736,949 people living in your city. What number would you round to, to tell a friend how many people lived in your city?

In a rainforest, there are exactly 1,997,382 trees. What would be a good approximation of this number? Explain your answer.

Ghulam wins £5,523,445 on the lottery. He spends £24,000 on a holiday of a lifetime to Australia. How much money has he left to spend? Use rounding to give an approximate answer.


Round any whole number to a required degree of accuracy

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The temperature is -6°C. It falls by 7 degrees. What is the temperature now?

The temperature is -12°C. It rises by 3 degrees. What is the temperature now?

The temperature in Alaska is -18°C. The temperature in Sweden is -2°C. How many degrees higher is the temperature in Sweden than in Alaska?

The temperature in Britain is 3°C. The temperature in Alaska is -18°C. What is the difference in temperatures between Britain and Alaska?

Jayden had a target long jump of 300 centimetres. He recorded each of his 7 attempts above and below his target as follows:

a) How far did he jump in his longest jump?

b) How far did he jump in his shortest jump?

c) How many more centimetres was his longest jump than his shortest jump?

d) Jayden’s 8th jump was 9 cm shorter than his longest jump. How far was his 8th jump?


Use negative numbers in context and calculate intervals across zero

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3 cm -2 cm 0 cm 4 cm -3 cm -1 cm 2 cm

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Gok is trying to put these numbers in order of size, starting from the smallest. Help him put them in the correct order.

7891 7982 7421 7834

Adam is practising throwing the cricket ball. He had a target of throwing the ball 25 metres. He recorded each of his 6 attempts above and below his target as follows.

a) Which attempt was closest to his target of 25 metres?

b) What was the difference between his longest and shortest throw?

The temperature in Moscow was -2°C. The temperature dropped by 5 degrees. What was the temperature after the drop?

Samantha has to think of a number that is smaller than 139,297 but larger than 139,295. What number should she choose?

Nadine rounded 2,872,785 to the nearest 10,000. What should her answer have been?

The number 9,873,033 was written on a ticket in words. What was written?


Solve problems involving number and place value

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A

+2m

C

-1m

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+3m

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-4m

D

+5m

F

-6m

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Addition, subtraction,multiplication and division

NUMBER

These are all about

addition, subtraction, multiplication and division

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NUMBER - Addition, subtraction, multiplication and division Year 6

There are 91 coloured pencils in a box. Katie shares them equally between 13 school friends. How many pencils does each friend receive?

Every morning, Ben bakes 384 cupcakes to sell in his shop. Each tray holds 16 cupcakes. How many trays of cupcakes will he have altogether?

The Zoo buys 868 bananas to feed 28 monkeys. The bananas are given out equally. What is each monkey’s share?

Logan is trying to work out the answer to 572 ÷ 26. What should his answer be?

Use a formal written method of long division to solve two thousand, one hundred and fifty six divided by fourteen.

Mr Williams saved £1456 for his holiday in a year. He saved an equal amount of money every week for 52 weeks. How much money did he save each week?

In preparation for the school party, Ashok was making bags of sweets. He shared 9146 sweets equally between 36 bags. How many sweets were in each bag? How many sweets were left over?

What is the quotient of 1053 and 39, where 39 is the divisor?

Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division

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There are 98 junior children in Daisyhill School. In assembly, children sit in rows of 18.

a) How many full rows can be made?

b) How many children are left over?

At the garden centre, Mr Poppy has 98 flower bulbs. He puts 16 bulbs in each plant pot. How many bulbs will he have left over?

There are 393 colouring pencils in Class 6. Selina divides them equally between the 29 children in her class.

a) How many colouring pencils does each child get?

b) How many pencils are left?

Yousef is trying to work out the answer to 856 divided by 29. What should his answer be? What is the remainder?

Springtree School buys 623 work books for the children. There are 96 children in the school.

a) How many work books does each child receive?

b) How many work books are left over as spares?

