white rose maths · 2021. 1. 30. · expression – a collection of terms involving mathematical...

Key Stage 3Student Book 1

White Rose Maths

Collin

sW

hite R

ose Math

s Key Stage 3 Student B

ook 1

Ian

Davies a

nd

Ca

roline H

am

ilton Ian Davies and Caroline Hamilton

The mastery programme for KS3 that lays the foundations for GCSE 9–1 success.

• Master topics and boost confi dence with concepts broken down into small steps

• Learn from visual examples and models that ensure a deep understanding of maths

• Build your fluency, reasoning and problem-solving skills with purposeful practice

978-0-00-840089-7 978-0-00-840090-3

8424_SB1 White Rose Maths Secondary.indd 1 18/12/2020 10:13

Support all students to become confident and capablemathematicians with a proven approach to mastery

Student Books

Follow the White Rose Maths schemes of learning for Y7-9 wheremathematical concepts are broken down into small steps to ensure allstudents make progress

Create a smooth and successful transition from KS2 to KS3

Support mastery with pictorial representations that ensure a deepunderstanding of maths

Ensure mathematical skills are interleaved and revisited in othercontexts throughout to aid memory and make connections

Build fluency, reasoning and problem-solving skills withpurposeful practice

Carefully planned and sequenced to support the White RoseMaths approach

The mastery programme for KS3 that isperfectly aligned to the White Rose Mathsschemes of learning and lays the foundationfor GCSE 9–1 success

collins.co.uk/WhiteRoseMaths

March 2021 May 2021 September 2021

Teacher Guides


Understand the White Rose Maths pedagogy behind the lessons

Support all teachers in the department to develop deep subjectknowledge and confidence to deliver high-quality maths lessons

Provide rationale for the lessons and practice exercises in thecorresponding Student Book with additional guidance on the models,representations and tasks

Carefully sequenced and planned lessons to support mastery and beinclusive for all students

Collins is delighted to be working in collaboration with White Rose Maths to create the only official White RoseMaths textbooks for Key Stage 3

March 2021 May 2021 September 2021

iii

Introduction v

Block 1 Sequences 1

1.1 Continuing sequences: shapes 21.2 Continuing linear sequences 81.3 Continuing non-linear sequences 151.4 Representing sequences 221.5 Finding missing terms in sequences 32

Block 2 Algebraic notation 40

2.1 Inputting to one-step function machines 412.2 Working backwards with one-step function machines 472.3 Using simple expressions 532.4 Inputs and outputs with two-step function machines 602.5 Using multi-step expressions 682.6 Decoding expressions 75

Block 3 Equality and equivalence 83

3.1 Using the equals sign 843.2 One-step linear equations 933.3 Like and unlike terms 1013.4 Using the equivalence sign 105

Block 4 Place value and ordering 114

4.1 Reading and writing integers 1154.2 Comparing and ordering integers 1234.3 Using ordered lists: the range and the median 1314.4 Rounding: powers of 10 1394.5 Understanding decimals 1454.6 Rounding to one significant figure 1544.7 Standard form: large numbers 1604.8 Standard form: small numbers 165

Block 5 Fractions, decimals and percentages 171

5.1 Tenths and hundredths 1725.2 Converting fractions and decimals 1805.3 Using percentages 1895.4 Pie charts 1975.5 Understanding fractions below 1 2045.6 Linking fractions and division 2145.7 Exploring fractions greater than 1 221

Block 6 Addition and subtraction 228

6.1 Linking addition and subtraction 2296.2 The column method of addition 2406.3 The column method of subtraction 2516.4 Addition and subtraction problems 2626.5 Adding and subtracting in standard form 278

Block 7 Multiplication and division 286

7.1 Linking multiplication and division 2877.2 Multiplying with powers of 10 2967.3 Multiplication: short and long 3067.4 Division: short and long 3187.5 Using the mean 327

