Oxford Mathematics — Primary Years Programme 4

The whole Year 4 student book in one page: all 10 units and 31 topics, each with the idea explained first and then four levels of practice — guided, independent, extended and extra. That is 1941 exercises, every one answered. Type straight into the page, check against the key, and your work is kept in this browser.

10 units · 31 topics 1941 exercises with answers Glossary of 150 words Progress tracker

How to use this book

Pick a unit from the list on the left. Every topic opens with a Learn it panel that explains the idea with a worked example, and then works through four stages: Guided practice with plenty of scaffolding, Independent practice as the scaffolding is taken away, Extended practice to stretch the thinking further, and Extra practice — a fuller set of questions for building fluency or for revision.

Type into any box and it is saved as you go. Use Mark complete at the top of a topic to fill in the progress bar, and the 🔑 Answers panel at the foot of each topic to check your work. The hundred charts and shading grids are clickable, and Print gives you a clean worksheet with the answers hidden.

The PYP mathematics course is taught in English, so the lessons and exercises are in English. Everything you type stays on this device — nothing is uploaded.

10units, from place value to chance

31topics, each in four stages

150glossary words with examples

100%of topics have a full answer key

Unit 1

Number and place value

Read, write, rename and round numbers past 100 000 — then build a toolkit of mental and written strategies for adding, subtracting, multiplying and dividing them.

Number & Algebra · 9 topics

Unit 1 · Topic 1

Place value

Every digit in a number has a value that depends on its place. We can rename the same number in lots of different ways by trading between the columns.

The number 23 854 is the same as:

23 854= 2 ten thousands · 3 thousands · 8 hundreds · 5 tens · 4 ones

= 23 thousands · 8 hundreds · 5 tens · 4 ones

= 238 hundreds · 5 tens · 4 ones

= 2385 tens · 4 ones

= 23 854 ones

When might it be useful to rename numbers?

Guided practiceWorked through with you

Show these numbers on the number expanders.
1. 34 926= ten thousands · thousands · hundreds · tens · ones
  
  = thousands · hundreds · tens · ones
  
  = hundreds · tens · ones
  
  = tens · ones
  
  = ones
2. 97 563= ten thousands · thousands · hundreds · tens · ones
  
  = thousands · hundreds · tens · ones
  
  = hundreds · tens · ones
  
  = tens · ones
  
  = ones

Independent practiceYour turn

Write these numbers on the expanders.
1. 17 329= ten thousands · thousands · hundreds · tens · ones
  
  = thousands · hundreds · tens · ones
  
  = hundreds · tens · ones
2. 80 154= ten thousands · thousands · hundreds · tens · ones
  
  = hundreds · tens · ones
  
  = tens · ones
3. 64 078= ten thousands · thousands · hundreds · tens · ones
  
  = thousands · hundreds · tens · ones
  
  = hundreds · tens · ones
4. 49 461= ten thousands · thousands · hundreds · tens · ones
  
  = hundreds · tens · ones
5. 28 935= ten thousands · thousands · hundreds · tens · ones
  
  = thousands · hundreds · tens · ones
Expand each number by place value.

a 51 345 = 50 000 + 1000 + 300 + 40 + 5
b 40 772 = + + +
c 87 024 = + + +
d 17 316 = + + + +
e 92 603 = + + +
f 55 555 = + + + +

Rewrite the collections from smallest to largest.

No.	Collection description	Number of items
1	Pairs of earrings	37 706
2	“Do not disturb” signs	11 570
3	Smart phones	1563
4	Dinosaur eggs	10 008
5	Rat and mouse memorabilia	47 398
6	Number plates	11 345
7	Toenail clippings	24 999
8	Magazines	50 953
9	Key chains	47 200
10	Olympic postage stamps	15 183

Collection number	Number of items

How can you tell if one number is larger than another?

Write these numbers in words.
1. 56 927
2. 80 401
3. 42 058
Write the numerals for these numbers.
1. Sixty-eight thousand, one hundred and forty-two
2. Twenty-four thousand and seventy
3. Ninety thousand and three

Extended practiceStretch your thinking

Round up or down to the nearest 10.
1. 73
2. 28
3. 1364
4. 62 147
Round up or down to the nearest 100.
1. 591
2. 1603
3. 21 977
Round up or down to the nearest 1000.
1. 6099
2. 24 270
3. 93 804
Round up or down to the nearest 10 000.
1. 19 878
2. 41 997
3. 83 025
Round up or down to the nearest 100 000.
1. 498 531
2. 628 197
3. 240 799
Write the numerals for:
1. 1 hundred thousand, 4 ten thousands, 44 hundreds and 2 tens.
2. 120 hundreds and 81 ones.
3. 61 thousands, 45 tens and 8 ones.
4. 402 thousands, 32 tens and 5 ones.
5. 49 thousands and 6 ones.
Rewrite the numbers from question 6 from smallest to largest.

Extra practiceMore questions to build fluency

Write the value of the digit 7 in each number.
1. 7412
2. 17 905
3. 3274
4. 70 168
5. 5127
Expand each number by place value.
1. 64 208 =
2. 90 516 =
3. 8043 =
Write these numbers in words.
1. 45 300
2. 60 017
3. 209 984
Write the numerals.
1. thirty-two thousand, four hundred and six
2. eighty thousand and ninety
3. one hundred and five thousand, two hundred
Rewrite these numbers from smallest to largest.

45 019 45 190 45 091 45 901
Round 27 486 to the nearest:
1. 10
2. 100
3. 1000
4. 10 000
Round 152 750 to the nearest:
1. 100
2. 1000
3. 10 000
4. 100 000
Rename 4500.
1. hundreds
2. tens
3. ones
A stadium holds 48 750 people. A newspaper wants to print the figure rounded to the nearest thousand. What will it print?
Kofi says 6 ten thousands is more than 60 thousands. Is he right? Explain your answer.

Answers · Topic 1 Place value

Guided practice

1a 34 926 = 3 ten thousands, 4 thousands, 9 hundreds, 2 tens, 6 ones = 34 thousands, 9 hundreds, 2 tens, 6 ones = 349 hundreds, 2 tens, 6 ones = 3492 tens, 6 ones = 34 926 ones.

1b 97 563 = 9 ten thousands, 7 thousands, 5 hundreds, 6 tens, 3 ones = 97 thousands, 5 hundreds, 6 tens, 3 ones = 975 hundreds, 6 tens, 3 ones = 9756 tens, 3 ones = 97 563 ones.

Independent practice

1a 1 ten thousand, 7 thousands, 3 hundreds, 2 tens, 9 ones = 17 thousands, 3 hundreds, 2 tens, 9 ones = 173 hundreds, 2 tens, 9 ones. b 8 ten thousands, 0 thousands, 1 hundred, 5 tens, 4 ones = 801 hundreds, 5 tens, 4 ones = 8015 tens, 4 ones. c 6 ten thousands, 4 thousands, 0 hundreds, 7 tens, 8 ones = 64 thousands, 0 hundreds, 7 tens, 8 ones = 640 hundreds, 7 tens, 8 ones. d 4 ten thousands, 9 thousands, 4 hundreds, 6 tens, 1 one = 494 hundreds, 6 tens, 1 one. e 2 ten thousands, 8 thousands, 9 hundreds, 3 tens, 5 ones = 28 thousands, 9 hundreds, 3 tens, 5 ones.

2b 40 000 + 700 + 70 + 2 c 80 000 + 7000 + 20 + 4 d 10 000 + 7000 + 300 + 10 + 6 e 90 000 + 2000 + 600 + 3 f 50 000 + 5000 + 500 + 50 + 5

3 3 (1563), 4 (10 008), 6 (11 345), 2 (11 570), 10 (15 183), 7 (24 999), 1 (37 706), 9 (47 200), 5 (47 398), 8 (50 953)

4a fifty-six thousand, nine hundred and twenty-seven b eighty thousand, four hundred and one c forty-two thousand and fifty-eight

5a 68 142 b 24 070 c 90 003

Extended practice

1 a 70 b 30 c 1360 d 62 150

2 a 600 b 1600 c 22 000

3 a 6000 b 24 000 c 94 000

4 a 20 000 b 42 000 c 83 000

5 a 500 000 b 600 000 c 200 000

6 a 144 420 b 12 081 c 61 458 d 402 325 e 49 006

7 12 081, 49 006, 61 458, 144 420, 402 325

Extra practice

1a 7000 b 7000 c 70 d 70 000 e 7

2a 60 000 + 4000 + 200 + 8 b 90 000 + 500 + 10 + 6 c 8000 + 40 + 3

3a forty-five thousand, three hundred b sixty thousand and seventeen c two hundred and nine thousand, nine hundred and eighty-four

4a 32 406 b 80 090 c 105 200

5 45 019, 45 091, 45 190, 45 901

6a 27 490 b 27 500 c 27 000 d 30 000

7a 152 800 b 153 000 c 150 000 d 200 000

8a 45 hundreds b 450 tens c 4500 ones

9 49 000

10 No — they are the same. 6 ten thousands is 6 × 10 000 = 60 000, and 60 thousands is 60 × 1000 = 60 000. Renaming a number does not change its size.

Unit 1 · Topic 2

Odd and even

The last digit of a number tells us if it is odd or even.

2365 is odd, because 5 is odd. 4792 is even, because 2 is even.

I wonder if 1 million is odd or even?

Guided practiceWorked through with you

Look at the last digit in each number, then write whether it is odd or even.
1. 573
2. 914
3. 1390
4. 8056
5. 23 474
6. 42 689
7. 95 005
8. 75 000
9. 10 101
10. 42 867
11. 57 838
12. 75 383
If you added 1 to each number in question 1, would each one be odd or even?

Independent practiceYour turn

Use these digits to make each number. Use every digit once.

23567
1. the largest odd number possible.
2. the smallest odd number possible.
3. the largest even number possible.
4. the smallest even number possible.
Use these digits to make:

00189
1. the largest even number possible.
2. the largest odd number possible.
3. the smallest even number possible.
4. the smallest odd number possible.
Use these digits to make:

04567
1. the largest odd number with 7 in the tens place.
2. the smallest even number with 0 in the thousands place.
3. the largest even number with 5 in the ten thousands place.
4. the smallest odd number with 4 in the hundreds place.

If you add an even number to an even number, the answer is always even. Fill in the other addition and subtraction rules.

Example	Operation	Answer
4 + 4 = 8	even + even	even
4 + 5 = 9	even + odd
5 + 4 = 9	odd + even
5 + 5 = 10	odd + odd
8 – 2 = 6	even – even
8 – 3 = 5	even – odd
9 – 4 = 5	odd – even
9 – 3 = 6	odd – odd

If you multiply an even number by an even number, the answer is always even. Fill in the other multiplication rules.

Example Operation Answer

2 × 2 = 4 even × even even

2 × 3 = 6 even ×

5 × 2 = 10 ×

5 × 3 = 15 ×
Write whether the answer will be odd or even.
1. 23 + 72
2. 456 − 97
3. 768 + 310
4. 803 − 549
5. 1765 + 9261
6. 8639 – 6223
7. 48 × 72
8. 83 × 46
You can use these rules to help check if your calculations are correct.

Example	Operation	Answer
2 × 2 = 4	even × even	even
2 × 3 = 6	even ×
5 × 2 = 10	×
5 × 3 = 15	×

Extended practiceStretch your thinking

Can you think of any examples that don’t fit these rules?

Solve the equations, then decide if the statements are true or false.
1. ÷ 2 = 14 ÷ 2 = 17 ÷ 2 = 50
  Only even numbers can be divided exactly by 2. True False
2. ÷ 3 = 5 ÷ 3 = 10 ÷ 3 = 100
  Only odd numbers can be divided exactly by 3. True False
3. ÷ 4 = 10 ÷ 4 = 4 ÷ 4 = 9
  Only even numbers can be divided exactly by 4. True False
Use your knowledge of odd and even numbers to sort these larger numbers.

34 17662 849123 456 987 654520 399471 002 1 098 7654 342 998 8 888 8817 676 767

Odd Even

Extra practiceMore questions to build fluency

Write whether each number is odd or even.
1. 3947
2. 12 580
3. 60 003
4. 8116
5. 105 555
6. 74 002
Without working out the answer, write whether each result is odd or even.
1. 27 + 45
2. 64 + 38
3. 51 + 26
4. 90 − 33
5. 7 × 9
6. 6 × 13
Complete each rule.
1. odd + odd =
2. even + even =
3. odd + even =
4. odd × odd =
5. even × any whole number =
Use the digits 3, 5, 6, 8 once each to make:
1. the largest even number
2. the smallest odd number
3. an even number between 5000 and 6000
True or false? Tick the true statements.
1. The sum of three odd numbers is always odd.
2. An even number can never end in 5.
3. Zero is an even number.
4. Every multiple of 3 is odd.
There are 47 children in a hall. Can they all be put into pairs with nobody left out? Explain.
I am an even number between 60 and 70. The sum of my digits is 10. What am I?

Answers · Topic 2 Odd and even

Guided practice

1 a odd b even c even d even e even f odd g odd h even i odd j odd k even l odd

2 a even b odd c odd d odd e odd f even g even h odd i even j even k odd l even

Independent practice

1 a 76 523 b 23 567 c 76 532 d 23 576

2 a 98 100 b 98 001 c 10 098 d 10 089

3 a 64 075 b 40 576 c 57 640 d 50 467

4 even + odd = odd; odd + even = odd; odd + odd = even; even – even = even; even – odd = odd; odd – even = odd; odd – odd = even

5 even × odd = even; odd × even = even; odd × odd = odd

6 a odd b odd c even d even e even f even g even h even

Extended practice

1a 28 ÷ 2 = 14, 34 ÷ 2 = 17, 100 ÷ 2 = 50 → True
b 15 ÷ 3 = 5, 30 ÷ 3 = 10, 300 ÷ 3 = 100 → False
c 40 ÷ 4 = 10, 16 ÷ 4 = 4, 36 ÷ 4 = 9 → True

2 Odd: 62 849, 520 399, 1 098 765, 8 888 881, 7 676 767.
Even: 34 176, 123 456, 987 654, 471 002, 4 342 998.

Extra practice

1a odd b even c odd d even e odd f even

2a even b even c odd d odd e odd f even

3a even b even c odd d odd e even

4a 8653 is the largest odd one, so the largest even number is 8536. b 3568 is even, so the smallest odd number is 3685. c 5368 or 5638 or 5836 or 5386.

5 Tick a, b and c. The last one is false — 6, 12 and 18 are all multiples of 3 and all even.

6 No. 47 is odd, so pairing them up leaves one child without a partner.

7 64 — the even numbers between 60 and 70 are 62, 64, 66 and 68, and only 64 has digits that add to 10.

Unit 1 · Topic 3

Addition mental strategies

Rearranging numbers can make them easier to add mentally.

23 + 36 + 17
= 23 + 17 + 36
= 40 + 36 = 76

Look for pairs of numbers that help you get to a 10.

Guided practiceWorked through with you

Rearrange the numbers to solve these sums.

a 2 + 35 + 18 = + + = + =
b 13 + 46 + 7 = + + = + =
c 38 + 51 + 32 = + + = + =
d 42 + 53 + 8 = + + = + =
e 16 + 92 + 4 = + + = + =
f 45 + 22 + 125 = + + = + =
g 17 + 42 + 13 + 28 = + + + = + =
h 19 + 44 + 16 + 21 = + + + = + =

Independent practiceYour turn

Rearrange the numbers in your head to solve these sums.
1. 29 + 23 + 1 =
2. 21 + 34 + 6 =
3. 62 + 17 + 23 =
4. 25 + 17 + 75 =
5. 86 + 243 + 14 =
6. 27 + 119 + 13 =
7. 21 + 28 + 9 + 32 =
8. 35 + 18 + 22 + 35 =
Use the jump strategy on the empty number line to solve.

Use jumps of hundreds, then tens, then ones.
1. 86 + 47 =
  Start at 86 · jump then
2. 251 + 26 =
  Start at 251 · jump then
3. 408 + 335 =
  jump · ·
4. 319 + 464 =
  jump · ·
5. 659 + 402 =
  jump ·
Split both numbers to solve.

a 572 + 215 = 500 + 200 + 70 + 10 + 2 + 5 = + + =
b 163 + 576 = + + + + + = + + =
c 815 + 462 = + + + + + = + + =
d 1625 + 3134 = + + + + + + + = + + + =
e 4328 + 2454 = + + + + + + + = + + + =
Try solving these sums in your head.
1. 172 + 23 =
2. 445 + 341 =
3. 532 + 229 =
4. 178 + 615 =
5. 340 + 555 =
6. 147 + 281 =
7. 758 + 205 =
8. 873 + 224 =
Which of these addition strategies could you also use for subtraction?

Extended practiceStretch your thinking

Use a mental strategy of your choice to solve.
1. 675 + 257 =
2. 3457 + 2342 =
3. 3466 + 4534 =
4. 1138 + 4214 + 2312 =
The table below shows weekly supermarket sales in different categories.

Item Cookies Doughnuts Cupcakes Apples Oranges Bananas Chocolate bars Cake mixes

Number sold 2371 630 7963 9317 3204 2426 5234 429

Solve these questions using a mental strategy of your choice.
1. What is the total of cookies, doughnuts and cake mixes sold?
2. What is the combined total of oranges and bananas sold?
3. Were more cookies and cupcakes, or oranges and chocolate bars, sold?
4. What is the total of the 2 items that sold the least?
5. What is the total of the 2 items that sold the most?

Item	Cookies	Doughnuts	Cupcakes	Apples	Oranges	Bananas	Chocolate bars	Cake mixes
Number sold	2371	630	7963	9317	3204	2426	5234	429

Extra practiceMore questions to build fluency

Rearrange to make friendly tens, then solve mentally.
1. 18 + 35 + 22 =
2. 47 + 26 + 13 =
3. 55 + 19 + 25 =
4. 34 + 48 + 16 =
Use the jump strategy. Write the jumps you made.
1. 68 + 35 → =
2. 147 + 26 → =
Split both numbers to solve.
1. 43 + 35 =
2. 126 + 52 =
3. 264 + 315 =
4. 418 + 271 =
Use compensation — round up, then take back off.
1. 56 + 29 =
2. 73 + 48 =
3. 134 + 99 =
4. 245 + 198 =
Solve these in your head.
1. 300 + 450 =
2. 1200 + 800 =
3. 2500 + 1500 =
4. 6400 + 3600 =
A shop sold 128 drinks on Friday, 96 on Saturday and 72 on Sunday. How many over the three days? Which two did you add first, and why?

Total
Find the missing number.
1. 37 + = 80
2. + 145 = 200
3. 68 + = 150
4. + 275 = 500

Answers · Topic 3 Addition mental strategies

Guided practice

1a 2 + 18 + 35 = 20 + 35 = 55 b 13 + 7 + 46 = 20 + 46 = 66
c 38 + 32 + 51 = 70 + 51 = 121 d 42 + 8 + 53 = 50 + 53 = 103
e 16 + 4 + 92 = 20 + 92 = 112 f 45 + 125 + 22 = 170 + 22 = 192
g 17 + 13 + 42 + 28 = 30 + 70 = 100 h 19 + 21 + 44 + 16 = 40 + 60 = 100

Independent practice

1 a 53 b 61 c 102 d 117 e 343 f 159 g 90 h 110

2 a 133 (86 → +40 → 126 → +7 → 133) b 277 (251 → +20 → 271 → +6 → 277)
c 743 (408 → +300 → 708 → +30 → 738 → +5 → 743) d 783 (319 → +400 → 719 → +60 → 779 → +4 → 783) e 1061 (659 → +400 → 1059 → +2 → 1061)

3a 700 + 80 + 7 = 787
b 100 + 500 + 60 + 70 + 3 + 6 = 600 + 130 + 9 = 739
c 800 + 400 + 10 + 60 + 5 + 2 = 1200 + 70 + 7 = 1277
d 1000 + 3000 + 600 + 100 + 20 + 30 + 5 + 4 = 4000 + 700 + 50 + 9 = 4759
e 4000 + 2000 + 300 + 400 + 20 + 50 + 8 + 4 = 6000 + 700 + 70 + 12 = 6782

4 a 195 b 786 c 761 d 793 e 895 f 428 g 963 h 1097

Extended practice

1 a 932 b 5799 c 8000 d 7664

2 a 3430 b 5630 c cookies and cupcakes (10 334 v 8438) d 1059 e 17 280

Extra practice

1a 75 (18 + 22 = 40, then + 35) b 86 (47 + 13 = 60, then + 26) c 99 (55 + 25 = 80, then + 19) d 98 (34 + 16 = 50, then + 48)

2a 103 — e.g. 68 + 30 = 98, then + 5 b 173 — e.g. 147 + 20 = 167, then + 6

3a 78 b 178 c 579 d 689

4a 85 (56 + 30 − 1) b 121 (73 + 50 − 2) c 233 (134 + 100 − 1) d 443 (245 + 200 − 2)

5a 750 b 2000 c 4000 d 10 000

6 296 drinks. Most people add 128 + 72 = 200 first, because those two make a friendly hundred, then add 96.

7a 43 b 55 c 82 d 225

Unit 1 · Topic 4

Addition written strategies

For larger numbers, it can be easier to add the smaller place value columns first.

3428 + 2647
= (8 + 20 + 400 + 3000) + (7 + 40 + 600 + 2000)
= 8 + 7 + 20 + 40 + 400 + 600 + 3000 + 2000
= 15 + 60 + 1000 + 5000
= 6075

When using a vertical algorithm, you add the smaller place value columns first, too!

Guided practiceSplit strategy

Solve using the split strategy, starting with the ones.

a 2376 + 5162
= ( + + + ) + ( + + + )
= + + + + + + +
= + + +
=

b 6284 + 8415
= ( + + + ) + ( + + + )
= + + + + + + +
= + + +
=

Independent practiceSplit strategy

Use the split strategy, starting with the ones.
1. 4935 + 1742 = = =
2. 13 428 + 32 517 = = =
3. 25 019 + 28 746 = = =
4. 44 754 + 35 632 = = =

In vertical addition you have to trade when the total of a place value column is more than 10.

	T	O
	1
	4	7
+	3	5
	8	2

7 + 5 = 12 ones. Trade 10 ones for 1 ten.

What would you need to do if the total in the tens column was 14?

Guided practiceVertical addition

Solve using trading in the ones column.

a
T O

3 4
+ 2 8

b
T O

7 6
+ 7 9

c
H T O

5 3 5
+ 2 4 7
Solve using trading in the tens column.

a
H T O

9 5
+ 7 2

b
H T O

4 6 5
+ 2 5 4

c
Th H T O

6 5 4 3
+ 2 3 7 1
Solve using trading in the hundreds or thousands column.

a
Th H T O

5 8 0 4
+ 2 6 9 3

b
Th H T O

3 6 1 7
+ 2 7 4 2

c
Tth Th H T O

9 2 5 6
+ 7 4 4 3

	T	O
	3	4
+	2	8

	T	O
	7	6
+	7	9

	H	T	O
	5	3	5
+	2	4	7

	H	T	O
	4	6	5
+	2	5	4

	Th	H	T	O
	6	5	4	3
+	2	3	7	1

	Th	H	T	O
	5	8	0	4
+	2	6	9	3

	Th	H	T	O
	3	6	1	7
+	2	7	4	2

	Th	H	T	O
	9	2	5	6
+	7	4	4	3

Independent practiceVertical addition

Remember to place the numbers in their correct place value columns.