Freddie picks 1067 strawberries. 22 strawberries are in a packet.

a) How many full packets of strawberries can be made?

b) How many strawberries will be left over?

Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division and interpret the remainder as a whole number

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Fourteen chairs fit in each row of a school hall. There are 91 chairs.

a) How many full rows will there be?

b) What fraction of another row can be filled?

A room in a library has 978 books. Each shelf can hold 18 books. How many shelves are used to hold the books? Give your answer as a mixed number, in its lowest terms.

A factory makes wheel nuts for lorries. Each lorry needs 28 wheel nuts.

a) How many lorries could be fitted with 819 wheel nuts?

b) What fraction of a lorry’s wheel nuts would be left?

931 chocolate drops are used to decorate 38 cakes. How many chocolate drops are used on each cake? Give your answer as a mixed number, in its lowest terms.

Zak thought of a mixed number and multiplied it by 60. The answer was 1605. What was the mixed number, in its lowest terms?

Sweets were packed into boxes of 42.

a) How many full boxes could be filled with 1568 sweets?

b) What fraction of another box could be filled with the remaining sweets? Give your answer in its lowest terms.

Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division and interpret the remainder as a fraction

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There are 1250 letters in Post Box A and 2350 letters in Post Box B.

a) How many letters are there altogether?

b) The postman delivers 2640 letters. How many letters still need to be delivered?

4400 Pottingham and 4700 Silverpool fans were watching the football match. 1550 fans left the game before the final whistle.

a) How many fans were watching altogether?

b) How many watched till the end?

Robert, the robot, takes 50 equal steps. He walks 2.5 metres. How many centimetres is each step?

There are 165 pieces of fruit for break time. The fruit is shared equally between 45 children.

a) How many pieces of fruit do they eat each?

b) How many pieces of fruit are left over?

Kelly’s average stride was 80 centimetres. She wanted to travel 10 metres. After 12 average strides, how far would she have left to travel?

There were 3800 black cars and 3600 blue cars made in the Liverchester factory. In the Manorpool factory, 1250 fewer cars were made. How many cars were made in Manorpool?

Perform mental calculations including with mixed operations and large numbers

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Fractions (including decimals and percentages)

NUMBER

These are all about

fractions, including decimals and percentages!

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Year 6NUMBER - Fractions (including decimals and percentages)

James ate three eighths of a cake and his brother, Dom, ate one half. Who ate more?

Abdallah and Scarlett played crazy golf. Abdallah won 5/12 of the holes and Scarlett won 1/3. Who has won more holes?

Mrs Bramble used eight ninths of a bag of flour and 12/6 bags of sugar to make her cakes. The bags of flour and the bags of sugar were the same size. Which ingredient did she use more of?

Which fraction is larger: 5/3 or 14/9?

Three friends are eating a pizza each. Zoey eats 3/4, Dawood eats 10/12 and Sierra has 5/8 of her pizza. Who eats the most and who eats the least amount of pizza?

Harriet has written down 13/5 and 23/10 on her whiteboard. Which fraction is smaller?

Caz says, “9/8 is smaller than 14/12”. Is he correct?

Put these fractions in order of size, from largest to smallest.

3/2 1/2 5/6 11/4 9/8

Compare and order fractions, including fractions greater than 1

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Year 6

Harry subtracts 1/2 from 3/4. What should his answer be?

Three fifths of the class are in the playground. Three tenths are on the field. How much more of the class are in the playground than on the field?

Zoya’s mum puts 5/8 of the vegetables on the table. One quarter are still on the hob. How much more of the vegetables are on the table than on the hob?

Riley eats 5/12 of the biscuits on Monday and 1/6 on Tuesday. Find the difference between the fraction of the biscuits he eats on Monday and Tuesday.

At the pet shop, a quarter of the rabbits are in hutches. One sixth are being held by customers. Find the difference between the fraction of rabbits in hutches and the fraction being held by customers.

Calculate 11/12 minus 1/8.

Find the difference between 10/11 and 3/5.

