using the tasks i see reasoning –y5 tasks to inspire mathematical thinking using the tasks i see...

CONTENTS

I SEE REASONING – Y5

Contents

Introduction

Using the Tasks

Place Value

Decimals

Negative Numbers

Rounding

Roman Numerals

Addition

Subtraction

Addition and Subtraction

Multiplication

Division

Multiplication and Division

Fractions

Fractions + – × ÷

Fractions, Decimals, Percentages

Measurement

I SEE REASONING

CONTENTS


Contents

Perimeter, Area, Volume

Angle

Shape

Position, Direction, Coordinates

Statistics

Answers

I See Maths Resources

I SEE REASONING

INTRODUCTION

I SEE REASONING – Y5Tasks to inspire mathematical thinking

Introduction

For use by the purchasing institution only. Copyright I See Maths ltd. Circulation is prohibited.

I See Reasoning – Y5 helps children build a strong

conceptual understanding of mathematics and gives a

wide range of tasks for deepening and extending

thinking.

Children grasp key concepts and ideas using Part-

Complete Examples and Spot the Mistakes questions.

Mathematical patterns and relationships are

highlighted by sequences of Small Difference Questions

and I know… so… tasks. Rank by Difficulty and Different

Ways prompts open up great debates, whilst

challenges are extended through Multi-Skill and How

Many Ways? questions.

The resource is comprised of 362 varied tasks, linked to

all different areas of England’s Year 5 mathematics

curriculum. This corresponds to US Grade 4 and

Australia Year 5. Screenshots of tasks can be used

within presentations or questions can be printed and

given to children.

I hope that I See Reasoning – Y5 enriches the maths

learning in your classroom, helping all children to build

knowledge and make rich connections. I also hope

that it gives you lots of great classroom moments!

Gareth Metcalfe

I SEE REASONING

USING THE TASKS


Using the TasksI See Reasoning – Y5 will help teachers to plan coherent

sequences of lessons. Some tasks help children to grasp key

ideas; others break down calculations. There are prompts for

generating discussion and challenges for deepen learning. This

section introduces the most common question types and

considers how they might be used.

Contexts: Context examples are used to connect maths

concepts to real world scenarios. For example, the Negative

Numbers section looks at contexts where negative numbers

are/are not used.

Explain the Mistakes: Common mistakes are shown so children

can understand key differences between correct and incorrect

thinking. For example, in the Angle section, children explain

mistakes when measuring an angle with a protractor.

Which Answer? Agree or Disagree? and Correct or Incorrect:

Here, children have to spot correct responses and explain

mistakes that have been made, like in the Multiplication and

Division example where different ways to complete a number

sentence are shown.

Estimate: These tasks show the thought process behind making

relatively accurate estimates, like in this Multiplication example.

Next Step and Part-Complete Examples: These tasks break

calculation procedures down into small steps. Children’s

attention is drawn to the key next step, like in this Division

example, or to complete the rest of a question like this

Multiplication task.

I SEE REASONING

USING THE TASKS


Using the TasksSmall Difference Questions and I know… so… In these tasks,

there are small differences between questions in a sequence.

This draws children’s attention to key patterns and ideas. If a

question changes but the answer stays the same, how can this

be explained? This technique is shown in the Rounding section.

Explain: For these prompts, children consider relationships,

explain patterns and make generalisations. Sentence stems may

be used to scaffold responses. This technique is used in the

Roman Numerals section.

Different Ways and Rank by Difficulty: These tasks encourage

children to use different calculation techniques and compare

strategies, deepening conversations. In this Adding Fractions

example, children are encouraged to use a range of methods.

Extend and Multi-Skill: To answer these questions, children have

to work through a range of steps or use knowledge from

different parts of the maths curriculum. This is shown in the Angle

section, with questions involving skills about reading clocks.

How Many Ways? The ultimate challenge on these tasks is for

children to find all the possible answers to the question. For this, a

system is needed to know that every solution has been

identified. There is an example in the Place Value section.

I SEE REASONING

How can 2410 be made using 10, 100 and 1000 counters?

What are the fewest counters that can be used?

What are the most counters that can be used?

Find a way of making 2410 using 16 coins.

PLACE VALUE

True or False?

392

3 hundreds

9 tens

2 ones

Different Ways

1 hundred

29 tens

2 ones

35 tens

32 ones

3 hundreds

7 tens

12 ones

or

Make 534

5 hundreds, tens, 4 ones.



tens, ones.

10010

1000

Different Ways

Extend: Find other ways of making 392

I SEE REASONING

PLACE VALUE

Which Answer?

Small Difference Questions

Three thousand and four30004

3004

Sixty thousand and two60002

600002

20006Two thousand and six

Twenty thousand and six

50070Fifty thousand and seventy

Five hundred and seventy

Write in words:

35000 ___________________________________________

30500 ___________________________________________

30050 ___________________________________________

3050 ____________________________________________

300005 __________________________________________

OrderIn each number, how many zeros?

Fifty-six thousand and twenty

Three hundred thousand, two hundred and seventy

Thirteen thousand and thirty-one

I SEE REASONING

PLACE VALUE

Spot the Pattern


Fill the gaps:

, 764, 774, 784, ,

963, , 983, ,

, 1206, 1106, ,

999 + 1 = 9999 + 1 =

999 + 100 = 9999 + 1000 =

999 + 10 = 9999 + 100 =

Estimate

Estimate the value of the numbers at the arrows:

0 10000

0 100000

Extend: Design a sequence question with missing numbers.

I SEE REASONING

PLACE VALUE

EstimateEstimate the position of 836 on each number line:

0 1000

800 850

700 1000

0 10000

EstimateEstimate the position of 3280 on each number line:

0 10000

3000 4000

3200 3300

0 5000

your example

your example

I SEE REASONING

PLACE VALUE

Different WaysWhat could the start and end numbers be?

194

Different WaysWhat could the start and end numbers be?

3070

0

Small Difference Questions5

0

500

0

25

I SEE REASONING

PLACE VALUE

How Many Ways?

Agree or Disagree?

A three-digit number is added to a two-digit number.

The sum is a 4-digit number.

Which digit must be in the blue box?

Explain

Jen thinks of a 2-digit number. The sum of its digits is 7.

Molly thinks of a 2-digit number. The sum of its digits is 15.

Jen’s number might be larger than Molly’s number

Example: The sum of the digits for 35 is 8

3 + 5 = 8

I think of an even 4-digit number. It is more than 4000.

The sum of its digits is 8. Each digit is different.

What could the number be?

Level 1: I can find a possible answer.

Level 2: I can find different possible answers.

Level 3: I know how many possible answers there are.

+ =

Different WaysWhich number with a digit sum of 16 is closest to 700?

There are two possible answers.

I SEE REASONING

DECIMALS


Show the position 0.36 on each number line:

0 1

0.3 0.4

0 0.5


Show the position 0.8 on each number line:

0 10

0 1

0.5 1

Agree or Disagree?

0 1

0.77

I SEE REASONING

I SEE REASONING - 5

Different Ways

Compare

Choose different start and end numbers. Position 3.74

on each number line:

Use < = > signs to compare the decimals:

Agree or Disagree?

140 is more than 80 as it has more digits

0.14 is more than 0.8 as it has more digits

0.64 0.52

0.64 0.9

0.64 0.614

0.64 0.644

0.7 0.70

0.55 0.505

0.08 0.088

0.915 0.92

>

DECIMALS

Question 1:

How many ways can 0.42 be made?

Question 2:

How many ways can 0.24 be made?

Explain the Mistakes

CompareComplete with the correct symbol: < = >

0.47 + 0.3 = 0.500.88 + 0.22 = 1

0.25 > 0.2 + 0.08

0.33 0.3 + 0.3

0.33 0.3 + 0.03

0.41 0.4 + 0.004

0.67 0.6 + 0.07

0.87 0.7 + 0.08

How Many Ways?

0.1

0.01

You have a pile of 0.1 and 0.01 counters.

Spot the PatternContinue the sequences:

0.37, 0.38, 0.39, ,

0.07, 0.08, 0.09, ,

40, 40, 4, ,

8, 4, 2, 1, ,

DECIMALS I SEE REASONING

Small Difference QuestionsOn each number line, which number is half-way?

20 21

2 2.1

800 810

0.8 0.81

Which Answer?

What is 53.48 to the nearest whole number?

Explain the mistakes.

(a) 54

(b) 53.5

(c) 53

38 39 40 41

Position 38.42, 39.08, 39.6, 40.27 and 40.82 on the number

line:

For each decimal, which is the nearest whole number?

Number Lines

23.5 23.6 23.7 23.8

Position 23.555, 23.64, 23.708 and 23.78 on the number

line:

For each decimal, which is the nearest tenth?


Different Ways

3.5 is half-way between…

3.5

Use 1 decimal place numbers:

3.5

Use 3 decimal place numbers:3.5

Use 2 decimal place numbers:


30 ÷ = 1

30 ÷ = 0.3

30 ÷ = 0.1

40 ÷ = 0.4

40 ÷ = 1

40 ÷ = 0.5

Different Ways2 = 0.2

2 = 0.2

0.8 = 80

0.8 = 80

= 0.1

= 0.1

= 0.1

Complete using different

symbols +

–

×

÷


0.05, , 0.15,

0.15, , 0.45,

Spot the Pattern

Different Ways


Continue the sequences:

0, 0.3, 0.6, 0.9, ,

0.02, 0.04, 0.06, ,

× = 3

× = 3

× = 3

× = 3

6 × 5 = 30

× = 2.4

× = 2.4

× = 2.4

× = 2.4

10 × 2.4 = 24

2 × = 6 0.2 × = 4 0.2 × = 2

4 × = 6 0.1 × = 4 0.25 × = 2

12 × = 6 0.5 × = 4 0.5 × = 2

× = 6 × = 4 × = 2


NEGATIVE NUMBERS

ContextsFor each example, are the negative numbers used

correctly?

There are 30 children. I

have 20 sweets. Each

child has a sweet. There

are -10 sweets left.

I was at level 8 in the car

park. I go down 10 levels

in the lift. Now I’m on

level -2.

I spent £15 on face paints

for the school fair. I made

£20 doing face paints.

Overall, the stall made -£5.

The temperature was 4˚c on

Monday. It was 7˚c colder on

Tuesday. The temperature on

Tuesday was -3˚c.

Think of other examples where negative numbers are used.

Question Number Sentence

I spent £6 on ribbon to make bracelets.

I made £4 at my stall selling bracelets.

What was my profit?

I was 2 floors below ground level at the

hotel. I got the lift. It took us 5 floors up.

Which floor am I on now?

5 – 7 = -2

Contexts

I SEE REASONING

Explain the Mistake

-2 -1 0 1 2 3

The difference

between -2 and 3

is 6. I counted all the numbers.

Different Ways

This question can be answered in two ways:

The difference between -20 and is 50


Show both answers on the number line:

-20 0

EstimatePosition the numbers on the number line: -8 -14 17 -3

-10 0-20 10 20

NEGATIVE NUMBERS

Position the numbers on the number line: -34 -58 65 -80

-50 0-100 50 100

Answer in two ways:



I SEE REASONING

Estimate

EstimatePosition 0 on each number line:

020

0

20

8

0

-10 10

-10 20

-20 40

-10 40

NEGATIVE NUMBERS

your own example

0

I SEE REASONING

Small Difference QuestionsOn each number line, which number is half-way?

2 10

-3 5 -5 7

-5 3

How are the questions similar? How are they different?


Half-way between -3 and 7 is


Half-way between -6 and -2 is

Half-way between -6 and is -2

Half-way between -6 and is 2


Half-way between 7 and 15 is


Half-way between -11 and is -4

Half-way between -11 and is 4

NEGATIVE NUMBERS I SEE REASONING

Which Answer?

What are the next 3 numbers in this sequence:

21, 16, 11, 6…

Explain the mistake

1, -6, -11 1, -4, -9

Rank by Difficulty

How Many Ways?

Different Ways

What is the first negative number in each sequence?

231, 229, 227…

120, 114, 108…

80, 73, 66…

There are 5 numbers in a sequence. 3 of the numbers are positive and 2 of the numbers are negative.

-5 and 7 are two of the numbers in the sequence.

What could the sequence be? Do in different ways.

The first number in the sequence is 17

The first negative number in the sequence is -3

To get the next number in the sequence subtract

Level 1: I can find a possible answer.



NEGATIVE NUMBERS I SEE REASONING

ROUNDING

Explain

Here are two examples where we use rounding:

To say how many people live in Scotland.

Why do we use rounding in these situations?

Think of other examples where we use rounding.

To give the time

of the day.

Explain

Would you use the exact number? Or the rounded number?

204 children in the school.

OR

200 children in the school.

Which Method?

Which questions would you use rounding to calculate?

198 – 57

You need 296g of flour.

OR

You need 300g of flour.

400m race time: 52.84 seconds

OR

400m race time: 53 seconds

59 × 4

249 + 148

49 × 49

503 + 246

Are there any questions where you would round

both numbers?

I SEE REASONING

Number Lines

Position the numbers on both number lines: 437, 446, 453

430 440 450 460

400 500

The closest 10 to 437 is The closest 100 to 437 is



Number Lines

Position 376, 419, 438 and 471 on the number line:

350 400 450 500

For each number, which is the nearest 50?

600 680 700620 640 660

Position 607, 632, 668 and 685 on the number line:

For each number, which is the nearest 20?

ROUNDING I SEE REASONING



245 rounded to the nearest 10 is





568 rounded to the nearest 10 =

568 rounded to the nearest = 600

568 rounded to the nearest 50 =

568 rounded to the nearest = 560

Agree or Disagree?

569 always rounds to a

bigger number because each digit is 5 or more

ROUNDING

Extend: round 534 to different numbers, getting as many different answers as possible.

I SEE REASONING


5082 is nearer to

6000 than 5000 as the 8 rounds up

4273 to the nearest 100 is 300

Which Answer?

The nearest 10 is 400

But 400 is not a 10, the nearest 10 is 410

What is 403 to the nearest 10?

Which Answer?

What is the largest whole number that, when rounded to the nearest 100, is 1500?

14991504 1549

Explain

Give examples to show that this statement is true:

‘Numbers can be close together but round to

different numbers.’


Different Ways

Multi-Skill

Question 1:

To the nearest 10, my number is 250. My number

is a multiple of 3. What could my number be?

Question 2:

To the nearest 10, my number is 300. My number

is a multiple of 7. What could my number be?

Extend: Design your own question.Your question must have two possible answers.


Put two numbers in each section of the Venn diagram:


258

Multi-StepTo the nearest 10p Tim has £1.50

To the nearest 50p Kam has £2.50

What is the largest possible difference between the

amount of money that Tim and Kam have?


ROMAN NUMERALS

Compare Questions

Rank by Difficulty

Write each number in Roman Numerals.

LXVI

LXIV

XLVI

XLIV

44

66

46

64

Draw lines to match numbers to Roman Numerals:

Which numbers were easiest/hardest

to write in Roman Numerals?

33 =

I = 1

V = 5

X = 10

L = 50

65 = 49 =

I = 1

V = 5

X = 10

L = 50

Compare Questions


Which of these questions are similar?

40 =

Writing 40 in Roman Numerals is similar to writing… because…

9 =

80 = 13 =

I = 1

V = 5

X = 10

L = 50

I SEE REASONING

ROMAN NUMERALS

Rank by Difficulty


Which numbers were easiest/hardest

to write in Roman Numerals?

230 = 780 =

490 =

I = 1

V = 5

X = 10

L = 50

C = 100

D = 500

Explain

Explain why both statements are sometimes true.

Order

Order from smallest to largest:

What do you notice?

I = 1

X = 10

C = 100

D = 500

M = 1000

DXX M DIII CM

I = 1

V = 5

X = 10

L = 50

C = 100

In Roman Numerals, numbers above 200 have more symbols than numbers below 20.

With Roman Numerals, adding another symbol increases the size of the number.

This is true for examples like…

However, if you compare the Roman Numerals… and…

I SEE REASONING

ROMAN NUMERALS

Spot the Pattern

These sequences increase in steps of 10.

Fill the gaps:

80 90 100 110 120

LXXX CX


210 220 230 240 250

CCX CCXX CCL

560 570 580 590 600

DLX DLXXX

I = 1

V = 5

X = 10

L = 50

1 less than XLIII is

1 more than XLIII is

10 more than XLIII is

10 less than XLIII is

1 less than XXX is

10 more than XXX is

50 more than XXX is

XLIII is 43

XXX is 30

I SEE REASONING

ROMAN NUMERALS

Extend

I think of a Roman Numeral.