Rewrite as vertical addition and solve.

a 6379 + 2115
Th H T O

+

b 3426 + 4832
Th H T O

+

c 17 245 + 24 531
Tth Th H T O

+

d 30 856 + 23 933
Tth Th H T O

+

e 52 394 + 11 240
Tth Th H T O

+

f 48 001 + 35 986
Tth Th H T O

+

g 43 764 + 15 482
Tth Th H T O

+

h 28 047 + 36 706
Tth Th H T O

+

Extended practiceStretch your thinking

Every student in Year 4 has a blog page. Here is a list of the most visited pages.

Name	Page hits	Name	Page hits
Rui	27 004	Waris	36 524
Mike	9865	Alice	8428
Torey	43 117	Vaheni	42 963
Frank	63 174	Patrick	5262
Sara	22 340	Bronte	31 808

Use a written strategy of your choice to find the total page hits for:
1. Alice and Patrick.
2. Rui and Frank.
3. Vaheni and Bronte.
4. Rui, Torey and Sara.
5. all the students with fewer than 10 000 page hits.
6. all the students with more than 40 000 page hits.
Rewrite as vertical addition and solve.

a 28 476 + 9214
Tth Th H T O

+

b 842 + 13 125 + 4702
Tth Th H T O

+

+

Extra practiceMore questions to build fluency

Solve using the split strategy, starting with the ones.
1. 2536 + 1342 =
2. 4271 + 3518 =
3. 1465 + 2324 =
4. 5213 + 2756 =
Rewrite as vertical addition and solve. These need trading.
1. 367 + 285 =
2. 1458 + 2673 =
3. 3849 + 4176 =
4. 5628 + 2495 =
Add three numbers.
1. 245 + 318 + 127 =
2. 1236 + 845 + 219 =
A website had 12 486 hits in May and 9 758 in June. How many hits in the two months?
Find the missing digits in this vertical addition.

4 7

+ 3 8

8 1 2
Estimate first by rounding to the nearest hundred, then work out the exact answer.
1. 682 + 439 → estimate exact
2. 1275 + 2846 → estimate exact
Jamila added 4 386 + 2 745 and got 6 021. Without adding it yourself, explain how you know she is wrong.

Answers · Topic 4 Addition written strategies

Guided practice · split strategy

1a 2376 + 5162 = (6 + 70 + 300 + 2000) + (2 + 60 + 100 + 5000) = 8 + 130 + 400 + 7000 = 7538
b 6284 + 8415 = (4 + 80 + 200 + 6000) + (5 + 10 + 400 + 8000) = 9 + 90 + 600 + 14 000 = 14 699

Independent practice · split strategy

1a 7 + 70 + 1600 + 5000 = 6677 b 15 + 30 + 900 + 5000 + 40 000 = 45 945
c 15 + 50 + 700 + 13 000 + 40 000 = 53 765 d 6 + 80 + 1300 + 9000 + 70 000 = 80 386

Guided practice · vertical addition

1 a 62 b 155 c 782
2 a 167 b 719 c 8914
3 a 8497 b 6359 c 16 699

Note: the printed answer key gives 95 for 1b. 76 + 79 = 155.

Independent practice · vertical addition

1 a 8494 b 8258 c 41 776 d 54 789 e 63 634 f 83 987 g 59 246 h 64 753

Extended practice

1 a 13 690 b 90 178 c 74 771 d 92 461 e 23 555 f 149 254

2 a 37 690 b 18 669

Extra practice

1a 3878 b 7789 c 3789 d 7969

2a 652 b 4131 c 8025 d 8123

3a 690 b 2300

4 22 244 hits

5 The missing digits are 2 and 5: 427 + 385 = 812.

6a estimate 700 + 400 = 1100; exact 1121 b estimate 1300 + 2800 = 4100; exact 4121

7 Rounding gives about 4400 + 2700 = 7100, so the answer should be a little over 7000. 6021 is smaller than 4386 + 2745 could possibly be — in fact it is smaller than the two numbers added even before the trading, so it must be wrong. (The correct answer is 7131.)

Unit 1 · Topic 5

Subtraction mental strategies

Rounding numbers can make mental subtraction easier. This is also called the compensation strategy.

83 – 28 → Think: 83 – 30 = 53, then + 2 = 55

So 83 – 28 = 55. I wonder why it is called compensation?

Guided practiceWorked through with you

Solve using the compensation strategy.

a 85 – 19 Think: 85 – 20 = , then + 1 = → So 85 – 19 =
b 73 – 22 Think: 73 – 20 = , then − 2 = → So 73 – 22 =
c 91 – 32 Think: 91 – = , then − 2 = → So 91 – 32 =

Independent practiceYour turn

Use the compensation strategy to solve these mentally.
1. 58 – 19 =
2. 76 – 18 =
3. 61 – 32 =
4. 98 – 41 =
5. 146 – 28 =
6. 281 – 39 =
7. 365 – 42 =
8. 217 – 38 =
You can also round to the nearest hundred.

574 – 397 → Think: 574 – 400 + 3 = 177

Try these.

a 423 – 198 Think: 423 – + 2 =
b 654 – 305 Think: 654 – − =
c 526 – 297 Think: 526 – + =
d 793 – 207 Think: 793 – − =
e 478 – 197 Think:
f 642 – 304 Think:
Rounding can also help you check your answers. Round to check whether these answers are correct or incorrect.

583 – 296 = 187? Round to 583 – 300 = 283. You would expect the answer to be close to 283, so the first answer needs checking!
1. 457 – 198 = 259 Correct Incorrect
2. 782 – 305 = 477 Correct Incorrect
3. 893 – 497 = 196 Correct Incorrect
4. 631 – 296 = 335 Correct Incorrect
When you are subtracting numbers that are close together, you can add on to find the difference.

1352 – 1348 → Think: 1348 + ? = 1352. The answer is 4.
1. 94 − 89 =
2. 82 − 78 =
3. 574 − 567 =
4. 698 − 685 =
5. 427 – 419 =
6. 653 – 647 =
Addition and subtraction are linked. You can check subtraction by adding.

What is 37 – 14? My answer: 23. Check by adding: 23 + 14 = 37. Correct!
1. What is 67 – 45? Check by adding:
2. What is 175 – 59? Check by adding:
3. What is 3408 – 98? Check by adding:
4. What is 8995 – 2004? Check by adding:

Extended practiceStretch your thinking

Year 4 were having a mathematics computer game championship. Sophia won with 3872 points. Work out how many points the others had by using a mental strategy of your choice.
1. Scarlet had 297 points less than Sophia. Score:
2. Duy had 1306 points less than Sophia. Score:
3. Aravinda had 3859 points less than Sophia. Score:
4. Alexis had 58 points less than Sophia. Score:
5. Harper had 601 points less than Sophia. Score:
Use the information in question 1 to work out the following.
1. Who came second?
2. Who came last?
3. How many more points did Scarlet have than Duy?
4. How many points did Scarlet beat Harper by?
5. How many more points would Aravinda have needed to beat Duy?
The Thomastown Tornadoes have 27 426 supporters. Below is the number of supporters who did not attend each game. Work out how many supporters did attend.
1. Game 1: 4103 absent. Attendance:
2. Game 2: 26 995 absent. Attendance:
3. Game 3: 597 absent. Attendance:
4. Game 4: 13 699 absent. Attendance:

Extra practiceMore questions to build fluency

Use compensation. Round the number you subtract, then adjust.
1. 74 − 29 =
2. 92 − 48 =
3. 63 − 19 =
4. 85 − 37 =
Round to the nearest hundred, then adjust.
1. 532 − 198 =
2. 746 − 299 =
3. 415 − 97 =
4. 860 − 397 =
Count on from the smaller number to find the difference.
1. 197 to 300 →
2. 385 to 500 →
3. 1250 to 2000 →
4. 4600 to 5000 →
Check each subtraction by adding. Write the check.
1. 146 − 78 = check:
2. 523 − 259 = check:
Solve mentally.
1. 1000 − 350 =
2. 2000 − 750 =
3. 5000 − 1200 =
4. 10 000 − 4500 =
A cyclist planned to ride 8000 km and has ridden 5 385 km. How far is left? Which strategy did you use?

km
Find the missing number.
1. 84 − = 47
2. − 36 = 58
3. 300 − = 165
4. − 250 = 475

Answers · Topic 5 Subtraction mental strategies

Guided practice

1a 85 – 20 = 65, 65 + 1 = 66. So 85 – 19 = 66
b 73 – 20 = 53, 53 – 2 = 51. So 73 – 22 = 51
c 91 – 30 = 61, 61 – 2 = 59. So 91 – 32 = 59

Independent practice

1 a 39 b 58 c 29 d 57 e 118 f 242 g 323 h 179

2 a 423 – 200 + 2 = 225 b 654 – 300 – 5 = 349 c 526 – 300 + 3 = 229
d 793 – 200 – 7 = 586 e 478 – 200 + 3 = 281 f 642 – 300 – 4 = 338

3 a correct b correct c incorrect (396) d correct

4 a 5 b 4 c 7 d 13 e 8 f 6

5 a 22 (22 + 45 = 67) b 116 (116 + 59 = 175) c 3310 (3310 + 98 = 3408) d 6991 (6991 + 2004 = 8995)

Extended practice

1 a 3575 b 2566 c 13 d 3814 e 3271

2 a Alexis b Aravinda c 1009 d 304 e 2554

3 a 23 323 b 431 c 26 829 d 13 727

Extra practice

1a 45 (74 − 30 + 1) b 44 (92 − 50 + 2) c 44 (63 − 20 + 1) d 48 (85 − 40 + 3)

2a 334 (532 − 200 + 2) b 447 (746 − 300 + 1) c 318 (415 − 100 + 3) d 463 (860 − 400 + 3)

3a 103 b 115 c 750 d 400

4a 68 — check 68 + 78 = 146 b 264 — check 264 + 259 = 523

5a 650 b 1250 c 3800 d 5500

6 2615 km. Counting on from 5385 to 8000 is quickest: 15 to 5400, 600 to 6000, then 2000 to 8000.

7a 37 b 94 c 135 d 725

Unit 1 · Topic 6

Subtraction written strategies

You can use the split strategy for written subtraction by splitting the number you are subtracting by place value.

4672 – 2145 = 4672 − 2000 − 100 − 40 − 5 = 2527

Write down the answer after each stage of the equation if it helps you.

Guided practiceSplit strategy

Solve using the split strategy.

a 6359 − 4243 = 6359 − − − − =
b 8946 − 3412 = 8946 − − − − =
c 7650 − 2517 = 7650 − − − − =
d 15 498 − 4057 = 15 498 − − − − =
e 28 575 − 14 324 = 28 575 − − − − − =

Independent practiceSplit strategy

Would the answer be the same if you subtracted the ones first?

Here is another way to set out the split strategy that works well for larger numbers. Solve using this method.

3782 – 2431 = 3782 – 2000 = 1782 → – 400 = 1382 → – 30 = 1352 → – 1 = 1351

a 7598 – 3471 = − = − = − = − =
b 15 537 – 13 116 = − = − = − = − = − =
c 58 926 – 32 604 = − = − = − = − = − =
d 94 589 – 62 719 = − = − = − = − = − =

In vertical subtraction, you have to trade when the number you are subtracting is bigger than the number you are taking away from.

	T	O
	6	1
	7	3
–	2	6
	4	7

You cannot take 6 from 3, so trade 1 ten. Now it is 13 ones.

What would you do if there were a zero in the column you needed to trade from?

Guided practiceVertical subtraction

Solve using trading from the tens to the ones column.

a
T O

4 1
– 2 4

b
T O

8 5
– 3 8

c
T O

7 4
– 6 5
Solve using trading from the hundreds to the tens column.

a
H T O

8 4 7
– 2 6 3

b
H T O

7 0 4
– 3 2 2

c
Th H T O

3 6 6 2
– 1 2 8 0
Solve using trading from the thousands to the hundreds column.

a
Th H T O

5 3 8 5
– 3 8 2 1

b
Th H T O

7 6 5 6
– 2 9 2 6

c
Tth Th H T O

2 3 2 5 7
– 1 1 5 4 6

	T	O
	4	1
–	2	4

	T	O
	8	5
–	3	8

	T	O
	7	4
–	6	5

	H	T	O
	8	4	7
–	2	6	3

	H	T	O
	7	0	4
–	3	2	2

	Th	H	T	O
	3	6	6	2
–	1	2	8	0

	Th	H	T	O
	5	3	8	5
–	3	8	2	1

	Th	H	T	O
	7	6	5	6
–	2	9	2	6

	Tth	Th	H	T	O
	2	3	2	5	7
–	1	1	5	4	6

Independent practiceVertical subtraction

Sometimes you have to trade from 2 columns in the same equation.

Rewrite as vertical subtraction and solve.

a 758 − 392
H T O

–

b 830 − 659
H T O

–

c 571 − 243
H T O

–

d 9949 – 1863
Th H T O

–

e 8237 – 3523
Th H T O

–

f 6845 – 4038
Th H T O

–

g 53 259 – 21 832
Tth Th H T O

–

h 78 146 – 77 624
Tth Th H T O

–

i 66 752 – 24 938
Tth Th H T O

–

j 98 901 – 64 728
Tth Th H T O

–

Extended practiceStretch your thinking

Yann planned to ride 30 000 km to raise money for charity.

Use a written subtraction method to work out how much further he has to go after each stop.

Day	Route	Total distance travelled so far
1	Banebridge to Sale	922 km
2–3	Sale to Melba to Newland	2526 km
4–6	Newland to Pindale	5223 km
7–9	Pindale to Broom	7463 km
10–17	Broom to Windar to Blue Springs to Stan Cove	12 740 km
18–22	Stan Cove to Brookefield	15 925 km
23–26	Brookefield to Cooktown	18 755 km
27–34	Cooktown to Hamsdale	22 747 km

Yann is aiming to raise $85 000. Complete the table to show how much he has left to raise after each day.

Day Total raised Left to raise

1 $834

9 $23 471

22 $65 023

34 $76 914
Yann receives a large donation at the end of his ride and ends up raising a total of $123 564. How much over his target does he raise?
How much more does Yann have to raise if he wants to meet a target of $150 000?

Day	Total raised	Left to raise
1	$834
9	$23 471
22	$65 023
34	$76 914

Extra practiceMore questions to build fluency

Solve using the split strategy. Write the answer after each stage.
1. 5847 − 2316 → =
2. 7695 − 3142 → =
Rewrite as vertical subtraction and solve. No trading needed.
1. 876 − 543 =
2. 4689 − 2357 =
3. 9875 − 4321 =
4. 6798 − 3456 =
Rewrite as vertical subtraction and solve. These need trading.
1. 724 − 386 =
2. 5063 − 2748 =
3. 8014 − 3567 =
4. 6200 − 1845 =
Fill in the missing digits.

6 3

− 2 8

3 6 5
Estimate by rounding to the nearest thousand, then find the exact answer.
1. 7 248 − 3 891 → estimate exact
2. 12 507 − 4 682 → estimate exact
A charity aimed to raise 25 000. So far it has raised 17 465. How much is still needed?
Check each answer by adding it back. Tick the ones that are correct.
1. 842 − 379 = 463
2. 1506 − 728 = 878
3. 3000 − 1455 = 1545

Answers · Topic 6 Subtraction written strategies

Guided practice · split strategy

1a 6359 – 4000 – 200 – 40 – 3 = 2116 b 8946 – 3000 – 400 – 10 – 2 = 5534
c 7650 – 2000 – 500 – 10 – 7 = 5133 d 15 498 – 4000 – 0 – 50 – 7 = 11 441
e 28 575 – 10 000 – 4000 – 300 – 20 – 4 = 14 251

Independent practice · split strategy

1a 7598 – 3000 = 4598 → – 400 = 4198 → – 70 = 4128 → – 1 = 4127
b 15 537 – 10 000 = 5537 → – 3000 = 2537 → – 100 = 2437 → – 10 = 2427 → – 6 = 2421
c 58 926 – 30 000 = 28 926 → – 2000 = 26 926 → – 600 = 26 326 → – 4 = 26 322
d 94 589 – 60 000 = 34 589 → – 2000 = 32 589 → – 700 = 31 889 → – 10 = 31 879 → – 9 = 31 870

Guided practice · vertical subtraction

1 a 17 b 47 c 9 2 a 584 b 382 c 2382 3 a 1564 b 4730 c 11 711

Independent practice · vertical subtraction

1 a 366 b 171 c 328 d 8086 e 4714 f 2807 g 31 427 h 522 i 41 814 j 34 173

Students may or may not include the zeroes at the start of some answers. Either way is acceptable at this point.

Extended practice

1a 29 078 km · 27 474 km · 24 777 km · 22 537 km · 17 260 km · 14 075 km · 11 245 km · 7253 km

1b $84 166 · $61 529 · $19 977 · $8086

1c $38 564 d $26 436

Extra practice

1a 3531 — e.g. 5847 − 2000 = 3847, − 300 = 3547, − 10 = 3537, − 6 = 3531 b 4553

2a 333 b 2332 c 5554 d 3342

3a 338 b 2315 c 4447 d 4355

4 The missing digits are 5 and 8: 653 − 288 = 365.

5a estimate 7000 − 4000 = 3000; exact 3357 b estimate 13 000 − 5000 = 8000; exact 7825

6 7535

7 Tick a and c. b is wrong: 1506 − 728 = 778, not 878 (778 + 728 = 1506 ✓).

Unit 1 · Topic 8

Multiplication written strategies

Extended multiplication is a written strategy for multiplying larger numbers.

4 × 53 = ?

	H	T	O
		5	3
×			4
		1	2
	2	0	0
	2	1	2

4 × 3 = 12, then 4 × 50 = 200. Add them: 212.

Extended multiplication works the same way as the split strategy or the grid method. You multiply by each place value column in turn.

Guided practiceExtended multiplication

Solve using extended multiplication.

a
T O

2 1
× 3

3 × 1, then 3 × 20

b
T O

4 2
× 2

2 × 2, then 2 × 40

c
T O

1 5
× 4

d
H T O

3 1
× 5

e
H T O

7 2
× 4

f
H T O

4 7
× 6

Independent practiceExtended multiplication

Rewrite as extended multiplication and solve.
1. 5 × 28 =
2. 6 × 43 =
3. 9 × 67 =
4. 7 × 66 =
5. 8 × 34 =
6. 6 × 89 =
7. Payal earned $74 a week for 7 weeks. How much does she have?
8. Tyler rode 35 km a day for 8 days. How far did he go?
Working-out space

Contracted multiplication is a shorter way to multiply larger numbers.

4 × 53 = ?

	H	T	O
		1
		5	3
×			4
	2	1	2

This method is similar to the addition vertical algorithm. Start at the ones and work left.

Guided practiceContracted multiplication

Solve using contracted multiplication.

a
T O

4 2
× 2

b
T O

1 9
× 5

c
T O

2 4
× 4

d
H T O

6 1
× 5

e
H T O

5 2
× 6

f
H T O

4 8
× 7
Solve 9 × 84 using extended multiplication, then using contracted multiplication.

Extended
H T O

8 4
× 9

Contracted
H T O

8 4
× 9

Independent practiceContracted multiplication

Rewrite as contracted multiplication and solve.
1. 4 × 32 =
2. 7 × 41 =
3. 6 × 54 =
4. 5 × 52 =
5. 9 × 46 =
6. 8 × 68 =
7. Namrita bought 8 games that each cost $99. How much did she spend?
8. Antony bought 9 boxes of marbles with 47 in each. How many does he have altogether?
Match the equations with their answers.

45 × 7 86 × 7 53 × 6 45 × 8 92 × 4

602368315318360

Extended practiceStretch your thinking

Use a written multiplication strategy of your choice to solve. Show your working.
1. Farmer Sam grew 48 carrots. Farmer Fred grew 6 times as many. How many did Farmer Fred grow?
  Working-out space
2. Farmer Sue harvested 32 carrots a day for 9 days. How many carrots did she have altogether?
  Working-out space
3. Which farmer had more — Fred or Sue?
Carlos was having 78 people to his party, including himself. Work out how many of each item he needs.

Item Number per guest Total needed

Hot dogs 4

Carrot sticks 7

Chocolate buttons 9

Mini pizzas 5
What if Carlos had 178 people, including himself? How many of each item would he need?

Item Number per guest Total needed

Hot dogs 4

Carrot sticks 7

Chocolate buttons 9

Mini pizzas 5

Extra practiceMore questions to build fluency

Solve using extended multiplication. Show both parts.
1. 6 × 47 → + =
2. 8 × 64 → + =
Rewrite as contracted multiplication and solve.
1. 3 × 58 =
2. 5 × 76 =
3. 7 × 89 =
4. 9 × 45 =
Multiply these three-digit numbers.
1. 4 × 236 =
2. 6 × 315 =
3. 8 × 427 =
4. 7 × 508 =
Multiply by 10, 100 and 1000.
1. 36 × 10 =
2. 36 × 100 =
3. 47 × 1000 =
4. 250 × 100 =
Estimate first by rounding, then work it out exactly.
1. 5 × 198 → estimate exact
2. 7 × 312 → estimate exact
A school orders 6 boxes of exercise books. Each box holds 144 books.
1. How many books altogether?
2. If each book costs 3, what is the total cost?
Ben worked out 4 × 68 and got 242. Use estimation to explain why that cannot be right, then give the correct answer.

Correct answer:

Answers · Topic 8 Multiplication written strategies

Guided practice · extended multiplication

1a 21 × 3 → 3 + 60 = 63 b 42 × 2 → 4 + 80 = 84 c 15 × 4 → 20 + 40 = 60
d 31 × 5 → 5 + 150 = 155 e 72 × 4 → 8 + 280 = 288 f 47 × 6 → 42 + 240 = 282

Independent practice · extended multiplication

1 a 140 b 258 c 603 d 462 e 272 f 534 g $518 h 280 km

Guided practice · contracted multiplication

1 a 84 b 95 c 96 d 305 e 312 f 336

2 9 × 84: extended 36 + 720 = 756; contracted 756

Independent practice · contracted multiplication

1 a 128 b 287 c 324 d 260 e 414 f 544 g $792 h 423

2 45 × 7 = 315, 86 × 7 = 602, 53 × 6 = 318, 45 × 8 = 360, 92 × 4 = 368

Extended practice

1 a 288 b 288 c Both farmers had the same.

2 Hot dogs 312 · Carrot sticks 546 · Chocolate buttons 702 · Mini pizzas 390

3 Hot dogs 712 · Carrot sticks 1246 · Chocolate buttons 1602 · Mini pizzas 890

Extra practice

1a 42 + 240 = 282 b 32 + 480 = 512

2a 174 b 380 c 623 d 405

3a 944 b 1890 c 3416 d 3556

4a 360 b 3600 c 47 000 d 25 000

5a estimate 5 × 200 = 1000; exact 990 b estimate 7 × 300 = 2100; exact 2184

6a 864 books b 2592

7 4 × 70 is 280, so the answer must be a bit under 280 — 242 is far too small. (Ben has probably forgotten to carry.) The correct answer is 272.

Unit 1 · Topic 9

Division written strategies

The number you start with (64) is called the dividend. The number you divide by (4) is the divisor.

6 tens ÷ 4 = 1 ten, with 2 tens left over. Trade them: 24 ones ÷ 4 = 6.

For division set out this way, start from the left and work your way right.