Emily spends 1/6 of her wage on a new dress and 4/9 on some shoes. Find the difference between the fraction of her wage she spends on shoes and the fraction she spends on a dress.

Subtract fractions with different denominators, using the concept of equivalent fractions

NUMBER - Fractions (including decimals and percentages)

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Year 6

Olivia writes down the number 78.3. What does the digit three represent?

Bishra partitions the number 728.9. How many tenths are there?

Mr Ambrose asks his class to partition 0.67 into tenths and hundredths. How many hundredths are there?

Lydia swam the length of the pool in 48.26 seconds. What proportion of a second is the digit 6?

Add the value of the digit eight in £45.89 to the value of the digit eight in £54.98.

Conrad measured out 4.652 kg of sugar. What proportion of a kilogram is the digit 2?

What is the value of the digit 6 in the number 1342.846?

For her homework, Shelby had to find the difference between pairs of numbers. She had to use one of the words tenths, hundredths or thousandths as part of her answer. What should her answers have been to the following?

a) 2.345 and 2.348

b) 10.362 and 10.462

c) 267.459 and 267.419

Identify the value of each digit in numbers to three decimal places


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35% of the crisps in a shop are salt and vinegar. There are 100 packets of crisps. How many packets of salt and vinegar crisps are there?

73% of 500 children go to school by car. How many children go to school by car?

There is 25% off in a sale. How much is knocked off a game which cost £48 before the sale?

What is 20% of £4.80?

Which is more: a) 75% of 300 or b) 30% of 800? Explain how you know.

A basketball team played 30 games. They won 70% of the games. How many games did they lose?

The cafe decided to give 25% of its takings to charity. How much would go to charity if the cafe sold one coffee, one sandwich and one cake?

The cafe sold 10 coffees and 8 sandwiches. It gave £6.90 to charity altogether. How many cakes must it also have sold?

Year 6

Solve problems involving percentage


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coffee - £1.60 sandwiches - £1.20

Cupcake Cafe

cakes - £1.00

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65% of the chocolate bars sold by a shop are milk chocolate. The shop sold 200 chocolate bars altogether. How many milk chocolate bars did the shop sell?

59% of the 400 children at the primary school are boys. How many children are boys?

There is 25% off in a sale. How much are the trainers reduced by, if they originally cost £56?

What is 75% of £62.20?

Which is more: a) 60% of £350 or b) 35% of £700? Explain how you know.

Southworth Primary has 30 school governors. 70% of the governors are women. How many of the governors are men?

25% of the money the shop takes is profit. How much profit is there if the shop sold 2 packets of cards, 3 packets of pencils and 10 bunches of flowers?

The shop decides to take 20% off everything in the sale. Write out the price of all the items in the sale.

Year 6

Solve problems involving percentage


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cards - £1.60(a packet)

flowers - £1.20(a bunch)

Creative Crafts

pencils - £1.00(a packet)

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After Amelia’s birthday party, one fifth of the food was left over. What percentage of the food was left over after Amelia’s birthday party?

Three quarters of Bobby’s birthday cake was eaten at his party. What percentage was left?

Laura had 15% of £300. Shabana had two fifths of £120. Who had more?

Rhys bought 7 toys costing £4.32 each and 3 games costing £12.26 each. How much did he spend altogether?

In the numeracy lesson, Niko had to change three fifths into a decimal. What should his answer be?

Which is bigger: thirteen twentieths or 0.69? Show your working out.

Marcus had two bank accounts. In one, he had £6432 and in another he had £2450. He spent 75% of his money from both bank accounts to buy a car. How much did he pay for the car?

At an athletics meeting, the first 5 long jumpers jumped as follows: 1.32 m, 1.15 m, 1.2 m, 1.33 m and 1 m. The sixth jumper jumped 10% further than the average of the first 5 jumps. What distance was the sixth jump?

Year 6

Solve problems involving fractions, decimals and percentages


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RATIO AND PROPORTION

These are all about ratio and proportion!