It has 3 symbols. They are all different.

It is less than 60.

What is my Roman Numeral?

How Many Ways?

I think of a Roman Numeral.

It has 3 symbols. They are all different.

It is more than 400 and less than 700.

What is my Roman Numeral?

I = 1

V = 5

X = 10

L = 50

C = 100

D = 500Level 1: I can find a possible answer.



I = 1

V = 5

X = 10

L = 50Level 1: I can find a possible answer.



How Many Ways?

I = 1

V = 5

X = 10

L = 50

What is the smallest number that can

be made using four different Roman

Numeral symbols?

Example: LXI is 61. It uses three

different symbols.

I SEE REASONING

ADDITION

Simplify


Mental or Written Method?

494 + 238 = 500 + 396 + 189 = 600 –

785 + 145 = 800 +

Extend: Create your own ‘simplify’ addition questions.

615 + 283 = 900 –

Simplify

2645 + 3978 = 4000 + 4950 + 1850 = 7000 –

2975 + 2060 = 5000 +

Extend: Create your own ‘simplify’ addition questions.

910 + 1200 = 2000 –

375 + 145 =

345 + 175 =

345 + 157 =

353 + 149 =

684 + 347 =

674 + 357 =

686 + 342 =

646 + 372 =

4731 + 5268 = 895 + 385 =

2480 + 2520 =463 + 278 =

I SEE REASONING

ADDITION


Correct or Incorrect?

Rank by Difficulty

Different Ways

In this calculation, two digits are hidden.

Circle the possible answers:

1523

2 6 5 5

+ 7 2 8 0

9 9 3 51

2 6 5 5

+ 7 2 8

3 4 8 311

2 6 5 5

+ 7 2 8

2 3 7 311

1 3.6

+ 6.7 2

8.0 81

8 3 9 5

+ 7 2 3 7

1 5 6 3 21 1

8 4 6 9

+ 78 3 7

9 2 9 61 1

4065 + 3205 =

744 + 579 =

2996 + 2995 =

473.6 + 516.2 =

3509 + 3444 =

16521442 1532

1 6 6 7

+ 78 3 5

I SEE REASONING

ADDITION

Which Answer?

6 1 8

+ 3 5

7 6 9

1 8 2 5

+ 7 3 6

1 6 6 4

Different Ways

Level 1: Answer each question.

Level 2: Spot the question(s) with more than one answer.

Level 3: Find all possible answers.

Extend: Design a question with one answer. Design a question with 2 or 3 answers.

1 2 4 5

+ 2 3 6

2 6 7 9

1 2 2 5

+ 1 3 8

2 6 8 6

How Many Ways?

As there are two blank

boxes in the tens

column, this question

can be answered in

different ways

This question

can only be

answered in

one way

Level 1: Complete using digits 0→9 (no repeated digits).

Level 2: Complete using digits 0, 1, 2, 4, 6, 7, 8

Level 3: I know how many ways it can be completed

using the digits 0, 1, 2, 4, 6, 7, 8

+ =

I SEE REASONING

SUBTRACTION

Simplify

525 – 384 = – 400 815 – 568 = 799 –

706 – 268 = – 261

Extend: Create your own ‘simplify’ subtraction questions.

732 – 475 = 757 –

Rank by Difficulty

43.2 – 16.8 =

601 – 337 =

869 – 321 =

845 – 297 =

Simplify

7127 – 2850 = – 3000 5362 – 3485 = 4999 –

3824 – 2476 = – 2451

Extend: Create your own ‘simplify’ subtraction questions.

4260 – 1790 = 4470 –

Rank by Difficulty

8765 – 2543 =

5432 – 2543 =

6672 – 3994 =

7002 – 3497 =

I SEE REASONING

SUBTRACTION

Different Methods

3728 – 1465 =

3012 – 2994 =

900 – 452 =

3000 – 30 =

764 – 296 =


50 – 35 =

500 – 35 =

5000 – 35 =

50000 – 35 =


558 – 233 =

564 – 239 =

584 – 219 =

600 – 235 =

6000 – 235 =

800 – 75 =

8000 – 75 =

8000 – 750 =

80000 – 750 =

5685 – 1675 =

5658 – 1675 =

5658 – 1995 =

5618 – 1965 =

5648 – 1961 =

I SEE REASONING

SUBTRACTION



6 0 5.8

– 3 2 8.5

3 2 3.3

Rank by Difficulty

Explain

14 2 5.6

– 2 7.2 5

1 5 3.1

31

6 0 4 4

– 2 0 8 1

3 0 6 3

5

8 9 0 3

– 3 2 6 7

5 6 3 6

81

9

1 5 8 0

– 8 5 7

7 3 3

011

2 6 3.0

– 4 4.8

2 1 8.2

125 1

Explain, using examples, why this statement is incorrect:

6 6 5

–72 3 4

1

The digit in the red box must be 4

Complete these calculations using the column method:

For each pair, which question is harder? Easier?

23.65 – 12.8 =

23.8 – 12.65 =

4600 – 1280 =

4006 – 1280 =

I SEE REASONING

SUBTRACTION

Explain

7 3– =5

Fill the gaps:

What do you notice?

5 3– =7

Difference between the digits in blue boxes =

Difference between the digits in red boxes =

How Many Ways?

Different Ways

Level 1: Answer each question.

Level 2: Spot the question(s) with more than one answer.

Level 3: Find all possible answers.

Extend: Design a question with 2 or 3 possible answers.

6 0 2

– 2 3 7

3 6 5

Level 1: I can find an answer.

Level 2: I can find different answers.

Level 3: I know how many answers there are.

Extend: Change one digit to create two

more possible answers.

8 3 9

– 2 8 5

5 5 4

9 1 6

– 1 6 0

7 5 6

4 7 9

– 1 8 6

2 9 3

I SEE REASONING

ADDITION AND SUBTRACTION

Broken Calculator‘The 7 and 5 keys on my calculator are broken!’

How can I use my calculator to work out:

750 + 850 =

505 – 367 =

866 – 597 =

Contexts

Question Which method?

Kara is 6 years old. Her Dad is 30 years old. How old will Kara’s Dad be

on her 18th birthday?

(a) 30 + 18

(b) 30 + 12

(c) 30 × 3

Ben and Tom meet. Then, Ben drives 15 miles south and Tom drives 7 miles

north. How far apart are they?

(a) 15 + 7

(b) 15 – 7

Max is given some money. He spends £15 and has £25 left. How

much money was Max given?

(a) £25 - £15

(b) £15 + £25

Three friends have 200 stickers. Kim has 55, Zara has 85. How many

stickers does Jen have?

(a) 200 – (55 + 85)

(b) 200 + 55 + 85

(c) 55 + 85 – 200

For each question, choose the correct method:

Extend: design your own ‘Broken Calculator’ questions.

7.5 – 5.5 =

63.5 + 79.5 =

1.4 – 0.7 =

I SEE REASONING


Building QuestionsDraw lines to join the blue information

to matching the red question.

Mo has £2.

How many oranges can he afford?

BananasApples

25p

Oranges

35p 15p

Ben has £2. He wants four oranges and three apples.

How much more money does he need?

Matt has £2. He wants four oranges and three bananas.

How much change does he get?

How much does it cost him?

Raja buys three apples and three bananas.

Which Bar Model?

Circle the bar model which correctly

shows the information in the question:

Apples Oranges

25p 35p

Sam buys four oranges and two apples. He pays with a £2 coin. How much change does he get?

35p 35p 35p 35p

£2

25p25p 35p 35p 35p 35p

£2

25p 25p

change

change

Explain

What is the largest number

that can go in the blue box?

Adam buys three apples and oranges.

He pays with a £2 coin and gets some change.

Apples Oranges

25p 35p

I SEE REASONING



Building Questions

Information Question Answer

How much

change did

she get?

75p

How much

more money

does he need?

15p

How many

melons can

she afford?

For each question, give the missing

information:

1. Zack bought a sandwich, an apple and a drink.

How much did it cost?

2. Nada wants a sandwich, an apple and a drink. She has £2.

How much more money does she need?

3. Jen bought a sandwich, an apple and a drink. She paid £3.

How much change did she get?

4. Tom has £2.

How many apples can he afford?

5. Joy has £2.

How many oranges can she afford?

6. Andy spent £2 on 3 items. He got 80p

change. What does he buy?

Sandwiches: £1.80

Oranges: 35p

Apples: 25p

Drinks: 50p

Melons: 90p

Pineapples: 75p

Mangoes: 55p

5 melons

I SEE REASONING


Multi-StepJohn bought 4 pieces of fruit and paid

with a £2 coin. He got 60p change.

What did he buy?

Oranges: 35p

Apples: 25p

Bananas: 20p

Kelly bought 5 pieces of fruit and paid with a £5 note.

She got £3.60 change. What did she buy?

Ben bought 6 pieces of fruit and paid with a £10 note.

He got £8.60 change. What did he buy?

Which Bar Model?A piece of cake and a drink costs £2. The cake costs 30p more than the drink. How much does the cake cost?

30pcake

85pdrink

85p

Cake = 85p

30pcake

85pdrink

85p

Cake = £1.15

£1.30

30p

cake

70pdrink

Cake = £1.30

cake

£2drink

Cake = £2.30

30p£2

Agree or Disagree?There are 30 children in the class.

There are 4 more boys than girls.

11 girls in the class 17 boys in the class

34 boys in the class

I SEE REASONING




1. There are 18 children at the park. There are 8 more boys

than girls. How many girls are there at the park?

2. There are 22 children at the park. There are 8 more boys

than girls. How many boys are there at the park?

3. Lisa and Kate have £22 in total. Lisa has £10 more than

Kate. How much money does Lisa have?

4. Kevin and Sam have £22 in total. Kevin has £9 more than

Sam. How much money does Kevin have?

1. A bucket and a spade cost 80p in total. The bucket costs

20p more than the spade. How much does the spade cost?

2. 80 people go to running club. There are 20 more women

than men. How many women are there at running club?

3. In total, Gavin and James have £80. Gavin has £30 more

than James. How much money does James have?

4. An apple and a kiwi weigh 240g. The apple weighs 90g

more than the kiwi. How much does the kiwi weigh?

5. I think of two numbers. The numbers have a sum of 240

and a difference of 70. What are my numbers?

Rank by Difficulty

A pear and an orange weigh 300g.

The pear is 60g lighter than the orange.

How much does the orange weigh?

There are 30 children in the class. There

are 6 more girls than boys.

How many boys are there in the class?

+ = 30

– = 5

=

=

I SEE REASONING


How Many Ways?

The sum of two numbers is more than 50 and less than 60.

The difference between the numbers is 8.

What could the numbers be?

Level 1: I can find an answer.

Level 2: I can find different answers.

Level 3: I know how many answers there are.

Small Difference QuestionsComplete with the correct symbol: < = >

55 + 45 < 100

75 + 45 < 120 – 10

75 + 65 < 150 – 10

85 + 75 < 160 – 20

85 + 75 < 160 + 20

95 + 85 < 155 + 25

230 – 165 < 83 – 8

230 – 165 < 83 – 18

230 – 165 < 93 – 28

230 – 185 < 93 – 48

230 – 85 < 93 + 48

230 – 85 < 103 + 48

= <

Which Answer?

275 + 165 = – 45

440


485395

I SEE REASONING


Agree or Disagree?

+ = –

The number in the green box is equal to the sum

of the numbers in the blue, purple and red boxes

The smallest number

must be the number

in the red box

Multi-SkillThe missing numbers are all multiples of 6.

Complete using the largest possible number:

90 – > 50

30 + < 70

– 20 < 30

Extend: Design your own ‘largest/smallest possible number’ questions.

Complete using the smallest possible number:

90 – < 50

30 + > 80

– 30 > 40

Multi-Step

>

+ = –

+ < 35

Complete using digits 1→9.

Position 5 and 3 as shown.

I started with… because…

The only digit that can go… is…

I SEE REASONING


Agree or Disagree?

Explore

Extend

You can calculate the

value of these shapes

+ = 32

= 45

You can’t calculate the

value of these shapes

+ = 40

+ = 35++

The shape I calculated first was… because…

35

38

39

35 32 45

19

25

16 12 16

The numbers show the sum for the columns and rows.

42

33

38

28 22 31 32

The numbers show the sum for the columns and rows.

Extend: design your own shape puzzle.

Use three different

shapes.

=

=

=

=

=

=

=

=

=

I SEE REASONING

MULTIPLICATION

Different Ways 14 × 9

Find 3 ways to calculate 15 × 8:

14

97×9= 63

7×9= 63

14

910×9= 90

14

9

14×3=42

14×3=42

14×3=42

63 × 2 = 126 42 × 3 = 126

4×9= 36

90 + 36 = 126

15

8

15

8

15

8


15 × 6 = 90

15 × 8 =

30 × 4 =

34 × 4 =

34 × 14 =

33 × 14 =

33 × 7 =

6 × 7 = 42

12 × 7 =

12 × 9 =

24 × 9 =

36 × 9 =

35 × 9 =

70 × 9 =

16 × 4 = 64

16 × 6 =

16 × 8 =

16 × 7 =

18 × 7 =

18 × 6 =

9 × 12 =

I SEE REASONING

Matching Number Sentences

+ or - number sentence × number sentence

4 + 8 + 12 4 × 6

6 + 18 + 30

24 + 32

24 + 36

12 + 18 + 27

Matching Number Sentences

+ or - number sentence × number sentence

27 + 15 3 ×

63 + 18 9 ×

72 + 48

49 + 35

I know… so…

24 × 18 = 432

24 × 19 =

30 × 15 = 450

40 × 15 =

35 × 12 = 420

38 × 12 =

MULTIPLICATION I SEE REASONING


25 × 6 =

23 × 6 =

18 × 6 =

18 × 12 =

36 × 24 =

41 × 24 =

21 × 30 =

21 × 15 =

21 × 16 =

26 × 16 =

39 × 16 =

39 × 24 =

Rank by Difficulty

37 × 6 =

804 × 6 =

45 × 6 =

MULTIPLICATION

98 × 6 =

To answer, did you use the same method?

Or different methods?

Rank by Difficulty

32 × 50 =

409 × 6 =

46 × 7 =

99 × 4 =

To answer, did you use the same method?

Or different methods?

I SEE REASONING

Different Methods

Different Ways

‘The 8 key on my calculator is broken!’


26 × 8

80 × 15

Broken Calculator

Ways to calculate 24 × 8:

less than 24 × 10

less than 25 × 8

Double ×

Double one number, halve the other number, the product

is the same. Example: 8 × 12 = 96 16 × 6 = 96

Which of these questions are made easier by a

doubling and halving strategy?

26 × 24 14 × 25

80 × 60 5 × 18

1.5 × 6


Estimation

293 × 4

The answer will be odd/even.

The answer will be a - digit number.

Of these numbers, the answer will be closest to:

800 950 1100 1300

Checking Possible AnswersDo not calculate the answer to the question.

Which numbers cannot be the answer to 493 × 6?

Explain how you know.

29413028 2958

369 × 7


300 × 7 = 2100 400 × 7 = 2800

Estimate for 369 × 7 is

Estimation

MULTIPLICATION

Estimation584 × 9


500 × 9 = 4500 600 × 9 = 5400

Estimate for 584 × 9 is

I SEE REASONING


Part-Complete Examples

2 6 1 × 4

8 2 4 4

Different Methods

500 20 7

6 3000 120

3 0 0 0

1 2 0

4 2

3 1 6 2

Two ways of calculating 527 × 6:

What’s the same? What’s different?

3

4 1 7 × 5

2 0 5 5

1

8 4 3 × 6

4 4 5 8

2

4 8 9 × 5

8 2 4 5

1

7 2 3 × 4

2 0 5 2

2

4 6 3 × 8

4 4 5 4

44

4

5 2 7

× 6

3 1 6 2

1

42


Estimation

53 × 19 The answer will be odd/even.

The answer will be a - digit number.

Of these numbers, the answer will be closest to:

900 1000 1100 1200

Checking Possible AnswersDo not calculate the answer to the question.

Which numbers cannot be the answer to 39 × 38?