Guided practiceWorked through with you

Solve the equations without trading.
1. 55 ÷ 5 =
2. 84 ÷ 4 =
3. 68 ÷ 2 =
4. 69 ÷ 3 =
5. 46 ÷ 2 =
6. 93 ÷ 3 =
Solve the equations with trading.
1. 75 ÷ 5 =
2. 84 ÷ 6 =
3. 96 ÷ 8 =
4. 54 ÷ 3 =
5. 91 ÷ 7 =
6. 92 ÷ 4 =

Independent practiceYour turn

Rewrite and solve.
1. 87 ÷ 3 =
2. 98 ÷ 2 =
3. 88 ÷ 8 =
4. 84 ÷ 7 =
5. 78 ÷ 3 =
6. 95 ÷ 5 =
7. 58 ÷ 2 =
8. 80 ÷ 4 =
9. 78 ÷ 6 =
Solve, then rewrite each one as a division equation.
1. 72 ÷ 6 =
2. 80 ÷ 5 =
3. 76 ÷ 4 =
4. 68 ÷ 4 =
5. 98 ÷ 7 =
6. 81 ÷ 3 =
7. 86 ÷ 2 =
8. 96 ÷ 3 =
9. 96 ÷ 4 =
Solve using a written division strategy.
1. 84 students were staying in rooms of 3 on their school trip. How many rooms did they need?
  Working-out space
2. 95 sheep were divided equally into 5 pens. How many were in each?
  Working-out space
3. Audrey divided her 96 basketball cards into 4 equal piles. How many cards in each?
  Working-out space
4. How many cards in each pile if Audrey divided them into 3 equal piles?
  Working-out space
5. 78 people in the audience sat in rows of 6. How many rows were there?
  Working-out space
6. Could the 78 people sit in rows of exactly 7? Why or why not?
  Working-out space

Extended practiceStretch your thinking

Remember to start from the left and write the answers above the correct place value columns.

Tick the numbers that can be divided exactly by:

243542 566075 8196108120
1. 2
2. 3
3. 4
4. 5
Calculate the answers.
1. 96 ÷ 3 =
2. 336 ÷ 3 =
3. 273 ÷ 3 =
4. 448 ÷ 4 =
5. 284 ÷ 4 =
6. 486 ÷ 2 =
7. 605 ÷ 5 =
8. 777 ÷ 7 =
9. 987 ÷ 7 =
Rewrite and solve.
1. Melinda was sharing 336 jelly beans into 6 bags. How many went in each?
2. Melinda realised she forgot to make a bag for herself. How many in each bag if she makes up another one?

Extra practiceMore questions to build fluency

Solve. These need no trading.
1. 84 ÷ 4 =
2. 693 ÷ 3 =
3. 848 ÷ 4 =
4. 966 ÷ 3 =
Solve. These need trading.
1. 72 ÷ 5 =
2. 136 ÷ 8 =
3. 245 ÷ 7 =
4. 432 ÷ 6 =
Solve. Write the remainder if there is one.
1. 95 ÷ 4 =
2. 167 ÷ 5 =
3. 250 ÷ 7 =
4. 523 ÷ 6 =
Tick the numbers that divide exactly by 4.
1. 116
2. 138
3. 224
4. 350
Write each division as a multiplication, then solve.
1. 144 ÷ 6 = → check: × 6 =
2. 203 ÷ 7 = → check: × 7 =
156 sweets are shared equally between 8 children.
1. How many does each child get?
2. How many are left over?
3. How many more sweets would be needed so there is no remainder?
A bus holds 48 passengers. 300 people need to travel. How many buses are needed? Explain why you cannot just ignore the remainder.

buses

Answers · Topic 9 Division written strategies

Guided practice

1 a 11 b 21 c 34 d 23 e 23 f 31

2 a 15 b 14 c 12 d 18 e 13 f 23

Independent practice

1 a 29 b 49 c 11 d 12 e 26 f 19 g 29 h 20 i 13

2 a 72 ÷ 6 = 12 b 80 ÷ 5 = 16 c 76 ÷ 4 = 19 d 68 ÷ 4 = 17 e 98 ÷ 7 = 14 f 81 ÷ 3 = 27 g 86 ÷ 2 = 43 h 96 ÷ 3 = 32 i 96 ÷ 4 = 24

3 a 28 b 19 c 24 d 32 e 13 f No.

Teacher: Look for students who understand that there would be leftovers or remainders, because 7 does not divide equally into 78.

Extended practice

1 a 24, 42, 56, 60, 96, 108, 120 b 24, 42, 60, 75, 81, 96, 108, 120
c 24, 56, 60, 96, 108, 120 d 35, 60, 75, 120

2 a 32 b 112 c 91 d 112 e 71 f 243 g 121 h 111 i 141

3 a 56 b 48

Extra practice

1a 21 b 231 c 212 d 322

2a 14 r 2 b 17 c 35 d 72

3a 23 r 3 b 33 r 2 c 35 r 5 d 87 r 1

4 Tick 116 (29 × 4) and 224 (56 × 4). 138 gives 34 r 2 and 350 gives 87 r 2.

5a 24 — check 24 × 6 = 144 b 29 — check 29 × 7 = 203

6a 19 each b 4 left over c 4 more would make 160, which is 20 each

7 7 buses. 300 ÷ 48 = 6 r 12, and those 12 people still need a seat — so a seventh bus is needed even though it will not be full. When the remainder is people (or anything that cannot be split), you always round the number of containers up.

Unit 2

Fractions and decimals

Fractions that look different can be exactly the same size. Learn to spot them, turn top-heavy fractions into mixed numbers, and write tenths and hundredths as decimals.

Number & Algebra · 3 topics

Unit 2 · Topic 1

Equivalent fractions

Equivalent fractions are the same size, even though they have different names.

12 = 24 = 36 = 48

The same amount of each bar is shaded every time.

Guided practiceWorked through with you

Tick the fraction that is equivalent to:
1. 14 23 28 48
2. 23 34 12 46
3. 68 34 36 810

Independent practiceYour turn

Label each pair of equivalent fractions.
1. One bar cut into thirds with 1 part shaded, next to a bar cut into sixths with 2 parts shaded.
  and
2. Fifths with 2 shaded, next to tenths with 4 shaded.
  and
3. Halves with 1 shaded, next to quarters with 2 shaded.
  and
4. Thirds with 2 shaded, next to ninths with 6 shaded.
  and
Shade and label an equivalent fraction for each of these. Click the squares to shade them.

12 → eighths

46 → thirds

810 → fifths

312 → quarters
Use the fraction wall to find equivalent fractions.

1 whole

1/2
1/2

1/3
1/3
1/3

1/4
1/4
1/4
1/4

1/5
1/5
1/5
1/5
1/5

1/6
1/6
1/6
1/6
1/6
1/6

1/8
1/8
1/8
1/8
1/8
1/8
1/8
1/8

1/10
1/10
1/10
1/10
1/10
1/10
1/10
1/10
1/10
1/10

1/12
1/12
1/12
1/12
1/12
1/12
1/12
1/12
1/12
1/12
1/12
1/12
1. 25 =
2. 812 =
3. 46 =
4. 14 =
5. 810 =
6. 68 =
7. 12 =
8. 1 =
What do you notice about all the fractions that are equivalent to 12?

Extended practiceStretch your thinking

This grid has 100 squares.
1. Shade 10 squares and write the fraction.
2. What is the equivalent fraction in tenths?
How many squares would you colour for:
1. 410?
2. 810?
3. 710?
4. 12?
Write an equivalent hundredths fraction for:
1. 410 =
2. 12 =
3. 310 =
4. 910 =
5. 1010 =
6. 14 =
Write >, < or =.
1. 12 510
2. 35 310
3. 58 34
4. 812 46

Extra practiceMore questions to build fluency

Complete each pair of equivalent fractions.
1. 13 = 9
2. 25 = 10
3. 34 = 12
4. 16 = 18
5. 58 = 16
6. 23 = 12
Find the missing denominator.
1. 12 = 6
2. 35 = 9
3. 27 = 8
4. 49 = 12
Write each fraction in its simplest form.
1. 48 =
2. 69 =
3. 1015 =
4. 1216 =
Write each fraction as hundredths.
1. 12 = 100
2. 14 = 100
3. 310 = 100
4. 45 = 100
Write >, < or = between each pair.
1. 12 35
2. 23 46
3. 34 58
4. 13 25
A pizza is cut into 12 slices. Ana eats 3 slices and Ben eats 14 of the pizza.
1. What fraction of the pizza did Ana eat, in its simplest form?
2. How many slices did Ben eat?
3. Did they eat the same amount?
Sam says 36 is bigger than 12 because 3 and 6 are bigger numbers than 1 and 2. Explain why he is wrong.

Answers · Topic 1 Equivalent fractions

Guided practice

1 a 2/8 b 4/6 c 3/4

Independent practice

1 a 1/3 and 2/6 b 2/5 and 4/10 c 1/2 and 2/4 d 2/3 and 6/9

2 a 4 sections shaded → 4/8 b 2 sections shaded → 2/3 c 4 sections shaded → 4/5 d 1 section shaded → 1/4

3 a 4/10 b 2/3 (or 4/6) c 2/3 (or 8/12) d 2/8 (or 3/12) e 4/5 f 3/4 (or 9/12)
g 2/4, 3/6, 4/8, 5/10, 6/12 h 2/2, 3/3, 4/4, 5/5, 6/6, 8/8, 10/10, 12/12

Extended practice

1 a 10/100 (or an equivalent) b 1/10

2 a 40 b 80 c 70 d 50

3 a 40/100 b 50/100 c 30/100 d 90/100 e 100/100 f 25/100

4 a = b > c < d =

Extra practice

1a 3 b 4 c 9 d 3 e 10 f 8

2a 12 b 15 c 28 d 27

3a ½ b ⅔ c ⅔ d ¾

4a 50 b 25 c 30 d 80

5a < b = c > d <

6a 3 out of 12 = ¼ b 3 slices c Yes — both ate ¼ of the pizza.

7 The size of the digits does not decide the size of a fraction — what matters is how many parts you have out of how many the whole was cut into. 3 out of 6 equal parts is exactly half the whole, and so is 1 out of 2. They are equivalent, not different.

Unit 2 · Topic 2

Improper fractions and mixed numbers

When the numerator is bigger than the denominator, it is called an improper fraction. You can change an improper fraction to a mixed number.

53 = 123

Why do you think they are called mixed numbers?

Guided practiceWorked through with you

Fill in the gaps.
1. Quarters, 0 to 2: 14, 24, , 44, 54, , , 84
2. Halves, 0 to 2: , 22, , 42
3. Thirds, 1 to 3: 33, 43, 53, , 73, , 93

Extended practiceStretch your thinking

Write an improper fraction and a mixed number for each set of shapes.

Diagram	Improper fraction	Mixed number
3 whole circles cut in sixths, plus 4 sixths
4 whole shapes cut in eighths, plus 3 eighths
2 whole shapes cut in fifths, plus 3 fifths
3 whole shapes cut in twelfths, plus 7 twelfths

Complete the number line counting by ninths, with fractions and mixed numbers.

Fractions: 0/9, 1/9, 2/9 … up to 26/9

Mixed numbers: 0, 1/9, 2/9 … 1, 1 1/9 … 2, 2 1/9 … 2 8/9
How many ninths in:
1. 1?
2. 249?
3. 189?
4. 4?
5. 359?
6. 519?

Extra practiceMore questions to build fluency

Change each improper fraction to a mixed or whole number.
1. 74 =
2. 113 =
3. 92 =
4. 165 =
5. 186 =
6. 238 =
Change each mixed number to an improper fraction.
1. 213 =
2. 325 =
3. 158 =
4. 412 =
How many are there in total?
1. How many quarters in 3?
2. How many thirds in 5?
3. How many halves in 7?
4. How many fifths in 4?
Tick the larger number in each pair.
1. 94 212
2. 145 235
3. 103 323
Count on in halves from 0. Write the next six numbers, using mixed numbers where you can.
A recipe uses 34 of a cup of flour per batch.
1. How much flour for 3 batches, as an improper fraction?
2. Write that as a mixed number.
3. How many whole cups would you need to buy?
Explain how you can tell, just by looking, that 154 is bigger than 3 but smaller than 4.

Answers · Topic 2 Improper fractions and mixed numbers

Guided practice

1a 3/4, 6/4, 7/4 b 1/2, 3/2 c 6/3, 8/3

Independent practice

1 a 1⅓ b 2⅓ c 3 d 1 2/4 (1½) e 2¾ f 2¼ g 2½ h 4½ i 3

2a ½, 1, 1½, 2, 2½, 3, 3½, 4, 4½
b 1/3, 2/3, 3/3, 4/3, 5/3, 6/3, 7/3, 8/3, 9/3, 10/3
c 1/4, 2/4, 3/4, 1, 1¼, 1 2/4, 1¾, 2
d 5, 4½, 4, 3½, 3, 2½, 2, 1½, 1

3 2/4 at the halfway point of 0–1; 1¼, 1¾ and 2 in order after 1.

4 2/3 before 1; 1⅔; 2⅓; 3⅓.

5 3/2, 5/2, 8/2 and 9/2 in order along the halves line.

6 a 2½ b 4½ c 4 d 1½

7 a 7/2 b 1⅔ c 3¼ d 12/4 e 10¼ f 7⅓ g 5½ h 9/3 i 4¼

Extended practice

1 a 22/6 = 3 4/6 b 35/8 = 4 3/8 c 13/5 = 2 3/5 d 43/12 = 3 7/12

2 Fractions 0/9 to 26/9; the matching mixed numbers run 0, 1/9 … 1, 1 1/9 … 2, 2 1/9 … up to 2 8/9.

3 a 9 b 22 c 17 d 36 e 32 f 46

Extra practice

1a 1¾ b 3⅔ c 4½ d 3⅕ e 3 f 2⅞

2a 7/3 b 17/5 c 13/8 d 9/2

3a 12 b 15 c 14 d 20

4a 9/4 = 2¼, so 2½ is larger. b 14/5 = 2⅘, so 14/5 is larger. c 10/3 = 3⅓, so 3⅔ is larger.

5 ½, 1, 1½, 2, 2½, 3

6a 9/4 b 2¼ c 3 whole cups

7 Four quarters make 1 whole, so 12 quarters make exactly 3 and 16 quarters make exactly 4. 15 is between 12 and 16, so 15/4 must sit between 3 and 4.

Unit 2 · Topic 3

Decimal fractions

You can write 110 as a decimal: 0.1
You can write 1100 as a decimal: 0.01

Ones		Tenths	Hundredths
0	.	0	1

Hundredths are 10 times smaller than tenths, and tenths are 10 times smaller than ones.

Guided practiceWorked through with you

Shade the grids and write each tenths fraction as a decimal.

a 210

b 510

c 810
Shade the grids and write each hundredths fraction as a decimal.
1. 45100 =
2. 26100 =
3. 53100 =
4. 82100 =
5. 99100 =
6. 60100 =
Use this grid to check one fraction at a time.

Independent practiceYour turn

Write the number on each numeral expander as a decimal and as a common fraction or mixed number.

Expander	Ones	Tenths	Hundredths
a	0	7	—
b	0	0	7
c	0	7	7
d	7	7	7
e	0	3	2
f	0	6	5
g	3	2	9
h	6	0	4

Complete the number lines.
1. 0, 0.1, 0.2, , , , 0.6, , , , 1, 1.1, , , , 1.5
2. 0, 0.01, , 0.03, , , , 0.07, , , 0.1, 0.11,
3. 1.7, 1.8, , , , 2.2, , , 2.5, , ,
4. 0.95, 0.96, 0.97, , , , 1.01, 1.02, , , ,

Write the numbers on the place value chart.

Number	Hundreds	Tens	Ones	.	Tenths	Hundredths
Thirty-six and four tenths				.
Five hundreds and twenty-two hundredths				.
Two hundred and twenty-two and twenty-two hundredths				.
Fourteen and fifty-eight hundredths				.
103 710				.
628 43100				.
946 4100				.

Extended practiceStretch your thinking

Tick the bigger number in each pair.
1. 0.90.09
2. 0.180.3
3. 0.250.52
4. 9.80.99
5. 0.50.05
6. 0.410.39
7. 0.780.87
8. 10.1

Mr Hoyne’s class had a long jump competition. Reorder the results from shortest to longest jump.

Name	Jump length	Order: name	Jump length
Silva	3.26 m
Raff	4.07 m
James	5.21 m
Elara	4.7 m
Lily	4.28 m
Dan	3.9 m
Nick	5.02 m

Extra practiceMore questions to build fluency

Write each fraction as a decimal.
1. 710 =
2. 3100 =
3. 45100 =
4. 910 =
5. 60100 =
6. 8100 =
Write each decimal as a fraction.
1. 0.4 =
2. 0.25 =
3. 0.07 =
4. 0.9 =
Write the value of the digit 6 in each number.
1. 3.62
2. 0.46
3. 16.05
4. 7.86
Write >, < or = between each pair.
1. 0.7 0.65
2. 0.3 0.30
3. 1.05 1.5
4. 2.4 2.04
Order these decimals from smallest to largest.

0.8 0.08 0.88 0.18
Complete the number line, counting in tenths.
1. 0.1, 0.2, , 0.4, , 0.6, , 0.8, , 1.0
A runner's times are 12.4 s, 12.04 s and 12.44 s.
1. Which is the fastest time?
2. Which is the slowest?
3. Explain why 12.04 is not bigger than 12.4 even though it has more digits.

Answers · Topic 3 Decimal fractions

Guided practice

1 a 20 squares shaded → 0.2 b 50 squares → 0.5 c 80 squares → 0.8

2 a 45 squares → 0.45 b 26 → 0.26 c 53 → 0.53 d 82 → 0.82 e 99 → 0.99 f 60 → 0.6 (or 0.60)

Independent practice

1 a 0.7 = 7/10 b 0.07 = 7/100 c 0.77 = 77/100 d 7.77 = 7 77/100
e 0.32 = 32/100 f 0.65 = 65/100 g 3.29 = 3 29/100 h 6.04 = 6 4/100

Accept equivalent fractions, such as 70/100 for 7/10.

2a 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3, 1.4, 1.5
b 0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1, 0.11, 0.12
c 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8
d 0.95, 0.96, 0.97, 0.98, 0.99, 1.00, 1.01, 1.02, 1.03, 1.04, 1.05, 1.06

3 36.4 · 500.22 · 222.22 · 14.58 · 103.7 · 628.43 · 946.04

Extended practice

1 a 0.9 b 0.3 c 0.52 d 9.8 e 0.5 f 0.41 g 0.87 h 1

2 Silva 3.26 m, Dan 3.9 m, Raff 4.07 m, Lily 4.28 m, Elara 4.7 m, Nick 5.02 m, James 5.21 m

Extra practice

1a 0.7 b 0.03 c 0.45 d 0.9 e 0.6 (or 0.60) f 0.08

2a 4/10 (= 2/5) b 25/100 (= ¼) c 7/100 d 9/10

3a 6 tenths (0.6) b 6 hundredths (0.06) c 6 ones d 6 hundredths (0.06)

4a > b = c < d >

5 0.08, 0.18, 0.8, 0.88

6 0.3, 0.5, 0.7, 0.9

7a 12.04 s b 12.44 s c Compare place by place, not by counting digits. Both have 12 ones; then 12.4 has 4 tenths while 12.04 has 0 tenths. Since 4 tenths beats 0 tenths, 12.4 is larger — the extra digit in 12.04 is only worth hundredths.

Unit 3

Money and financial mathematics

When the smallest coin is 5c, every cash total has to be rounded. Learn which amounts round up, which round down, and how to work out the change.

Number & Algebra · 1 topic

Unit 3 · Topic 1

Money and money calculations

Imagine there are no 1c and 2c coins and 5c coins have the lowest value. To give change in cash, everything is rounded to the nearest 5 cents.

$9.91 rounds down to $9.90 → you would get 10c change from $10. $9.94 rounds up to $9.95 → you would get 5c change from $10.

If you paid $10 for an item that cost $9.94, would you get 6c change? Why not?

Guided practiceWorked through with you

Write the digits 1, 2, 3, 4, 6, 7, 8 and 9 in the correct boxes below.

Rounds up to 0 Rounds down to 0 Rounds up to 5 Rounds down to 5

Complete the table.

Amount	Rounds up or down?	Rounds to
$1.62	down	$1.60
$3.58
$7.86
$15.32
$23.01
$99.99
$85.43
$48.04
$59.97

Independent practiceYour turn

A	B	C	D	E
Mints	Choc bar	Item C	Item D	Item E
$1.47	$3.52	$2.98	$2.01	$3.23

1. How much change would you get from $5 for:
  A? B? C? D? E?
2. How much change would you get from $10 for:
  A? B? C? D? E?
Choose 3 items from above.
1. Calculate the total cost.
  
  Items Cost
  
  +
  
  Total
2. Round the total to the nearest 5c.
3. How much change would you get from $20?
4. How much change would you get from $100?
Would you round the total for each pair of items up or down?
1. A and B
2. C and E
3. B and D
4. A and D
Use a calculator to work out which 2 books together would give you:

A B C D E F

The Kite Bear Why? Alien Dancer Little Train My Pony Book F

$9.53 $5.68 $12.82 $9.39 $14.31 $17.63
1. no change from $20.
2. $1.10 change from $20.
3. $1.50 change from $20.
4. $23 change from $50.
5. $22.85 change from $50.
Which digits round up to the nearest 5 and which round down?

You have $5 to spend at Dean’s Ice-creams.

Ice-cream cones	Price	Mix-ins	Price
1 scoop	$2.75	Crushed cookies	$1.14
2 scoops	$3.50	Sprinkles	32c
		Honeycomb	$1.38
		Choc chips	95c
		Crushed doughnut	$1.46
		Caramel pieces	64c
		Strawberries	89c

Choose which ice-creams and mix-ins you want and calculate the total cost.
Working-out space
How much change will you get?

Extended practiceStretch your thinking

Many countries use decimal currency. In South Africa, 1 Rand = 100 cents, or R1 = 100c. Coins come in these denominations: 5c, 10c, 20c, 50c, R1, R2 and R5.

Round each amount to the nearest 5c.
1. 33c
2. R1.76
3. R5.63
4. R3.07
5. R8.99
6. R7.02
How much change would you get from R10 for:
1. R4.98
2. R2.51
3. R9.22
4. 45c
5. R7.36
6. R5.74
How many 50c coins in:
1. R1?
2. R2?
3. R5?
How many of each item could you buy with R20?
1. An item costing R1.50
2. An item costing R2.52
3. An item costing R6.68

Extra practiceMore questions to build fluency

Round each amount to the nearest 5c.
1. $4.62 →
2. $7.38 →
3. $12.91 →
4. $0.47 →
5. $25.13 →
6. $9.99 →
Add these amounts.
1. $3.45 + $2.30 =
2. $12.75 + $6.80 =
3. $8.95 + $4.55 =
4. $19.20 + $30.85 =
Work out the change from $20.
1. Spent $13.40 →
2. Spent $8.65 →
3. Spent $17.95 →
4. Spent $4.05 →
How many of each coin make $2?
1. 10c coins
2. 20c coins
3. 50c coins
4. 5c coins
A book costs $14.60 and a pen costs $3.85.
1. What is the total?
2. Rounded to the nearest 5c, what would you pay in cash?
3. What change from $20?
Four friends share a bill of $37.60 equally.
1. How much does each pay?
2. If a fifth friend joins and they split $47.00 five ways, how much each?
Priya buys 3 drinks at $2.40 each and pays with a $10 note. Work out her change, and explain how you could check it quickly.