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Year 6

Mrs Beattie, made lemon cupcakes every day, except Sundays. Each day she had to make a different number. Rewrite the recipe for every day, as shown below:

Monday - 60 cupcakes

Tuesday - 10 cupcakes

Wednesday - 40 cupcakes

Thursday - 15 cupcakes

Friday - 5 cupcakes

Saturday - 100 cupcakes

Solve problems involving the relative size of quantities using division and multiplication

RATIO AND PROPORTION

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Ingredients to make 30 Lemon Cupcakes

3 cups of flour

1 packet of butter

1.5 cups of white sugar

6 eggs

3 teaspoons of lemon zest

1 cup of milk

3 tablespoons of lemon juice

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swimming - 20% football - 30%netball - 16% athletics - 24%did not know - 10%

In Class 6B, 40% of children came to school by car. There were 30 children in 6B. How many children came to school by car?

Louisa needed to save £120 for a trip in June. By April, she had saved 60%. How much had she saved?

Nameeta was also going on the trip. She had saved 75% by April. How much more had she left to save?

There were 150 children in Key Stage 2 at Pear Tree Primary School. Year 6 carried out a survey to find the favourite sport of each child in Key Stage 2. Look at their results and then answer the questions, which follow.

How many children liked swimming or football the best?

How many children knew which sport they liked the best?

How many children did not like netball the best?

How many children did not like netball or football the best?

Jawaad said, “The survey has found that 1 in every 5 children like swimming the best.” Was he correct? Explain your answer.

Solve problems involving the calculation of percentages

Year 6RATIO AND PROPORTION

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Oscar drew a pie chart to represent the proportions of ingredients for his favourite fruit punch, shown below:

What was the angle size of the orangeade section?

Which of the ingredients was represented by the section with a 90° angle?

The first angle Oscar drew was to represent the lemonade. What was the size of the reflex angle he had left?

What was the angle size of the section representing Oscar’s secret ingredient?

Oscar made 6 litres of his favourite punch. How many millilitres of each ingredient did he need?

How many more millilitres of his secret ingredient would Oscar need to make 4 litres than 3 litres?

One day, Oscar wanted to make 4.5 litres of punch, but he only had 400 ml of lemonade. How much more did he need?

Lucy only liked a third as much orangeade as Oscar. If she drew a pie chart, how many degrees would that section have?

Solve problems linking percentages, angles and pie charts


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orangeade – 60%

lemonade – 10%

lime juice – 25%

secret ingredient – 5%

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Solve problems involving scaling of shapes


The perimeter of a square was 12 cm. A new square 25 times bigger was drawn. What was the size of one side on the new square?

Each side of regular hexagon-shaped play area measured 3 m. A scale drawing was made, where 1 cm represented 0.5 metre. What was the length of each side on the scale drawing?

The area of rectangle A was 12 cm2. Rectangle B’s length and width were 3 times as big. What were the possible lengths and widths of rectangle B?

Year 6 made a scale model of their school. The scale used was 1: 50.

a) The actual dimensions of a window were 1 m x 0.75 m. What were the dimensions of the window on the model? Give your answer in centimetres.

b) The area of the yard (which was square) on the model was 1 m². What was the actual length of the yard?

c) There was also a triangular-shaped play area in the yard. One side on the model was 25 cm, another was 30 cm and its perimeter was 70 cm. What was the actual length of the sides of the play area? Give your answer in metres.

d) One wall in the school was 30 metres by 25 metres. What was the area of this wall on the scale model?

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ALGEBRA

These are all about algebra!

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Year 6ALGEBRA

Express missing number problems algebraically

Cucumber Café needed to put five more tea cups than coffee mugs on the counter. To make sure they always had the correct amount, this table was on the wall:

What are the missing numbers in the table?

Oscar had 12 football cards, but he lost some on the bus. When he got home, he only had 9 left. Which of the following formulae would show how many football cards (f) Oscar lost?