Contents

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iv

Contents

7.6 Order of operations 3337.7 More complex multiplication and division 339

Block 8 Fractions and percentages of amounts 347

8.1 Working with fractions of amounts 3488.2 Working with percentages of amounts 3558.3 Over 100% 363

Block 9 Directed number 370

9.1 Understanding directed number 3719.2 Adding and subtracting directed numbers 3789.3 Multiplying and dividing directed numbers 3839.4 Substitution with directed numbers 3909.5 Two-step equations 3959.6 Order of operations 4009.7 Powers and roots 406

Block 10 Fractional thinking 413

10.1 Converting fractions 41410.2 Adding and subtracting fractions (1) 42010.3 Adding and subtracting fractions (2) 42810.4 Adding and subtracting fractions (3) 43510.5 Fractions and algebra 442

Block 11 Construction and measuring 451

11.1 Geometric notation 45211.2 Working with angles 46111.3 Lines and shapes 46811.4 Constructing shapes 48111.5 Pie charts 490

Block 12 Geometric reasoning 500

12.1 Points and lines 50112.2 Angles in triangles 50912.3 Angles in quadrilaterals 51712.4 Angles in polygons 52512.5 Angles and parallel lines 53012.6 Geometric proof 537

Block 13 Developing number sense 543

13.1 Making connections with integers 54413.2 Making connections with fractions and decimals 55213.3 Making connections and estimating 55813.4 Using known facts 564

Block 14 Sets and probability 571

14.1 Representing sets 57214.2 Intersections and unions 58014.3 Introducing probability 58914.4 Calculating probability (1) 59614.5 Calculating probability (2) 606

Block 15 Primes and proof 612

15.1 Factors and multiples 61315.2 Special numbers 61915.3 HCF and LCM 62515.4 Using prime numbers 63115.5 Conjectures and proof 640Glossary 645Answers 649

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40

In this block, I will learn…

how to find inputs and outputs of one-step function machines

15 × 7 ?

2 Algebraic notation

how to represent expressions in a variety of ways

4d + 3

d d d d 3

1 1 1

the meaning of algebraic expressions

3x x + 3 30 – x

30 – 2x x2 x3

how to substitute numbers

into algebraic expressions

When a = 7

3a – 2 = 3 × 7 – 2

= 21 – 2 = 19

When b = 4.6,

b 2 = 4.6 × 4.6

= 21.26

how to generate sequences given an algebraic rule

When n = 1, 5n – 2 = 5 × 1 – 2 = 5 – 2 = 3When n = 2, 5n – 2 = 5 × 2 – 2 = 10 – 2 = 8When n = 3, 5n – 2 = 5 × 3 – 2 = 15 – 2 = 13When n = 4, 5n – 2 = 5 × 4 – 2 = 10 – 2 = 18

how to find inputs and outputs

of a series of function machines

3.5 + 6 square ?

? ÷ 3 – 7 41

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53

2.3 Using simple expressions

Small steps ■ Use diagrams and letters to generalise

number operations

■ Substitute values into single operation expressions

■ Generate sequences given an algebraic rule

Key words

Expression – a collection of terms involving mathematical operations

Substitute – to replace letters with numerical values

Evaluate – to work out the numerical value of

Are you ready?

1 Explain the meaning of these expressions

a 6x b 6 + x c x6

d x – 6 e 6 – x

2 Describe these sequences.

a 10 15 20 25 30 …

b 4 40 400 4000 …

3 Explain the difference between y2 and 2y.

4 Find the missing numbers in these bar models.

a 10

? 3

b 60

? ?

Models and representationsCubes and place value counters

1 1 1 1 1

can be used to represent “x + 5”.

If you want to work out the value of x + 5 when x = 4, you can replace with 4 ones.

= 1 1 1 1

So 1 1 1 1 1 =

1 1 1 1 1 1 1 1 1

This shows x + 5 = 9

Bar models

a a a a

This represents 4a.