14821602 1543

29 × 26


20 × 20 = 400 30 × 30 = 900

Estimation for 29 × 26 is

Estimation

MULTIPLICATION

Estimation89 × 43


90 × 40 = 3600 90 × 50 = 4500

Estimation for 89 × 43 is

I SEE REASONING


Different Methods


50

3

2000

What is the same? Different?

40

2

150 6

80

2000

150

80

+ 6

2236

1 5 62 0 8 0

2 2 3 6

5 2

× 4 3

61 5 0

8 02 0 0 0

2 2 3 6

5 2

× 4 3

2 1 62 8 8

5 0 4

7 2 × 4 3

1 6 84 0 2 0 0

4 0 3 6 8

8 4 × 5 2

1

1

1 8 9

0 2 5 0

4 3 6 9

6 3

× 5 3

1

4

2 4 0

2 4

× 1 6

2

4 8 6

0 2

4 3 6

8 1

× 4 6

3

8 21 2 3 0

1 3 1 2

4 1 × 2 3

11 1


Missing Digits

8 2 5

× 6

2 5 6 8

How Many Ways?

Level 1: I can find a way

Level 2: I can find different ways

Level 3: I know how many ways there are

× =

Complete using digits 0-9.

The digit in the box with a border must be odd.

5 2 4

× 6

4 1 9 2

8 9 5

× 5

3 4 7 5

Different Answers

8 3 5

× 6

3 7 2 4

8 3 5

× 6

3 7 2 4

This question can be answered in

two ways.

MULTIPLICATION

Tip: think about which digits can and cannot be used in the ones columns.

I SEE REASONING

DIVISION

Different MethodsWhat is different about how you answer each question?

480 ÷ 240 =

480 ÷ 10 =

480 ÷ 6 =480 ÷ 4 =

480 ÷ 20 =

I know… so…

72 ÷ 3 = 24

78 ÷ 3 =

98 ÷ 7 = 14

91 ÷ 7 =

72 ÷ 4 = 18

144 ÷ 8 =

84 ÷ 6 = 14

168 ÷ 6 =

112 ÷ 4 = 28

192 ÷ 4 =

48 ÷ 6 = 8

108 ÷ 6 =


48 ÷ 3 = 16

54 ÷ 3 =

108 ÷ 3 =

108 ÷ 6 =

216 ÷ 12 =

64 ÷ 8 = 8

64 ÷ 4 =

104 ÷ 4 =

144 ÷ 4 =

144 ÷ 8 =

I SEE REASONING


56 ÷ 4 = 14

112 ÷ 4 =

224 ÷ 8 =

304 ÷ 8 =

344 ÷ 8 =

108 ÷ 3 = 36

216 ÷ 6 =

216 ÷ 3 =

246 ÷ 3 =

261 ÷ 3 =

Which Operation?

For each question, do you multiply or divide to find

the missing number?

100 ÷ = 20

÷ 100 = 20

40 × = 200

4 = ÷ 20

= 20 ÷ 4

Contexts

Question number sentence

16 children share some sweets. They get 4 sweets each. How many sweets?

90 eggs are packed into 15 boxes. How many eggs in each box?

÷ 16 = 4

Write a question for and 60 ÷ = 5÷ 5 = 60

DIVISION I SEE REASONING

Estimate

165 ÷ 6

Whole number answer?

Yes Possibly No

120 ÷ 6 = 20 180 ÷ 6 = 30

Estimate for 165 ÷ 6

344 ÷ 8


Yes Possibly No

320 ÷ 8 = 40 400 ÷ 8 = 50


Estimate

705 ÷ 3


Yes Possibly No

600 ÷ 3 = 200 900 ÷ 3 = 300


ExplainFor this task, do not work out the answers to the questions.

For each question, the answer has how many digits?

70 ÷ 5

digit(s)

435 ÷ 5

digit(s)

3000 ÷ 4

digit(s)

3075 ÷ 3

digit(s)

615 ÷ 5

digit(s)

719 ÷ 4


Yes Possibly No

400 ÷ 4 = 100 800 ÷ 4 = 200


DIVISION


I SEE REASONING

Which Answer?

Next StepIn each calculation, what’s the remainder? 8 46

1 42

Next StepIn each calculation, what’s the remainder? 5 8 23

1 9 42 1

8 6 44

2 1

9 6 44

2 41

6 6 44

1 62

2 6 13

0

6 7 53

2 2

7 7 43

2 51

92 ÷ 4 = To answer, split 92

into 90 and 2

To answer, split 92

into 80 and 12 To answer, split 92

into 88 and 4

4 53

1

5 53

1

6 53

2

7 53

2

7 64

1

8 64

2

9 64

2

9 68

1



The calculations have been started. Finish them:

5 24

11

9 24

2

8 9 24

2

8 9 28

1

4 9 28

04

7 9 24

1


The calculations have been started. Finish them:

8 5 23

22

8 6 23

22

7 6 23

2

4 4 86

04

8 4 86

12

8 3 86

1

Which Answer?

745 ÷ 4

Find the correct calculation.

Spot the mistakes.

7 4 543 2

1 9 6 r 1

7 4 543 2

1 8 5 r 5

7 4 543 2

1 8 6 r 1


Explain the Mistakes 5258 ÷ 6

5 2 5 865 4

0 8 8 6 r 23

5 2 5 865 4

0 8 7 63

5 2 5 865 4

0 8 7 31

Question Answer

Eggs are put in boxes of 6. The farmer has 51

eggs. How many boxes does he need for all

the eggs?9 boxes

A sunflower grows to a height of 51cm in 6

weeks. On average, how many centimetres

does it grow each week?

51 children turn up for a 6-a-side football

tournament. How many teams can be made?Teams can have substitutes.

An artist works on a masterpiece for 51 hours

over 6 days. On average, how long does she

work each day?

Form of Answer 5 16

0 8 r 35

Form of Answer

Jess sleeps for 29 hours over 4 nights.

On average, how long is she asleep each night?

7 hours 10 minutes

7.1 hours

7 hours 15 minutes


Mental or Written?

320 ÷ 3 =

Which questions use the same/different calculation

methods?

320 ÷ 6 = 320 ÷ 9 =

320 ÷ 5 = 320 ÷ 8 = 320 ÷ 10 =

Mental or Written?

540 ÷ 4 =

Which questions use the same/different calculation

methods?

540 ÷ 6 = 540 ÷ 7 =

550 ÷ 55 = 550 ÷ 25 = 550 ÷ 15 =


560 ÷ 4 =

560 ÷ 8 =

560 ÷ 7 =

560 ÷ 9 =

567 ÷ 9 =

1134 ÷ 9 =

441 ÷ 3 =

4410 ÷ 30 =

441 ÷ 6 =

501 ÷ 6 =

501 ÷ 3 =

1503 ÷ 9 =


Rank by Difficulty340 ÷ 20 =

550 ÷ 30 =

425 ÷ 5 =

546 ÷ 6 =

500 ÷ 3 =

How Many Ways?

‘The 6 and 8 keys on my calculator are broken!’


522 ÷ 6 =

624 ÷ 8 =

Broken Calculator




6 ÷ =

Complete using digits 0-9.

Position the digit 6 as shown.

DIVISION

My system for knowing I have found all of the answers is…

I SEE REASONING

MULTIPLICATION AND DIVISION

Contexts

Which operation(s) does each question involve?

addition subtraction multiplication division

(a) Tim is years old. Jack is years old.

When Tim is , how old will Jack be?

(b) Zara has t-shirts, pairs of trousers and hats.

How many different outfits can Zara wear?

(c) Holly is saving for a bike that costs £ . She has £ .

Holly earns £ per week.

How many weeks will Holly need to save up for?

Extend

At work, Mr James wears a pair of trousers and a shirt. Sometimes he wears a tie.

He has 4 pairs of trousers, 5 shirts and 8 ties.

How many combinations of outfits does Mr James have?

Agree or Disagree?

Kam has 3 pairs of jeans, 6 t-shirts and 2 caps.

Pete has 3 pairs of jeans, 5 t-shirts and 3 caps.

Kam and Pete have the same number of outfits as they own the

same amount of clothing

I SEE REASONING


Spot the Difference

What’s the same? What’s different?

(a) Ruth earns £15 each day delivering newspapers. How much money does Ruth earn each week?

(b) Kate plays the piano for 15 minutes every day. How long, in hours and minutes, does Kate spend playing the piano each week?

Answer the questions:

Explore

At the market, apples cost 25p each.

At the shop, it costs £1.10 for a bag of 6 apples.

Using this information, think of a question that involves:

(a) Multiplication

(b) Multiplication and addition

(c) Multiplication and subtraction

(d) Division

Explore

1500 people are travelling from Sheffield to Leeds to

go to the match. They travel by car or by coach.

200 people fit in a coach. 5 people fit in a car.

Using this information, think of a question with the answer:

(a) 6 coaches

(b) 140 cars

I SEE REASONING



Explore

(a) 18 people camping. Each tent fits 3 people.

How many tents are needed?

(b) 36 people camping. Each tent fits 6 people.


(c) 31 people camping. Each tent fits 6 people.


(d) 25 people camping. They use 7 tents.

How many people fit in each tent?

(e) Some people camping. They use 8 tents that fit 4 people.

How many people could be camping?

(f) Some people camping. They use 4 tents that fit 8 people.

How many people could be camping?

There are children in the class.

The teacher has 2-litre bottles of orange juice.

Add information to create a question with the answer:

(a) 250ml each

(b) 1 litre left

(c) 12 children

Extend

Adam is seven times as old as his daughter, Lara.

Next year, Adam will be six times as old as Lara.

How old is Adam? Explain how you know.

I SEE REASONING



480 ÷ 100 = _____48


I know… so…

7.6 × 10 = _____7.60

3600 ÷ 36 = _____10 3.05 × 1000 = ______3005

360 ÷ ______ = 3.61000 0.34 ÷ 10 = _____3.4

1200 ÷ 12 = _____100 3.02 × ______ = 30201000

17 × 6 = 102

17 × 60 =

170 × 60 =

42 ÷ 6 = 7

420 ÷ 60 =

420 ÷ 6 =

160 ÷ 10 = 16

160 ÷ 16 =

1600 ÷ 16 =


1050 ÷ ______ = 10105 400 ÷ 50 = _____80

60 × 70 = _____420 320 ÷ _____ = 1032

403 ÷ 100 = _____4.3 90 × _____ = 450050

I SEE REASONING


Which Answers?

Circle the factors of 12

4 36 60 0.5 6

Circle the multiples of 30

90 6 200 15 120

Which Answers?Circle the prime numbers:

21 31 41 51 61 71

Spot the Mistakes

Which Answers?

Circle the prime number(s):

87 121 137 375

Which numbers do you immediately know are not prime?

5cm × 5cm × 5cm

= 125cm²9cm × 9cm =

81cm³

Which Answers?

Circle the square numbers.

Underline the cube numbers:

24 27 36 64 121

I SEE REASONING


Mental or Written Calculation?

Spot the Pattern

Which of the digits from 1 to 9 are factors of 128?

1 2 3 4 5 6 7 8 9

Which digits could you work out mentally?

Which written calculations did you do?

Mental or Written Calculation?

Which of the digits from 1 to 9 are factors of 624?

1 2 3 4 5 6 7 8 9

Which digits could you work out mentally?

Which written calculations did you do?

Complete using positive whole numbers greater than 1:

× = 420

× × = 420

× × × = 420

× × × × = 420

Example:

12 × 5 = 60

6 × 5 × 5 = 60

6 × 5 × 5 × 5 = 60

Extend: Create your own example.

I SEE REASONING


I know… so…

360 ÷ 15 = 24

List different factors of 360

How does the calculation help you to find lots of

other single-digit factors of 360?

Order

Order from the number with the least factors

to the number with the most factors:

35 36 45

Explain the Mistake


Every multiple of is also a multiple of 4

Every multiple of is also a multiple of 6

Every multiple of 15 is also a multiple of

Every multiple of 5 is also

a multiple of 10

Change the statement to make it correct.

The number… has more factors because…

I SEE REASONING


Different Answers

multiples of 4 multiples of 6


Different Answers

multiples of 4 factors of 24


Which Answers?

Circle the common

factors of 18 and 42:

2 3 6 7 9

Circle the common

factors of 27 and 45:

3 5 7 9 90

I SEE REASONING


How Many Ways? Level 1: I can find a way


Level 3: I know how many ways

there are

60 ÷ = 3 ×


Small Difference QuestionsComplete with the correct symbol: < = >

6 × 32 < 200

6 × 32 < 200 – 20

6 × 36 < 200 + 20

<

4 and 320

Which pairs of numbers can go in the boxes?

80 × = ÷ 2

5 and 800

3 and 480

6 and 240

Different Ways Answer in 4 different ways:

20 × = × 4 20 × = × 4

20 × = × 4 20 × = × 4

What’s the relationship between the numbers in the red

and blue boxes?

6 × 40 < 240 ÷ 4

6 × 20 < 240 ÷ 2

12 × 20 < 120 × 2

I SEE REASONING

FRACTIONS

Read the Picture

Which fractions do you see?

Read the PictureMatch the pictures to the fractions.

𝟏

𝟒

𝟏

𝟑

𝟏

𝟓

ExplainWhich is the larger fraction?


I SEE REASONING

FRACTIONS

Explain

For each pair, circle the larger fraction:

True or False?

EstimateFor each rectangle, estimate the fraction that is blue.

The fingers as a fraction of a hand

The hands as a fraction of the body

OR

The toes as a fraction of a foot

The fingers as a fraction of a hand

OR

The brain as a fraction of the head

The bones as a fraction of the body

OR

or

𝟏

𝟓

𝟓

𝟖𝟑

𝟖

I SEE REASONING

FRACTIONS

Context Question8 children share 5 pizzas equally.

Show the amount of pizza that each child gets.

Which Answer?9 boiled eggs shared between 4 people.

How many eggs each?

Context Question4 people share 11 pieces of toast equally.

Show the amount of toast that each person gets.

(a) 2 eggs each

(b)𝟒

𝟗of an egg each

(c) 2𝟏

𝟒eggs each

Explain your choice.

I SEE REASONING

FRACTIONS

What Fraction?

Different Number Lines

0 1

0 1

0 2

0 1

2

1

Draw arrows to show the position of 𝟑

𝟒on each line

0 2

0 1

1

1

2

1

4

I SEE REASONING

FRACTIONS

Explain

0 1

0 1

0 1

Which equivalent fractions are shown by the number lines?

Agree or Disagree?

0 1

0 1

These number lines show that thirds and sixths are equivalent

Read the Picture

0 1

0 1

0 1

Use the number lines to order these fractions: 4

5

5

6

6

8

I SEE REASONING

FRACTIONS

Which Answer?

Odd One Out

0 1

0 1

0 1

0 1

Odd One Out

0 1

6

(a)𝟏

𝟏𝟐

(b)𝟏

𝟑

The fraction at the

red arrow is…

I SEE REASONING

FRACTIONS

Agree or Disagree?


+4

3

4=

8

+47To make equivalent

fractions, do the same

thing to the denominator

and the numerator.

Explain the Mistake

Both shapes

are 𝟏

𝟔blue

=

21 4

Complete each question by positioning the digits:

8

Digits:

=

21 4

8

Digits:

<

21 4

8

Digits:

Do in different ways.

I SEE REASONING

FRACTIONS

How Many Ways?

Explain the Mistake

2

1 3

Make equivalent fractions using these numbers:

=

Rank by Difficulty




6

4 12

Note:

𝟏

𝟐= 𝟐

𝟒is not an

answer because

2 is used twice.

I know 𝟑

𝟒is more than

𝟓

𝟔because

quarters are bigger than sixths

For each pair, circle the larger fraction.

Which question was hardest? Which was easiest?

𝟏

𝟒or

𝟏

𝟓

𝟑

𝟔or

𝟒

𝟔

𝟑

𝟒or

𝟓

𝟖

𝟑

𝟒or

𝟗

𝟏𝟎

I SEE REASONING

FRACTIONS

Read the Pictures

Which is more?

Explain

Order the improper fractions from smallest to largest:

What do you notice?

𝟏𝟎

𝟑

Finish the Pictures

3𝟏

𝟐=

23𝟏

𝟐=

4

𝟏𝟎

𝟒

𝟏𝟏

𝟓

𝟏𝟖

𝟔

Finish the pictures. Fill in the missing numbers:

I SEE REASONING

FRACTIONS


Different WaysAnswer this question in two different ways:

11= 2

11= 2

4 =

54 𝟑

𝟓=

55 𝟑

𝟓=

5

4

12=

4

15=

4

19=

Agree or Disagree?