Change:

Answers · Topic 1 Money and money calculations

Guided practice

1 Rounds up to 0: 8, 9 · Rounds down to 0: 1, 2 · Rounds up to 5: 3, 4 · Rounds down to 5: 6, 7

2 $3.58 up $3.60 · $7.86 down $7.85 · $15.32 down $15.30 · $23.01 down $23.00 · $99.99 up $100.00 · $85.43 up $85.45 · $48.04 up $48.05 · $59.97 down $59.95

Independent practice

1a A $3.55 · B $1.50 · C $2.00 · D $3.00 · E $1.75
1b A $8.55 · B $6.50 · C $7.00 · D $8.00 · E $6.75

2 a–d Teacher to check.

Teacher: Students may add the 3 money amounts using a vertical algorithm, then apply their understanding of rounding and change-giving to identify the rounded amount and calculate the change required.

3 a up b down c up d up

4 a B and E b A and D c B and C d D and F e C and E

5 a Teacher to check. Look for the ability to accurately add the chosen amounts and to show an understanding of the financial concepts by not going over the given amount.
b Teacher to check: the answer will depend on the student's response to part a.

Extended practice

1 a 35c b R1.75 c R5.65 d R3.05 e R9 (R9.00) f R7 (R7.00)

2 a R5 b R7.50 c 80c d R9.55 e R2.65 f R4.25

3 a 2 b 4 c 10

4 a 13 b 7 c 2

Extra practice

1a $4.60 b $7.40 c $12.90 d $0.45 e $25.15 f $10.00

2a $5.75 b $19.55 c $13.50 d $50.05

3a $6.60 b $11.35 c $2.05 d $15.95

4a 20 b 10 c 4 d 40

5a $18.45 b $18.45 (already a multiple of 5c) c $1.55

6a $9.40 each b $9.40 each

7 3 × $2.40 = $7.20, so the change is $2.80. A quick check: $2.40 is nearly $2.50, and 3 × $2.50 = $7.50, so the change should be a little more than $2.50 — $2.80 fits.

Unit 4

Patterns and algebra

Find the rule, predict what comes next, and turn tricky word problems into number sentences you can actually solve.

Number & Algebra · 2 topics

Unit 4 · Topic 1

Number patterns

Recognising patterns can help to solve number problems. What is the next number in this sequence?

Term	1	2	3	4	5	6	7	8	9	10
Value	12	17	22	27	32	37	42	47	52	?

Rule: Add 5.

The ones pattern is 2, 7. What is the tens pattern?

Guided practiceWorked through with you

Write the rule and find the 10th term.
1. 1 2 3 4 5 … 10
  
  7 9 11 13 15 …
  Rule:
2. 1 2 3 4 5 … 10
  
  99 88 77 66 55 …
  Rule:
3. 1 2 3 4 5 … 10
  
  50 47 44 41 38 …
  Rule:
4. 1 2 3 4 5 … 10
  
  13 23 33 43 53 …
  Rule:
5. 1 2 3 4 5 … 10
  
  9 18 27 36 45 …
  Rule:

Independent practiceYour turn

Write the rule for each function machine.
1. In: 3, 5, 8 → Out: 21, 35, 56 Rule:
2. In: 20, 34, 51 → Out: 11, 25, 42 Rule:
3. In: 15, 42, 60 → Out: 35, 62, 80 Rule:
4. In: 4, 12, 25 → Out: 40, 120, 250 Rule:
Machines redrawn for this edition — the rules match the printed answer key.
Find the outputs for each machine.

In Out

34

88

97

Rule: Add 12.

In Out

24

300

48

Rule: Divide by 3.
A multiple is the result of multiplying one number by another. The numbers 4, 6, 8 and 10 are multiples of 2.

12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899100
1. Mark all the multiples of 2 on the hundred chart.
2. Now list the multiples of 4.
3. Which numbers are both multiples of 2 and of 4?
4. List the multiples of 8.
5. How many of the multiples of 8 are also multiples of 2 and 4?
What do you notice about all the multiples of 2 and 4?
Use a second hundred chart for multiples of 5 and 2.

12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899100
1. Mark all the multiples of 5.
2. List the multiples of 2.
3. What do you notice about numbers that are multiples of both 2 and 5?
4. Which of those numbers are also multiples of 10?

Extended practiceStretch your thinking

Write the rule and complete the pattern.
1. 1 2 4 7 11 29
  Rule:
2. 3 5 9 15 23 59
  Rule:
3. 1 2 4 8 16 128
  Rule:
Create your own rule for each function machine, then show 3 inputs and outputs.

In Out

Rule:

In Out

Rule:
1. Write the first 10 multiples of 7.
2. Which of these are also multiples of 2?
3. Which are also multiples of 5?
4. Which are also multiples of 3?

Extra practiceMore questions to build fluency

Write the rule and the next three terms.
1. 4, 11, 18, 25, … rule next
2. 90, 81, 72, 63, … rule next
3. 3, 6, 12, 24, … rule next
4. 1, 4, 9, 16, … rule next
A machine's rule is × 3 then + 2. Find the outputs.
1. in 4 → out
2. in 7 → out
3. in 10 → out
4. in 0 → out
Work backwards. The rule is × 4 then − 1. Find the inputs.
1. out 11 → in
2. out 23 → in
3. out 39 → in
4. out 3 → in
Find the 10th term of each pattern.
1. 5, 10, 15, 20, …
2. 2, 5, 8, 11, …
3. 100, 90, 80, …
4. 7, 14, 21, …
Fill in the gaps in each pattern.
1. 6, , 18, , 30, 36
2. 64, 56, , 40, , 24
3. 1, 2, 4, , 16,
A pattern of tiles grows: 4 tiles, 7 tiles, 10 tiles, 13 tiles.
1. What is the rule?
2. How many tiles in the 8th shape?
3. Which shape uses 31 tiles?
Two patterns start at 2. One adds 4 each time, the other doubles each time. Write the first five terms of each, then say which grows faster and why.

Adds 4:

Doubles:

Answers · Topic 1 Number patterns

Guided practice

1 a 25, Add 2 b 0, Subtract 11 c 23, Subtract 3 d 103, Add 10 e 90, Add 9

Independent practice

1 a Multiply by 7 b Subtract 9 c Add 20 d Multiply by 10

2 a 46, 100, 109 b 8, 100, 16

3 a, b & d Teacher to check.
c Students may list the individual numbers or observe that all the multiples of 4 are both circled and shaded.
e All of them.

4 a & b Teacher to check.
c They all end in zero.
d The numbers that are multiples of both 2 and 5 are also multiples of 10.

Extended practice

1a 1, 2, 4, 7, 11, 16, 22, 29, 37, 46 — Rule: Add 1 more each time
b 3, 5, 9, 15, 23, 33, 45, 59, 75, 93 — Rule: Add 2 more each time
c 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 — Rule: Multiply the previous number by 2

Accept any answer that accurately describes the patterns.

2 a & b Teacher to check. Look for the ability to apply knowledge of number patterns to create an appropriate rule and formulate 3 examples that demonstrate that rule.

3 a 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 b 14, 28, 42, 56, 70 c 35, 70 d 21, 42, 63

Extra practice

1a add 7 → 32, 39, 46 b subtract 9 → 54, 45, 36 c double → 48, 96, 192 d square numbers (add 3, 5, 7, 9 …) → 25, 36, 49

2a 14 b 23 c 32 d 2

3a 3 b 6 c 10 d 1

4a 50 b 29 c 10 d 70

5a 12, 24 b 48, 32 c 8, 32

6a add 3 each time b 25 tiles c the 10th shape

7 Adds 4: 2, 6, 10, 14, 18. Doubles: 2, 4, 8, 16, 32. Doubling grows faster — adding 4 makes the same size step every time, but doubling makes a bigger step each time because the step is the whole number so far.

Unit 4 · Topic 2

Problem solving

When this number is subtracted from 30, the answer is the same as 16 plus 7. What is the number?

To solve a word problem:

Change the word problem into a number sentence: 30 – ☐ = 16 + 7
Then complete the calculation: 30 – ☐ = 23 The answer is 7.

You can check by doing opposites: 23 + 7 = 30 and 30 – 7 = 23. It’s correct!

Which words tell you the operation you need to use to solve the problem?

Guided practiceWorked through with you

Change to number sentences and solve.
1. When this number is added to 15, the answer is the same as 48 minus 12. What is the number?
  Number sentence: 15 + ☐ = 48 – 12 Answer:
2. When this number is added to 42, the answer is the same as 31 plus 27. What is the number?
  Number sentence: 42 + ☐ = Answer:
3. When this number is subtracted from 73, the answer is the same as 26 + 23. What is the number?
  Number sentence: Answer:

Independent practiceYour turn

Write number sentences to solve.
1. What number subtracted from 100 gives the same answer as 31 added to 27?
2. What number added to 56 gives the same answer as 108 minus 21?
3. When this number is added to 98, the answer is the same as 200 minus 72. What is the number?
4. There were 43 boys and 54 girls at the party. 72 guests chose pizza; the rest had burgers. How many had burgers?
5. Of the total guests in question d, 18 left to play in a cricket match. Of those still there, 61 had cake. How many didn’t have cake?
Fill in the gaps to complete the number sentences.
1. + 17 = 32
2. 58 – = 44
3. – 23 = 61
4. 35 + = 89
5. × 8 = 48
6. 7 × = 56
7. 63 ÷ = 9
8. ÷ 5 = 11
9. 26 + 34 = 100 –
10. 78 – 46 = 19 +
11. 147 – = 96 + 15
12. + 83 = 180 – 32
Write number sentences to solve.
1. Jeremy had 12 boxes with 6 eggs in each. How many eggs in total?
2. Scarlet wrote a poem of 8 lines with 9 words in each line. How many words altogether?
3. Ben collected 15 football cards. Cruz has 6 times more cards than Ben. How many cards does Cruz have?
4. Each classroom shelf holds 7 books. If the teacher puts 49 books away, how many shelves has he filled?
5. The chef made 54 grams of meringue mix. How many meringues can she make if each one is 6 grams?
6. Maggie completes 28 pieces of a puzzle on Sunday and 32 on Monday. She still has 10 times as many pieces left. How many pieces has she got to go?
Can you think of more than one way to solve each problem?
Write your own word problem for:
1. 110 ÷ 11 = 10
2. 6 × 32 = 192

Extended practiceStretch your thinking

Li has a total of 106 green, red and blue marbles. How many of each colour might he have? Show 3 different options.

Option 1 Option 2 Option 3
Marley has 48 cookies. Show different ways she could share them equally with her friends.

Working-out space
Enrica has $75. Show some different combinations of notes and coins that she could have.

Working-out space

Extra practiceMore questions to build fluency

Change each word problem into a number sentence, then solve.
1. A number added to 27 gives 64. → answer
2. When 18 is subtracted from a number, the answer is 45. → answer
3. A number multiplied by 6 gives 54. → answer
4. A number divided by 7 gives 8. → answer
Fill in the gaps.
1. 45 + = 30 + 40
2. − 26 = 12 × 3
3. 9 × = 100 − 28
4. 144 ÷ = 6 × 2
Solve these two-step problems.
1. A box holds 24 pencils. A school buys 15 boxes and gives 80 pencils away. How many are left?
2. Ravi had $75. He bought 4 shirts at $12 each. How much is left?
3. A bus carries 52 people. Three buses are full and 17 people walk. How many people in total?
Which operation do these words tell you to use? Write + , − , × or ÷ .
1. altogether
2. shared equally
3. how many more
4. groups of
5. difference
6. total
Check each answer by doing the opposite. Tick the correct ones.
1. 96 ÷ 8 = 12
2. 235 − 87 = 158
3. 17 × 6 = 102
A cinema has 18 rows of 24 seats. 63 seats are broken.
1. How many seats in total?
2. How many seats can be used?
3. If tickets are $9, how much is taken when every usable seat is sold?
Write your own two-step word problem whose answer is 36. Then write the number sentence that solves it.

Answers · Topic 2 Problem solving

Guided practice

1a 15 + 21 = 48 – 12 b 42 + 16 = 31 + 27 c 73 – 24 = 26 + 23

Independent practice

1a 100 – 42 = 31 + 27 b 56 + 31 = 108 – 21 c 98 + 30 = 200 – 72
d 43 + 54 = 72 + 25 (25 had burgers) e 97 – 18 = 61 + 18 (18 didn’t have cake)

2 a 15 b 14 c 84 d 54 e 6 f 8 g 7 h 55 i 40 j 13 k 36 l 65

3 a 12 × 6 = 72 b 8 × 9 = 72 c 15 × 6 = 90 d 49 ÷ 7 = 7 e 54 ÷ 6 = 9 f (28 + 32) × 10 = 600

These are the most likely responses; accept any response that shows an understanding of what the question requires.

4 a & b Teacher to check. Look for students who demonstrate an understanding of the relationship between word problems and number sentences by being able to write scenarios that fit the given equations.

Extended practice

1 There are multiple answers possible — e.g. 40 green, 40 red and 26 blue; 100 green, 3 red and 3 blue; or 35 green, 35 red and 36 blue.

2 Possible answers: 1 each for 48 people, 2 each for 24 people, 3 each for 16 people, 4 each for 12 people, 6 each for 8 people, 8 each for 6 people, 12 each for 4 people, 16 each for 3 people.

3 Teacher to check — there are multiple possible answers for this question.

Extra practice

1a 27 + ☐ = 64, answer 37 b ☐ − 18 = 45, answer 63 c ☐ × 6 = 54, answer 9 d ☐ ÷ 7 = 8, answer 56

2a 25 b 62 c 8 d 12

3a 280 pencils (15 × 24 = 360, then − 80) b $27 c 173 people (3 × 52 = 156, then + 17)

4a + b ÷ c − d × e − f +

5 Tick a and c. b is wrong: 235 − 87 = 148, and the check 148 + 87 = 235 confirms it.

6a 432 seats b 369 seats c $3321

7 Teacher to check — any sensible two-step problem, for example "A box holds 6 eggs. I buy 8 boxes and drop 12 eggs. How many are left?" with 8 × 6 − 12 = 36.

Unit 5

Using units of measurement

Millimetres to metres, square centimetres to square metres, millilitres to litres, grams to kilograms, degrees, seconds and centuries — choose the right unit and use it well.

Measurement & Geometry · 7 topics

Unit 5 · Topic 1

Length and perimeter

10 millimetres = 1 centimetre · 10 mm = 1 cm 100 centimetres = 1 metre · 100 cm = 1 m

Perimeter is the distance around the edges of a shape.

If the spider crawled all the way around the rectangle, it would have gone around the perimeter.

Guided practiceWorked through with you

Find the length of each worm in mm. (Measure the pictures in your printed book, or use the lengths given in the answers to check.)
1. Worm a mm
2. Worm b mm
3. Worm c mm
4. Worm d mm
Use the worms in question 1 to work out:
1. how much longer b is than a.
2. how much longer c is than d.
3. the combined length of b and c.
Find the length of each snake in cm.
1. Snake a cm
2. Snake b cm
3. Snake c cm

Independent practiceYour turn

Would you use mm, cm or m to measure these items in real life?
1. A pencil
2. A classroom
3. A book
4. A safety pin
5. A swimming pool
6. A grain of rice
How many mm in:
1. 2 cm?
2. 10 cm?
3. 512 cm?
4. 23 cm?
5. 2.5 cm?
6. 3.8 cm?
7. 38 cm?
8. 12 cm?
9. 1.2 cm?
How many cm in:
1. 2 m?
2. 10 m?
3. 512 m?
4. 1.25 m?
5. 3.5 m?
6. 4.75 m?
7. 30 mm?
8. 35 mm?
9. 100 mm?
How many m in:
1. 100 cm?
2. 500 cm?
3. 250 cm?
Estimate which shape in question 6 has the greatest perimeter.
Find the perimeter of each shape in cm.

a cm

b cm

c cm

d cm
Find the perimeter of each shape in mm.
1. Shape a mm
2. Shape b mm
3. Shape c mm
4. Shape d mm

Extended practiceStretch your thinking

Choose 2 objects in the classroom that you would measure in mm, then repeat for cm and m. Estimate first, then measure.

Unit Object Estimated length Actual length Difference

mm

cm

m
1. Which of your items was the longest?
2. Which of your items was the shortest?
3. What is the difference between the lengths of the two items you measured in mm?
4. What is the difference between the lengths of the two items you measured in cm?
5. What is the difference between the lengths of the two items you measured in m?
6. What is the difference between the lengths of your longest item and your shortest item?

Extra practiceMore questions to build fluency

Convert these lengths.
1. 4 cm = mm
2. 12 cm = mm
3. 70 mm = cm
4. 3 m = cm
5. 250 cm = m
6. 6.5 cm = mm
Which unit would you use — mm, cm or m?
1. the length of a pencil
2. the height of a door
3. the thickness of a coin
4. the length of a football pitch
Find the perimeter of each rectangle.
1. 8 cm by 5 cm
2. 12 cm by 7 cm
3. 20 m by 15 m
4. 9 cm by 9 cm
Find the perimeter of each shape.
1. A triangle with sides 6 cm, 8 cm and 10 cm
2. A regular pentagon with sides of 7 cm
3. A regular hexagon with sides of 40 mm, in cm
Work backwards.
1. A rectangle has a perimeter of 30 cm and is 9 cm long. How wide is it?
2. A square has a perimeter of 48 cm. How long is each side?
A garden bed is 4 m long and 250 cm wide.
1. Write both measurements in metres.
2. What is the perimeter in metres?
3. Fencing costs $8 per metre. What is the total cost?
Two rectangles both have a perimeter of 20 cm but they are not the same shape. Give the sides of two such rectangles.

Answers · Topic 1 Length and perimeter

Guided practice

1 a 8 mm b 25 mm c 43 mm d 37 mm

2 a 17 mm (or 1 cm and 7 mm) b 6 mm c 68 mm (or 6 cm and 8 mm)

3 a 13 cm b 5 cm c 9 cm

Independent practice

1 a cm b m c cm d mm e m f mm

These are the most likely answers. Accept alternatives if students can offer adequate justification — e.g. “I would measure the safety pin in centimetres using decimals.”

2 a 20 mm b 100 mm c 55 mm d 230 mm e 25 mm f 38 mm g 380 mm h 120 mm i 12 mm

3 a 200 cm b 1000 cm c 550 cm d 125 cm e 350 cm f 475 cm g 3 cm h 3.5 cm i 10 cm

4 a 1 m b 5 m c 2.5 m

5 Teacher to check. Look for an appropriate rationale using the language of length.

6 a 20 cm b 18 cm c 20 cm d 23 cm

7 a 80 mm b 75 mm c 120 mm d 168 mm

Due to the small size of the unit, allow for slight variations in results.

Extended practice

1 a–k Teacher to check. Look for students who can match appropriate units of measurement to the items they choose, make reasonable estimates, measure accurately, and convert units to find the difference between their shortest and longest items.

Extra practice

1a 40 b 120 c 7 d 300 e 2.5 f 65

2a cm b m (or cm) c mm d m

3a 26 cm b 38 cm c 70 m d 36 cm

4a 24 cm b 35 cm c 24 cm (6 × 4 cm)

5a 6 cm — the two lengths use 18 cm, leaving 12 cm for the two widths b 12 cm

6a 4 m by 2.5 m b 13 m c $104

7 Any two different pairs whose sides add to 10, for example 1 cm by 9 cm and 4 cm by 6 cm. (Also 2 by 8, 3 by 7, 5 by 5.)

Unit 5 · Topic 2

Area

Square centimetres (cm²) are used to measure smaller areas. Square metres (m²) are used to measure larger areas.

A book cover: Area = 588 cm² A basketball court: Area = 420 m²

Does the book or the basketball court have the greater area?

Guided practiceWorked through with you

Match the items with their likely areas in real life.

600 cm²2 m²19 cm²465 m²81 cm²
1. Matchbox lid
2. Netball court
3. Smart phone
4. Chopping board
5. Table top

Independent practiceYour turn

Circle the unit you would use to measure the area of each item or place in real life.
1. A postcard cm²m²
2. A playground cm²m²
3. A page in a book cm²m²
4. A classroom floor cm²m²
Use grid paper to draw 4 different shapes with an area of 8 cm².

Describe your 4 shapes
Record the area of each shape in cm².
1. Shape a cm²
2. Shape b cm²
3. Shape c cm²
4. Shape d cm²
Choose 2 places in the school that you could measure in square metres. Estimate the area of each place, then measure and record the actual area.

Place Estimated area Actual area

Extended practiceStretch your thinking

A quick way to find the area of a rectangle is to multiply the length by the width. Find the area of each rectangle.
1. 4 cm × 4 cm = cm²
2. 8 cm × 2 cm = cm²
3. 4 cm × 10 cm = cm²
4. 3 cm × 5 cm = cm²
Draw a shape that:
1. is 4 cm wide and has an area of 12 cm².
  Describe or sketch it
2. has an area of 16 cm² and one side that is 5 cm long.
  Describe or sketch it

Extra practiceMore questions to build fluency

Find the area of each rectangle.
1. 6 cm by 4 cm
2. 9 cm by 7 cm
3. 12 m by 5 m
4. 8 cm by 8 cm
Would you measure these in cm² or m²?
1. a postage stamp
2. a classroom floor
3. a phone screen
4. a tennis court
Work backwards.
1. A rectangle has an area of 24 cm² and is 6 cm long. How wide is it?
2. A square has an area of 49 cm². How long is each side?
3. A rectangle has an area of 60 m² and is 12 m wide. How long is it?
Find the area of each compound shape, made from two rectangles.
1. A 10 cm by 4 cm rectangle joined to a 5 cm by 3 cm rectangle
2. A 7 m by 6 m rectangle with a 2 m by 3 m rectangle cut out of it
Write the sides of three different rectangles with an area of 36 cm².
A wall is 5 m long and 3 m high. Tiles are 1 m².
1. What is the area of the wall?
2. How many tiles are needed?
3. Tiles come in packs of 4. How many packs?
Two rectangles both have a perimeter of 16 cm. One is 2 cm by 6 cm and the other is 4 cm by 4 cm. Do they have the same area? What does this tell you?

Answers · Topic 2 Area

Guided practice

1 matchbox lid 19 cm² · netball court 465 m² · smart phone 81 cm² · chopping board 600 cm² · table top 2 m²

Independent practice

1 a cm² b m² c cm² d m²

2 Teacher to check. Look for students who demonstrate fluency with the concept of area by being able to draw 4 different shapes with the same area.

3 a 24 cm² b 18 cm² c 16 cm² d 12½ cm²

4 Teacher to check. Look for the ability to choose areas for which square metres are an appropriate unit, and to make a reasonable calculation of chosen areas.

Extended practice

1 a 16 cm² b 8 cm × 2 cm = 16 cm² c 4 cm × 10 cm = 40 cm² d 3 cm × 5 cm = 15 cm²

2 Teacher to check. Look for students who demonstrate an understanding of area by being able to draw shapes that meet the given specifications.

Extra practice

1a 24 cm² b 63 cm² c 60 m² d 64 cm²

2a cm² b m² c cm² d m²

3a 4 cm b 7 cm c 5 m

4a 55 cm² (40 + 15) b 36 m² (42 − 6)

5 Any three of: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6.

6a 15 m² b 15 tiles c 4 packs (3 packs give only 12)

7 No — 2 × 6 = 12 cm² but 4 × 4 = 16 cm². Shapes with the same perimeter can have very different areas, and the squarer the rectangle the bigger its area.