The value of 4a when a = 20 can be shown as

10 10 10 10 10 10 10 10

so 4a = 80


54


Example 1

Work out the value of each of these expressions when n = 40.

a 3n b n + 90 c n2

d n – 27

a 3 × 40 = 120

b 40 + 90 = 130

c 40 ÷ 2 = 20

d 40 – 27 = 13

3n means “multiply the value of n by 3”. You can think of this in several ways.

n n n

40 40 40 40 + 40 + 40 = 3 × 40 = 120

n n n

10 10 10 10 10 10 10 10 10 10 10 10

40 + 40 + 40 = 3 × 40 = 120

n + 90 means add 90 to the value of n.

n 90

40 90

40 + 90 = 130

n 10 10 10 10 10 10 10 10 10

10 10 10 10 10 10 10 10 10 10 10 10 10

90 + 40 = 130

n2

means “divide the value of n by 2”.

n 4040 ÷ 2 = 20

n2

n2

20 20

n – 27 means subtract 27 from the value of n.

Start with n

n

Subtract 27

n27

This section represents n – 27

When n = 40

40

27

This section represents 40 – 27 = 13

Do you think it matters whether the bar models are to scale?


55


Example 2

Substitute d = 11.4 into each of these expressions.

a d – 4.17 b 6d c d3

d 58 – d

a d – 4.17 = 11.4 – 4.17

= 7.23

b 6d = 6 × 11.4

= 68.4

c d3 = 11.43

= 3.8

d 58 – d = 58 – 11.4

= 46.6

Substitute means replace the letter with the given number.

You could use bar models or other representations to help you.

Replace d with 11.4 and then subtract 4.17

Remember that 6d means “multiply d by 6”.

Replace d with 11.4 and then multiply by 6

Remember that d3

means “divide d by 3”.

Replace d with 11.4 and divide by 3

Replace d with 11.4 and subtract from 58

Practice 2.3A

1 What expression is shown by each question mark?

a ?

n n n n nb ?

q q q

c x? ? ?

d ?

t 7

e t? 12

f 30

y ?

2 Draw bar models to represent these expressions.

a b + 4 b b – 4 c b4

d 4b

3 Substitute b = 12.5 into each of these expressions.

a b + 4 b b – 4 c b4

d 4b

4 Evaluate these expressions when g = 400

a g10

b 10g c

g100

d 100g e

g20

f 20g

g g

200 h 200

g i g

400 j 400

g k g

1000 l 1000

g


56


5 Work out the value of each of these expressions when c = 5.2

a c + 5.2 b c5.2

c c – 5.2 d 5.2c

e 5.2 – c f 5.2c g 5.2 + c

6 a Match the expressions that are equal when x = 10

10x

x10

x + 10

10 – x

10x

10 + x

x – 10

b Will any of the pairs of expressions in part a be equal for all values of x?

7 a What does x2 mean?

b Work out x2 for each of these values of x.

i x = 8 ii x = 80 iii x = 0.8 iv x = 800

8 a What does t mean?

b Work out t for each of these values of t.

i t = 324 ii t = 3.24 iii t = 0.004 iv t = 400

9 Are the expressions on each pair of cards always equal, sometimes equal or never equal in value?

a

p + 7

7 + p

b

5p

p5

c

p – 2

2 – p

What do you think?

1 There is only one value of n for which the following expressions are equal. Find the value.

4n

n4

2 How many values of n can you find so that these pairs of expressions are equal in value?

a

n

n2

b

2n

n2

c

n + 3

2n

d

n + 3

3n

3 p5

and 5p are equal when p = 0

a Show that Chloe is wrong.

b Find a value for p for which p5

and 5p are equal.

If you substitute the whole numbers in orderinto the same expression, you get a sequence.

Remember that a sequence is a list of items in a given order, usually following a rule.


57


Example 3

a Substitute n = 1, n = 2, n = 3, n = 4 and n = 5 into the expression 4n and describe the sequence formed by your answers.

b What is the tenth term of the sequence?

a When n = 1, 4n = 4 × 1 = 4

When n = 2, 4n = 4 × 2 = 8

When n = 3, 4n = 4 × 3 = 12

When n = 4, 4n = 4 × 4 = 16

When n = 5, 4n = 4 × 4 = 20

The sequence formed is 4, 8, 12, 16, 20 – the first five multiples of 4.