𝟖

𝟑is equivalent to

𝟏𝟔

𝟔

Extend: Design your own question by changing the denominator and the whole number. Your question must have 2 possible answers.

I SEE REASONING

FRACTIONS


Mistake A:

𝟑

𝟒𝐨𝐟 𝟑𝟔 =

36

10 10 10 6

30

Mistake B:

𝟐

𝟑𝐨𝐟 𝟐𝟒 =

24

9 9 9

18

Mistake C:

𝟑

𝟓𝐨𝐟 𝟐𝟎 =

20

4 4 4 4 416

Finish the Pictures

𝟐

𝟑𝐨𝐟 =

There are different possible answers.

𝟑

𝟒𝐨𝐟 =

𝟑

𝟓𝐨𝐟 =

I SEE REASONING

FRACTIONS

Two WaysAnswer each question in two different ways:

1of 45 =

1of 45 =

2of 24 =

2of 24 =

I know… so…𝟑

𝟒𝐨𝐟 𝟐𝟒𝟎 = 1801

4of 240 =

3

4of 248 =

I know… so…𝟐

𝟑𝐨𝐟 𝟏𝟒𝟒 = 961

3of 144 =

2

3of 288 =

4

6of 144 =

3

8of 480 =

3

8of 240 =

2

9of 144 =

Give other examples

Give other examples

3of 60 =

3of 60 =

I SEE REASONING

FRACTIONS

Small Difference Questions𝟏

𝟑𝐨𝐟 𝟒𝟓 =

𝟐

𝟑𝐨𝐟 𝟒𝟓 =

𝟐

𝟑𝐨𝐟 𝟒𝟖 =

𝟐

𝟑𝐨𝐟 𝟗𝟔 =

𝟏

𝟓𝐨𝐟 𝟒𝟎 =

𝟏

𝟓𝐨𝐟 𝟖𝟎 =

𝟒

𝟓𝐨𝐟 𝟖𝟎 =

𝟒

𝟓𝐨𝐟 𝟖𝟓 =

Small Difference Questions𝟏

𝟐𝐨𝐟 𝟑𝟐 =

𝟏

𝟒𝐨𝐟 𝟑𝟐 =

𝟑

𝟒𝐨𝐟 𝟑𝟐 =

𝟑

𝟒𝐨𝐟 𝟔𝟒 =

𝟑

𝟖𝐨𝐟 𝟔𝟒 =

𝟏

𝟒𝐨𝐟 𝟐𝟒 =

𝟏

𝟔𝐨𝐟 𝟐𝟒 =

𝟏

𝟑𝐨𝐟 𝟐𝟒 =

𝟐

𝟔𝐨𝐟 𝟐𝟒 =

𝟐

𝟑𝐨𝐟 𝟐𝟒 =

Small Difference Questions𝟑

𝟖𝐨𝐟 𝟏𝟐𝟎 =

𝟔

𝟖𝐨𝐟 𝟐𝟒𝟎 =

𝟓

𝟖𝐨𝐟 𝟐𝟒𝟎 =

𝟓

𝟖𝐨𝐟 𝟐𝟖𝟎 =

𝟐

𝟑𝐨𝐟 𝟏𝟖𝟎 =

𝟒

𝟔𝐨𝐟 𝟏𝟖𝟎 =

𝟐

𝟔𝐨𝐟 𝟑𝟔𝟎 =

𝟓

𝟔𝐨𝐟 𝟑𝟔𝟎 =

I SEE REASONING

FRACTIONS + – ×


1

4=+

3

8

4

12

Mistake A:

1

4=+

3

8

4

8

Mistake B:

Small StepsFor each calculation, choose a common denominator:

𝟑

𝟓+

𝟑

𝟓Common denominator:

𝟐

𝟑+

𝟓

𝟏𝟐Common denominator:

𝟓

𝟏𝟎+

𝟐

𝟒Common denominator:

ExplainCircle the calculations where the sum is a mixed number:

𝟑

𝟒+

𝟐

𝟖


𝟓

𝟏𝟎+

𝟑

𝟒

𝟏

𝟐+

𝟑

𝟖

𝟕

𝟏𝟎+

𝟐

𝟓

𝟕

𝟏𝟎+

𝟏𝟎

𝟐𝟓Common denominator:

𝟕

𝟐𝟎+

𝟑

𝟓

𝟓

𝟗+

𝟑

𝟔

𝟏

𝟒+

𝟏

𝟔Common denominator:

I SEE REASONING


𝟏

𝟑+

𝟏

𝟔=

Rank by Difficulty

𝟏

𝟑+

𝟑

𝟔=

𝟐

𝟑+

𝟑

𝟗=

𝟐

𝟒+

𝟏𝟎

𝟐𝟎

𝟏

𝟑+

𝟏

𝟏𝟐

𝟒

𝟓+

𝟑

𝟓

𝟗

𝟐𝟎+

𝟕

𝟐𝟎

𝟏

𝟑+

𝟓

𝟏𝟐


𝟏

𝟑+

𝟐

𝟗=

𝟏

𝟑+

𝟑

𝟗=

𝟐

𝟑+

𝟐

𝟗=

𝟑

𝟏𝟎+

𝟑

𝟐𝟎=

𝟑

𝟓+

𝟑

𝟐𝟎=

𝟐

𝟓+

𝟑

𝟏𝟎=

𝟑

𝟒+

𝟏

𝟖=

𝟑

𝟒+

𝟏

𝟏𝟔=

𝟑

𝟖+

𝟕

𝟏𝟔=

FRACTIONS + – ×

𝟏

𝟔+

𝟏

𝟗

𝟐

𝟓+

𝟏𝟑

𝟐𝟎=

𝟐

𝟓+

𝟏𝟑

𝟒𝟎=

𝟑

𝟓+

𝟏𝟐

𝟒𝟎=

𝟐

𝟒+

𝟔

𝟏𝟐=

𝟏

𝟒+

𝟓

𝟏𝟐=

𝟏

𝟒+

𝟓

𝟏𝟔=

𝟐

𝟑+

𝟐

𝟔=

𝟐

𝟓+

𝟑

𝟒𝟎=

𝟑

𝟒+

𝟑

𝟏𝟔=

I SEE REASONING

Different Ways

How Many Ways?

Different Ways




there are

4

3=+

8

Find different possible answers.

6=+

3

1

Ways to calculate 𝟑

𝟒+

𝟓

𝟖

Convert 𝟑

𝟒into

Split 𝟓

𝟖into and

8 8

4 8+++

2

1

2

1

The answer is a proper fraction

FRACTIONS + – × I SEE REASONING

FRACTIONS + – ×


7

10=–

1

2

6

8

Mistake A:

7

10=–

1

2

6

10

Mistake B:


𝟓

𝟔−

𝟏

𝟔=

𝟓

𝟔−

𝟏

𝟐=

𝟓

𝟔−

𝟏

𝟏𝟐=

𝟕

𝟖−

𝟏

𝟒=

𝟕

𝟖−

𝟑

𝟒=

𝟕

𝟖−

𝟏𝟐

𝟏𝟔=


𝟏 −𝟏

𝟖=

𝟏𝟏

𝟒−

𝟏

𝟖=

𝟏𝟏

𝟒−

𝟑

𝟖=

𝟏𝟏

𝟐−

𝟑

𝟖=

𝟏𝟑

𝟖−

𝟏

𝟐=

𝟐𝟑

𝟖− 𝟏

𝟏

𝟐=

I SEE REASONING

FRACTIONS + – ×

ExplainCircle the calculations where the answer is a

mixed number:


𝟏𝟑

𝟏𝟎−

𝟏

𝟓 𝟏𝟏

𝟖−

𝟏

𝟒

How Many Ways?Level 1: I can find a way



there are4=–

8

1

Extend

=+2

1–

4

1Fill the gaps.

1

Different Methods

𝟑𝟑

𝟖− 𝟏

𝟑

𝟒 𝟐𝟐

𝟔− 𝟏

𝟏

𝟑

𝟏𝟑

𝟓−

𝟕

𝟏𝟎𝟏𝟏

𝟐−

𝟑

𝟒𝟏𝟏

𝟑−

𝟓

𝟔

Answer these questions using different methods:

Example methods: find the difference, take away,

using knowledge about halves

I SEE REASONING

FRACTIONS + – ×

Which Answer?

2

3=× 6

12

18

Answer A:

2

3=× 6

12

3

Answer B:

2

3=× 6

2

18

Answer C:

Different Methods

as an

improper

fraction

less than 310

Ways to calculate7

10× 3

as a

mixed

number

Spot the Pattern

Fill the gaps using single-digit numbers.

What do you notice?

2

5=×

3

8=×


I SEE REASONING

FRACTIONS + – ×

I know… so…

Rank by Difficulty

Different WaysAnswer each question in three different ways.

× = 1𝟏

𝟐× = 1

𝟏

𝟐× = 1

𝟏

𝟐

× = 3𝟏

𝟑× = 3

𝟏

𝟑× = 3

𝟏

𝟑

𝟑

𝟒× 𝟒 = 𝟑

3

4× 6 =

𝟒

𝟓× 𝟔 = 𝟒

𝟒

𝟓

4

5× 7 =

𝟐

𝟕× 𝟏𝟐 = 𝟑

𝟑

𝟕

2

7× 14 =

𝟓

𝟔× 𝟔

𝟏𝟏

𝟒× 𝟕

𝟓

𝟕× 𝟒

𝟐𝟑

𝟒× 𝟑

I SEE REASONING

FRACTIONS, DECIMALS, PERCENTAGES

True or False?

Which Answers?

Which fractions have been positioned correctly?

0 0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

𝟏

𝟓𝟓

𝟏𝟎𝟎𝟎

𝟔𝟓

𝟏𝟎𝟎

𝟏

𝟒

0.05 is the same as… circle the correct answers

𝟓𝟎

𝟏𝟎𝟎𝟓

𝟏𝟎𝟎

𝟏

𝟓𝟎

Five thousandths Five hundredths

Odd One Out

0.77

10

25

1000.4

0.006 is the same as… circle the correct answers

𝟔

𝟏𝟎𝟎𝟎𝟔

𝟏𝟎𝟎

𝟔𝟎𝟎

𝟏𝟎𝟎𝟎

Six thousandths Six hundredths

1

7

1

4

I SEE REASONING


Two WaysFor each arrow, give the fraction and the decimal:

0 1

𝟎. 𝟖 𝐨𝐫𝟖

𝟏𝟎

How Many Ways?

0

0.010

1

10




0 0.50.25 1

Make all the fractions that are more than 0.25

and less than 0.5 using these numbers:

1 2 3 4 5 10

My system for knowing I have found all of the answers is…

I SEE REASONING


Explain

I know… so…

The green section is 20%,

the blue section is 40%

and the red section is 30%

of the square

Explain why this statement must be incorrect.

Estimate: % green % blue % red

1= 25%

1= 50%

1= 5%

1= 10%

1= %

50

1= %

25

1= %

5

50 × 2 = 100

25 × 4 = 100

20 × 5 = 100

10 × 10 = 100

Agree or Disagree? or

1= 20%

20

1= 10%

10= 5%

1

5

I SEE REASONING

Odd One Out

Different Ways

1= %

539

= %50

0.52 = %

0.6 = %

Rank by Difficulty

4= %

8

6= %

107

= %25

36= %

50

Make fractions that are more than 50% and less

than 75% using the digits 1, 2, 3, 4 and 5.

1

2

3

4

5


Rank by Difficulty

30%3

10

1

205%

1

3

1

5

I SEE REASONING

Contexts

MEASUREMENT

Match what is being measured with the correct type of measure and the appropriate unit of measure:

Measuring: Type: Unit:

flour for baking

adult footprint

water in cup

skipping rope

football pitch

length

weight

volume

area

Centimetres (cm)

Metres (m)

Square centimetres (cm²)

Square metres (m²)

Grams (g)

Millilitres (ml)

Litres (l)

Which Answer?

Light years measure time

Light years measure length

Which Answer?For each item, circle the appropriate unit(s) of measure:

Size of a carpet cm cm² m m² kg

Size of a bottle cm cm² litres (l) millilitres (ml)

Size of lamp post kg cm cm² m m²

Size of a puddle cm m litres (l) millilitres (ml)

I SEE REASONING

Odd One Out

MEASUREMENT

Extend: Can you think of a reason why each one

could be the odd one out?

Centimetres (cm)

Explore

Agree or Disagree?

100 metres is approximately

the same length as 110 yards.

Ounces (oz) Kilograms (kg)

Which unit of measure is the odd one out?

This means a yard is longer than a metre

Metric measures Measures of length

Write these measures in the correct section of the

Venn diagram: litres, inches, kilometres, stones, yards

Extend: Add another unit of measure to each section.

I SEE REASONING


MEASUREMENT

800m = _____km8

25mm = _____cm2503.8m = _____cm308

Agree or Disagree?

𝟏

𝟐litre = ______ml5000

408cm = ______m4.8 4090g = ______kg4.09

True or False?

(a) 75cm is more than 0.6m and less than 800mm

(b) 0.7m is more than 55cm and less than 600mm

(c) 410mm is more than 40cm and less than 𝟏

𝟐metre

Compare

1 yard = 36 inches 1 metre (m) = 100centimetres (cm)

6m = cm

6 yards = inches

2.5m = cm

2.5 yards = inches

It is harder/easier converting between… because…

𝟑

𝟒metre = ______cm75

I SEE REASONING


MEASUREMENT

Which Answer?


minutes = 1 hour and 40 minutes

minutes = 3 hours and 20 minutes

minutes = 5 hours

150 minutes = hours and minutes

250 minutes = hours and minutes

The time is 2:10pm. What was the time 𝟐𝟏

𝟐hours ago?

4:40pm 11:40pm11:20am

11

4hours ago the time was 4:50pm. What is the time now?

(a) 3:25pm (b) 3:35pm (c) 6:05pm (d) 6:15pm

The train left at 6:45 and arrived at 9:20.

How long was the journey?

(a) 3 hours 25 minutes (b) 2 hours 35 minutes

Extend: create two questions converting between minutes and hours and minutes

I SEE REASONING

Compare

MEASUREMENT

Use the symbols < = > to compare the lengths of time:

6 weeks 44 days

100 hours 4 days

120 hours 5 days

120 hours 5 days

3 months 2400 hours

3 months 12 weeks

3 months 63 days

3 months 63 days

Multi-Step

weeks is more than 2 months and less than 65 days.

days is less than than𝟏

𝟒year and more than 12 weeks.

Extend: Write your own example. Make the difference between the lengths of time small.

Multi-Step

Order the lengths of time from shortest to longest:

Question A:

180 days 27 weeks 7 months 1

2year

Question B:

140 days 18 weeks 3 months 1

3year

I SEE REASONING

Different Ways

PERIMETER, AREA, VOLUME

Which method(s) correctly

calculate the perimeter of

the rectangle?

3cm

5cm

5cm × 3cm Double 5cm + 3cm

5cm + 3cm5cm + 3cm + 5cm + 3cm

ExplainFor each shape, how many sides need to be measured to calculate the perimeter?

rectanglesquare

Estimate

isosceles

trianglecross

Explain why you don’t have to measure every edge.

7cm

Estimate the perimeter of the rectangle.

OrderOrder these shapes from smallest to largest perimeter without measuring them:

I SEE REASONING

Explain


Here are two identical rectangles. They are put together to make one rectangle.

I know… so…Each shape is made using identical rectangles.

Calculate the perimeter of each shape.

How can this be done to make the perimeter of the new rectangle as small as possible?

Or as large as possible?

1.5cm

2.5cm

Perimeter = 8cm Perimeter = cm Perimeter = cm

Perimeter = cm Perimeter = cm

Perimeter = cm

I SEE REASONING



Explain

Each shape is made using squares.

Calculate the perimeter of each shape.

Perimeter = cm

12cm 12cm

Perimeter = cm

12cm

Perimeter = cm

Complete the sentences:

purple + orange =

black – blue =

Write your own.

Agree or Disagree? 6cm

To calculate the

perimeter, more

information is needed

9cm

8cm

I SEE REASONING

Which Answer?


Which answer is correct?


Agree or Disagree?

Estimate

7cm

12cm

7cm

12cm

Calculate the area:

12 × 7 = 84cm²

12 × 7 × 12 × 7 = 7056cm²

12 + 7 + 12 + 7 = 38cm

If the lengths of the sides of a rectangle are doubled, the rectangle

becomes four times bigger.