Unit 5 · Topic 3

Volume and capacity

1 cubic centimetre = 1 cm³ 5 mL = 5 millilitres 5 L = 5 litres

There are 1000 millilitres in 1 litre.

Guided practiceWorked through with you

Write the volume of each object in cm³.
1. Object a cm³
2. Object b cm³
3. Object c cm³
Which has the greatest volume?
Colour the containers to show:
1. 3 L on the 5 L jug
2. 100 mL on the 400 mL jug
3. 350 mL on the 400 mL jug
Where would each level sit on the scale?
Which container has the smallest capacity?

Independent practiceYour turn

1. Make and draw a cube with a volume of 8 cm³.
  Describe your cube
2. How many layers?
3. How many cm³ in each layer?
1. Make and draw a cube with a volume of 27 cm³.
  Describe your cube
2. How many layers?
3. How many cm³ in each layer?
Estimate the volume of each object in cm³.

Your estimates
Match the measuring jugs with the containers that filled them.

Your matches
Match the measuring jugs (A–E, each 5 L) with the items you think filled them.

Your matches

Which of the items do you think has a capacity closest to your drink bottle?
1. Which of the containers in question 5 have a capacity of less than 1 litre?
2. What is the capacity of the largest container?
3. How much larger is the capacity of the largest container than the smallest?

Extended practiceStretch your thinking

Each of these jugs had 1 litre of water in it before a rock was put in. Order the rocks from smallest to largest based on the water they have displaced.

Jug A B C D E

Water level 1500 mL 1750 mL 1100 mL 1250 mL 1600 mL

Levels redrawn for this edition — the order matches the printed answer key.

Smallest volume Largest volume
Rewrite as millilitres and litres.
1. 1400 mL = litre millilitres
2. 2500 mL = litres millilitres
3. 3859 mL =
4. 7643 mL =
Rewrite in millilitres.
1. 3 litres 25 millilitres = mL
2. 5 litres 340 millilitres = mL
3. 7 litres 654 millilitres = mL
4. 19 litres 999 millilitres = mL

Extra practiceMore questions to build fluency

Convert these capacities.
1. 3 L = mL
2. 2500 mL = L
3. 0.5 L = mL
4. 7500 mL = L
5. 1.25 L = mL
6. 400 mL = L
Find the volume of each cuboid built from 1 cm³ cubes.
1. 3 by 2 by 4
2. 5 by 5 by 2
3. 6 by 3 by 3
4. 10 by 4 by 2
Would you measure these in mL or L?
1. a teaspoon of medicine
2. a bucket of water
3. a carton of juice
4. a bath
Add or subtract these capacities.
1. 1 L 250 mL + 750 mL =
2. 3 L − 1200 mL =
3. 500 mL × 6 =
A jug holds 2 L. Cups hold 250 mL each.
1. How many cups can be filled from the full jug?
2. After filling 5 cups, how much is left in the jug?
A box is 8 cm long, 5 cm wide and 3 cm high.
1. What is its volume?
2. How many 1 cm³ cubes would fill it?
3. If the height doubled, what would the new volume be?
Two containers have the same capacity but very different shapes. Is that possible? Explain.

Answers · Topic 3 Volume and capacity

Guided practice

1 a 6 cm³ b 12 cm³ c 16 cm³ 2 c 3 Teacher to check — look for students who can accurately mark the correct level on the scale and interpret both litre and millilitre measurements. 4 b

Independent practice

1 a Teacher to check b 2 layers c 4 cm³ in each layer

2 a Teacher to check b 3 layers c 9 cm³ in each layer

3 a–c Teacher to check. Look for the ability to make a rectangular prism with the same number of cubic centimetres in each layer.

4 & 5 Teacher to check the matching.

6 a A & C b 4 litres (4 L) c 3 litres 700 millilitres (3.7 L)

Extended practice

1 C, D, A, E, B

2 a 1 litre 400 millilitres b 2 litres 500 millilitres c 3 litres 859 millilitres d 7 litres 643 millilitres

3 a 3025 mL b 5340 mL c 7654 mL d 19 999 mL

Extra practice

1a 3000 mL b 2.5 L c 500 mL d 7.5 L e 1250 mL f 0.4 L

2a 24 cm³ b 50 cm³ c 54 cm³ d 80 cm³

3a mL b L c mL (or L) d L

4a 2 L b 1 L 800 mL (1800 mL) c 3 L (3000 mL)

5a 8 cups b 750 mL

6a 120 cm³ b 120 cubes c 240 cm³

7 Yes. Capacity is how much a container holds, not how tall or wide it is. A tall narrow bottle and a short wide bowl can both hold exactly 1 litre — the only way to be sure is to fill one and pour it into the other.

Unit 5 · Topic 4

Mass

Mass = 1.25 kg, or 1 kg and 250 g Mass = ½ kg, or 500 g

Which of the 2 items has the smaller mass?

Guided practiceWorked through with you

Write the mass shown on each dial in 2 ways.

Scale As a decimal (kg) As kg and g

a

b

c

d

e

f

Independent practiceYour turn

You will need a set of scales. Choose a classroom item for each category, estimate its mass, then measure it.

Category Item Estimated mass Actual mass Difference

About 500 g

About 1 kg

About 2 kg

More than 2 kg
1. Which of your items has the greatest mass?
2. Which has the smallest mass?
3. What is the difference between the mass of the heaviest and lightest items?
4. What is the total mass of your items?
5. Write the mass of your heaviest item in two different ways.
6. Write the mass of your lightest item in two different ways.
Show where the arrow would point on a 5 kg dial for each mass.
1. 3.3 kg
2. 900 g
3. 1.6 kg
4. 1 kg 200 g
5. 3.7 kg
6. 0.75 kg
How much heavier is the heaviest item than the lightest?
Use the scales in question 2 to work out the mass of:
1. 1 phone book.
2. 1 banana.
3. 1 cricket ball.
4. 1 cake.
5. 1 pumpkin.
6. 1 remote control.

Extended practiceStretch your thinking

Complete the table.

kg kg and g g

1.7 kg 1 kg 700 g

4 kg 500 g

314 kg

620 g

7 kg 750 g

5.03 kg

Here are the ingredients for blueberry muffins.

Ingredient	Mass	Mass in grams
Blueberries	0.125 kg	g
Eggs (a dozen)	0.84 kg	g
Flour	2 kg	g
Sugar	112 kg	g
Milk	1.65 kg	g
Butter	14 kg	g

How much more is the mass of the eggs than the blueberries?
What is the total mass of the muffin ingredients (blueberries, eggs, flour, sugar and butter)?
The recipe for blueberry muffins only needs 2 eggs. What is their mass?

Extra practiceMore questions to build fluency

Convert these masses.
1. 2 kg = g
2. 3500 g = kg
3. 0.25 kg = g
4. 1.75 kg = g
5. 500 g = kg
6. 4200 g = kg
Write each mass in two ways, as in the Learn it panel.
1. 1500 g = kg =
2. 2250 g = kg =
Would you measure these in g or kg?
1. an apple
2. a bag of rice
3. a feather
4. a person
Add or subtract these masses.
1. 1 kg 400 g + 800 g =
2. 5 kg − 1750 g =
3. 250 g × 8 =
A muffin recipe needs 300 g of flour and makes 12 muffins.
1. How much flour for 24 muffins?
2. How much for 6 muffins?
3. How many muffins could you make with 1.5 kg of flour?
A parcel weighs 2.4 kg. Postage costs $3 per kilogram, rounded up to the next whole kilogram.
1. How many kilograms will be charged?
2. What is the cost?
Order these from lightest to heaviest: 1.2 kg, 950 g, 1 kg 50 g, 890 g.

Answers · Topic 4 Mass

Guided practice

1 a 1.3 kg, 1 kg and 300 g b 3.2 kg, 3 kg and 200 g c 2.5 kg, 2 kg and 500 g (2½ kg) d 5.5 kg, 5 kg and 500 g (5½ kg) e 4.2 kg, 4 kg and 200 g f 26.7 kg, 26 kg and 700 g

Independent practice

1 a–j Teacher to check. Look for reasonable estimates of the masses of familiar objects, and fluency with recording and calculating with masses.

2 Teacher to check the arrow positions.

3 a 1.1 kg b 150 g c 160 g d 600 g e 1.85 kg f 150 g

Accept equivalents — e.g. 1100 g for 1.1 kg.

Extended practice

1 1.7 kg = 1 kg 700 g = 1700 g · 4.5 kg (4½ kg) = 4 kg 500 g = 4500 g · 3¼ kg = 3 kg 250 g = 3250 g · 0.62 kg = 0 kg 620 g = 620 g · 7.75 kg (7¾ kg) = 7 kg 750 g = 7750 g · 5.03 kg = 5 kg 30 g = 5030 g

2a 125 g · 840 g · 2000 g · 1500 g · 1650 g · 250 g

2b 715 g c 4715 g, 4 kg 715 g or 4.715 kg d 140 g (0.14 kg)

Extra practice

1a 2000 g b 3.5 kg c 250 g d 1750 g e 0.5 kg f 4.2 kg

2a 1.5 kg = 1 kg and 500 g b 2.25 kg = 2 kg and 250 g

3a g b kg c g d kg

4a 2 kg 200 g (2200 g) b 3 kg 250 g (3250 g) c 2 kg (2000 g)

5a 600 g b 150 g c 60 muffins

6a 3 kg b $9

7 890 g, 950 g, 1 kg 50 g (1050 g), 1.2 kg (1200 g)

Unit 5 · Topic 5

Temperature

Temperature can be measured in degrees Celsius (°C).

Can you think of something warmer than the margarine but colder than the soup?

Guided practiceWorked through with you

Record the temperature shown on each thermometer.
1. Thermometer a °C
2. Thermometer b °C
3. Thermometer c °C
4. Thermometer d °C
5. Thermometer e °C
6. Thermometer f °C

Independent practiceYour turn

Mark these temperatures on the thermometers: 7 °C, 10 °C, 35 °C, 36 °C, 49 °C and 74 °C.

Where does each one sit on the scale?
1. Which temperature in question 1 is the highest?
2. Which is the lowest?
3. What is the difference between the highest and lowest temperatures?
4. Which 2 temperatures have a difference of exactly 25 °C?
5. Which 2 temperatures have the smallest difference?
6. Which temperature might be the maximum for a winter’s day where you live?
7. Which temperature might be the maximum for a summer’s day where you live?
Tick the colder item or place in each pair.
1. Snow sceneDesert
2. Glass of waterCup of tea
3. Cupcake out of the ovenOven
4. Person in the shadePerson in the sun
Do you know what the temperature of the human body is?
Choose an adjective to describe each temperature: icy · cold · cool · warm · hot · boiling.
1. A hot summer’s day
2. Ice cubes
3. Bath water
4. A cloudy winter morning

Extended practiceStretch your thinking

Match the items with their likely temperatures.

2 °C42 °C12 °C100 °C65 °C
1. Boiling kettle
2. Hot bath
3. Cup of tea
4. Cold winter morning
5. Inside a fridge
Use a thermometer to find the temperature of the places listed, plus 2 more at school. Rank them from 1 (hottest) to 4 (coldest).

Place Temperature Ranking

Classroom

Playground
1. What is the difference in temperature between the coldest and the hottest place you measured?
2. Imagine the forecast for today is 25 °C. By how much is your classroom hotter or colder than the forecast?
3. By how much is the playground hotter or colder than the forecast?

Extra practiceMore questions to build fluency

Write a sensible temperature in °C for each.
1. ice
2. a hot summer day
3. boiling water
4. a fridge
Which is colder? Write the colder one.
1. 12 °C or 21 °C
2. 0 °C or 5 °C
3. −3 °C or 2 °C
4. −8 °C or −1 °C
Work out the change in temperature.
1. From 8 °C to 19 °C
2. From 25 °C to 13 °C
3. From −4 °C to 6 °C
4. From 3 °C to −5 °C
Order these temperatures from coldest to warmest.

7 °C −2 °C 15 °C 0 °C −9 °C
A week of midday temperatures: 14, 17, 12, 19, 16, 21, 15 °C.
1. What was the highest?
2. What was the lowest?
3. What is the difference between them?
The temperature at 6 am was 4 °C. By midday it had risen 11 degrees. By 9 pm it had fallen 8 degrees from the midday reading.
1. What was the midday temperature?
2. What was the temperature at 9 pm?
Why does a thermometer need numbers below zero?

Answers · Topic 5 Temperature

Guided practice

1 a 30 °C b 60 °C c 0 °C d 44 °C e 89 °C f 100 °C

Independent practice

1 Teacher to check the marked positions.

2 a 74 °C b 7 °C c 67 °C d 10 °C and 35 °C e 35 °C and 36 °C
f–g Answers will vary depending on the students’ location. Likely answers are f 7 °C and 10 °C, g 35 °C, 36 °C and 49 °C.

3 Circled: a Snow scene b Glass of water c Cupcake d Person in shade

4 The most likely answers are a hot b freezing c warm or hot d cold or cool.

Answers may vary depending on students’ perceptions. This can be the basis for a discussion on how a particular temperature may be considered hot in one context but warm in another.

Extended practice

1 Boiling kettle 100 °C · Hot bath 42 °C · Cup of tea 65 °C · Cold winter morning 12 °C · Inside a fridge 2 °C

2 a–f Teacher to check. Look for the ability to accurately measure and record temperature and to understand how thermometers are used to compare places.

Extra practice

1 Sensible answers: a 0 °C b about 30–35 °C c 100 °C d about 4 °C

2a 12 °C b 0 °C c −3 °C d −8 °C

3a up 11 degrees b down 12 degrees c up 10 degrees d down 8 degrees

4 −9 °C, −2 °C, 0 °C, 7 °C, 15 °C

5a 21 °C b 12 °C c 9 degrees

6a 15 °C b 7 °C

7 Because temperatures can fall below the freezing point of water, which is marked as zero. Without negative numbers there would be no way to record how much colder than freezing it is — −5 °C and −15 °C are both "below zero" but very different.

Unit 5 · Topic 6

Time

To change minutes to seconds, multiply by 60. 5 minutes = 5 × 60 = 300 seconds To change hours to minutes, multiply by 60. 10 hours = 10 × 60 = 600 minutes

How would you change minutes to hours?

Guided practiceWorked through with you

How many:
1. seconds in 1 minute?
2. minutes in 1 hour?
3. hours in 1 day?
4. days in 1 week?
5. days in 1 year?
6. weeks in 1 year?
Fill in the gaps.
1. 2 minutes = seconds
2. 6 minutes = seconds
3. 3 hours = minutes
4. 5 hours = minutes
5. 112 minutes = seconds
6. 212 hours = minutes
7. 48 hours = days
8. 3 days = hours
9. 49 days = weeks
10. 5 weeks = days

Independent practiceYour turn

Below are the race times for 6 students from a class in Year 4. Complete the times in the table, then rank the students from fastest (1) to slowest (6).

Name	Time in seconds	Time in minutes and seconds
Todd	75 seconds
Harper		2 mins 20 seconds
Jessica		1 min 40 seconds
Mario	90 seconds
Stirling	120 seconds
Anthony		1 min 10 seconds

Tick the longer time period in each pair.
1. 3 weeks27 days
2. 97 minutes2 hours
3. 700 days2 years
4. 660 minutes10½ hours
5. 3 days70 hours
6. 10 years4000 days
7. 3½ hours200 mins
8. 1 hour400 seconds
How many:
1. days in 5 weeks?
2. minutes in 5 hours?
3. seconds in 5 minutes?
4. months in 5 years?
5. days in 2 years?
6. hours in 2 days?
We use am for times before midday and pm for times after midday. Write am or pm for each description.
1. School starts at 9
2. School ends at 3:15
3. Lunch is at 1
4. The mail arrived at 11
5. I went to bed at 9:30
6. An owl woke me up at 2
Rewrite the times in question 4 from earliest to latest in the day.

Is midnight am or pm?
Write each time as an am or pm time.
1. 1 minute to 7 in the morning
2. 26 past 8 in the evening
3. 10 minutes past midnight
4. 47 minutes past midday

Extended practiceStretch your thinking

Use the cinema timetable to answer the questions.

Movie	Length	Morning session	Afternoon session	Evening session
Marshmallow Attack	90 mins	10:00 am	1:35 pm	8:15 pm
My Mother the Plumber	83 mins	11:15 am	2:00 pm	9:00 pm
Cop Capers	92 mins	9:45 am	12:30 pm	7:20 pm
Cakes on a Train	76 mins	10:30 am	1:45 pm	6:40 pm

What time does the morning session of Marshmallow Attack finish?
How much longer is Cop Capers than Cakes on a Train?
How much later is the evening session of Cop Capers than the morning session?
Which movie will finish at 3:23 pm?
What time does the afternoon session of Cakes on a Train end?
Which movie is longer than 1½ hours?
Will the afternoon session of Marshmallow Attack or Cakes on a Train finish earlier?
Show the start and finish times for the evening session of Cop Capers. Start Finish

Extra practiceMore questions to build fluency

Convert these times.
1. 4 minutes = seconds
2. 3 hours = minutes
3. 180 seconds = minutes
4. 240 minutes = hours
5. 2 days = hours
6. 1 week = days
Write each as an am or pm time.
1. 14:00
2. 07:30
3. 19:45
4. 00:15
How long between these times?
1. 9:15 am to 11:45 am
2. 1:30 pm to 4:15 pm
3. 10:40 am to 1:10 pm
4. 8:50 pm to 11:05 pm
Tick the longer time period in each pair.
1. 90 minutes 1½ hours
2. 200 seconds 4 minutes
3. 50 hours 2 days
A film starts at 6:40 pm and lasts 1 hour 55 minutes.
1. What time does it finish?
2. How many minutes long is it?
A train leaves at 08:25 and arrives at 11:10.
1. How long is the journey?
2. If it leaves 20 minutes late but still takes the same time, when does it arrive?
Ana says 2.5 hours is the same as 2 hours 50 minutes. Explain her mistake and give the correct time.

Answers · Topic 6 Time

Guided practice

1 a 60 b 60 c 24 d 7 e 365 (or 366) f 52

2 a 120 b 360 c 180 d 300 e 90 f 150 g 2 h 72 i 7 j 35

Independent practice

1 Todd 75 s = 1 min 15 secs, rank 2 · Harper 140 s = 2 mins 20 secs, rank 6 · Jessica 100 s = 1 min 40 secs, rank 4 · Mario 90 s = 1 min 30 secs, rank 3 · Stirling 120 s = 2 mins, rank 5 · Anthony 70 s = 1 min 10 secs, rank 1

2 a 27 days b 2 hours c 2 years d 660 minutes e 3 days f 4000 days g 3½ hours h 1 hour

3 a 35 b 300 c 300 d 60 e 730 (or 731) f 48

4 a am b pm c pm d am e pm f am

5 2 am, 9 am, 11 am, 1 pm, 3:15 pm, 9 pm

6 a 6:59 am b 8:26 pm c 12:10 am d 12:47 pm

Extended practice

1 a 11:30 am b 16 minutes c 9 hours 35 minutes d My Mother the Plumber e 3:01 pm f Cop Capers g Cakes on a Train h start 7:20 pm, finish 8:52 pm

Extra practice

1a 240 b 180 c 3 d 4 e 48 f 7

2a 2:00 pm b 7:30 am c 7:45 pm d 12:15 am

3a 2 h 30 min b 2 h 45 min c 2 h 30 min d 2 h 15 min

4a They are equal — 1½ hours is 90 minutes. b 4 minutes (240 s) c 50 hours (2 days is 48 hours)

5a 8:35 pm b 115 minutes

6a 2 h 45 min b 11:30

7 The .5 in 2.5 hours means half an hour, not 50 minutes. Half an hour is 30 minutes, so 2.5 hours = 2 hours 30 minutes. Decimals of an hour are out of 60, not out of 100.

Unit 5 · Topic 7

Timelines

A timeline shows the order in which events occurred over a particular period of time. It could be a timeline for an hour, a day or a thousand years.

The first two years of Audrey’s life.

Periods of time need to be spaced evenly on a timeline.

Guided practiceWorked through with you

Look at the timeline above.
1. Did Audrey start to walk before or after she said her first word?
2. How old was Audrey when she got her first tooth?
3. For how long did Audrey suck her thumb?
Imagine the timeline now runs to 5 years. Where would you add each event?
1. Audrey broke her arm when she was three and a half.
2. Just before she turned five, Audrey started school.
3. A year and a half after Audrey started to walk, she learned to swim.

Independent practiceYour turn

This timeline spans one year, from 1 January to 1 December. Use it to complete the activities.
1. Write the first of each month on the timeline.
2. Tran’s birthday is on 7 April. His friend Ben has a birthday exactly three months after. When is Ben’s birthday?
3. Tran’s first school day of the year is towards the end of January. Add this to the timeline.
4. Estimate the date of the music festival.
5. The school play is two months before December 25. When is it?
6. On New Year’s Eve, Tran and his family watch a fireworks show. When is this?
7. On 15 May, Tran received a special award at school. Add this to the timeline.
Samira created this timeline after an excursion to a wildlife park. It runs from 9 am to 3 pm. She left school, arrived at the park, and left the wildlife park.
1. At what time did Samira and her class leave school?
2. How long did it take to get to the wildlife park?
3. They ate lunch at 12:30 pm. Add this to the timeline.
4. At what time did they all leave the wildlife park?
5. In the morning, Samira saw wombats followed by koalas. Add this to the timeline.
6. The class went to the gift shop one hour before they left. When was this?
This timeline of Australian history spans over 200 years, from 1780 to 2000. Place these events on the timeline by writing the letter in order, earliest first.
1. 1851: Gold was discovered in Australia (A)
2. 1956: The Olympic Games were held in Melbourne (B)
3. 1977: The flag of the indigenous people of Australia was first flown (C)
4. 1788: White settlement of Australia occurred (D)
5. 1967: Indigenous people were allowed to become Australian citizens (E)
6. 2000: The Olympic Games were held in Sydney (F)
7. 1901: Australia became a nation (Federation) (G)
8. 2008: The Australian Government said sorry to the indigenous people (H)
9. 1817: Governor Macquarie recommended changing the name from New Holland to Australia (I)
Order, from left to right:

Extended practiceStretch your thinking

William got his first tooth when he was 3 months old. He started to talk at 18 months old. He learned to swim on his third birthday and rode a two-wheeler bike when he was four and a half. He started school just after his fifth birthday.

Timeline A spaces the events out evenly. Timeline B uses an even scale of time, so the gaps match how long each period really was.

Look at the timelines and answer the following questions.
1. What is the problem with the position of the events on Timeline A?
2. In what way does Timeline B represent the information more accurately?
3. Why is it important to have a scale for Timeline B?
Make a timeline of some important events in your life. Begin by deciding on a suitable scale.

My timeline

Extra practiceMore questions to build fluency

A timeline runs from 2015 to 2025, with a mark every year.
1. How many marks are there altogether?
2. How many years does the timeline cover?
3. Which year is exactly halfway?
Work out how long ago each event was, if this year is 2025.
1. 2010
2. 1998
3. 1975
4. 1900
Put these events in order on a timeline, earliest first.

Started school (2019) · Born (2014) · Learned to ride a bike (2020) · First tooth (2015) · Moved house (2022)
A timeline of a single day runs from 6 am to 6 pm.
1. How many hours does it cover?
2. If each hour is 1 cm, how long is the timeline?
3. Where would midday sit?
Use this school timeline to answer: 1998 school opened · 2005 library built · 2012 hall built · 2020 new playground.
1. How many years between the library and the hall?
2. How old was the school when the playground was built?
3. Which gap was the longest?
Why must the spaces on a timeline be evenly sized?
Design a timeline for your own week, with one mark per day. Write two events and which day each belongs to.