The difference between the terms in the sequence is 4, and the sequence is linear and ascending.

b 4 × 10 = 40

If you carried on substituting whole number values of n, you would get more multiples of 4

For the 10th term in the sequence, n has the value 10

Can you remember what “difference” means? Can you remember what a linear sequence is?

Example 4

a Substitute n = 1, n = 2, n = 3, n = 4 and n = 5 into the expression 4 – n and describe the sequence formed by your answers.

b What is the tenth term of the sequence?

a When n = 1, 4 – n = 4 – 1 = 3

When n = 2, 4 – n = 4 – 2 = 2

When n = 3, 4 – n = 4 – 3 = 1

When n = 4, 4 – n = 4 – 4 = 0

When n = 5, 4 – n = 4 – 5 = –1

The sequence formed is 3, 2, 1, 0, –1

The difference between the terms in the sequence is 1 and the sequence is linear and descending.

b 4 – 10 = –6

n is often used in expressions to represent sequences.

It is OK to get a negative answer. We will look at negative numbers in Block 9.

For the 10th term in the sequence, n has the value 10


58


Practice 2.3B

1 a Substitute n = 1, n = 2, n = 3, n = 4 and n = 5 into each of these expressions.

i n + 4 ii n + 0.4 iii 40 + n iv 40 – n

b Describe the sequences formed by your answers.

What’s the same and what’s different?

c Find the 10th term and the 100th term of each sequence in part a.

2 a Substitute n = 1, n = 2, n = 3, n = 4 and n = 5 into each of these expressions.

i 6n ii n + 6 iii 6n iv n

6b Describe the sequences formed by your answers.

What’s the same and what’s different?

c Find the 10th term and the 100th term of each sequence in part a.

3 a Substitute n = 1, n = 2, n = 3, n = 4 and n = 5 into the expression n2.

b Describe the sequence formed by your answers.

c Find the 10th term and the 100th term of the sequence in part a.

d Now substitute n = 1, n = 2, n = 3, n = 4 and n = 5 into the expression n3.

e Describe the sequence formed by your answers.

f Find the 10th term and the 100th term of the sequence in part d.

g What’s the same and what’s different about the sequences formed by substituting into n2 and n3?

4 a Find the 10th term of the sequences given by these expressions.

i n + 10 ii n – 10 iii 10n iv n10

v 10n vi 10 + n

b Predict whether the 100th term will be greater than or smaller than the 10th term for each sequence.

c Now work out the 100th term of each sequence in part a to see if you were right.

5 a Substitute n = 1, n = 2, up to n = 10 into the expression 2n.

b What do you notice about the last digits of the numbers in this sequence?

c Can you predict the last digit of the 20th term of the sequence?

d Can you predict the last digit of the 20th term of the sequence given by 10n?

What do you think?

1 a Find a value of n for which Abdullah is correct.

b Find a value of n for which Abdullah is incorrect.

c Which sequence is greater more often?

2 Compare the sequences given by 2n and n2.

2n means 2 multiplied by itself n times, for example 24 = 2 × 2 × 2 × 2

The terms of the sequence

3n are greater than those of the sequence n3


59


Consolidate – do you need more?

1 Draw a bar model to represent each of these expressions.

a x + 5 b x − 3 c 5x d 50 − x

2 Substitute k = 49 into each of these expressions.

a k + 81 b 81 − k c 7k d k7

e k2 f k

3 Find the value of p40

when

a p = 4 b p = 0 c p = 400 d p = 0.4 e p = 40

4 a Find the first four terms of the sequences given by each of these expressions.

i n + 8 ii n8

iii 8n iv 8 − n

b Which of the sequences are linear?

5 Find the difference between the 15th terms of each pair of sequences.

a 60 − n, 60n b n + 40, 40 + n c 2n, n2

d 30n , n

30 e 20 + n, 20 − n

Stretch – can you deepen your learning?