12 ÷ f = 912 – f = 912 + 9 = f

Daniel bought seven packets of biscuits. Each packet had the same number of biscuits. Daniel had 112 biscuits altogether. Which of the following formulae could be used to work out how many biscuits (b) were in each packet?

b = 7 x 112b = 112 + 7b = 112 ÷ 7

Write down a formula which could be used to solve this problem: There were 14 boxes. Each box had 12 sweets. How many sweets (s) were there altogether?

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Tea

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12

Coffee

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Year 6ALGEBRA

Enumerate possibilities of combinations of two variables

There were five chocolates in a bag. Some were plain (p) and some were nutty (n). Use the equation p + n = 5 and list all the possible combinations of plain and nutty chocolates in the box.

Year 6 had been learning about factors and multiples. They had the equation a x b = 36. What could a x b represent? List all the possibilities.

The next equation said 30 ÷ x = y. What could x and y represent? List all the possibilities.

Manny’s class had thirty children. The number of girls (g) was between ten and fifteen and the number of boys (b) was between fifteen and twenty. Use the equation g + b = 30 to find all the possible combinations of boys and girls.

There were some salt and vinegar (v) and cheese and onion (o) crisps in a box. The shopkeeper said that v x o = 42. List all the possible combinations of salt and vinegar and cheese and onion crisps in the box.

Zac was given this equation: A x B + 3 = 27. His teacher asked him to list all the possible combinations for A and B. What should his answer have been?

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MEASUREMENT

These are all about

measurement!

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Year 6MEASUREMENT

Alice’s garden was square-shaped. Each side measured 6 m. Work out the perimeter and area of Alice’s garden.

John had a piece of rectangular paper. The length was 26 cm and the width was 10 cm. Calculate the perimeter and area of the paper.

John said, “If I cut my paper in half along its length and then put the two new rectangles back together to make a rectangle 13 cm by 20 cm, the area will be the same as in the original rectangle.” Is he correct? Explain your answer.

A rectangle has an area of 36 cm2. What could the length and width of the rectangle be?

Two rectangular gardens each have a perimeter of 28 m. Each garden has different dimensions. What could the area of each garden be? Explain your answer.

Look at the compound shape shown below. Luke drew a rectangular shape with exactly the same area. What could be the length and width of Luke’s shape? (drawing not to scale)

4 cm

4 cm6 cm

12 cm

Solve problems, recognising that shapes with the same areas can have different perimeters and vice versa

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Year 6MEASUREMENT

Solve problems using formula for volume

Write down the formula to find the volume of a cuboid.

What is the volume of a cuboid with dimensions: 4.5 m x 3 m x 1.2 m?

A swimming pool had a length of 20 m, a width of 10 m and a depth of 6.5 m. What was the volume of the pool?

What is the volume of a cube with a height of 3 m?

The volume of a cuboid was 24 cm3. Its length was 3 cm and its width was 2 cm. What was the height of the cuboid?

The volume of a cuboid-shaped tub was 6 cm3. What could the possible dimensions be?

The swimming pool in Question 3 could also be converted to a diving pool. Five metres of its length could be lowered to a depth of 15 m. What was the volume of the pool when it was converted to a diving pool?

Some Year 6 children were collecting rainwater in a cuboid shaped container. The volume of the water they collected after a week was 72 cm3. The base of the container was 6 cm by 4 cm and its height was 5 cm. During the next week, the tub became full. What volume of water did they collect in the second week?

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Year 6MEASUREMENT

Elana is making a toy necklace with 5 mm cubes and cuboids measuring 4 mm x 5 mm x 6 mm. She uses 4 cubes and 3 cuboids. What is the total volume of the necklace?

Ebony is building a tower with cuboid-shaped bricks. The blue bricks are 30 mm x 50 mm x 20 mm. The red bricks are 10 mm x 20 mm x 20 mm. She uses 2 blue bricks and 5 red bricks. What is the total volume of the tower?