Answers · Topic 7 Timelines

Guided practice

1 a after b 6 months c 2 years

2 a Students to add a label, e.g. “I broke my arm”, pointing to three and a half years.
b Students to add a label, e.g. “I started school”, pointing to just before 5 years.
c Students to add a label, e.g. “I learned to swim”, and an arrow just after two and a half years.

Independent practice

1 Teacher to check the completed timeline. Ben’s birthday is 7 July; the school play is 25 October; the fireworks are on 31 December.

2 a 9:30 am b 30 minutes c Students to add a label, e.g. “Lunch”, at 12:30 pm d Any time around 2:45 pm e Students to add “Wombats” and “Koalas” in the first and second boxes respectively f Students to add “Gift shop” before 2 pm

3 From left to right: D, I, A, G, B, E, C, F, H

Extended practice

1a The arrows are spread out evenly, but the time gaps are not all the same.
b It is easier to tell the length of time between each event on the timeline.
c If there were no scale, we would not be able to tell the length of time between each event.

2 This task could be as simple or as complex as desired. Students could, for example, be encouraged to make a digital display of the timeline including photographs.

Extra practice

1a 11 marks b 10 years c 2020

2a 15 years b 27 years c 50 years d 125 years

3 Born (2014), First tooth (2015), Started school (2019), Learned to ride a bike (2020), Moved house (2022)

4a 12 hours b 12 cm c exactly halfway, 6 cm along

5a 7 years b 22 years old c The gaps are 7 years (1998–2005), 7 years (2005–2012) and 8 years (2012–2020), so the longest is 2012 to 2020.

6 So that the distance along the line always means the same amount of time. If the spaces were uneven, a long gap could look short and you could not compare periods just by looking — which is the whole point of a timeline.

7 Teacher to check — any week timeline with 7 evenly spaced marks and events placed on the correct days.

Unit 6

Shape

Count the sides and angles, split shapes apart and put them back together, and learn to draw solid shapes and the views you get from the top, the front and the side.

Measurement & Geometry · 2 topics

Unit 6 · Topic 1

2D shapes

A square: 4 sides · 4 angles · regular

This shape: 4 sides · 4 angles · irregular

A regular shape has sides of the same length and angles of the same size. An irregular shape does not.

Guided practiceWorked through with you

Complete the table.

Shape name Sides Angles Sketch it

square 4 4

octagon

5 5

trapezium

kite

6

Independent practiceYour turn

Look at the big shape below, which is split into smaller shapes.
1. What is the big shape?
2. What shapes is it split into?
3. Draw a line to split the shape into a parallelogram and a triangle.
4. Draw a line to split the shape into 2 triangles.
1. Split a parallelogram into 3 shapes.
2. What shapes is it split into?
3. Draw a line to split the shape into 2 triangles.
1. Draw a line to show how this shape is made from 1 square and 1 triangle.
2. Use a different colour to show how it can be made into 3 triangles.
1. Draw a new shape that can be made from 4 right-angled triangles.
  Describe or sketch it
2. Name the shape.
Flipping or turning a shape around does not make it a different shape.
1. Draw a new shape that can be made from 1 rectangle and 2 equilateral triangles.
  Describe or sketch it
2. Name the shape.
1. Complete the table for the shapes A–E drawn on a centimetre grid.
  
  Shape Name Angles Area
  
  A Triangle
  
  B 15 cm²
  
  C
  
  D 4
  
  E
2. Which two shapes are similar?

Extended practiceStretch your thinking

1. Draw a regular shape with an area of 9 cm².
  Describe or sketch it
2. Name your shape.
3. What is the area of the hexagon?
4. Is it regular or irregular?
5. Divide the hexagon into 2 triangles and 1 rectangle.
6. What is the area of each triangle?
2D shapes can be used to construct 3D shapes. Which 2D shapes do you need to make:
1. a rectangular prism?
2. a pentagonal pyramid?
3. a cylinder?

Extra practiceMore questions to build fluency

Complete the table for each regular shape.

Shape Sides Angles

triangle

pentagon

hexagon

octagon
Write regular or irregular.
1. A square
2. A rectangle that is 8 cm by 3 cm
3. An equilateral triangle
4. A triangle with sides 3 cm, 4 cm and 5 cm
Name the quadrilateral from its properties.
1. 4 equal sides and 4 right angles
2. 2 pairs of parallel sides, no right angles, all sides equal
3. Exactly one pair of parallel sides
4. 2 pairs of equal adjacent sides, one line of symmetry
How many lines of symmetry?
1. square
2. rectangle
3. equilateral triangle
4. regular hexagon
True or false? Tick the true statements.
1. All squares are rectangles.
2. All rectangles are squares.
3. A regular shape has all sides and all angles equal.
4. A shape with 6 sides is always regular.
A large rectangle is split into 4 equal smaller rectangles.
1. If the large one is 12 cm by 8 cm and is cut into 4 equal strips across its length, what size is each strip?
2. What is the perimeter of one strip?
Can a triangle be regular and also have a right angle? Explain.

Answers · Topic 1 2D shapes

Guided practice

1 square 4 sides, 4 angles · octagon 8, 8 · pentagon 5, 5 · trapezium 4, 4 · kite 4, 4 · hexagon 6, 6

Students may draw different versions of certain shapes — e.g. an irregular pentagon rather than a regular one. This is acceptable if they show the correct properties. Alternative names are also acceptable — e.g. “quadrilateral” for kite.

Independent practice

1 a trapezium b 1 rectangle and 2 triangles c & d Teacher to check the lines drawn.

2 b 1 rectangle and 2 triangles (these are examples — students may choose a different way to split the shape; check that their description matches their diagram).

3 a & b Teacher to check.

4 Teacher to check. Look for the ability to combine the 4 triangles into a new polygon and accurately identify the new shape.

5 Teacher to check. Look for the ability to combine the rectangle and triangle into a new polygon and accurately identify it.

6a A Triangle, 3 angles, 8 cm² · B Rectangle, 4 angles, 15 cm² · C Hexagon, 6 angles, 20 cm² · D Parallelogram, 4 angles, 8 cm² · E Hexagon
6b The two hexagons (C and E).

Extended practice

1 a & b Teacher to check. The most likely answer is a 3 cm by 3 cm square.
c & d The hexagon is irregular — teacher to check the area from the diagram.
e Teacher to check the division. f 4 cm²

2 a 2 smaller rectangles and 4 larger rectangles b 1 pentagon and 5 triangles c 2 circles and 1 rectangle

Extra practice

1 triangle 3 and 3; pentagon 5 and 5; hexagon 6 and 6; octagon 8 and 8.

2a regular b irregular c regular d irregular

3a square b rhombus c trapezium d kite

4a 4 b 2 c 3 d 6

5 Tick a and c. Not all rectangles are squares, and a hexagon is only regular if all six sides and angles match.

6a 3 cm by 8 cm b 22 cm

7 No. In a regular triangle all three angles are equal, and the three angles of any triangle add to 180°, so each must be 60°. A right angle is 90°, so it cannot appear in a regular triangle.

Unit 6 · Topic 2

3D shapes

To draw prisms or pyramids:

Start with the bases.
Then draw lines to join the corners of the prism bases, or the pyramid point with the base corners.

Rectangular prism

Square pyramid

What is the difference between a prism and a pyramid?

Guided practiceWorked through with you

Join the corners to complete each prism, then name it.
1. Two rectangles joined at the corners
2. Two pentagons joined at the corners
Join the base corners to the point of each pyramid, then name it.
1. A triangular base with a point above it
2. A pentagonal base with a point above it

Independent practiceYour turn

Try drawing these 3D shapes on your own: a cube, a triangular prism, a cone and a cylinder.

Notes about your drawings
Complete the top, front and side views of the 3D shapes.

Shape Top view Front view Side view

a

b

c

To see the front and side views, it helps to view the shape at eye level.
Label the top, front and side views of the 3D shapes.
1. Shape 1: view view view
2. Shape 2: view view view
3. Shape 3: view view view
Draw top, front and side views for two more shapes.

Shape a

Shape b

Extended practiceStretch your thinking

Draw and name the 3D shapes with the properties described below.

Description	Drawing	Name
2 rectangular bases · 8 corners · 12 edges
1 triangular base · 4 corners · 6 edges
2 hexagonal bases · 12 corners · 18 edges

Make 2 different 3D shapes using 8 cubes. Draw each shape, then show the top, front and side views.

Shape 1 — top / front / side views

Shape 2 — top / front / side views

Extra practiceMore questions to build fluency

Complete the table.

Shape Faces Edges Vertices

cube

rectangular prism

square pyramid

triangular prism
Name the 3D shape from its description.
1. 2 triangular bases and 3 rectangular faces
2. 1 square base and 4 triangular faces
3. 2 circular faces and 1 curved surface
4. 6 identical square faces
What shape is the cross-section if you slice straight through:
1. a cylinder, parallel to its base
2. a cube, parallel to a face
3. a triangular prism, parallel to its base
Write prism or pyramid.
1. Two identical bases
2. One base and a point
3. Same cross-section all the way through
4. Faces meeting at an apex
A shape has 5 faces and 5 vertices. Name it and say how many edges it has.

Name Edges
Describe the top, front and side views of a cylinder standing upright.
1. Top
2. Front
3. Side
For a cube, add faces + vertices, then subtract edges. Try it again for a square pyramid. What do you notice?

Cube: Square pyramid:

Answers · Topic 2 3D shapes

Guided practice

1 a rectangular prism b pentagonal prism

2 a triangular pyramid b pentagonal pyramid

Independent practice

1 a–d Teacher to check. Look for the ability to draw the objects with a reasonable degree of accuracy and an understanding of their properties, such as the base shapes.

2 Teacher to check the completed views.

3 a front view · side view · top view
b side view · top view · front view
c front view · side view · top view

4 a & b Teacher to check.

Extended practice

1 2 rectangular bases, 8 corners, 12 edges → rectangular prism
1 triangular base, 4 corners, 6 edges → triangular pyramid
2 hexagonal bases, 12 corners, 18 edges → hexagonal prism

2 Teacher to check. Look for students’ ability to make a reasonable representation of their 2 objects and accurately draw front, top and side views.

Extra practice

1 cube 6, 12, 8; rectangular prism 6, 12, 8; square pyramid 5, 8, 5; triangular prism 5, 9, 6.

2a triangular prism b square pyramid c cylinder d cube

3a a circle b a square c a triangle

4a prism b pyramid c prism d pyramid

5 A square pyramid, with 8 edges.

6a a circle b a rectangle c a rectangle (the same as the front)

7 Cube: 6 + 8 − 12 = 2. Square pyramid: 5 + 5 − 8 = 2. The answer is always 2 for these solids — a famous result called Euler's rule. Try it on the triangular prism: 5 + 6 − 9 = 2 as well.

Unit 8

Location and transformation

Flip, slide and turn shapes to build symmetrical and tessellating patterns — then read maps, grid references and scales to find your way around.

Measurement & Geometry · 2 topics

Unit 8 · Topic 1

Symmetry

You can make symmetrical patterns by reflecting, translating or rotating.

Reflecting is flipping

Translating is sliding

Rotating is turning

Guided practiceWorked through with you

Finish the symmetrical patterns a, b and c in your book, then describe what you did.

Pattern a

Pattern b

Pattern c

Independent practiceYour turn

1. Colour the squares to make a symmetrical pattern using 3 colours.
2. Draw a line of symmetry on your pattern. Where is it?
1. Colour the shapes to make a pattern with 2 lines of symmetry.
  Describe your pattern
2. Draw in the lines of symmetry. Where are they?
1. Draw 4 lines of symmetry on the pattern in your book.
2. Circle the shape that shows reflection, translation and rotation.
Make a pattern by rotating the shape:
1. a 12 turn clockwise.
2. a 14 turn anticlockwise.
3. a 12 turn anticlockwise, then a 14 turn clockwise.
4. If you made a pattern by rotating a shape through a full turn, would it be the same as reflecting or translating the shape?
1. Make your own rotating pattern.
  Describe your pattern
2. Describe your pattern.

Extended practiceStretch your thinking

Shapes tessellate if they can be rotated, translated or reflected to fit together without any gaps. Squares tessellate by themselves. Regular pentagons do not tessellate by themselves.

Use diagrams to show which of these regular shapes tessellate by themselves.
1. Regular triangles — tessellates?
2. Regular octagons — tessellates?
3. Regular hexagons — tessellates?
Make a tessellating pattern that has at least 1 line of symmetry.

Describe your pattern

Extra practiceMore questions to build fluency

Name the transformation: reflecting, translating or rotating.
1. Sliding a shape 3 squares to the right
2. Flipping a shape over a mirror line
3. Turning a shape a quarter turn about a point
4. Moving a shape without turning or flipping it
How many lines of symmetry does each have?
1. a regular pentagon
2. a rhombus
3. an isosceles triangle
4. a parallelogram
5. a circle
6. a regular octagon
Which capital letters have a vertical line of symmetry? Tick them.
1. A
2. F
3. M
4. T
5. R
6. W
Which of these regular shapes tessellate by themselves? Write yes or no.
1. equilateral triangle
2. square
3. regular pentagon
4. regular hexagon
A shape has rotational symmetry of order 4.
1. How many times does it look the same in a full turn?
2. How many degrees between each match?
3. Name a shape with order 4.
Give the order of rotational symmetry for each.
1. equilateral triangle
2. rectangle
3. regular hexagon
4. the letter S
Why do regular pentagons leave gaps when you try to tessellate them, while regular hexagons do not?

Answers · Topic 1 Symmetry

Guided practice

1 a, b & c Teacher to check the completed symmetrical patterns.

Independent practice

1 a & b Teacher to check. Look for the ability to apply an understanding of symmetry to create a pattern with at least one identifiable line of symmetry.

2 a & b Teacher to check. Look for a pattern with two identifiable lines of symmetry.

3 a Teacher to check the 4 lines of symmetry. b The trapezium should be circled.

4 a, b & c Teacher to check the rotations. d Yes — a full turn brings the shape back to its starting position, so the pattern would look the same as translating it.

5 a & b Teacher to check. Look for the ability to apply an understanding of the rotation transformation and to describe the pattern made.

Extended practice

1 a Tessellates — check students have shown how regular triangles tessellate.
b Doesn’t tessellate — check students have shown that regular octagons do not tessellate with each other.
c Tessellates — check students have shown how regular hexagons tessellate.

2 Teacher to check. Look for an understanding of both symmetry and tessellation.

Extra practice

1a translating b reflecting c rotating d translating

2a 5 b 2 c 1 d 0 e infinitely many f 8

3 Tick A, M, T and W. F and R have none.

4a yes b yes c no d yes

5a 4 times b 90° c a square

6a 3 b 2 c 6 d 2

7 The angles at each corner have to add up to exactly 360° to close the gap around a point. A regular hexagon's angles are 120°, and three of them make 360°. A regular pentagon's angles are 108°, and three make only 324° while four make 432° — so there is always a gap or an overlap.

Unit 8 · Topic 2

Scales and maps

A map’s scale tells you how big each centimetre on the map is in real life. A legend is a key that tells you what the symbols on a map mean.

Hillcrest Fairgrounds · Scale: 1 cm = 10 m

The fairground map shows: Through Road and McKenzie Lane around the edge; the animal nursery, food stalls and carnival rides along the top; the main stage and Grand Arena in the middle; the horse pavilion, car parking area and showbag hall below. The legend includes first aid, information and toilets.

Guided practiceWorked through with you

Use the map to find:
1. the length of the main stage.
2. the number of toilets at the Hillcrest Fairgrounds.
3. where first aid is located.
4. the width of the fairgrounds.
1. Draw and label a 10 m by 15 m picnic area below the animal nursery. How big is that on the map?
2. How far is your picnic area from the car parking area?
3. Add your own police symbol to the legend. Describe it.
4. Choose a place to draw your police symbol on the map.
5. Describe where your police station is.

Independent practiceYour turn

This is O’Brien’s Farm.

Using a scale of 1 cm = 5 m, draw and label:
1. a field that is 30 m long and 20 m wide. On the map:
2. a barn that is 10 m long and 5 m wide. On the map:
3. a farmhouse that is 15 m wide and 20 m long. On the map:
4. an orchard that is 15 m long and 10 m wide. On the map:
How will you decide where to place each item?
Create symbols in the legend and add the following items to the map.
1. 5 trees
2. 2 water tanks
3. a windmill
4. 7 cows
1. Draw a track the length of the farm.
2. How long is your track in metres?
If the scale was 1 cm = 10 m, what would be the dimensions of:
1. the field? long and wide
2. the barn? long and wide
3. the farmhouse? long and wide
The City Fun Run course is drawn on a grid map with columns A–L and rows 1–6. Scale: 1 cm = 50 m. The map shows City Road, Johns Street, Riverside Boulevard, Bow River, Boundary Road and Bingo Road, plus the city square, sports stadium, botanical gardens, art gallery and station.
1. About how long is the Fun Run course?
2. Describe where the course goes.
3. Write directions from the city square to the sports stadium.
What is at:
1. E2?
2. D4?
3. C3?
What is the grid reference for:
1. first aid?
2. the station?
3. the finish line?

Extended practiceStretch your thinking

A pirate map is drawn on a grid with columns A–J and rows 1–6, and a compass rose showing N, NE, E, SE, S, SW, W and NW. It shows Castaway Island, Skull Island, Shark Alley, Coconut Island, Volcano Island, Shipwreck Cliffs and Mystery Island.

What is:
1. west of Coconut Island?
2. southwest of Skull Island?
3. northwest of Shipwreck Cliffs?
4. northeast of Mystery Island?
1. Draw the best way for the pirate ship to sail to the treasure.
2. Describe the route using grid references.
3. Describe the route using compass directions.
4. Draw another way for the pirate ship to reach the treasure.
5. Which route is longer? How can you tell?

Extra practiceMore questions to build fluency

A map has a scale of 1 cm = 10 m. Find the real distance.
1. 3 cm on the map
2. 7.5 cm on the map
3. 12 cm on the map
4. 0.5 cm on the map
Same scale. Find the map distance.
1. 80 m in real life
2. 150 m in real life
3. 45 m in real life
4. 5 m in real life
A different map uses 1 cm = 5 m. A garden is 9 m by 6 m.
1. What size is the garden on the map?
2. What is the real perimeter of the garden?
Use grid references. A grid is labelled A–E across and 1–5 up.
1. Write the reference for the square 3 across and 2 up.
2. How many squares are in the whole grid?
What is a legend on a map for?
A path on a map is 8 cm long. The scale is 1 cm = 20 m. A person walks the path at 80 m per minute.
1. How long is the path in real life?
2. How long does the walk take?
Two maps show the same park. One uses 1 cm = 10 m, the other 1 cm = 50 m. Which map shows the park bigger, and why?

Answers · Topic 2 Scales and maps

Guided practice

1 a 24 metres b 4 c In between the horse pavilion and the animal nursery d 115 metres

2a Teacher to check. Look for students who understand that 1 cm = 10 m and therefore draw a 1 cm by 1.5 cm area.
b Teacher to check, based on the location of the student’s picnic area.
c–e Teacher to check. Look for the understanding that a symbol on a legend needs to represent the place in some way, and for the use of the language of location.

Independent practice

1 a–d Teacher to check. Look for the ability to apply an understanding of scale — e.g. the field should be 6 cm long and 4 cm wide.

2 a & b Teacher to check. Look for an understanding of how to use symbols to represent places on maps, and a justification of why items are placed in particular locations.

3 Teacher to check the track. The width of the farm is 65 metres.

4 a 3 cm long and 2 cm wide b 1 cm long and 0.5 cm wide c 1.5 cm wide and 2 cm long

5 a 2000 m or 2 km b & c Teacher to check. Look for accurate descriptions of directions using the language of location.

6 a water station b Bow River c Start and/or Information

7 a I2 b L4 c C5

Extended practice

1 a Shark Alley b Coconut Island c Castaway Island d Shipwreck Cliffs

2 a–e Teacher to check. Look for the ability to accurately interpret maps using both grid references and compass directions, and an awareness of why one route may be a better choice than another.

Extra practice

1a 30 m b 75 m c 120 m d 5 m

2a 8 cm b 15 cm c 4.5 cm d 0.5 cm

3a 1.8 cm by 1.2 cm b 30 m

4a C2 b 25 squares

5 A legend (or key) explains what the symbols and colours on the map stand for, so that a small picture can replace a lot of writing and everyone reads the map the same way.

6a 160 m b 2 minutes

7 The 1 cm = 10 m map. On that map each centimetre covers less ground, so the park needs more centimetres to be drawn — it appears larger and shows more detail. The 1 cm = 50 m map squeezes the same park into a fifth of the space.

Unit 9 · Data handling

Data representation and interpretation

Ask the right question, record the answers carefully, then choose a graph that tells the truth about what you found.

Data Handling · 2 topics

Unit 9 · Topic 1

Collecting data

Different survey questions give you different information.

“Do you like cats?” — that’s a yes/no question, so I only have 2 answer options. “What do you think about cats?” — that’s a more open question. I could give lots of different answers.

When is it useful to ask yes/no questions?

Guided practiceWorked through with you

1. Write a survey question about sport that has a yes/no answer.
2. Ask 10 people your question and record the answers with tally marks.
  
  Yes No
1. Write a question about sport that doesn’t give limited options.
2. Ask 2 people your question and record their responses.

Independent practiceYour turn

Tick the survey question that would be best to find out:
1. how many people in your class like chocolate.
  When did you last eat chocolate? What chocolate do you like?
  What is your favourite dessert? Do you like chocolate?
2. the most popular ice-cream flavour.
  Where do you buy ice-cream? How popular is ice-cream?
  What is your favourite ice-cream flavour? Do you like ice-cream?
1. Write a survey question with the following possible responses: 1 = dislike a lot · 2 = dislike a bit · 3 = not sure · 4 = like a bit · 5 = like a lot
2. What do you think will be the most common response from your class?
3. Ask 10 classmates your question and record their answers below.
  
  Response 1 2 3 4 5
  
  Number of people
4. What was the most common response?
5. Write a statement about how the results compared with what you expected.
Nakeil checked the pencil cases of some of his friends and recorded how many pens they each had.

2012 7332 4132
1. Record the information in a table.
  
  Number of pens 0 1 2 3 4 5 6 7
  
  Tally
2. Make a bar graph with the data. How tall would each bar be?
1. Count the number of pens 6 classmates have and record this in a table.
  
  Classmate Number of pens
2. Make and label a bar graph of the results. Remember to label both axes.

Extended practiceStretch your thinking

Write 3 survey questions about food.
1. Choose a question with limited options to ask 15 people. Which one?
2. Record their responses.
  Results
What information will you need to record?
Make a pictograph or bar graph of the results.

Describe your graph — scale, labels, title

Extra practiceMore questions to build fluency

Write open or closed for each survey question.
1. "Do you walk to school?"
2. "How do you get to school?"
3. "What is your favourite sport?"
4. "Do you like football — yes or no?"
Which question would be better for finding out each thing? Write A or B.
1. How many children own a bike. A: "Do you own a bike?" B: "What do you think of bikes?"
2. Which flavours to sell. A: "Do you like ice-cream?" B: "Which flavour of ice-cream do you like best?"
What is wrong with each survey question? Write a better version.
1. "You agree that maths is the best subject, don't you?"
2. "How old are you? Under 8 / 8–10 / 10–12"
A class of 30 is surveyed about pets. The tally shows dog 12, cat 9, fish 5, none 4.
1. Do the numbers add up correctly?
2. Which pet was most common?
3. How many more dogs than fish?
A survey about school lunches is carried out only in the canteen queue. Why might the results be misleading?
Write three survey questions you could ask about how classmates spend their weekends. Make at least one of them a closed question.
Why is it usually better to survey 60 people than 6?