Tip: Use an example to support your opinion.

Estimate the lengths shown by the arrows:

Area = 24cm² Area = 24cm²

not to scale

I SEE REASONING

Spot the Mistake


Calculate the area of the shape:

3cm

6cm

10cm

9cm

7cm

3cm × 10cm = 30cm²

7cm × 9cm = 63cm²

30cm² + 63cm² = 93cm²

Different Ways

Calculate the area:

4cm

10cm

8cm

5cm

3cm 6cm

Method 1

Do 10cm × 5cm

Add cm × cm

Method 2

Do 8cm × 4cm

Add cm × cm

Method 3

Do 10cm × 8cm

Subtract cm × cmArea = cm²

I SEE REASONING

Spot the Pattern


Calculate the area and perimeter of each shape:

What do you notice?

Shape A6cm

6cm

Shape B4cm

8cm

Shape C2cm

10cm

Same and Different

Part 1 Draw a rectangle with

the same perimeter and a larger area.2cm

8cm

Part 2

Draw a rectangle with

the same perimeter and a smaller area.5cm

6cm

ExplainA rectangle has a perimeter of 28cm.

What is the largest possible area for the shape?


I SEE REASONING



Part 1Draw a rectangle with

the same area and a larger perimeter.4cm

6cm


the same area and a smaller perimeter.4cm

10cm

Extend

Design a rectangle with a smaller perimeter and a larger area.

3cm

9cm


the same area and a larger perimeter.4cm

10cm

I SEE REASONING

True or False?


True or False?

Explore

This cuboid is made

using 20 cubes

A cube can be made using 16 smaller cubes

Make a cuboid using

18 cubes.

Your cuboid must have two square faces.

How Many Ways?

Make a cuboid using 16 to 18 cubes.

There must be at least 4 squares on each face of the cuboid.




6 squares on this face

square face

rectangular face

A cube can be made using 8 smaller cubes

I SEE REASONING

Order

ANGLE

Tip: Make each angle with a pair of scissors.

Order the angles from smallest to largest without using a protractor:

Explore

Find all the acute angles, right angles and obtuse angles:

ExplainIs each angle acute, a right angle or obtuse?

Explain how you know. Don’t use a protractor.

I SEE REASONING


ANGLE

84° 122°

40°

Which Answer?

47° 127°

133°


I SEE REASONING

Which Answer?

ANGLE

(a) 104˚

(b) 76˚

(c) 86˚

For each example, what is the missing angle?

(a) 245˚

(b) 255˚

(c) 65˚115˚61˚

43˚

Explain

Angle A is larger than angle D.

A

Angle B is larger/smaller than angle C.

C DB


Calculate the missing angles:

G

68˚

F

E

64˚

64˚ 68˚

H

E =

F = G = H =

I SEE REASONING


ANGLE

Calculate the missing angles:

J76˚ K41˚

35˚

L41˚

65˚

J = K = L =

M

41˚

65˚

M =

90˚

N

51˚

75˚

N =

90˚

True or False?

(a) A 360˚ turn can be made with four acute angles.

(b) A 360˚ turn can be made with a reflex angle and two acute angles.

(c) A 360˚ turn can be made with a reflex angle and two obtuse angles.

Extend: A 360˚ turn can be made with obtuse angles.

How many ways can this question be answered?

Always, Sometimes or Never?

Triangles have smaller angles than quadrilaterals

I SEE REASONING

Multi-Skill

ANGLE

12 12

3

4567

8

9

10

11

At each time, is the smaller angle between the minute hand and the hour hand acute, obtuse or a right-angle?

10:30am

obtuse

7:10am 4:00pm 9:00am

9:30pm 8:55pm

Which Answer?

12 12

3

4567

8

9

10

11

Draw the times on the clocks.

At each time, what is the smaller angle between the minute hand and the hour hand?

5:00pm

(a) 150˚

(b) 25˚

(c) 120˚

12 12

3

4567

8

9

10

11

6:30am

(a) 0˚

(b) 30˚

(c) 15˚

12 12

3

4567

8

9

10

11

4:30pm

Multi-Skill

12 12

3

4567

8

9

10

11

At each time, what is the smaller angle between the minute hand and the hour hand?

1:00pm

12 12

3

4567

8

9

10

11

8:00pm

12 12

3

4567

8

9

10

11

5:30pm

12 12

3

4567

8

9

10

11

3:20pm

I SEE REASONING


SHAPE

How many lines of symmetry does each shape have?

Explain

All four shapes… Three shapes have…

Two shapes have… One shape has…

Odd One OutThink of a reason why each shape could be the

odd one out:

Use different properties of shape in your answers.

Use different properties of shape in your answers.

I SEE REASONING

Explain

SHAPE

Draw shapes in different sections of the Venn diagram:

Two Acute Angles No Lines of Symmetry

At Least Two Pairs of Parallel Lines

Explore

Circle the regular shapes.

For each irregular shape, explain why they are not regular.

I SEE REASONING

SHAPE

cuboidsquare-based

pyramidtriangular prism

Think of a reason why each shape could be

the odd one out:

Odd One Out

A cuboid has more face(s),

more edge(s) and more vertices

than a triangular prism.

For each pair of shapes, calculate the difference

between the number of faces, edges and vertices:


A square based pyramid has more face(s),

more edge(s) and more vertices than

a triangular pyramid.

Extend: Design your own question to compare the faces, edges and vertices of different prisms or pyramids.

I SEE REASONING

SHAPE


One more square needs adding to this diagram to make the net of a cube.

Which diagrams have been completed correctly to make the net of a cube?


Which nets can be folded into a cuboid?

Extend: Design your own net for a cuboid.

I SEE REASONING


POSITION, DIRECTION, COORDINATES

Which Answer?For each example, has the black shape

been reflected, translated or rotated?

Reflect the shape in the

mirror line.

mirror

Mistake 1

mirror

Mistake 3

mirror

Mistake 2

I SEE REASONING



Translate 2 squares right and 3 squares up.

Mistake 1

Explain

The shape has been reflected

The shape has been translated

The shape may have been translated or reflected

The shape has been reflected

The shape has been translated

The shape may have been translated or reflected

Start position End position Start position End position

Tick the correct box for each example.

Explain your choice.

Mistake 2

I SEE REASONING


Different Ways

Estimate

Could the coordinates of the blue dot be:

(5,3) (6,10)

(30,50) (4,5)

Estimate the coordinate points of the red and blue dots.

Think of possible coordinates for the blue dot:

(8,0)

Estimate

Estimate the coordinate points of the green and purple dots.

(18,24)

I SEE REASONING


Agree or Disagree?

Explain the Mistake

Which Answer?

(9,6)The blue dot is (9,0)

The red dot is (6,0)

(0,4)Point B is (7,4)

Point B is half-way between point A and point C.

(14,8)

A

B

C

(3,2)

Inside, on the edge or outside the rectangle?

(10,7) Inside Edge Outside

(3,5)

(9,3)

(2,6)

(8,7)

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Different Answers

(0,12)

Each square is the same size.

Find coordinates that

are on the edge, at

the corner and on

the inside of the red

square.

Different Answers

Both rectangles are the same size.

Find coordinates that are on the edge, at the

corner and on the inside of the blue rectangle.

(5,8)

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STATISTICS

What’s the Question?

Here is a train timetable:

1. The answer is 8:08. What’s the question?

2. The answer is 9:05. What’s the question?

3. The answer is 11 minutes. What’s the question?

Macclesfield 7:45 8:27 9:05 9:52

Poynton 7:57 8:38 9:17 10:04

Cheadle Hulme 8:03 8:45 9:24 10:10

Stockport 8:08 8:50 9:30 10:16

Manchester Piccadilly 8:19 9:01 9:41 10:27

Multi-StepHere are two train timetables:

Helen lives in Crewe.

She is travelling to

Cannock.

Helen needs to arrive

by 16:30 at the latest.

What time does Helen

need to arrive at

Crewe railway station?

Crewe 13:56 14:20 14:52

Stafford 14:15 14:39 15:11

Wolverhampton 14:38 15:05 15:37

Birmingham 14:55 15:23 15:54

Wolverhampton 14:12 14:56 15:45

Dudley 14:26 15:10 15:58

Walsall 14:58 15:41 16:30

Cannock 15:14 15:58 16:45

Why does this question use two train timetables?

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STATISTICS

Fill the GapsHere are the results from six football matches:

Won Drawn Lost Points Goals

For

Goals Against

Ashton 7 5

0 1 6 6

1 8

Duddon 0 5

Contexts

Fill the gaps in the league table:

Kelsall 2 – 3 Tarvin

Ashton 5 – 2 Duddon

Duddon 2 – 4 Kelsall

Tarvin 1 – 3 Ashton

Ashton 2 – 2 Kelsall

Tarvin 5 – 1 Duddon

3 points for a win, 1 point from a draw, 0 points for a loss.

For each example, should the data be presented as a bar graph or a line graph?

Temperature in Garden on 1st July

Number of Different Vegetables Grown in Garden

Rainfall per Month in Garden

Think of other contexts where we use bar graphs.

Think of other contexts where we use line graphs.

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STATISTICS

Which Graph?Draw lines to match the heading to the correct graph.

Favourite Fruit

for Children in Y5/6 Class

Number of Children in Each Class in School

Shoe Size for Children in Y5/6

Age of Children in Y5/6 Class

Which Graph?Draw lines to match the heading to the correct graph.

Speed of

Runner in Hill Run Race

Speed of Runner in Marathon Race

Speed of Runner in 100m Race

Explain the differences in the graphs.

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STATISTICS

Act the Graph

Speed of Running on the Spot

spe

ed

Time (seconds)

very

slow 15 30

Fear in Facial Expression:

Rollercoaster Ride

Act the Graph

exc

ite

me

nt

Time (seconds)

100%

0%

50%

10 20

very

fast

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STATISTICS

Which Answer?This graph shows the speed of a 400m runner.

What is happening at the point showed by the arrow?

(a) The runner’s fastest speed

(b) The runner finishes

(c) The runner slows down

Speed of Runner

Sp

ee

d (

mp

h)

Time (seconds)

20

50

10

00

Which Answer?This graph shows the distance travelled by a cyclist.

What is happening at the point showed by the arrow?

(a) The cyclist has stopped

(b) The cyclist is riding slower

(c) The cyclist is riding at the

same speed

Distance Travelled by Cyclist

Dis

tan

ce

(m

ph

)

Time (hours)

50

3

25

00

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ANSWERS

Place Value, page 7:

True or False? True examples: 3 hundreds, 9 tens, 2 ones1 hundred, 29 tens, 2 ones

Different Ways 3 tens 23 tens 12 tens Example answers: 53 tens & 4 ones 43 tens & 14 ones

Different Ways Fewest counters = 7 (2 thousands, 4 hundreds, 1 ten)

Most counters = 241 (241 tens)2410 with 16 counters: 2 thousands, 3 hundreds, 11 tens2410 with 16 counters: 1 thousand, 14 hundreds, 1 ten


Which Answer? 3004 fifty thousand and seventy 60002 twenty thousand and six

Small Difference Questions: Thirty-five thousandThirty thousand five hundred Thirty thousand and fiftyThree thousand and fifty Three hundred thousand and five

Order: 56020 (2 zeros) 300270 (3 zeros) 13031 (1 zero)


Spot the Pattern: Line 1 gaps: 754 794 804Line 2 gaps: 973 993 1003 Line 3 gaps: 1306 1006 906

Small Difference Questions: Left column: 1000 1099 1009Right column: 10000 10999 10099

Estimate: Estimates: blue = 800 red = 8400 purple = 3000 green = 78000


Estimate: Example reasoning, line 1: 500 is half-way, split 500→1000 into 5 steps of 100, position 836. Line 2: 836 slightly less than one-tenth along number line. Line 3: split 800 → 850 into 5 steps of 10, position 836. Line 4: 850 is half-way, position 836.

Estimate: Example reasoning, line 1: 5000 is half-way, split 0→5000 into 5 steps of 1000, position 3280. Line 2: Double the distance of line 1.Line 3: 3500 is half-way, split 3000→3500 into 5 steps of 100, position 3280.

Line 4: 3280 is 4

5along the number line.


Answers

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Small Difference Questions: 100 10000 500

Different Ways: Example answers: 0 & 450 190 & 200 100 & 240

Different Ways: Example answers: 0 & 5000 2000 & 3600 3000 & 3100


Explain: Answer = 9. Less than 100 is added to a 3-digit number to make 1000 or more. Therefore, the 3-digit number must be more than 900.

Agree or Disagree? Agree: Jen’s number could be 70 and Molly’s number could be 69.

Different Ways: 691 or 709

How Many Ways? 6 ways: 5210, 5120, 5102, 5012, 4310, 4130

Decimals, page 13:

Small Difference Questions: 0.8: Small Difference Questions: 0.36

Agree or Disagree? Disagree, the arrow shows the position of 0.7

Decimals, page 14:

Different Ways: Example answers for 3.74:

Agree or Disagree? Blue is true, red is false. Look at the most significant column. 140 has more 100s than 80; 0.8 has more tenths than 0.14.

Compare: 0.64 < 0.9 0.64 > 0.614 0.64 < 0.6440.7 = 0.70 0.55 > 0.505 0.08 < 0.088 0.915 < 0.92

ANSWERS


Answers

3.7 3.8

0 10

3 4

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ANSWERS

Decimals, page 15:

Explain the Mistakes: Blue example: 0.3 should be added to the tenths, answer = 0.77. Red example: 0.8 + 0.2 = 1, 0.08 + 0.02 = 0.1, answer = 1.1

Compare: 0.33 < 0.3 + 0.3 0.33 = 0.3 + 0.030.41 > 0.4 + 0.004 0.67 = 0.60 + 0.07 0.87 > 0.7 + 0.08

How Many Ways? Q1: 5 ways: 4 × 0.1 & 2 × 0.01 3 × 0.1 & 12 × 0.01

2 × 0.1 & 22 × 0.01 1 × 0.1 & 32 × 0.01 42 × 0.01Q2: 3 ways: 2 × 0.1 & 4 × 0.01 1 × 0.1 & 14 × 0.01 24 × 0.01

Spot the Pattern: 0.4, 0.41 0.1, 0.11 0.4, 0.04 0.5, 0.25

Decimals, page 16:

Small Difference Questions: 20.5 2.05 805 0.805

Which Answer? Correct answer = 53. Mistakes: 54 is incorrect because the number should be rounded down; 53.5 is not a whole number.

Number Lines: 38.42 rounds to 38 39.08 rounds to 39 39.6 rounds to 4040.27 rounds to 40 40.82 rounds to 41

23.555 rounds to 23.6 23.64 rounds to 23.6 23.708 rounds to 23.723.78 rounds to 23.8

Decimals, page 17:

Different Ways: Example answers: 3.3 & 3.7 3.45 & 3.55 3.499 & 3.501

Small Difference Questions: 30 ÷ 30 = 1 30 ÷ 100 = 0.3 30 ÷ 300 = 0.1

40 ÷ 100 = 0.4 40 ÷ 40 = 1 40 ÷ 80 = 0.5

Different Ways: Example answers: 0.07 + 0.03 = 0.1 10 ÷ 100 = 0.10.01 × 10 = 0.1 2 ÷ 10 = 0.2 2 – 1.8 = 0.2 0.8 + 79.2 = 80 0.8 × 100 = 80


Answers

38 39 40 41

38.42 39.08 39.6 40.27 40.82

23.5 23.6 23.7 23.8

23.555 23.64 23.708 23.78

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ANSWERS


AnswersDecimals, page 18:

Spot the Pattern: 1.2, 1.5 0.08, 0.1 0.1, 0.2 0.3, 0.6

Different Ways: Example answers: 30×0.1 = 3 10×0.3 = 3 6×0.5 = 3Example answers: 6×0.4 = 2.4 0.4×6 = 2.4 24×0.1 = 2.4

Small Difference Questions: Left column: 3 1.5 0.5Middle column: 20 40 8 Right column: 10 8 4

Negative Numbers, page 19:

Contexts: Sweets example incorrect (negative numbers don’t describe quantities). Face paints is incorrect as the profit is £5. The other examples are correct.

Contexts: £4 - £6 = -£2 -2 + 5 = 3 Example question: The temperature was 5˚c. Then the temperature fell by 7˚c. What is the temperature now?