1 Will any of the terms of the sequence given by the rule 2n end in the number 5? Explain how you know.

2 Will any of the terms of the sequence given by the rule n2 end in the number 5? Explain how you know.

3

Is Seb right? What other patterns can you see in the last digits of the numbers in the sequence?

Reflect

1 Explain the meaning of the words

substitute

evaluate

expression

2 If you are given an algebraic rule, explain how you can find the value of any term in the sequence.

The only even numbers in the sequence given by the

rule n2 end in 4 or 6


251251

Key words

Difference – the result of subtracting a smaller number from a larger number

Debit – an amount of money taken out of an account

Range – the difference between the greatest value and the smallest value in a set of data

Small steps ■ Use formal methods for subtraction

of integers

■ Use formal methods for subtraction of decimals

■ Solve problems in the context of perimeter

■ Solve financial maths problems

■ Choose the most appropriate method: mental strategies, formal written or calculator

6.3 The column method of subtraction

Are you ready?

1 Use the fact that 8 – 5 = 3 to work out

a 80 – 30

c 0.08 − 3100

b 0.8 – 0.3

d thirty thousand less than 80 000

2 Work out these subtractions. Compare the answers.

a 7 – 1 b 7 – 0.1 c 7 – 0.01 d 7 – 0.001 e 7 – 0.0001

3 Solve these equations.

a a + 8 = 20 b 0.6 + b = 0.9 c 600 = c + 200

4 Use a mental method or jottings to work out

a 600 – 100

e 600 – 200

b 600 – 10

f 600 – 250

c 600 – 1

g 600 – 251

d 600 – 0.1

h 600 – 251.1

Models and representationsBase 10 blocks

= 247

Place value counters

100 100 10 10

10 10

1 1 1 1

1 1 1= 247

1 1 1

1 1= 5.27

0.1 0.1 0.01 0.01 0.01 0.01

0.01 0.01 0.01

Place value charts

Ones tenths

1 1 1 1 1

1 1 1

0.1 0.1 0.1 0.1 0.1

0.1 0.1

Column method

O t

8 7

− 2 3

6 4


252


Example 1Work out these subtractions. Compare the calculations.

a 67 – 32 b 67 – 38 c 6.7 – 3.2 d 0.67 – 0.38

a 67 – 32

Tens Ones

10 10 10 10 10

10

1 1 1 1 1

1 1

So 67 – 32 = 35

T O

6 7− 3 2

3 5

b 67 – 38

Tens Ones

10 10 10 10 10

10

1 1 1 1 1

1 1

Tens Ones

10 10 10 10 10 1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1

Tens Ones

10 10 10 10 10 1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1

So 67 – 38 = 29

This calculation does not involve exchanging or regrouping.

There are enough tens and enough ones to subtract the numbers from each column

This time you want to subtract 8 ones but there are only 7 ones in the ones column.

You need to exchange one of the tens for 10 ones.

This uses the fact that you can partition 67 as 50 + 17 instead of 60 + 7

You can now subtract 38 by removing 3 tens and 8 ones.

You can also work this out using the column method.

T O

6 7

− 3 8

2 9

5 1 67 is partitioned into 50 and 17


253


c 6.7 – 3.2

Ones tenths

1 1 1 1 1

1

0.1 0.1 0.1 0.1 0.1

0.1 0.1

So 6.7 – 3.2 = 3.5

d 0.67 – 0.38

Ones tenths hundredths

0.1 0.1 0.1 0.1 0.1

0.1

0.01 0.01

0.01

0.01

0.01

0.01 0.01


0.1 0.1 0.1 0.1 0.1 0.01 0.01

0.01

0.01

0.01

0.01 0.01

0.01

0.01

0.01

0.01 0.01

0.01

0.01

0.01

0.01 0.01


0.1 0.1 0.1 0.1 0.1 0.01 0.01

0.01

0.01

0.01

0.01 0.01

0.01

0.01

0.01

0.01 0.01

0.01

0.01

0.01

0.01 0.01

So 0.67 – 0.38 = 0.29

This decimal subtraction is similar to a – there is no need to exchange.

You subtract decimals in the same way as you subtract integers.