Answers · Topic 1 Collecting data

Guided practice

1 a & b Teacher to check. Look for students who can write a yes/no question on the topic and accurately record the responses.

2 a Teacher to check — look for an understanding of the difference between open and closed questions. b Teacher to check the recording of 2 responses.

Independent practice

1 a Do you like chocolate? b What is your favourite ice-cream flavour?

2a Teacher to check — the question can only have limited responses.
b–e Teacher to check. Look for the ability to justify predictions about the survey outcome and to accurately record the results using numbers, names, ticks or tally marks.

Number of pens	0	1	2	3	4	5	6	7
Tally	1	2	4	3	1	0	0	1

3b Bars of height 1, 2, 4, 3, 1, 0, 0, 1 above 0–7 on the horizontal axis, with “Number of pencils” along the bottom and “Number of students” up the side.

4 a & b Teacher to check. Look for the ability to use observation as a data collection method and to accurately represent the data in a table and matching bar graph. Students should be able to label the x- and y-axes as well as graphing the data.

Extended practice

1 Teacher to check. Look for a variety of questions on the topic that show knowledge of survey construction and language.

2 a & b Teacher to check. Students should have exactly 15 responses recorded.

3 Teacher to check. Look for an understanding of the conventions of graph construction, and data that matches the previous question.

Extra practice

1a closed b open c open d closed

2a A b B

3a It is a leading question — it pushes people to agree. Better: "Which subject do you like best?" b The groups overlap (a 10-year-old fits two boxes) and there is no option above 12. Better: "Under 8 / 8–9 / 10–11 / 12 or over".

4a Yes — 12 + 9 + 5 + 4 = 30 b dog c 7 more

5 Everyone in the queue has already chosen to buy a school lunch, so children who bring a packed lunch — the very people whose opinion would change the answer — are never asked. The sample is biased.

6 Teacher to check — for example "Do you play sport at the weekend? (yes/no)", "What do you enjoy doing most at the weekend?", "How many hours do you spend outside?"

7 A larger sample gives a more reliable picture. With only 6 people, one unusual answer changes the result completely; with 60, the pattern in the data is much more likely to match the whole group.

Unit 9 · Topic 2

Displaying and interpreting data

Survey question: What do you think of peas?

Dislike a lot	Dislike a little	Don’t know	Like a little	Like a lot
8	24	1	10	12

Pictograph

Key: each ● stands for 2 people.

How many people does each face on the pictograph represent?

Guided practiceWorked through with you

1. Use the data above to complete the bar graph. How tall is each bar?
  
  Dislike a lot · Dislike a little · Don’t know · Like a little · Like a lot
2. Which response was the most popular?
3. Which was the least popular?
4. Do more people like or dislike peas overall?
5. How many more people dislike peas a little than like them a little?

Independent practiceYour turn

Here are the results of the survey “What do you think of action movies?”

Dislike a lot Dislike a little Not sure Like a little Like a lot

4 2 3 7 11

Counts redrawn for this edition — the totals match the printed answer key (27 people surveyed, 3 “Not sure”, “Like a lot” most popular).
1. Choose an appropriate way to display the data.
  Sketch or describe your display
2. What type of display did you choose?
3. Why?
Use your graph to answer these questions.
1. What was the most popular response?
2. How many people were surveyed?
3. How many people answered “Not sure”?
4. Write two of your own statements about the data.
This graph shows the average homework time per night in Year 4.

Time 15 min 30 min 45 min 60 min 75 min 90 min 105 min

Number of students 6 10 14 9 5 3 1

Counts redrawn for this edition, keeping the shape of the printed graph (vertical scale 0–14).

Do you think the results would be a lot different for Year 4 students at your school?

Write 3 questions that can be answered by the data.
Does the data tell you:
1. how students feel about homework?
2. how many students do more than 60 minutes of homework on average?
3. who does the least homework?
4. how many students responded to the question?
5. the shortest average time spent on homework?
6. the average age of the students?

Extended practiceStretch your thinking

A survey was done about favourite crisp flavours. Two graphs were made from the same responses — but they use different scales, so one makes barbecue look far more popular than the other.
1. Why do the results look different?
2. Looking at the first graph, would you say barbecue is: a lot more popular? a bit more popular? not popular?
3. Looking at the second graph, would you say barbecue is: a lot more popular? a bit more popular? not popular?
4. Which graph do you think the makers of barbecue crisps would prefer people to see?
5. Why?
6. How many people were surveyed in total?

Extra practiceMore questions to build fluency

A pictograph uses ● = 4 people. How many people are shown by:
1. 3 symbols
2. 5 symbols
3. half a symbol
4. 2½ symbols
Using the same key, how many symbols would you draw for:
1. 16 people
2. 28 people
3. 6 people
4. 2 people
A bar graph of favourite fruit shows: apple 14, banana 20, mango 8, orange 6.
1. How many were surveyed?
2. Which fruit was most popular?
3. How many more chose banana than mango?
4. Which two fruits together equal the banana total?
Homework times per night for Year 4: 0–15 min 5 children, 16–30 min 12, 31–45 min 9, over 45 min 4.
1. How many children in total?
2. Which group was largest?
3. How many did more than 30 minutes?
Answer yes or no. Can this data tell you:
1. which child did the most homework?
2. how many children did under 16 minutes?
3. what subject the homework was?
Which graph would you choose, and why?
1. Showing how a plant's height changed over 8 weeks
2. Comparing how many children chose each of 5 sports
A pictograph leaves the key off. Explain why the graph then becomes impossible to read properly.

Answers · Topic 2 Displaying and interpreting data

Guided practice

1a The completed bar graph:

1b Dislike a little c Don’t know d Dislike (32 people dislike peas; 22 like them) e 14

Independent practice

1a Teacher to check. Look for a display method that allows accurate representation of the data — e.g. a bar graph or pictograph. Students should include all the relevant elements, such as titles and scales.
b & c Teacher to check. Look for the ability to correctly identify the type of graph used and justify the choice — e.g. a bar graph because none of the categorical values are very high and it was easy to make the scale.

2 a Like a lot b 27 c 3 d Teacher to check.

3 Teacher to check. Look for students’ ability to demonstrate an understanding of data interpretation by writing questions that can be answered by the given information.

4 a No b Yes c No d Yes e Yes f No

Extended practice

1a Because the scale is different.
b a bit more popular — accept “a lot more popular” if students can justify this, e.g. by explaining that they used the scale to draw the conclusion.
c a lot more popular — accept “a bit more popular” if students can justify this, e.g. by quantifying how many more people prefer barbecue.
d & e Teacher to check — e.g. Graph 1 makes barbecue look more popular than Graph 2 because of the scale used.
f 48

Extra practice

1a 12 b 20 c 2 d 10

2a 4 b 7 c 1½ d ½

3a 48 b banana c 12 more d apple and orange (14 + 6 = 20)

4a 30 children b 16–30 minutes c 13 children

5a no b yes (5) c no

6a A line graph — it shows change over time. b A bar graph or pictograph — it compares separate categories.

7 Without the key you cannot tell what one symbol is worth. Five symbols might mean 5 people, 20 people or 500 — the picture alone carries no information about scale.

Unit 10 · Data handling

Chance

From impossible to certain — put events on a likelihood scale, spot events that can’t happen together, and run experiments to see how close your predictions get.

Data Handling · 2 topics

Unit 10 · Topic 1

Chance events

“I’m more likely to have takeaway tonight than eat in a restaurant.” “But I’m more likely to have a home-cooked meal tonight than takeaway.”

So on a likelihood scale, these three events would sit in this order:

Eat in a restaurant

Have takeaway

Eat a home-cooked meal

Very unlikelyVery likely

Where would you place each item on the likelihood scale?

Guided practiceWorked through with you

Write a letter for each statement in the boxes, from very unlikely to very likely.
1. A I will write in my mathematics book today.
2. B I will be away from school today.
3. C We will have a fire drill today.
4. D I will spend time with my friends today.
Very unlikely Very likely
Order these statements on the scale.
1. A I will have homework today.
2. B I will go shopping after school.
3. C I will have pasta for dinner.
4. D I will see the principal today.
Very unlikely Very likely

Independent practiceYour turn

Order the likelihood terms on the scale from very unlikely to most likely.

likelyequally likely possibleimpossible most likelyvery unlikely unlikelyprobable
Choose a word from question 1 to describe the likelihood of:
1. you walking home from school today.
2. you going on a plane tonight.
3. you watching TV today.
4. you drinking water today.
5. your class going on an excursion this term.
6. you having a sandwich for lunch.
7. having school assembly today.
Write something that:
1. is unlikely to happen to you today.
2. will probably happen to you today.
Are you more likely, less likely or equally likely to:
1. select a queen rather than a king from a full deck of cards?
2. select a king after already selecting and removing a king from a full deck of cards?
3. toss a coin and land on heads rather than tails?
4. toss a coin a second time and land on heads rather than tails?
5. draw a yellow marble from a bag that holds mostly yellow marbles, without looking?
Will everyone in your class have the same answers to these questions?
Match the pairs of events that cannot happen at the same time.

A coin lands on heads.Simon has a cold. School is starting.Simon is on a train. Simon likes vegetables.

Simon is at home.School is ending. A coin lands on tails. Simon dislikes beans and carrots.Simon is well.
Finish the sentences with events that cannot happen at the same time.
1. If I travel home by car, I can’t
2. If I go to the park after school, I can’t
3. If I do my homework at 4, I can’t
4. If it is raining right now, it can’t be
5. If I am playing cricket right now, I can’t

Extended practiceStretch your thinking

The students of Year 4 have put forward a proposal to build a minigolf course in the playground. Complete the sentences to show how you think different people respond.
1. The Year 6 students will probably because
2. The principal is likely to because
3. The parents are unlikely to because
4. It is possible the younger students will because
The statements below are about your Year 4 teacher.
1. A Teaching Year 1 this year
2. B Male
3. C Likes movies
4. D Older than you
5. E Drives a car
6. F Likes mathematics
1. Order the statements from impossible to certain by placing the corresponding letter on the scale.
  Impossible Certain
2. Write 2 more of your own statements and add them to the scale.
  G H

Extra practiceMore questions to build fluency

Write certain, likely, unlikely or impossible.
1. The sun will rise tomorrow.
2. You will roll a 7 on an ordinary dice.
3. It will rain at some point next month.
4. A coin will land on heads.
5. You will grow younger next year.
Order these on a likelihood scale, least likely first.

A: rolling an even number on a dice · B: rolling a 6 · C: rolling a number under 7 · D: rolling a 0
A bag holds 5 red, 3 blue and 2 green counters. Write more likely, less likely or equally likely.
1. Red compared with blue
2. Green compared with red
3. Blue compared with green
Match the pairs of events that cannot happen at the same time. Write yes if they cannot, no if they can.
1. Rolling a 3 and rolling an even number
2. Rolling a 4 and rolling an even number
3. It is Monday and it is the weekend
4. It is raining and it is sunny
Write an event that is:
1. certain
2. impossible
3. about equally likely to happen or not
A spinner has 8 equal sections: 4 red, 2 blue, 1 green, 1 yellow.
1. Which colour is most likely?
2. Which two colours are equally likely?
3. Is red more likely than all the others put together?
Ben says "I did not win the raffle last time, so I am more likely to win this time." Explain why that reasoning is wrong.

Answers · Topic 1 Chance events

Guided practice

1 & 2 Teacher to check. Look for the ability to offer appropriate justification for the placement of each event.

Independent practice

1 impossible · very unlikely · unlikely · possible · equally likely · likely · probable · most likely

Answers may vary slightly — e.g. students may think “possible” is closer to “very unlikely”.

2 a–g Teacher to check. Look for the ability to show an understanding of the language of chance and to use reasoning to justify responses.

3 a & b Teacher to check.

4 a equally likely b less likely c equally likely d equally likely e more likely

5 A coin lands on heads ↔ A coin lands on tails · Simon has a cold ↔ Simon is well · School is starting ↔ School is ending · Simon is on a train ↔ Simon is at home · Simon likes vegetables ↔ Simon dislikes beans and carrots

6 Teacher to check. Look for the ability to understand the language of probability and identify mutually exclusive events.

Extended practice

1 Teacher to check. Look for the ability to offer appropriate justifications for choices and to attribute likely events to each of the people represented.

2 a & b Teacher to check. Look for students’ ability to make reasonable guesses about the probability of their Year 4 teacher having specific attributes, and to put forward their own speculations and rank the likelihood of them occurring.

Extra practice

1a certain b impossible c likely d equally likely (an even chance) e impossible

2 D (impossible), B, A, C (certain)

3a more likely b less likely c more likely

4a yes — 3 is odd b no — 4 is even, so both happen at once c yes d no — it can rain while the sun shines

5 Teacher to check — for example certain: "tomorrow will be a day of the week"; impossible: "I will breathe underwater without equipment"; even chance: "a tossed coin shows tails".

6a red b green and yellow c No — red is 4 out of 8 and the others together are also 4 out of 8, so they are equally likely.

7 Each raffle is a separate event. The tickets do not remember what happened last time, so losing before does not change this draw's chances at all. This mistake is common enough to have a name — the gambler's fallacy.

Unit 10 · Topic 2

Chance experiments

A spinner

The spinner is:

most likely to land on green.
equally likely to land on red as on purple.
very unlikely to land on blue.

How likely is it that the spinner lands on yellow?

Guided practiceWorked through with you

True or false? For a spinner that is mostly green, with equal blue and green wedges being untrue, a small yellow slice, equal red and pink slices, and a small purple slice — decide whether each statement is true or false.
1. The spinner is most likely to land on red.
2. It is equally likely to land on green as on blue.
3. It is unlikely to land on yellow.
4. It is equally likely to land on red as on pink.
5. It is unlikely to land on purple.
6. It is very likely to land on green.
Colour a spinner so that it is:
1. most likely to land on red.
2. equally likely to land on green as on pink.
3. impossible to land on orange.
4. unlikely to land on blue.
5. more likely to land on green than on yellow.

Independent practiceYour turn

1. There are 4 ice-creams in a box — red, green, yellow and blue. List the 6 possible outcomes if you draw out 2 and the order is not important.
2. You decide that the first to come out is yours and the second is for your friend. Show the possible outcomes if the order does matter.
3. How would you describe the likelihood of drawing out:
  1. red and blue?
  2. yellow and green?
  3. pink and blue?
List the possible outcomes if you roll 2 dice and the order matters.

How many outcomes are there altogether?
Put the following counters in a bag: 13 green · 8 red · 8 blue · 1 yellow.
1. Which colour are you most likely to draw out?
2. Which colour are you very unlikely to draw out?
3. Which 2 colours are you equally likely to draw out?
Conduct 20 trials with your counters, drawing out 1 each time. Replace the counters in the bag after you draw them out.
1. Record the results.
  
  Colour Tally Total
  
  Green
  
  Red
  
  Blue
  
  Yellow
2. Which colour did you draw out most?
3. Were red and blue drawn the same number of times? Why do you think this is?
4. Which colour did you draw out least?
5. Write 2 statements that show whether or not your results were as you expected.
Are your results what you expected?

Extended practiceStretch your thinking

If you were to draw out 2 counters at a time from the bag in the last activity, what are the possible outcomes if the order is not important?

List the possible outcomes across the top of the table. Conduct 20 trials drawing out 2 counters. Record the results, returning the counters to the bag after each trial.

Possible outcomes
Results

Which outcome was most common?
Which outcome was least common?
Write 2 statements about your results.
If you conducted another 20 trials, do you think the results would be the same? Why or why not?

Extra practiceMore questions to build fluency

A fair dice is rolled once. How many outcomes give:
1. an even number
2. a number over 4
3. a multiple of 3
4. a number under 7
A coin is tossed twice. List all the possible outcomes.

How many are there?
Two dice are rolled and the numbers are added.
1. What is the smallest possible total?
2. What is the largest possible total?
3. How many different ways can you make a total of 7?
4. How many ways can you make a total of 2?
A spinner is spun 100 times and lands on red 48 times, blue 27 times and green 25 times.
1. Which colour is the largest section likely to be?
2. Roughly what fraction of the spinner is red?
3. Would you expect exactly the same result if you spun it again?
Design a spinner with 6 equal sections so that:
1. red is twice as likely as blue, and green is impossible. Write how many sections of each colour.
A bag has 10 counters. You draw one 40 times, replacing it each time, and get red 32 times.
1. Roughly what fraction of the draws were red?
2. How many red counters do you estimate are in the bag?
Why do we replace the counter each time in an experiment like question 6?

Answers · Topic 2 Chance experiments

Guided practice

1 a False b True c True d False e False f False

2 a–e Teacher to check. The spinner should have more red segments than any other colour, the same number of green and pink, no orange, few blue and fewer yellow than green — e.g. 6 red, 2 each of green and pink, 1 blue and 1 yellow segment.

Independent practice

1a red and green · red and yellow · red and blue · green and yellow · green and blue · yellow and blue

1b red and green, green and red, red and yellow, yellow and red, red and blue, blue and red, green and yellow, yellow and green, green and blue, blue and green, blue and yellow, yellow and blue

1c Teacher to check. Look for students who can select appropriate language to describe the probabilities and offer reasonable explanations for their choices. (Note that pink is not one of the four colours, so drawing pink and blue is impossible.)

2 These 36 outcomes are possible: 6 and 6, 6 and 5, 5 and 6, 6 and 4, 4 and 6, 6 and 3, 3 and 6, 6 and 2, 2 and 6, 6 and 1, 1 and 6, 5 and 5, 5 and 4, 4 and 5, 5 and 3, 3 and 5, 5 and 2, 2 and 5, 5 and 1, 1 and 5, 4 and 4, 4 and 3, 3 and 4, 4 and 2, 2 and 4, 4 and 1, 1 and 4, 3 and 3, 3 and 2, 2 and 3, 3 and 1, 1 and 3, 2 and 2, 2 and 1, 1 and 2, 1 and 1.

3 a green b yellow c red and blue

4 a Teacher to check — look for accurate recording of the trials using an efficient method such as tally marks.
b Teacher to check — the response should match the data collected in part a.
c–e Teacher to check. Look for a reasonable explanation of the results — e.g. if the numbers were the same, students may point out that there was an equal chance of drawing out the colours because there was the same number of each in the bag. If the results were different, they may discuss the fact that chance plays a role in the results and they will therefore vary from predictions.

Extended practice

1a 2 green · 2 red · 2 blue · 1 green and 1 yellow · 1 green and 1 blue · 1 green and 1 red · 1 red and 1 blue · 1 red and 1 yellow · 1 blue and 1 yellow

1b Teacher to check. Look for students’ ability to accurately record the results of their 20 trials using an appropriate method.

1c–e Teacher to check. Look for students’ ability to accurately interpret their experiment results and use the language of chance and mathematical reasoning to make statements that reflect their data.

1f Teacher to check. Look for students who demonstrate an understanding that, although you can predict the likelihood of certain outcomes of the experiment, the actual outcomes will vary because chance plays a part.

Extra practice

1a 3 (2, 4, 6) b 2 (5, 6) c 2 (3, 6) d 6 — all of them

2 HH, HT, TH, TT — 4 outcomes.

3a 2 b 12 c 6 ways (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) d 1 way (1+1)

4a red b about a half (48 out of 100) c No — the results would be close but almost certainly not identical, because each spin is chance.

5 For example 4 red and 2 blue, with no green sections. (Any answer where red is double blue and green is 0 is correct: 2 red and 1 blue with 3 of a fourth colour also works.)

6a about ⅘ (32 out of 40) b about 8 red counters

7 So that every draw has the same chance. If counters were kept out, the contents of the bag would change after each draw and the later results would not be comparable with the earlier ones.

Reference

Glossary

Every mathematical word used in this book, explained in plain language. Type in the search box to jump straight to a word.

No words match that search. Try a shorter word.

A

acute angle: An angle that is smaller than a right angle, or 90 degrees.

addition: The joining or adding of two numbers together to find the total. Also known as adding, plus and sum. See also vertical addition.3 and 2 is 5

algorithm: A process or formula used to solve a problem in mathematics.Horizontal algorithm: 24 + 13 = 37. Vertical algorithm: 24 above 13, added in columns.

analogue time: Time shown on a clock or watch face with numbers and hands to indicate the hours and minutes.

angle: The space between two lines or surfaces at the point where they meet, usually measured in degrees.A 75-degree angle

anticlockwise: Moving in the opposite direction to the hands of a clock.

area: The size of an object’s surface.It takes 12 tiles to cover this poster.

area model: A visual way of solving multiplication problems by constructing a rectangle with the same dimensions as the numbers you are multiplying, and breaking the problem down by place value.6 × 10 = 60 and 6 × 8 = 48, so 6 × 18 = 108

array: An arrangement of items into even columns and rows to make them easier to count.

B

balance scale: Equipment that balances items of equal mass; used to compare the mass of different items. Also called a pan balance or equal arm balance.

bar graph: A way of representing data using bars or columns to show the values of each variable.

base: The bottom edge of a 2D shape or the bottom face of a 3D shape.

C

capacity: The amount that a container can hold.The jug has a capacity of 4 cups.

Cartesian plane: A grid system with numbered horizontal and vertical axes that allow for exact locations to be described and found.

categorical variables: The different groups that objects or data can be sorted into based on common features.Within the category of ice-cream flavours, variables include vanilla, chocolate and strawberry.

centimetre or cm: A unit for measuring the length of smaller items.Length is 80 cm.

circumference: The distance around the outside of a circle.

clockwise: Moving in the same direction as the hands of a clock.

common denominator: Denominators that are the same. To find a common denominator, you need to identify a multiple that two or more denominators share.½ + ¼ + ⅛ = 4/8 + 2/8 + 1/8 = 7/8

compensation strategy: A way of solving a problem that involves rounding a number to make it easier to work with, then paying back or “compensating” the same amount.24 + 99 = 24 + 100 – 1 = 123

composite number: A number that has more than two factors — that is, a number that is not a prime number.

cone: A 3D shape with a circular base that tapers to a point.

coordinates: A combination of numbers, or numbers and letters, that show a location on a grid map.

corner: The point where two edges of a shape or object meet. Also known as a vertex.

cross-section: The surface or shape that results from making a straight cut through a 3D shape.

cube: A rectangular prism where all six faces are squares of equal size.

cubic centimetre or cm³: A unit for measuring the volume of smaller objects.This cube is exactly 1 cm long, 1 cm wide and 1 cm deep.

cylinder: A 3D shape with two parallel circular bases and one curved surface.

D

data: Information gathered through methods such as questioning, surveys or observation.

decimal fraction: A way of writing a number that separates any whole numbers from fractional parts expressed as tenths, hundredths, thousandths and so on.1.9 is the same as 1 whole and 9 parts out of 10, or 1 9/10.

degrees Celsius: A unit used to measure temperature against the Celsius scale, where 0 °C is the freezing point and 100 °C is the boiling point.

denominator: The bottom number in a fraction, which shows how many pieces the whole or group has been divided into.

diameter: A straight line from one side of a circle to the other, passing through the centre point.

digital time: Time shown on a clock or watch face with numbers only to indicate the hours and minutes.

division / dividing: The process of sharing a number or group into equal parts, with or without remainders.

dot plot: A way of representing pieces of data using dots along a line labelled with variables.

double / doubles: Adding two identical numbers, or multiplying a number by 2.2 + 2 = 4 · 4 × 2 = 8

duration: How long something lasts.Most movies have a duration of about 2 hours.