A cube has a volume of 27,000 mm3. A cuboid with dimensions of 30 mm x 30 mm x 20 mm is placed inside the cube. If another cuboid was placed in the cube so that there was no space left, what would be the dimensions of the other cuboid?

Year 6 were using a computer programme to estimate the volume of lakes. The programme measured the volume by adjusting the dimensions of the lakes into cuboid shapes. The class adjusted the dimensions of Lake A to be 7 km x 2 km x 1.5 km. Lake B had a length of 6.5 km, a width of 3 km and a depth of 1 km. What was the estimated difference in the volume of the lakes? Show your working.

Next, they used the programme to estimate the volume of icebergs. They adjusted the dimensions of an iceberg, so that it had a length of 1 km and a width of 0.5 km. The total volume of the iceberg was 1 km3. The height of the iceberg above the water was 0.8 km. What was the estimated depth of the iceberg below the water?

Solve problems by calculating and comparing the volume of cubes and cuboids extending to other units (mm³and km³)

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STATISTICS

These are all about statistics!

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STATISTICS Year 6

Interpret pie charts and use these to solve problems

Look at the favourite subjects of the pupils in Alyssa’s class and answer the questions below.

a) What percentage of the class like P.E. lessons best?

b) What fraction is this?

a) What fraction of the class like English best?

b) What percentage of the class like English best?

a) What percentage of the class did not choose maths as their favourite subject?

b) What fraction is this?

If there were 32 children in the class, how many children liked P.E. best?

If there were 29 children in Alyssa’s class, could this chart represent the class? Explain your answer.

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English

Maths135o

45o

180oP.E.

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STATISTICS Year 6

Interpret line graphs and use these to solve problems

Approximately, how many kilometres is equivalent to 20 miles?

Dayna’s family went on holiday in France. They saw a road which said, ‘Calais - 60 km’. Approximately, how many miles was it to Calais?

Tabassum went for a cycle ride. She cycled 5 miles. Approximately, how many kilometres did Tabassum cycle?

What is the total approximate distance in kilometres of 20 miles, 10 miles and 50 miles?

Ruthie lives 80 kilometres away from London. Joe lives 120 kilometres away from London. Approximately, how many more miles does Joe live away from London than Ruthie?

50

0 10 30 50 7020 40 60 80

40

30

20

10

0

kilometres

miles

Conversion graph: kilometres - miles

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How far had the car travelled by 11:30 am?

Look at the vertical (y) axis. How many miles do points A and B represent?

Estimate how many miles the car had travelled at 1 pm.

Five miles is equivalent to approximately eight kilometres. How many km did the car travel altogether?

How many miles had the car travelled between 12 noon and 3 pm? Approximately how many kilometres is this?

What time do you think the driver of the car had lunch? Explain your answer, using the line graph.

STATISTICS Year 6

Interpret line graphs and use these to solve problems

0

40

120

B

12 noon11 am10 am 1 pm 2 pm 3 pm

80

A

time

A graph to show how far a car travelled on a trip

distance(miles)

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STATISTICS Year 6

Calculate and interpret the mean as an average

What would you do to find Demi’s average score?

Who has the highest average score?

In the next test, Demi scored 39. What was her new average?

Linford scored 18, 17, 13, 19 and 18 in some mental arithmetic tests. What was his average score?

Each test had 20 questions. Linford says, “I want to increase my average score to 19, in the next test.” Is this possible? Explain your answer.

Lily wants to find out her average high jump over 6 jumps. Would she find the mean, mode or median?

The average of Frank’s five high jumps was 1.2 metres. He jumped 1 m, 1.2 m, 1.25 m and 1.15 m for his first four jumps. How high must he have jumped in his fifth jump?

The average for Zara’s 5 high jumps was 1.3 m. She jumped a different height for each of her jumps. How high might she have jumped on each jump?