O t6 7

− 3 23 5

This decimal subtraction is similar to b. You need to make an exchange between the tenths and the hundredths columns.

The relationship between adjacent columns is the same as for integers, so you can use the same the process.


O t h0 6 7

− 0 3 80 2 9

6 tenths and 7 hundredths is partitioned into 5 tenths and 17 hundredths.

5 1


254


Example 2Find the difference between 812 and 357.

H T O

8 1 2− 3 5 7

4 5 5

1 17

H T O

4 5 5+ 3 5 7

8 1 21 1

Remember that “difference” means the result of subtracting the two numbers.

You need to exchange a ten for 10 ones and a 100 for 10 tens.

You can check the answer by adding 455 to 357

As 455 + 357 = 812, it means that 812 – 357 = 455, so the answeris correct.

You can always check addition by subtraction and subtraction by addition as these are inverse operations.

Example 3Work out

a 47 – 23 b 47 – 2.3 c 47 – 0.23

Use the column method for each calculation.

a 4 7− 2 3

2 4

b 4 7 0− 2 3

4 4 7

6 1

c 4 7 0 0− 0 2 3

4 6 7 7

6 1 9 1

You do not need to exchange as 7 > 3

Write 47 as 47.0 and line up the decimal points.As 0 < 3, you need to exchange 1 one for 10 tenths.

Write 47 as 47.00 and line up the decimal points.Then make two exchanges.

0


255


Practice 6.3A

1 Use the column method to find the difference between each pair of numbers.

a 63 and 21

c 603 and 201

e 603 and 207

g 630 and 27

i 630 and 217

b 63 and 27

d 603 and 210

f 630 and 7

h 630 and 57

j 630 and 347

Which questions were most difficult? Why? Would it have been easier to use a mental method for any of the questions?

2 Work out

a 8.7 – 3.4 b 8.7 – 3.8 c 8.17 – 3.4 d 8.07 – 3.4 e 8.7 – 3.04

3 Find the missing numbers in these diagrams.

a

165

427

c 76.528.7

b

27.2

− ?

85.6

d 472 386584

4 Work out

a 6.72 − 810

b 6.72 − 8100

c 6.72 − 35

d 6.72 − 34

5 Given that w = 2.8, x = 0.28, y = 2.08 and z = 8.2, find the value of

a z − w

d y − x

b w − x

e z − x

c w − y

f z − y

6 a Subtract seven thousand five hundred from 9236.

b Subtract seven thousand and fifty from 9236.

c Subtract seven thousand and five from 9236.


256


What do you think?

1

Explain why Jackson is wrong.

2

134

416

219

Compare these methods for finding the missing number.

■ Subtract 134 from 416 and then subtract 219 from the result.

■ Add 134 and 219 and then subtract the result from 416.

Which method do you prefer? Give a reason why.

3 Solve these equations.

a a + 307 = 410 b b + 370 = 410 c c + 37 = 410 d d + 3.7 = 410

4 Find the value of each letter.

a w + 35.2 = 71.7 − 18.3 b x − 294 = 2000 − 378

c 582 − y = 217 + 159 d 13.8 − z + 6.7 = 17.4 − 2.53

Many real-life problems involve subtractions, or additions and subtractions.

Example 4Complete this bank statement by filling in the balances.

Date Description Credit(£)

Debit(£)

Balance (£)

Mar 1 Opening balance 846.72

Mar 2 Rent 520.00 A

Mar 6 Car service 88.53 B

Mar 7 Electricity bill 74.87 C

H T O t h

8 4 6 7 2− 5 2 0 0 0

3 2 6 7 2

Remember: debits are amounts that are taken out of an account.

To find the balance A after the rent is taken out of the account, subtract £520.00 from £846.72

You can use the column method.

As no exchange is needed, you could also do this mentally.

326.72

− 500

846.72346.72

− 20

You cannot find the difference between 672 and 921 because 921 is greater than 672


257


H T O t h

3 2 6 7 2− 8 8 5 3

2 3 8 1 9

H T O t h

2 3 8 1 9− 7 4 8 7

1 6 3 3 2

To find the balance at B, subtract £88.53 from £326.72

To find the balance at C, subtract £74.87 from £238.19

Example 5Find the length of the unknown side of this triangle.