Explain the Mistake: Counting all the numbers is incorrect because you don’t include the first number in a count to calculate the difference.

Different Ways: 30 and -70 8 and -42

Estimate:


Estimate: Approximate values: -5 -7 and 13 -5 and 3

Estimate: First line 0 is 1

2along. Second line 0 is

1

3along.

Third line 0 is 1

3along. Fourth line 0 is

1

5along.


Small Difference Questions: top left: 6 bottom left: 1 top right: -1 bottom right: 1

Small Difference Questions: 2 -2 -4 2 10

Small Difference Questions: 11 4 3 19

-20 0-8-14 -3 1710 20-10

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ANSWERS


AnswersNegative Numbers, page 23:

Which Answer? Red answer is correct. Blue is incorrect as the pattern in the ones value when subtracting 5 changes when bordering 0.

Rank by Difficulty: Blue = -1 (highest start number, - 2 so odd number pattern); Red = -6 (120 is a multiple of 6 so zero will be in the sequence); Green = -4 (lowest start number but calculation bordering 0 is harder).

How Many Ways? 4 ways: 4 5 10 20

Different Ways: Example sequences: -5, -1, 3, 7, 11 31, 19, 7, -5, -17

Rounding, page 24:

Explain: Example answers: The population of Scotland is impossible to measure exactly so it is rounded. It may be rounded to the nearest thousand or the nearest million depending on the context. We typically give the time to the nearest 5 minutes.

Explain: Typically, a school would say the exact number of children in their school. Recipe quantities are rounded. A 400m race time would be

rounded to the nearest second for normal races, but given to 2 decimal places for professional races.

Which Answer? 198 – 57 can be calculated by rounding 198 to 200. 503 + 246 doesn’t require any carries so rounding may not make this easier. 249 + 148 can be rounded to 250 + 150 – 3. When multiplying, typically rounding is only used when one number is rounded. 60 × 4 can be used to derive 59 × 4.

Rounding, page 25:

Number Lines: Nearest 20: 607 closest to 600 632 closest to 640

668 closest to 660 685 closest to 680Nearest 50: 376 closest to 400 419 closest to 400 438 closest to 450 471 closest to 450

Number Lines: 437: closest 10 is 440, closest 100 is 400. 446: closest 10 is 450, closest 100 is 400.453: closest 10 is 450, closest 100 is 500.

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ANSWERS


AnswersRounding, page 26:

Small Difference Questions: 250, 200, 300, 500, 500

Small Difference Questions: 570 100 550 20

Agree or Disagree: This is true if the number is rounded to 10, 100 or 1000. However, if the number is rounded to the nearest 20 or the nearest 50 then it would round to a smaller number.

Rounding, page 27:

Explain the Mistakes: Blue answer: When rounding to the nearest 1000, the hundreds value determines how the number is rounded.Red answer: The hundreds value has been rounded correctly, but the nearest 100 is 4300 (the 4000 has not been included in the answer).

Which Answer? The purple answer is correct as 400 is a multiple of 10.

Which Answer? 1549

Explain: Example answer: 349 and 350 are consecutive but rounded to the nearest 100 349 is 300 and 350 is 400.

Rounding, page 28:

Different Ways: Left: 245→249 Centre: 250→254 Right: 255→349

Multi-Step: The least Tim could have is £1.45, the most Kam could have is £2.74, therefore the maximum possible difference is £1.29

Multi-Skill: Question 1: 246, 249, 252 Question 2: 301Example question: To the nearest 10, my number is 200. My number is a multiple of 6. What could my number be? (198 and 204).

Roman Numerals, page 29:

Compare Questions: LXVI = 66 LXIV = 64 XLVI = 46 XLIV = 44

Rank by Difficulty: 33 = XXXIII 49 = XLIX (note that for 40 and 9 symbols are used for less than 10 & 50) 65 = LXV

Compare Questions: XL = 40, IX = 9 (similar structures, both use a symbol for 1/10 less than 10/50). XIII = 13, LXXX = 80 (similar adding structures).

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ANSWERS


Rank by Difficulty: 230 = CCXXX 490 = CDXC (symbols needed for 100 less than 500 and 10 less than 100) 780 = DCCLXXX (7 symbols needed).

Order: DIII = 503 DXX = 520 CM = 900 M = 1000 (note that with these examples, the numbers with more symbols are larger).

Explain: Red: symbols can be used for less than. Example: in XL, the X

represents 10 less than 50, making the number smaller.Blue: Some numbers above 200 have fewer symbols than numbers below 20. Examples: 18 = XVIII (5 symbols), 201 = CCI (3 symbols).


Spot the Pattern: LXXX, XC, C, CX, CXX

CCX, CCXX, CCXXX, CCXL, CCL DLX, DLXX, DLXXX, DXC, DC

Small Difference Questions: XLII XLIV LIII XXXIII XXIX XL LXXX


Extend: XLIV (44)

How Many Ways? 7 answers: XIV XVI XLI XLV LIV LVI LIX

How Many Ways? 15 answers: CDI CDV CDX CDL DIV DVI DIX DXI DLI DLV DLX DCI DCV DCX DCL

Addition, page 33:

Simplify: Top row: 232 15 Bottom row: 130 2


Small Difference Questions: Left column: 520, 520, 502, 502Right column: 1031, 1031, 1028, 1018

Mental or Written Method? 4731 + 5268 = 9999 (can add each column)463 + 278 = 741 (may suit column method)895 + 385 = 1280 (can change to 900 + 380)2480 + 2520 = 5000 (equal to 2500 + 2500)


Answers

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ANSWERS


AnswersAddition, page 34:

Explain the Mistakes: Left example: the numbers are not lined up correctly. Middle example: error in addition of hundreds column. Right example: the digits that have been carried haven’t been added.

Correct or Incorrect? Left example is incorrect: the numbers are not lined up correctly (see the decimal point). Middle example is correct.

Right example is incorrect: error in tens value addition.

Rank by Difficulty: 4065 + 3205 = 7270 (the same as 4070 + 3200).744 + 579 = 1323 (may suit column method). 473.6 + 516.2 = 989.8 (can add each column).2996 + 2995 = 5991 (can do 3000 × 2 – 9).3509 + 3444 = 7003 (the same as 3503 + 3500).

Different Ways: 1442 and 1532. 1523 has an incorrect ones value. The hundreds value can’t be 6 so the answer can’t be 1652.

Addition, page 35:

Which Answer? The red answer is correct. In this case, the blue statement is incorrect as one more hundred needs to be added, so the missing tens value in the addend must be 9.

How Many Ways? 4 ways: 28 + 76 = 104 26 + 78 = 104

68 + 72 = 140 62 + 78 = 140

Different Ways: First question has 3 answers: 1025 + 739 = 17641125 + 739 = 1864 1225 + 739 = 1964Second question has 2 answers: 1243 + 836 = 2079 1243 + 936 = 2179Third question has 1 answer: 1928 + 158 = 2086

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ANSWERS


AnswersSubtraction, page 36:


Rank by Difficulty: 43.2 – 16.8 = 26.4 (using the column method, two regroups are needed so relatively challenging)845 – 297 = 548 (equivalent to 848 – 300)601 – 337 = 264 (equivalent to 599 – 335 which can be done using the

column method without regrouping)869 – 321 = 548 (can use the column method without regrouping)


Rank by Difficulty: 8765 – 2543 = 3222 (can use the column method without regrouping)7002 – 3497 = 3505 (5 more than 7000 – 3500)5432 – 2543 = 2889 (using the column method, two regroups are needed so relatively challenging)6672 – 3994 = 2678 (equal to 6678 – 4000)

Subtraction, page 37:

Different Methods: 3728 – 1465 = 2263 (strategy: column method)3012 – 2994 = 18 (strategy: calculate the difference)3000 – 30 = 2970 (strategy: count back)900 – 452 = 448 (strategy: near doubles)764 – 296 = 468 (strategy: equivalent to 768 – 300)


Small Difference Questions: Left column: 325, 325, 365, 365, 5765Right column: 4010, 3983, 3663, 3653, 3687

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ANSWERS


AnswersSubtraction, page 38:

Explain the Mistakes: Left example: 5 – 8 ≠ 3 and 0 – 2 ≠ 2Middle example: the numbers are not lined up correctly.Right example: Should be regrouped into 5000, 900 & 100 (so the 0 in the hundreds column should be crossed out and a 9 positioned above)Correct or Incorrect: Left and right examples are correct. The middle example is incorrect because to get the extra 10 for the ones column the 8 tens must become 7 tens.

Rank by Difficulty: 4600 – 1280 = 3320 4006 – 1280 = 2726 Note that the second questions requires more regrouping. This is more challenging with zeros: three columns are adjusted on the first regroup.23.64 – 12.8 = 10.84 23.8 – 12.64 = 11.16 Note that the second calculation in particular is made simpler when 23.8 is written as 23.80

Explain: If the tens value of the minuend is smaller than the tens value of the subtrahend, then the digit in the red box will be 3. This can be shown using the column method, because in this case a regroup would be

required. Example: 615 – 234 = 381

Subtraction, page 39:

Explain: Example answers: 47 – 15 = 32 55 – 17 = 32Difference between digits in blue box = 3, difference between digits in red box = 4 Explanation: in the second example, the ones value of the subtrahend is smaller than the ones value of the minuend. Therefore, for the difference to be 30+, the difference between the tens values must be more than 30.Different Ways: First question has one answer: 839 – 285 = 554Second question has one answer: 916 – 160 = 756Third question has two answers: 479 – 186 = 293 479 – 196 = 283

How Many Ways? Four possible answers: 602 – 237 = 365 612 – 237 = 375 622 – 237 = 385 632 – 237 = 395

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ANSWERS


AnswersAddition and Subtraction, page 40:

Broken Calculator: Example methods: 750 + 850 = 800 × 2 505 – 367 = 499 – 361 866 – 597 = 869 – 600 7.5 – 5.5 = 8 – 6 63.5 + 79.5 = 63 + 80 1.4 – 0.7 = 1.4 ÷ 2

Contexts: (b) 30 + 12 (a) 15 + 7 (b) £15 + £25 (a) 200 – (55 + 85)

Addition and Subtraction, page 41:

Building Questions: Raja buys three apples and three bananas. How much does it cost him?

Mo has £2. How many oranges can he afford?

Ben has £2. He wants four oranges and three apples. How much more money does he need?

Matt has £2. He wants four oranges and three bananas. How much change does he get?

Which Bar Model: The right-side bar model is correct: £2 is more than the cost of the fruit so change is given.

Explain: Largest number = 3 oranges. 3 apples cost 75p. £2 – 75p = £1.25. With £1.25, 3 is the maximum number of oranges that can be bought (3 × 35p = £1.05).


Small Difference Questions: 1. £2.55 2. 55p 3. 45p 4. 8 apples5. 5 oranges 6. A drink and 2 oranges

Building Questions: Example line 1: Lucy buys 3 pineapples. She pays £3. Example line 1: Kelly buys 3 melons and a mango. She pays £4Example line 2: Tom want to buy 3 mangos. He has £1.50Example line 2: Jim has £3. He wants to buy 3 pineapples and a melon.Example line 3: Zara has £4.50 (any value in the range £4.50→£5.39)


Multi-Step: John bought 4 oranges.Kelly bought 2 apples, 2 oranges and 1 banana.Ben bought 4 apples and 2 bananas.

Which Bar Model? The bottom left bar model is correct.

Agree or Disagree? The one correct answer is 17 boys.

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ANSWERS



Small Difference Questions: 1. 5 girls 2. 15 boys 3. £16 4. £15.50

Small Difference Questions: 1. 30p 2. 50 women 3. £25 4. 75g 5. 85 and 155

Rank by Difficulty: Orange = 180g 12 boys circle = 17.5 square = 12.5Note that the oranges/pears calculation is the same as the boys/girls

except it’s 10 times bigger and the answer asks for a different part. The shape puzzle is different in that the answer is a decimal. This happens when the sum is even and the difference is odd and vice versa.


How Many Ways? 4 ways: 22 & 30 23 & 31 24 & 32 25 & 33Which Answer? 485 is correct. The mistake for 395 is to subtract the 45, ignoring that the subtraction is on the other side of the = sign. The mistake for 440 is to ignore the – 45.

Small Difference Questions: 75 + 45 > 120 – 10 75 + 65 = 150 – 10

85 + 75 > 160 – 20 85 + 75 < 160 + 20 95 + 85 = 155 + 25230 – 165 = 83 – 18 230 – 165 = 93 – 28 230 – 185 = 93 – 48230 – 85 > 93 + 48 230 – 85 < 103 + 48


Agree or Disagree: The red statement is false. Example: 50 + 5 = 70 – 15The green statement is true.

Multi Step: Example answer: 8 > 6 4 + 2 = 7 – 1 5 + 3 < 9Note that the only digit that can go in the bottom box is 9. The middle line can only be completed using the 7 as shown in the example above.

Multi Skill: Left box: 36 36 48 Right box: 42 48 72

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ANSWERS



Agree or Disagree: The blue statement is true: the circle = 13, one additional circle increases the sum by 13. Therefore the star = 19The red statement is true: the pentagon is 5 more than the diamond but the value of those shapes will differ depending on the value of the oval.

Explore: = 15 as three hexagons = 45. Therefore = 8 (middle row)

and = 12 (middle column).

= 7 (compare top row to middle column). Therefore = 9 = 5

Extend: = 10 (compare the top row to the 4th column).

Therefore = 11 (use 4th column) and = 6

Multiplication Calculation, page 48:

Different Ways: Examples: partition into two sections of 15×4; partition into three sections of 5×8; partition into 11×8 and 4×8

Small Difference Questions: Left column: 84, 108, 216, 324, 315, 630Middle column: 96, 128, 112, 126, 108, 108Right column: 120, 120, 136, 476, 462, 231


Matching Number Sentences: 6×9 (also could be 3×19 or 2×27). 8×7 (also could be 4×14 or 2×28). 6×10 (also could be 3×20 or 2×30). 3×19

Matching Number Sentences: 3×14 9×9 8×15 (also could be 4×30 or 2×60) 7×12

I know… so… 24×19 = 456 (24 more) 40×15 = 600 (150 more)38×12 = 456 (36 more)


Small Difference Questions: Left column: 150, 138, 108, 216, 864, 984Right column: 630, 315, 336, 416, 624, 936

Rank by Difficulty: 37×6 partitioned and multiplied gives 180 and 42. 804×6 partitioned and multiplied gives 4800 and 24. Easier addition.98×6 can be calculated using rounding. 45×6 = 90×3

Rank by Difficulty: 32×50 = 16×100. Use rounding for 99×4. Partitioning and multiplying 46×7 gives 280 and 42. Partitioning and multiplying 409×6 gives 2400 and 54. Easier addition.

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ANSWERS

Multiplication, page 51:

Different Methods: Examples: 14×25 becomes 7×501.5×6 becomes 3×3 5×18 becomes 10×9

Different Ways: Blue: 8 Red: 48 Green: 12×8 or 24×4

Broken Calculator: 26×8 example ways: 52×4 or 26×10 – 5280×15 example ways: 40×30 or 79×15 + 15


Estimation: Odd. Estimate slightly closer to 2800 than 2100.

Estimation: Even. Estimate significantly closer to 5400 than 4500.

Estimation: Even. 4-digit number. Closest to 1100: 300×4 = 1200,the answer is less than this so is closer to 1100 than 1300.

Checking Possible Answers: Not 3028: 500×6 = 3000, answer must be less than 3000. Not 2941, answer must be even.


Explain the Mistakes: The 2 hundreds from 60×4 = 240 should be positioned below the line and added to 200×4 = 800. The 3 tens from 7×5 = 35 have not been added to the 10×5 = 50. Error in the calculation of 800×6.

Part-Complete Examples: 2445 2892 3704

Different Methods: The same calculations. For the grid method, the multiplication and addition are done separately. For the short method, the multiplication and addition calculation are done simultaneously.


Estimation: Even. Estimate nearer to 900 than 400.

Estimation: Odd. Estimate slightly more than 3600.

Estimation: Odd. 4-digit number. Closest to 1000: close to 50×20 = 1000.

Checking Possible Answers: Not 1602: 40×40 = 1600, answer must be less than 1600. Not 1543 as the answer must be even.


Answers

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ANSWERS


Different Methods: For the grid method, the multiplication and addition are done separately. This is the same for the expanded method but the numbers are aligned in columns. In the long multiplication method, the multiplication and addition calculation are done simultaneously so only two numbers need to be added as the final step.