Demi: 26, 32, 24, 39, 32, 21, 24, 32, 18, 32, 17

Test Scores

Jaz: 27, 29, 23, 24, 39, 24, 23, 33, 23, 30, 33

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© Copyright HeadStart Primary Ltd 2016 1

ANSWERS Year 6

Year 6: NUMBER - Number and place value

Page 1: 1)eighty nine 2) 522 3) £1299 4) ten thousand, seven hundred and fifty six 5) 53,611 6) £28,665 7) no; 2,415,361 8) nine million, four hundred and twenty three thousand, six hundred and three poundsPage 2: 1) forty two 2) 384 3) six hundred and seventy three 4) 3554 5) £2599 6) four hundred and forty thousand, nine hundred and twenty seven 7) no; one million, two hundred and eighty five thousand, three hundred and twenty two 8) 8,502,639Page 3: 1) 7432, 7452, 7482, 8452 2) 53,949 3), 532,998, 532,873, 532,200 4) 854,324 5) 1,263,892 6) 2,999,998 7) 5,841,989 8) 3,500,089, 5,389,000, 8,938,030, 10,900,500Page 4: 1) 1823; appropriate explanation 2) 5405, 5050, 5005, 5000 3) 7343 4) 33,735, 33,532, 33,275, 33,253 5) 562,453 6) 242,873, 442,837, 738,242, 873,422 7) 1,852,446 by 26,982 8) 9,899,999Page 5: 1) 3 2) 1 3) 5 4) thirty thousand 5) six thousand 6) 8 in £180,532; appropriate explanation 7) 770,000 8) 2 in £7,243,689; appropriate explanationPage 6: 1) 2 2) 7 3) 8000 4) no; 3000, appropriate explanation 5) no; 50,000, appropriate explanation 6) 2,000,000 7) 8 in £2,843,993; appropriate explanation 8) 500,000Page 7: 1) 4 2) 40,000 3) 8 4) 10,000 5) 600,000 6) 1,480,000 7) 8,400,000 8) 1000; appropriate explanationPage 8: 1) 500 2) 4400 3) £18,600 4) no; appropriate explanation 5) 300,000 6) 2,700,000 7) 2,000,000; appropriate explanation 8) £5,500,000Page 9: 1) -7°C 2) 13°C 3) -23°C 4) -17 5) 19 6) 4°C 7) no; appropriate explanation 8) -6 Page 10: 1) -13°C 2) -9°C 3) 16°C 4) 21°C 5) a) 304 cm b) 297 cm c) 7 cm d) 295 cmPage 11: 1) 2100 2) 5 3) 900 4) 300,000 5) no; 21, appropriate explanation 6) no; 213,000 appropriate explanation 7) two million, eight hundred and thirty two thousand, two hundred and forty nine pounds 8) 6,943,999Page 12: 1) 700 2) -4 3) 372,489 4) 980,000 5) 308,257 6) 1O°C 7) 1000 8) no; appropriate explanationPage 13: 1) 7421, 7834, 7891, 7982 2) a) c b) 11 m 3) -7°C 4) 139,296 5) 2,870,000 6) nine million, eight hundred and seventy three thousand and thirty three

Year 6: NUMBER - Addition, subtraction, multiplication and division

Page 14: 1) 552 2) 810 3) 270 4) 3000 5) £7030 6) £19,125 7) 56,025 8) 330,358Page 15: 1) 368 2) 744 3) 1876 4) 2496 5) 13,379 6) 7882 7) 118,272 8) 71,050Page 16: 1) 6 2) 14 3) 23 4) 8 5) 12 6) 126 7) 126 8) 129Page 17: 1) 7 2) 24 3) 31 4) 22 5) 154 6) £28 7) 254; 2 left over 8) 27

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Year 6

Primary

problems for all appropriate objectives of the 2014 Curriculum

This book includes:

built-in differentiation

editable versions of the pages on a CD-ROM

one-step, two-step and multi-step problems

suggested whole school procedure for problem solving

MATHEMATICS CURRICULUM

Word Problems

T. 01200 423405E. [email protected]

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