7.6 m

8.9 m

Perimeter = 20 m

x m

7.6 + 8.9 = 7.6 + 9 – 0.1

= 16.6 – 0.1

= 16.5

20 – 16.5 = 3.5

The missing side is 3.5 m

First, find the total of the two sides you know.

You could use the column method, but because the number you are adding is nearly a whole number, compensation is quicker.

Then subtract the total of the first two sides from the perimeter.

You could use the column method or counting on.

16.5

+ 3

20

+ 0.5

17

Give the unit with the final answer.

2 1 1 1 6 1

1 1 7 1


258


Practice 6.3B

1 Ali buys old furniture and then resells it. The table shows how much he paid for some items and how much he sold them for. Find the profit or loss he makes on each item.

Item Bought for Sold for

a Table £35 £80

b Chest of drawers £20 £60

c Sideboard £110 £47

d Chair £12 £40

2 All of these shapes have perimeter 50 cm. Find the lengths of the missing sides. All lengths are given in centimetres.

a

19 17

a

d

18.9

17.3d

b 14.2

14.3

b

12.9

e7.9

e

7.9

7.9

11.6

c

18

c

18

c

3 Copy and complete this bank statement by filling values A to E.


Debit(£)

Balance(£)

May 1 Opening balance 546.23

May 2 Wages 600.00 A

May 6 Phone bill 18.65 B

May 7 Shopping C 1038.86

May 11 TV subscription 39.99 D

May 13 Refund E 1043.87

4 Find the range of each set of values. You studied the range in chapter 4.3.

a 12 15 15 18 23 27 31 40

b 127 cm 141 cm 183 cm 152 cm 160 cm

c 8.13 7.2 6.9 9.8 12.4 18.7 21.3

d 3.07 5 2.98 1.64 3 1.92

Think carefully about whether you need to add or subtract, and whether you can work the amounts out mentally.


259


5 a Beca is 146.3 cm tall. Her sister is 12.5 cm shorter. How tall is Beca’s sister?

b Samira has £1200 to spend on a holiday. The flights cost £385.28 and accommodation costs £468. How much money does Samira have left?

c A book has 314 pages. Jackson reads 168 pages of the book. How many more pages does he have to read?

d June 20th is the 171st day of 2022. How many days are there after 20th June in 2022?

6 Decide whether to use a mental or written method for these calculations. Then work out each answer using your chosen method.

a 5 billion − 2.6 billion

d 3.61 − 2.05

b £137.56 − £19.99

e 20 000 − 13 568

c £74.68 − £45.23

f 2400 − 657

What do you think?

1 Explain the mistakes that have been made in each of these calculations.

a 4 9 2 b 9 4 c 6

− 3 5 − 2 6 3 – 3 0 7

4 5 7 7 2 3 3 0 7

2 Find the missing numbers in these subtractions.

a 4 5 b 3 c 3− 2 − 3 6 − 8 2

2 8 6 2 3 5 8 7

3 a = 82, b = 39 and c = 14.5

Work out the value of

a a − b − c b a − c − b c a + b − c d a − b + c

e a − c + b f a + c − b g a − (b + c)

Which calculations give the same answers? Why is this?

8 1


260


Consolidate – do you need more?

1 a Find the difference between each pair of numbers.

i Hundreds Tens Ones

100 100 100 100 10 10 10 10 10

10 10 10

1 1 1 1 1

1

100 100 100 1 1 1 1

ii Hundreds Tens Ones

100 100 100 100 10 10 10 10 10

10 10 10

1 1 1 1 1

1

100 100 10 1 1 1 1 1

1 1 1

iii 486 and 391

v 486 and 74

iv 486 and 52

vi 486 and 658

b Work out

i 45 – 23

iv 435 – 83

ii 45 – 28

v 485 – 263

iii 435 – 218

vi 405 – 263

2 Find the missing number in each of these diagrams.

a

687

945b

82.3 101.8

c 12.3 7.865.93

3 a Subtract eighty-five from 6837

b Subtract eight hundred and five from 6837

c Subtract five thousand and eight from 6837

d Subtract 6837 from eight thousand and five.

e Subtract 6837 from eight thousand and fifty.

f Subtract 6837 from eight thousand five hundred.