Explain the Mistakes: 72×43: no zero in the ones column in the second line of the calculation. 41×23: the calculation done is 41×32. 84×52: The second line of the calculation should be 4200, not 40200.

Part-Complete Examples: 63×53: line 1 is 189, line 2 is 3150, line 3 is 3339.24×16: line 1 is 144, line 2 is 240, line 3 is 384. 81×46: line 1 is 486, line 2 is 3240, line 3 is 3726.


Missing Digits: 428×6 = 2568 524×8 = 4192 695×5 = 3475

Different Answers: 931×4 = 3724 or 532×7 = 3724

How Many Ways? 3 ways: 27×3 = 81 19×3 = 57 29×3 = 87

Division, page 57:

Different Methods: Possible methods: 480 ÷ 4 = 120 (halve twice)480 ÷ 6 = 80 (6 × 8 × 10) 480 ÷ 10 = 48 (move digits one column)480 ÷ 20 = 24 (480 ÷ 2 ÷ 10) 480 ÷ 240 = 2 (counting the 240s in 480)

I know… so… 78 ÷ 3 = 26 (2 more 3s) 168 ÷ 6 = 28 (double)108 ÷ 6 = 18 (10 more 6s) 91 ÷ 7 = 13 (1 less 7) 144 ÷ 8 = 18 (doubling the dividend and the divisor = same quotient) 192 ÷ 4 = 48 (20 more 4s)

Small Difference Questions: First column: 18, 36, 18, 18

Second column: 16, 26, 36, 18


Answers

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ANSWERS

Division, page 58:

Small Difference Questions: First column: 28, 28, 38, 43Second column: 36, 72, 82, 87

Which Operation? 100 ÷ 5 = 20 (division) 2000 ÷ 100 = 20 (multiplication) 40 × 5 = 200 (division) 5 = 20 ÷ 4 = 120 (division) 4 = 80 ÷ 20 (multiplication)

Contexts: 64 ÷ 16 = 4 Number sentence: 90 ÷ = 15

Example question: Some people are going to the match. 5 people travel in each car and there are 60 cars. How many people are going to the match?Example question: The bill for a group of friends at the café is £60. Each person pays £5. How many friends at the café?

Division, page 59:

Estimate: 165 ÷ 6 can’t give a whole number answer. A reasonable estimate will be significantly nearer to 30 than to 20.344 ÷ 8 could give a whole-number answer but it isn’t immediately obvious if it does. A reasonable estimate will be nearer to 40 than to 50.

Estimate: 705 ÷ 3 could give a whole-number answer although it isn’t immediately obvious that it does. A reasonable estimate will be nearer to 200 than to 300.719 ÷ 4 can’t give a whole number answer. A reasonable estimate will be significantly nearer to 200 than to 100.

Explain: 70 ÷ 5: 2-digits (more than 10 × 5). 435 ÷ 5: 2-digits (less than 100 × 5). 3000 ÷ 4: 3-digits (less than 1000 × 4). 3075 ÷ 3: 4-digits (more than 1000 × 4). 615 ÷ 5: 3-digits (more than 100 × 5).

Division, page 60:

Which Answer? The green answer is incorrect because 90 and 2 are not multiples of 4. The blue and red responses are correct. When using written calculation we partition as shown by the blue response.

Next Step: Top row: 1 2 0 1 Bottom row: 3 0 1 1

Next Step: Top row: 2 0 2 Bottom row: 2 1 2


Answers

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ANSWERS

Division, page 61:

Part-Complete Examples: Top row: 13 23 223 Bottom row: 111 r 4 61 r 4 198

Part-Complete Examples: Top row: 284 287 r 1 254Bottom row: 74 r 4 141 r 2 139 r 4

Which Answer? The middle answer is correct. Left-hand calculation

incorrect because the remainder is larger than the divisor. Right-hand calculation incorrect because there are not 9 lots of 4 in 34.

Division, page 62:

Explain the Mistakes: Left example: there are 7 lots of 6 in 45, not 8 lots. Middle example: the answer has a remainder of 2Right example: the final remainder carried should be 3 rather than 1 (the difference between 7 × 6 and 45 is 3).

Form of Answer: Sunflower = 8.5cm 8 teams (3 teams have a substitute)8 hours 30 minutes

Form of Answer: The red answer is correct. 29 ÷ 4 = 7 r 1 and that represents quarter of an hour.

Division, page 63:

Mental or Written? 320 ÷ 3 = 106 r 2, partitioning 320 into 300 and 20 so may not require written method. 320 ÷ 5 = 64, can be done as 320 ÷ 10 × 2. 320 ÷ 6 = 53 r 2, partitioning 320 into 300 and 20 so may not require written method. 320 ÷ 8 = 40, can be done mentally. 320 ÷ 9 = 35 r 5, likely to require written method. 320 ÷ 10 = 32, can be done mentally.

Mental or Written? 540 ÷ 4 = 135, can calculate by halving twice.

540 ÷ 6 = 90, can be done mentally. 540 ÷ 7 = 77 r 1, likely to require written method. 550 ÷ 55 = 10, can be done mentally using place value knowledge. 550 ÷ 25 = 22, can be done mentally if we know 100 ÷ 25 = 4. 550 ÷ 15 = 36 r 10, likely to require a written method.

Small Difference Questions: Left column: 140 70 80 62 r 2 63 126Right column: 147 147 73.5 83.5 167 167


Answers

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ANSWERS

Division, page 64:

Rank by Difficulty: Example methods: 340 ÷ 20 = 17, same as 34 ÷ 2546 ÷ 6 = 91, partitioning 546 into 540 and 6 which may be done mentally.500 ÷ 3 = 166 r 2, likely to require a written method.425 ÷ 5 = 85, can be done by 425 ÷ 10 × 2550 ÷ 30 = 18 r 10, can use 55 ÷ 3 = 18 r 1Broken Calculator: Example methods: 522 ÷ 2 ÷ 3 = 87624 ÷ 8 = 312 ÷ 4 = 78

How Many Ways? 6 ways: 36 ÷ 2 = 18 76 ÷ 2 = 38 86 ÷ 2 = 43 96 ÷ 2 = 48 76 ÷ 4 = 19 96 ÷ 4 = 12

Multiplication and Division, page 65:

Contexts: (a) subtraction to calculate the difference between their ages and addition to calculate Jack’s future age. (b) multiplication (c) subtraction to calculate the amount that needs saving and division to work out how many weeks she needs to save for.

Agree or Disagree? Disagree: Kam has 3 × 6 × 2 = 36 outfits, Pete has 3 × 5 × 3 = 45 outfits.

Extend: Mr James has 4 × 5 × 8 = 160 outfits including a tie and 4 × 5 = 20 outfits without a tie, a total of 180 outfits.


Spot the Difference: (a) £105 (b) 1 hour 45 minutes

Explore: Example questions: (a) How much does it cost for 5 apples at the market? (b) Tim buys ten apples at the market. Mo buys 6 apples at the shop. How much do they spend in total? (c) Kerry bought 12 apples at the market. She paid with a £5 note. How much change did she get? (d) Tom has £2. How many apples can he afford from the market?

Explore: Example questions: (a) 60 cars are being driven to the match. How many coaches are needed? (b) 4 coaches are going to the match. How many cars are needed?


Answers

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ANSWERS


Small Difference Questions: (a) 6 tents (b) 6 tents (c) 6 tents(d) 4 person tents (e) 29→32 people (f) 25→32 people

Explore: Example questions: (a) There are 32 children in the class. The teacher has 4 2-litre bottles of orange juice. How much juice can each child have?

(b) There are 20 children in the class. The teacher has 3 2-litre bottles of orange juice. Each child has 250ml juice. How much juice is left?(c) The teacher has three 2-litre bottles of orange juice. Each child drinks 500ml of juice. How many children are there in the class?

Extend: Adam is 35, Lara is 5 (next year Adam = 36, Lara = 6).


Explain the Mistakes: 480÷100 = 4.8, digits need to move two columns.3600÷36 = 100, 36 is 100 times smaller than 3600.7.6×10 = 76, the digits need to move one column.3.05×1000 = 3050, each digit must move 3 places.

Correct or Incorrect? Correct responses to incorrect examples:360÷100 = 3.6 0.34÷10 = 0.034

I know… so… 17×60 = 1020 170×60 = 10200160÷16 = 10 1600÷16 = 100 420÷60 = 7 420÷6 = 70

Correct or Incorrect? Correct responses to incorrect examples:60×70 = 4200 403÷100 = 4.03 400÷50 = 8


Which Answers? Factors of 12: 4 and 6 Multiples of 30: 90 and 120

Which Answers? Prime numbers: 31 41 61 71

Which Answers? The only prime number is 137. Factors of 87 include 3, factors of 121 include 11, factors of 375 include 5

Spot the Mistakes: 9cm×9cm = 81cm² (not cubed).5cm×5cm×5cm = 125cm³ (not squared).

Which Answers? Square numbers: 36, 64, 121 Cube numbers: 27, 64


Answers

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ANSWERS


Mental or Written Calculation? Factors of 128: 1 2 4 8

Mental or Written Calculation? Factors of 624: 1 2 3 4 6 8

Spot the Pattern: Example answers: 70×6 = 420 35×2×6 = 420 7×5×2×6 = 420 7×5×2×3×2 = 420


I know… so… Note that factors of 24 and 15 must also be factors of 360. Factors of 360: 1 2 3 4 5 6 8 9 10 12 15 18 20 24 30 36 40 45 60 72 90 120 180 360

Order 35 has 4 factors: 1 5 7 35 45 has 6 factors: 1 3 5 9 15 4536 has 7 factors: 1 2 3 6 12 18 36

Explain the Mistake Some multiples of 5 are not multiples of 10, for example 25. However, every multiple of 10 is also a multiple of 5.

Small Difference Questions: Example answers: Every multiple of 8 is also a multiple of 4. Every multiple of 12 is also a multiple of 6. Every multiple of 15 is also a multiple of 5.


Which Answers? Left question: 2, 3, 6 Right question: 3, 9

Different Answers: Example answers: Left oval: 16, 32. Right oval: 18, 30. Middle section: 12, 24. Outside: 7, 10.

Different Answers: Example answers: Left oval: 16, 20. Right oval: 3, 6. Middle section: 8, 12. Outside: 5, 14.


Correct or Incorrect? The blue and green statements are correct.

Different Ways: Example answers: 20×2 = 10×4 20×3 = 15×420×5 = 25×4 20×10 = 50×4 The number in the red box is five times as large as the number in the blue box.

Small Difference Questions: 6×32 > 200–20 6×36 < 200+20 6×40 > 240÷4 6×20 = 240÷2 12×20 = 120×2

How Many Ways? Six possible answers: 60÷1 = 3×20 60÷2 = 3×1060÷4 = 3×5 60÷5 = 3×4 60÷10 = 3×2 60÷20 = 3×1


Answers

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ANSWERS

Fractions, page 74:

Read the Picture: Red = 𝟏

𝟒Green =

𝟏

𝟒Yellow =

𝟏

𝟖(half the size of red)

Blue = 𝟑

𝟖(the rest of the shape and same size as red + yellow)

Read the Picture: Blue = 𝟏

𝟓Red =

𝟏

𝟒Green =

𝟏

𝟑

Explain: The red fraction is smaller, despite the red piece being a smaller

area than the blue piece. The red part is 𝟏

𝟒of its rectangle whilst the blue

part is 𝟏

𝟔of its rectangle.

Fractions, page 75:

Explain: Example 1: fingers as a fraction of a hand. Example 2: fingers as a fraction of a hand. Example 3: the brain as a fraction of the head. For each example, compare relative sizes of parts and wholes.

True or False? 𝟏

𝟓and

𝟓

𝟖are false,

𝟑

𝟖is true

Estimate: Top left: 𝟏

𝟑Top right:

𝟏

𝟓Bottom left:

𝟑

𝟒Bottom right:

𝟒

𝟓

Fractions, page 76:

Context Question: 𝟓

𝟖pizza each. Could show by dividing each pizza into

eighths; could divide four pizzas into halves and one pizza into eighths.

Context Question: 2𝟑

𝟒pieces of toast each. Could show by dividing each

piece of toast into quarters giving 𝟏𝟏

𝟒or could share 8 pieces of toast and

share the remaining 3 pieces of toast, e.g. 2 whole pieces, one half and one quarter per person.

Estimate: Correct answer is 2 eggs each if you don’t think you can share

a boiled egg or 2𝟏

𝟒eggs each if you think you can share a boiled egg.

Fractions, page 77:

What Fraction? 𝟏

𝟑

𝟏

𝟒

𝟏

𝟐

𝟏

𝟖

Different Number Lines:


Answers

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ANSWERS

Fractions, page 78:

Explain: Example pairs of equivalent fractions: 4

10&

2

5

5

10&

2

4

Agree or Disagree? All thirds have an equivalent in sixths, for example2

3&

4

6. Some fractions in sixths don’t have an equivalent in thirds, e.g.

5

6

Read the Picture: Smallest to largest: 𝟔

𝟖

𝟒

𝟓

𝟓

𝟔

Fractions, page 79:

Which Answer? 1

12which is half of

1

6

Odd One Out: All are 𝟑

𝟒apart from the top-right square which is

𝟑

𝟖

Odd One Out: All are 𝟐

𝟑apart from the top-left number line which is

𝟑

𝟒

Fractions, page 80:

Agree or Disagree? Agree,1

6is equivalent to

2

12

Explain the Mistake: To find an equivalent fraction you must multiply or divide the numerator and denominator by the same number.

Small Difference Questions: 2

8=

1

4

4

8=

1

2

1

8<

2

4

Fractions, page 81:

How Many Ways? 9 ways: 1

2=

3

6

1

2=

6

12

2

4=

6

12

1

3=

2

6

1

3=

4

122

6=

4

12

1

4=

3

12

1

6=

2

12

2

3=

4

6

Explain the Mistake: 3

4is

1

4less than 1, whereas

5

6is

1

6less than 1. Therefore

5

6

is larger as it is a smaller amount less than 1.

Rank by Difficulty: Larger fractions (left to right): 1

4

4

6

9

10

3

4

Note the processes for comparing fractions. When the numerator is the same, fractions with a larger denominator are smaller. When the denominator is the same, fractions with a larger numerator are larger. For example 3, consider how much less than 1 each fraction is. For the

last example, compare the fractions by converting 3

4into eighths.


Answers

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ANSWERS

Fractions, page 82:

Read the Pictures: Blue is 21

2circles. Yellow is 2

1

4circles. More blue.

Finish the Pictures: 31

2=

7

2which is shown by splitting circles into halves.

31

2=

14

4which is shown by splitting circles into quarters.

Explain: Order (smallest to largest): 11

5

10

4

18

6

10

3Note that the largest

fraction has the smallest denominator.

Fractions, page 83:

Small Difference Questions: 4 = 20

543

5=

23

553

5=

28

512

4= 3

15

4= 3

3

4

19

4= 4

3

4

Agree or Disagree? Agree. Note the same method for calculating

equivalent fractions applies to mixed numbers. Both fractions = 2𝟐

𝟑

Different Ways:11

4= 2

3

4and

11

5= 2

1

5Extend example:

15= 2

Fractions, page 84:

Explain the Mistakes: Mistake A: the quarters are not equal. Mistake B: 24 ÷ 3 = 8 Mistake C: the answer given is four-fifths

Finish the Pictures: Each question has multiple possible answers. Check

answers match bar models. Example answers: 2

3of 12 = 8

3

4of 40 = 30

3

5of 20 = 12

Fractions, page 85:

Two Ways: Example answers line 1: 1

3of 45 = 15

1

9of 45 = 5

Example answers line 2: 2

3of 24 = 16

2

4of 24 = 12

Example answers line 3: 3

4of 60 = 45

3

10of 60 = 18

I Know… so… Left to right: 60 186 90 180

I Know… so… Left to right: 48 192 32 96


Answers

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ANSWERS

Fractions, page 86:

Small Difference Questions: Left column: 8, 16, 64, 68

Right column: 15, 30, 32, 64

Small Difference Questions: Left column: 6, 4, 8, 8, 16Right column: 16, 8, 24, 48, 24


Fractions + – ×, page 87:

Explain the Mistakes: Mistake A: the denominators have been added and a common denominator has not been found.