4 a An isosceles triangle has two equal sides that are 47 mm long. The perimeter of the triangle is 125 mm. Find the length of the third side.

b The length of a rectangle is 6.7 cm. Its perimeter is 22.6 cm. Find the width of the rectangle.


261


5 Find the missing values A to F in this bank statement.


Debit(£)

Balance(£)

Aug 18 Opening balance 136.75

Aug 23 BACS transfer A 386.75

Aug 25 Bookshop 9.99 B

Aug 27 Withdrawal 75.00 C

Aug 29 Interest 1.47 D

Aug 30 Gas bill 56.88 E

Aug 31 Wages F 1011.80

Stretch – can you deepen your learning?

1 a Huda says, “10 – 7.43 is the same as 9.99 – 7.42”. Explain why Huda is right.

Use Huda’s method to find

b 10 – 7.43

c 20 – 15.72

d the change from a £50 note when you spend £34.83

2 In how many ways can you use the digits 0 to 9, once only, to produce a correct subtraction with this format?

−

(You cannot use 0 as the first digit of a number.)

Reflect

1 How can you tell whether a problem involves addition or subtraction?

2 How do you decide whether to use a mental method, a written method or a calculator to work out the answer to a subtraction?

3 What’s the same and what’s different about subtracting integers and decimals?

Are there any values you can work out mentally?


I have become fluent in…■ Xxxx

I have developed my reasoning skills by…■ Xxxxx

I have been problem-solving through…■ Xxxxx

285

I have become fluent in…■ Using the column

method for addition

■ Using the columnmethod for subtraction

■ Reading and interpretingtables, timetables,charts and graphs

■ Calculating with time

■ Adding and subtractingwith standard form

I have developed my reasoning skills by…■ Extracting relevant

information from tables and charts

■ Choosing whether to doa calculation usingmental or writtenmethods

■ Deciding whether aproblem needs to besolved using addition,subtraction or both

I have been problem-solving through…■ Designing tables and

frequency trees to find required information

■ Working out profit, loss,credit and debit

■ Working out perimetersand missing lengths inshapes

■ Representing equationsin a variety of forms

6 Addition and subtractionChapters 6.1–6.5

Check my understanding

1 There are 16 842 adults and 7548 children at a football match.

a How many people are there altogether at the football match?

b How many more adults are there than children?

2 Find the third term of the linear sequence that starts 36.7, 52.3 …

3 A merchant has £600. She sells her stock, making a profit of £784. She then buys more stock at a cost of £927. How much money does the merchant have now?

4 The perimeter of the triangle is equal to the perimeter of the parallelogram.

Work out the value of a.

5 Will any of these three calculations be equal in value? Explain your reasoning.

1000 – 567 999 – 566 999 – 568

6 Beca thinks 8 – 5.23 = 3.23. Show that Beca is wrong.

7 Represent the information in this two-way table as a completed frequency tree.

Prefer tea Prefer coffee No preference Total

Over 50s 27

Under 50s 46 17 120

Total 40 300

24.6

18.7

a

18.8

14.414.4

18.8


Key Stage 3White Rose Maths

Ian Davies and Caroline Hamilton

Behind the QuestionsTeacher Guide 1

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ose Math

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d the Q

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The mastery programme for KS3 that is perfectly aligned to the White Rose Maths schemes of learning and lays the foundations for GCSE 9–1 success.

• Provides the rationale for the lessons andpractice exercises in the corresponding studentbook with additional guidance on the models,representations and tasks

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Evaluation packStudent Book 1 Student Book 2Student Book 3 Teacher Guide 1 Teacher Guide 2 Teacher Guide 3 Complete Digital

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