Mistake B: 1

4becomes

2

8so the numerator of the answer is 5

Small Steps: Common denominators: 5 12 20 or 2 50 24

Explain:3

4+

2

8= 1 as

2

8is equivalent to

1

4(not a mixed number)

5

10+

3

4is a mixed number as

5

10is equivalent to

1

21

2+

3

8is not a mixed number as

3

8is less than half

7

10+

2


2

5is equivalent to

4

10and

7

10+

4

10= 1

1

107

20+

3


3

5is equivalent to

12

20and

7

20+

12

20=

19

20


Small Difference Questions: Left column: 3

6

5

61

Middle column: 7

8

13

16

13

16Right column: 1

1

20

29

40

36

40


9

6

9

8

91

Middle column: 9

20

15

20

7

10

19

40Right column: 1

8

12

9

16

15

16

Rank by Difficulty: Answers (left to right): 5

1212

51

5

18

9

12

16

20

Areas to discuss: the relative difficulty of finding a common denominator; questions which produce an answer of more than 1; note

that 2

4+

10

20is equivalent to

1

2+

1

2


Answers

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ANSWERS


Different Ways: Convert 3

4into

6

8

1

2+

1

2+

1

4+

1

8Split

5

8into

2

8and

3

8

Different Ways: Example answers: 2

8+

1

2=

3

4

4

8+

1

4=

3

4

5

8+

1

8=

3

4


3+

1

6=

3

6

1

3+

1

3=

4

6

1

3+

1

2=

5

6

2

3+

1

6=

5

6


Explain the Mistakes: Mistake A: the denominators have been subtracted. Also, a common denominator has not been found.Mistake B: a common denominator needs to be found before subtracting the numerators.


6

2

6

9

12

Right column: 5

8

1

8

1

8or

2

16


811

8

7

8

Right column: 11

8

7

8

7

8


Explain: 13

10−

1


1

5is less than

3

10

11

8−

1


1

4is more than

1

8

33

8− 1

3

4is a mixed number as less than 2 is being subtracted from more

than 3 so the answer will be more than 1

22

6− 1

1

3= 1 (not a mixed number)

Different Methods: 13

5−

7

10=

9

10can be done by subtracting

6

10then

1

10

11

2−

3

4=

3

4and can be done by halving

11

3−

5

6=

3

6and can be done by finding the difference


8−

1

2=

1

4

4

8−

1

4=

1

4

3

8−

1

8=

1

4

Extend: 1

2+

1

8=

3

4−

1

8


Answers

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ANSWERS


Which Answer? Answer B is correct, the answer can be simplified to 4. The mistake in answer A is multiplying the denominator as well as the numerator. The mistake in answer C is only multiplying the denominator.

Different Methods: 21

10as an improper fraction. 2

1

10as a mixed number.

9

10less than 3.

Spot the Pattern:2

5× 5 = 2

3

8× 8 = 3 Multiplying by the denominator

makes the product the same as the numerator of the fraction.


I know… so… 3

4× 6 = 4

1

2(1

1

2more)

4

5× 7 = 5

3

5(4

5more)

2

7× 14 = 4 (

4

7more)

Rank by Difficulty: Answers (left to right): 5 83

481

426

7

Areas to discuss: the relative difficulty of multiplying mixed numbers and converting improper fractions into mixed numbers.

Different Ways: Example answers: 1

2× 3 = 1

1

2

1

4× 6 = 1

1

2

3

4× 2 = 1

1

2

Example answers: 1

3× 10 = 3

1

3

1

6× 20 = 3

1

3

2

3× 5 = 3

1

3

Fractions, decimals, percentages, page 94:

True or False? Correct answers: 1

5and

65

100

Which Answers: 0.05 = five hundredths and 5

100

0.006 = six thousandths and 6

1000

Odd One Out: 1

70.4


Two Ways: 1st number line: 0.05 or 5

1002nd number line: 0.03 or

3

100and

0.065 or 65

10003rd number line: 0.001 or

1

1000and 0.007 or

7

1000


3

2

5

3

10

4

10


Answers

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ANSWERS


Explain: Incorrect because the three sections must add up to 100% (ensure that children’s estimates total 100%).

I know… so… 1

2= 50%

1

25= 4%

1

10= 10%

1

50= 2%

1

20= 5%

1

5= 20%

1

4= 25%

Agree or Disagree? Only true statement is 1

10= 10%, because 100 ÷ 10 = 10

Percentages, page 97:

Odd One Out: 1

3

1

5

Rank by Difficulty: 1

5= 20% (calculate by doing 100 ÷ 5) 0.6 = 60% (note

misconception 0.6 = 6%) 0.52 = 52% 39

50= 78% (discuss strategies for

doubling 39)

Rank by Difficulty: 36

50= 72% (discuss strategies for doubling 36)

4

8= 50%

(4 is half of 8, therefore 50%) 6

10= 60%

7

25= 28%

Different Ways: 2 ways: 2

3

3

5

Measurement, page 98:

Contexts: Adult footprint, area, cm² Water in cup, volume, mlSkipping rope, length, cm Football pitch, area, m²

Which Answer? Suggested answers: size of carpet m², size of lamp post m, size of a bottle millilitres or litres, size of puddle litres.

Which Answer? Light years measure length (the distance light travels in one year).


Odd One Out: Centimetres are a measure of length. Ounces are an imperial measure. Kilograms measure relatively larger objects.

Agree or Disagree? Disagree: there are more yards than metres in the same length, so one yard must be smaller than one metre.

Explore: Left oval: litres. Centre: kilometres. Right oval: yards, inches. Outside: stones.


Answers

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ANSWERS


Explain the Mistakes: To convert 25mm into cm, divide by 10. 3.8m = 380cm, the 8 needs to be in the tens column. To convert metres into km, divide by 1000. In this example, 800 is divided by 100.

Agree or Disagree? Correct answers for mistakes: 408cm = 4.08m1

2litre = 500ml

True or False? (a) True (b) False (0.7m is more than 600mm) (c) True

Compare: 6m = 600cm 6 yards = 216 inches 2.5m = 250cm2.5 yards = 90 inches Note: conversion between units is easier using metric measures.


Explain the Mistakes: Blue answer: it was before 2:10pm, not after. Red answer: error in calculation, possibly thinking half an hour = 50 minutes. Green answer: the time will be am, not pm.

Which Answer: (c) 6:05pm (b) 2 hours 35 minutes

Small Difference Questions: 100 minutes 200 minutes 300 minutes2 hours 30 minutes 4 hours 10 minutes


Compare: 6 weeks < 44 days 100 hours > 4 days 120 hours = 5 days3 months < 2400 hours 3 months > 12 weeks 3 months < 63 days

Multi-Step: 9 weeks Answers in the range 85→91 days

Multi-Step: QA (shortest to longest): 180 days, 1

2year, 27 weeks, 7 months

QB (shortest to longest): 3 months, 1

3year, 18 weeks, 140 days

Perimeter, Area, Volume, page 103:

Different Ways: 5cm + 3cm + 5cm + 3cm Double 5cm + 3cm

Order: Smallest to largest perimeter: square, rectangle, star.

Explain: Rectangle: 2 sides Square: 1 side Cross: 1 sideIsosceles triangle = 2 sides

Estimate: Actual perimeter = 20cm


Answers

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Explain: Small as possible: Large as possible:

Long sides matching short sides matching

I know… so… Top line (left to right): 11cm 13cmMiddle line (left to right): 16cm 18cm Bottom line: 18cm


Small Difference Questions: Left to right: 40cm 48cm 60cm

Explain: purple + orange = red black – blue = green Note that black + red = purple + blue + orange + green

Agree or Disagree? Perimeter = 46cm The bottom length = 9cm + 6cm and the two missing vertical lengths add up to 8cm.


Which Answer? 12 × 7 = 84cm² is correct. Blue answer is the perimeter. Green answer multiplies all the side lengths rather than length × width.

Agree or Disagree? True. Examples: a rectangle with dimensions 4cm ×3cm has an area of 12cm². 48cm² is four times bigger than 12cm².

Estimate: Blue arrow = 6cm Red arrow = 3cm


Spot the Mistake: The right-hand section of the shape is an area of6cm × 7cm = 42cm², not an area of 9cm × 7cm. Area of shape = 72cm²

Different Ways: Method 1: Add 4cm × 3cm Method 2: add 6cm × 5cm

Method 3: Subtract 6cm × 3cm Area = 62cm²


Spot the Pattern: All three shapes have a perimeter of 24cm. The narrower the rectangle, the smaller the area.

Shape A area = 36cm² Shape B area = 32cm² Shape C area = 20cm²

Same and Different: Part 1: Example rectangle 5cm × 5cmPart 2: Example rectangle 9cm × 2cm

Explain: Largest possible area = 49cm² from a 7cm × 7cm square (which

is a type of rectangle).

ANSWERS


Answers

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Same and Different: Part 1: Example rectangle 8cm × 3cmPart 2: Example rectangle 8cm × 5cmPart 3: Example rectangle 20cm × 2cm

Extend: A 6cm × 5cm rectangle has an area of 30cm² (larger than

27cm²) and a perimeter of 22cm (smaller than 24cm).


True or False? False: there are 16 cubes (4 × 2 × 3). Note that 20 is the

number of squares that can be counted on the faces that are visible.

True or False? The blue statement is true. The red statement is false.

Explore: Dimensions of completed cuboid: 3 × 3 × 2

How Many Ways? 3 ways: cube dimensions 4 × 4 × 1 4 × 2 × 2 3 × 3 × 2

Angle, page 111:

Order: Smallest to largest: green, blue, red. Note: the green angle is the smallest even though the lines are longer and the angle marking is wide.

Explore: Acute angles = red, right-angles = blue, obtuse angles = green

Explain: Acute angles = red, right-angles = blue, obtuse angles = green

Angle, page 112:

Explain the Mistakes: 84˚: read the wrong scale.122˚: the angle is 2˚ less than 120˚, not 2˚ more. 40˚: protractor not lined up with the bottom line.

Which Answer? 133˚ is correct. The mistake with 47˚ is reading the wrong side of the protractor. The mistake with 127˚ is thinking the angle is 3˚ less than 130˚.

ANSWERS


Answers

From vertex, lines 3← 3↑ and 3→ 3↑, creates right-angle

From vertex, lines 1← 4↑ and 5→ 1↑, creates obtuse angle

From vertex, lines 4← 1↑ and 2 ← 4↓, creates acute angle

From vertex, lines 4← 1↑ and 4→ 1↑, creates right-angle

I SEE REASONING

Angle, page 113:

Which Answer? (a) 245˚ (b) 76˚

Explain: Angle B is smaller than angle C. There is more to add to angle D to make 180˚ so angle C must be larger.

Small Difference Questions: E = 26˚ F = 116˚ G = 112˚ H = 292˚

Angle, page 114:

Small Difference Questions: J = 104˚ K = 104˚ L = 74˚ M = 164˚ N = 144˚

True or False? (a) False, 90˚ × 4 = 360˚, acute angles are less than 90˚

(b) True, sum of two acute angles is less than 180˚, 3rd angle is reflex. (c) False, sum of two obtuse angles is more than 180˚, 3rd angle is obtuse. Extend: 3 obtuse angles.

Always, Sometimes or Never? Individual angles of a triangle could be larger than individual angles of quadrilaterals (triangles can have obtuse angles). However, the sum of the angles in a triangle is always 180˚ and the sum of the angles in a quadrilateral is always 360˚.

Angle, page 115:

Multi-Skill: 7:10am = obtuse 4pm = obtuse 9am = right-angle9:30pm = obtuse 8:55pm = acute

Which Answer? (a) 150˚ (5 × 30˚)

(c) 15˚ (the hour hand is half-way between 6 and 7 which is 15˚)45˚, the hour hand is half-way between 4 and 5 which is 15˚, plus the 30˚ between 5 and 6

Multi-Skill: Left to right: 30˚ 120˚ 15˚ 20˚ (the hour hand is 1

3the

turn between 3 and 4, which is 10˚, so 20˚ between the hands)

Shape, page 116:

Small Difference Questions: Top line (left to right): 4 2 1Bottom line (left to right): 1 (diagonal bottom-left to top-right) 0 0

Explain: Example answers: All four shapes are hexagons. Three shapes have at least one line of symmetry (not 4th shape). Two shapes have at least one reflex angle (1st and 3rd). One shape is a regular hexagon (2nd).

Odd One Out: Example answers (left to right): Has acute angles. Is a hexagon. No lines of symmetry. No reflex angles.

ANSWERS


Answers

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Shape, page 117:

Explain: Left to right: the triangle isn’t: it has unequal angles and one unequal side length. The diamond isn’t: it has unequal angles. The next 3 shapes are regular. The pentagon isn’t: is has unequal angles and side lengths.

Explore: Examples:

Shape, page 118:

Odd One Out: Example answers: A cuboid doesn’t have 5 faces. A triangular prism doesn’t have a square face. The triangular pyramid is not a prism.

Small Difference Questions: A cuboid has 1 more face, 3 more edges and 2 more vertices than a triangular prism. A square-based pyramid has 1 more face, 2 more edges and 1 more vertex than a triangular pyramid.

ANSWERS


Answers

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Shape, page 119:

Correct or Incorrect? Correct correct correct incorrect


Position, Direction, Coordinates, page 120:

Explain the Mistakes: Mistake 1: translated, not reflected. Mistake 2: incorrect distance from the mirror line. Mistake 3: middle vertex different

Which Answer? Top left: rotated Top right: translatedBottom left: reflected Bottom right: either reflected or translated


Explain: Left example: may have been translated or reflectedRight example: reflected

Explain the Mistakes: Mistake 1: translation is 3 squares right, not 2Mistake 2: left vertex not positioned correctly


Different Ways: The blue dot is positioned in a ratio 3 along, 5 up. Possible coordinates include (6,10) and (30,50) and realistic estimates. Not correct examples are (5,3) as y coordinate is more than x coordinate and (4,5) as the ratio of the lengths is not correct.

Estimate: Blue = (8,5), red = (4,2), accept realistic estimates.

Estimate: Purple = (18,8), green = (36,18), accept realistic estimates.

ANSWERS


Answers

faces overlap

3 square faces

I SEE REASONING


Agree or Disagree? Blue coordinate correct. Red coordinate incorrect, the correct coordinate is (0,6).

Explain the Mistake: The x coordinate is correct (7 is half-way between 0 and 14). The y coordinate is incorrect. The mistake comes from halving the second y coordinate of 8. To calculate the correct y coordinate,

find half-way between 4 and 8 which is 6. Correct coordinate = (14,6).

Which Answer? (9,3) inside (2,6) outside (8,7) edge


Different Answers: Corners: (4,4) (8,4) (8,8) (4,8) Examples for edges: (6,4) (8,6) Examples for inside: (6,6) (7,6)

Different Answers: Corners: (13,8) (13,3) (5,3) Examples for edges: (10,8) (13,5) Examples for inside: (9,5) (6,7)

Statistics, page 125:

What’s the Question? 1. What time does the first train arrive at Stockport? 2. At what time does the third train leave Macclesfield? 3. How long does the journey take from Stockport to Manchester Piccadilly?

Multi-Step Question: Helen needs to travel to Cannock on the second train, arriving 15:58. This train leaves Wolverhampton at 14:56. She therefore needs to get on the first train from Crewe, leaving at 13:56. There are two train timetables because there isn’t a direct train from Crewe to Cannock.


Fill the Gaps:

Explain: Temperature in garden: line graph. Number of different vegetables: bar graph. Rainfall per month: line graph.

ANSWERS


Answers

6

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Which Graph? Children in each class: bottom right (bars are similar sizes). Shoe size: top left (some children have large/small feet; most have shoe sizes nearer the average). Age of children in Y5/6 class: top right graph (all children either 9, 10 or 11). Favourite fruit: bottom left (no pattern in size of bars).

Which Graph? Speed of 100m runner: top graph (fastest speed, acceleration from standing start). Speed of runner in hill race: left graph (variation in speed). Speed of marathon runner: right graph (relatively constant speed).


Act the Graph: Increase in speed at approximately 10 seconds and 30 seconds to reflect rise in graph.

Act the Graph: Facial expression to show increased fear at approximately 7 seconds and 14 seconds to reflect rise in graph. The last 5 seconds are more relaxed.


Which Answer? (c) this is not the top speed but the athlete continues running afterwards so it’s part-way through the race.

Which Answer? (a) the distance travelled stays the same so the cyclist has stopped.

ANSWERS


Answers

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I SEE MATHS RESOURCES

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