E

edge: The side of a shape, or the line where two faces of an object meet.

equal: Having the same number or value.

equation: A written mathematical problem where both sides are equal.4 + 5 = 6 + 3

equilateral triangle: A triangle with three sides and angles the same size.

equivalent fractions: Different fractions that represent the same size in relation to a whole or group.½ = 2/4 = 3/6 = 4/8

estimate: A thinking guess.

even number: A number that can be divided equally into 2.4 and 8 are even numbers.

F

face: The flat surface of a 3D shape.

factor: A whole number that will divide evenly into another number.The factors of 10 are 1 and 10, 2 and 5.

financial plan: A plan that helps you to organise or manage your money.

flip: To turn a shape over horizontally or vertically. Also known as reflection.

fraction: An equal part of a whole or group.One out of two parts, or ½, is shaded.

G

grams or g: A unit for measuring the mass of smaller items.1000 g is 1 kg.

graph: A visual way to represent data or information.

GST or Goods and Services Tax: A tax, such as 10%, that applies to most goods and services bought in many countries.Cost + GST (10%) = Amount you pay · $10 + $0.10 = $10.10

H

hexagon: A 2D shape with six sides.

horizontal: Parallel with the horizon, or going straight across.

I

improper fraction: A fraction where the numerator is greater than the denominator, such as 3/2.

integer: A whole number. Integers can be positive or negative.–5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5

inverse operations: Operations that are the opposite or reverse of each other. Addition and subtraction are inverse operations.6 + 7 = 13 can be reversed with 13 – 7 = 6.

invoice: A written list of goods and services provided, including their cost and any GST.

isosceles triangle: A triangle with two sides and two angles of the same size.

J

jump strategy: A way to solve number problems that uses place value to “jump” along a number line by hundreds, tens and ones.16 + 22 = 38, jumping +10, +10, +1, +1

K

kilograms or kg: A unit for measuring the mass of larger items.

kilometres or km: A unit for measuring long distances or lengths.

kite: A four-sided shape where two pairs of adjacent sides are the same length.

L

legend: A key that tells you what the symbols on a map mean.

length: The longest dimension of a shape or object.

line graph: A type of graph that joins plotted data with a line.

litres or L: A unit for measuring the capacity of larger containers.The capacity of this bucket is 8 litres.

M

mass: How heavy an object is.4.5 kilograms

metre or m: A unit for measuring the length of larger objects.

milligram or mg: A unit for measuring the mass of lighter items, or to use when accuracy of measurements is important.

millilitre or mL: A unit for measuring the capacity of smaller containers.1000 mL is 1 litre.

millimetre or mm: A unit for measuring the length of very small items, or to use when accuracy of measurements is important.There are 10 mm in 1 cm.

mixed number: A number that contains both a whole number and a fraction.2¾

multiple: The result of multiplying a particular whole number by another whole number.10, 15, 20 and 100 are all multiples of 5.

N

near doubles: A way to add two nearly identical numbers by using known doubles facts.4 + 5 = 4 + 4 + 1 = 9

net: A flat shape that, when folded up, makes a 3D shape.

number line: A line on which numbers can be placed to show their order in our number system, or to help with calculations.

number sentence: A way to record calculations using numbers and mathematical symbols.23 + 7 = 30

numeral: A figure or symbol used to represent a number.1 – one 2 – two 3 – three

numerator: The top number in a fraction, which shows how many pieces you are dealing with.

O

obtuse angle: An angle that is larger than a right angle, or 90 degrees, but smaller than 180 degrees.

octagon: A 2D shape with eight sides.

odd number: A number that cannot be divided equally into 2.5 and 9 are odd numbers.

operation: A mathematical process. The four basic operations are addition, subtraction, multiplication and division.

origin: The point on a Cartesian plane where the x-axis and y-axis intersect.

outcome: The result of a chance experiment.The possible outcomes if you roll a dice are 1, 2, 3, 4, 5 or 6.

P

parallel lines: Straight lines that are the same distance apart and so will never cross.

parallelogram: A four-sided shape where each pair of opposite sides is parallel.

pattern: A repeating design or sequence of numbers.Number pattern: 2, 4, 6, 8, 10, 12

pentagon: A 2D shape with five sides.

per cent or %: A fraction out of 100.62/100, or 62 out of 100, is also 62%.

perimeter: The distance around the outside of a shape or area.Perimeter = 7 m + 5 m + 10 m + 3 m + 6 m = 31 m

pictograph: A way of representing data using pictures so that it is easy to understand.

place value: The value of a digit depending on its place in a number.In 2 748, the 7 has a value of 700.

polygon: A closed 2D shape with three or more straight sides.

polyhedron (plural polyhedra): A 3D shape with flat faces.

power of: The number of times a particular number is multiplied by itself.4³ is 4 to the power of 3, or 4 × 4 × 4.

prime number: A number that has just two factors — 1 and itself. The first four prime numbers are 2, 3, 5 and 7.

prism: A 3D shape with parallel bases of the same shape and rectangular side faces.Triangular prism · rectangular prism · hexagonal prism

probability: The chance or likelihood of a particular event or outcome occurring.There is a 1 in 8 chance this spinner will land on red.

protractor: An instrument used to measure the size of angles in degrees.

pyramid: A 3D shape with a 2D shape as a base and triangular faces meeting at a point.Square pyramid · hexagonal pyramid

Q

quadrant: A quarter of a circle, or one of the four quarters on a Cartesian plane.

quadrilateral: Any 2D shape with four sides.

R

radius: The distance from the centre of a circle to its circumference or edge.

reflect: To turn a shape over horizontally or vertically. Also known as flipping.

reflex angle: An angle that is between 180 and 360 degrees in size.

remainder: An amount left over after dividing one number by another.11 ÷ 5 = 2 r1

rhombus: A 2D shape with four sides, all of the same length, and opposite sides parallel.

right angle: An angle of exactly 90 degrees.

right-angled triangle: A triangle where one angle is exactly 90 degrees.

rotate: Turn around a point.

rotational symmetry: A shape has rotational symmetry if it fits into its own outline at least once while being turned around a fixed centre point.

round / rounding: To change a number to another number that is close to it, to make it easier to work with.229 can be rounded up to 230 (nearest 10) or down to 200 (nearest 100).

S

scale: A way to represent large areas on maps by using ratios of smaller to larger measurements.1 cm = 5 m

scalene triangle: A triangle where no sides are the same length and no angles are equal.

sector: A section of a circle bounded by two radius lines and an arc.

semi-circle: Half a circle, bounded by an arc and a diameter line.

skip counting: Counting forwards or backwards by the same number each time.By fives: 5, 10, 15, 20, 25, 30 · By twos: 1, 3, 5, 7, 9, 11, 13

slide: To move a shape to a new position without flipping or turning it. Also known as translate.

sphere: A 3D shape that is perfectly round.

split strategy: A way to solve number problems that involves splitting numbers up using place value to make them easier to work with.21 + 14 = 20 + 10 + 1 + 4 = 35

square centimetre or cm²: A unit for measuring the area of smaller objects. It is exactly 1 cm long and 1 cm wide.

square metre or m²: A unit for measuring the area of larger spaces. It is exactly 1 m long and 1 m wide.

square number: The result of a number being multiplied by itself. The product can be represented as a square array.3 × 3, or 3², = 9

straight angle: An angle that is exactly 180 degrees in size.

strategy: A way to solve a problem. In mathematics, you can often use more than one strategy to get the right answer.32 + 27 = 59, by jump strategy or split strategy

subtraction: The taking away of one number from another number. Also known as subtracting, take away, difference between and minus. See also vertical subtraction.5 take away 2 is 3

survey: A way of collecting data or information by asking questions.

symmetry: A shape or pattern has symmetry when one side is a mirror image of the other.

T

table: A way to organise information that uses columns and rows.

tally marks: A way of keeping count that uses single lines, with every fifth line crossed to make a group.

term: A number in a series or pattern.The sixth term in the pattern 3, 6, 9, 12, 15, 18, 21, 24 is 18.

tessellation: A pattern formed by shapes that fit together without any gaps.

thermometer: An instrument for measuring temperature.

three-dimensional or 3D: A shape that has three dimensions — length, width and depth. 3D shapes are not flat.

time line: A visual representation of a period of time with significant events marked in.

translate: To move a shape to a new position without flipping or turning it. Also known as slide.

trapezium: A 2D shape with four sides and only one set of parallel lines.

triangular number: A number that can be organised into a triangular shape.The first four are 1, 3, 6 and 10.

turn: Rotate around a point.

two-dimensional or 2D: A flat shape that has two dimensions — length and width.

U

unequal: Not having the same size or value.

V

value: How much something is worth.This coin is worth 5c. This coin is worth $1.

vertex (plural vertices): The point where two edges of a shape or object meet. Also known as a corner.

vertical: At a right angle to the horizon, or straight up and down.

vertical addition: A way of recording addition so that the place-value columns are lined up vertically to make calculation easier.36 + 21 = 57

vertical subtraction: A way of recording subtraction so that the place-value columns are lined up vertically to make calculation easier.57 – 21 = 36

volume: How much space an object takes up.This object has a volume of 4 cubes.

W

whole: All of an item or group.A whole shape · a whole group

width: The shortest dimension of a shape or object. Also known as breadth.

X

x-axis: The horizontal reference line showing coordinates or values on a graph or map.

Y

y-axis: The vertical reference line showing coordinates or values on a graph or map.

Reference

Answers

Full answers for all 31 topics. Where a question is open-ended, a note for the teacher explains what to look for instead.

Tip: each topic also carries its own answer key at the foot of its page — look for the 🔑 Answers panel. Use 🔑 Show all answers in a unit’s toolbar to open them all at once.

Unit 1 · Topic 1 — Place value

Guided practice

1a 34 926 = 3 ten thousands, 4 thousands, 9 hundreds, 2 tens, 6 ones = 34 thousands, 9 hundreds, 2 tens, 6 ones = 349 hundreds, 2 tens, 6 ones = 3492 tens, 6 ones = 34 926 ones.

1b 97 563 = 9 ten thousands, 7 thousands, 5 hundreds, 6 tens, 3 ones = 97 thousands, 5 hundreds, 6 tens, 3 ones = 975 hundreds, 6 tens, 3 ones = 9756 tens, 3 ones = 97 563 ones.

Independent practice

1 Each number renamed the same way: a 17 329, b 80 154, c 64 078, d 49 461, e 28 935.

2 b 40 000 + 700 + 70 + 2 · c 80 000 + 7000 + 20 + 4 · d 10 000 + 7000 + 300 + 10 + 6 · e 90 000 + 2000 + 600 + 3 · f 50 000 + 5000 + 500 + 50 + 5

3 3 (1563), 4 (10 008), 6 (11 345), 2 (11 570), 10 (15 183), 7 (24 999), 1 (37 706), 9 (47 200), 5 (47 398), 8 (50 953)

4 a fifty-six thousand, nine hundred and twenty-seven · b eighty thousand, four hundred and one · c forty-two thousand and fifty-eight

5 a 68 142 b 24 070 c 90 003

Extended practice

1 a 70 b 30 c 1360 d 62 150 2 a 600 b 1600 c 22 000

3 a 6000 b 24 000 c 94 000 4 a 20 000 b 42 000 c 83 000

5 a 500 000 b 600 000 c 200 000

6 a 144 420 b 12 081 c 61 458 d 402 325 e 49 006

7 12 081, 49 006, 61 458, 144 420, 402 325

Unit 1 · Topic 2 — Odd and even

Guided practice

1 a odd b even c even d even e even f odd g odd h even i odd j odd k even l odd

2 a even b odd c odd d odd e odd f even g even h odd i even j even k odd l even

Independent practice

1 a 76 523 b 23 567 c 76 532 d 23 576

2 a 98 100 b 98 001 c 10 098 d 10 089

3 a 64 075 b 40 576 c 57 640 d 50 467

4 even + odd = odd · odd + even = odd · odd + odd = even · even – even = even · even – odd = odd · odd – even = odd · odd – odd = even

5 even × odd = even · odd × even = even · odd × odd = odd

6 a odd b odd c even d even e even f even g even h even

Extended practice

1a 28 ÷ 2 = 14, 34 ÷ 2 = 17, 100 ÷ 2 = 50 → True
b 15 ÷ 3 = 5, 30 ÷ 3 = 10, 300 ÷ 3 = 100 → False
c 40 ÷ 4 = 10, 16 ÷ 4 = 4, 36 ÷ 4 = 9 → True

2 Odd: 62 849, 520 399, 1 098 765, 8 888 881, 7 676 767. Even: 34 176, 123 456, 987 654, 471 002, 4 342 998.

Unit 1 · Topic 3 — Addition mental strategies

Guided practice

1 a 2 + 18 + 35 = 20 + 35 = 55 · b 13 + 7 + 46 = 20 + 46 = 66 · c 38 + 32 + 51 = 70 + 51 = 121 · d 42 + 8 + 53 = 50 + 53 = 103 · e 16 + 4 + 92 = 20 + 92 = 112 · f 45 + 125 + 22 = 170 + 22 = 192 · g 17 + 13 + 42 + 28 = 30 + 70 = 100 · h 19 + 21 + 44 + 16 = 40 + 60 = 100

Independent practice

1 a 53 b 61 c 102 d 117 e 343 f 159 g 90 h 110

2 a 133 b 277 c 743 d 783 e 1061

3 a 787 b 739 c 1277 d 4759 e 6782

4 a 195 b 786 c 761 d 793 e 895 f 428 g 963 h 1097

Extended practice

1 a 932 b 5799 c 8000 d 7664

2 a 3430 b 5630 c cookies and cupcakes d 1059 e 17 280

Unit 1 · Topic 4 — Addition written strategies

Guided practice · split strategy

1a 8 + 130 + 400 + 7000 = 7538 b 9 + 90 + 600 + 14 000 = 14 699

Independent practice · split strategy

1 a 6677 b 45 945 c 53 765 d 80 386

Guided practice · vertical addition

1 a 62 b 155 c 782 2 a 167 b 719 c 8914 3 a 8497 b 6359 c 16 699

The printed answer key gives 95 for 1b; 76 + 79 = 155.

Independent practice · vertical addition

1 a 8494 b 8258 c 41 776 d 54 789 e 63 634 f 83 987 g 59 246 h 64 753

Extended practice

1 a 13 690 b 90 178 c 74 771 d 92 461 e 23 555 f 149 254

2 a 37 690 b 18 669

Unit 1 · Topic 5 — Subtraction mental strategies

Guided practice

1a 85 – 20 = 65, 65 + 1 = 66 → 66 b 73 – 20 = 53, 53 – 2 = 51 → 51 c 91 – 30 = 61, 61 – 2 = 59 → 59

Independent practice

1 a 39 b 58 c 29 d 57 e 118 f 242 g 323 h 179

2 a 423 – 200 + 2 = 225 · b 654 – 300 – 5 = 349 · c 526 – 300 + 3 = 229 · d 793 – 200 – 7 = 586 · e 478 – 200 + 3 = 281 · f 642 – 300 – 4 = 338

3 a correct b correct c incorrect (396) d correct

4 a 5 b 4 c 7 d 13 e 8 f 6

5 a 22 b 116 c 3310 d 6991

Extended practice

1 a 3575 b 2566 c 13 d 3814 e 3271

2 a Alexis b Aravinda c 1009 d 304 e 2554

3 a 23 323 b 431 c 26 829 d 13 727

Unit 1 · Topic 6 — Subtraction written strategies

Guided practice · split strategy

1 a 2116 b 5534 c 5133 d 11 441 e 14 251

Independent practice · split strategy

1 a 4127 b 2421 c 26 322 d 31 870

Guided practice · vertical subtraction

1 a 17 b 47 c 9 2 a 584 b 382 c 2382 3 a 1564 b 4730 c 11 711

Independent practice · vertical subtraction

1 a 366 b 171 c 328 d 8086 e 4714 f 2807 g 31 427 h 522 i 41 814 j 34 173

Students may or may not include the zeroes at the start of some answers. Either way is acceptable at this point.

Extended practice

1a 29 078 km · 27 474 km · 24 777 km · 22 537 km · 17 260 km · 14 075 km · 11 245 km · 7253 km

1b $84 166 · $61 529 · $19 977 · $8086 c $38 564 d $26 436

Unit 1 · Topic 7 — Multiplication and division facts

Guided practice

1 a 9 × 5 = 45; 45 ÷ 9 = 5 · b 8 × 5 = 40; 40 ÷ 8 = 5 · c 3 × 7 = 21; 21 ÷ 7 = 3 · d 5 × 8 = 40; 40 ÷ 5 = 8 · e 8 × 7 = 56; 56 ÷ 7 = 8 (turnaround facts also accepted)

Independent practice

1b 6, 2, 8, 4, 0 c 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 e 9, 8, 7, 6, 5, 4, 3, 2, 1, 0 f 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 g 99, 108, 117 h 66, 72, 78

2a 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 b 4 × 2 = 2 × 4, and so on to 4 × 10 = 10 × 4 c 4 ÷ 1 = 4; 8 ÷ 4 = 2, 8 ÷ 2 = 4; 12 ÷ 4 = 3, 12 ÷ 3 = 4; 16 ÷ 4 = 4; 20 ÷ 4 = 5, 20 ÷ 5 = 4; 24 ÷ 4 = 6, 24 ÷ 6 = 4; 28 ÷ 4 = 7, 28 ÷ 7 = 4; 32 ÷ 4 = 8, 32 ÷ 8 = 4; 36 ÷ 4 = 9, 36 ÷ 9 = 4; 40 ÷ 4 = 10, 40 ÷ 10 = 4

3 a 32 b 24 doubled = 48 c 36 doubled = 72

Extended practice

1 a 36 b 360 c 63 d 153 2 a 10 b 100 c 6 d 60

3 a 7, 9 b 4, 6 c 4, 6, 9 d 4, 6, 7

Unit 1 · Topic 8 — Multiplication written strategies

Guided practice · extended multiplication

1 a 63 b 84 c 60 d 155 e 288 f 282

Independent practice · extended multiplication

1 a 140 b 258 c 603 d 462 e 272 f 534 g $518 h 280 km

Guided practice · contracted multiplication

1 a 84 b 95 c 96 d 305 e 312 f 336 2 9 × 84 = 36 + 720 = 756 both ways

Independent practice · contracted multiplication

1 a 128 b 287 c 324 d 260 e 414 f 544 g $792 h 423

2 45 × 7 = 315 · 86 × 7 = 602 · 53 × 6 = 318 · 45 × 8 = 360 · 92 × 4 = 368

Extended practice

1 a 288 b 288 c Both farmers had the same.

2 Hot dogs 312 · Carrot sticks 546 · Chocolate buttons 702 · Mini pizzas 390

3 Hot dogs 712 · Carrot sticks 1246 · Chocolate buttons 1602 · Mini pizzas 890

Unit 1 · Topic 9 — Division written strategies

Guided practice

1 a 11 b 21 c 34 d 23 e 23 f 31 2 a 15 b 14 c 12 d 18 e 13 f 23

Independent practice

1 a 29 b 49 c 11 d 12 e 26 f 19 g 29 h 20 i 13

2 a 72 ÷ 6 = 12 · b 80 ÷ 5 = 16 · c 76 ÷ 4 = 19 · d 68 ÷ 4 = 17 · e 98 ÷ 7 = 14 · f 81 ÷ 3 = 27 · g 86 ÷ 2 = 43 · h 96 ÷ 3 = 32 · i 96 ÷ 4 = 24

3 a 28 b 19 c 24 d 32 e 13 f No.

Look for students who understand that there would be leftovers or remainders, because 7 does not divide equally into 78.

Extended practice

1 a 24, 42, 56, 60, 96, 108, 120 · b 24, 42, 60, 75, 81, 96, 108, 120 · c 24, 56, 60, 96, 108, 120 · d 35, 60, 75, 120

2 a 32 b 112 c 91 d 112 e 71 f 243 g 121 h 111 i 141

3 a 56 b 48

Unit 2 · Topic 1 — Equivalent fractions

Guided practice

1 a 2/8 b 4/6 c 3/4

Independent practice

1 a 1/3 and 2/6 · b 2/5 and 4/10 · c 1/2 and 2/4 · d 2/3 and 6/9

2 a 4 sections → 4/8 · b 2 sections → 2/3 · c 4 sections → 4/5 · d 1 section → 1/4

3 a 4/10 b 2/3 (4/6) c 2/3 (8/12) d 2/8 (3/12) e 4/5 f 3/4 (9/12) g 2/4, 3/6, 4/8, 5/10, 6/12 h 2/2, 3/3, 4/4, 5/5, 6/6, 8/8, 10/10, 12/12

Extended practice

1 a 10/100 b 1/10 2 a 40 b 80 c 70 d 50

3 a 40/100 b 50/100 c 30/100 d 90/100 e 100/100 f 25/100

4 a = b > c < d =

Unit 2 · Topic 2 — Improper fractions and mixed numbers

Guided practice

1 a 3/4, 6/4, 7/4 · b 1/2, 3/2 · c 6/3, 8/3

Independent practice

1 a 1⅓ b 2⅓ c 3 d 1 2/4 e 2¾ f 2¼ g 2½ h 4½ i 3

2a ½, 1, 1½, 2, 2½, 3, 3½, 4, 4½
b 1/3, 2/3, 3/3, 4/3, 5/3, 6/3, 7/3, 8/3, 9/3, 10/3
c 1/4, 2/4, 3/4, 1, 1¼, 1 2/4, 1¾, 2
d 5, 4½, 4, 3½, 3, 2½, 2, 1½, 1

3–5 Teacher to check the positions marked on each number line.

6 a 2½ b 4½ c 4 d 1½

7 a 7/2 b 1⅔ c 3¼ d 12/4 e 10¼ f 7⅓ g 5½ h 9/3 i 4¼

Extended practice

1 a 22/6 = 3 4/6 · b 35/8 = 4 3/8 · c 13/5 = 2 3/5 · d 43/12 = 3 7/12

2 Fractions 0/9 to 26/9, with matching mixed numbers 0 up to 2 8/9.

3 a 9 b 22 c 17 d 36 e 32 f 46

Unit 2 · Topic 3 — Decimal fractions

Guided practice

1 a 0.2 b 0.5 c 0.8 2 a 0.45 b 0.26 c 0.53 d 0.82 e 0.99 f 0.6 (0.60)

Independent practice

1 a 0.7 = 7/10 · b 0.07 = 7/100 · c 0.77 = 77/100 · d 7.77 = 7 77/100 · e 0.32 = 32/100 · f 0.65 = 65/100 · g 3.29 = 3 29/100 · h 6.04 = 6 4/100

Accept equivalent fractions, such as 70/100 for 7/10.

2a 0, 0.1 … 1.5 in tenths b 0, 0.01 … 0.12 in hundredths c 1.7 … 2.8 in tenths d 0.95 … 1.06 in hundredths

3 36.4 · 500.22 · 222.22 · 14.58 · 103.7 · 628.43 · 946.04

Extended practice

1 a 0.9 b 0.3 c 0.52 d 9.8 e 0.5 f 0.41 g 0.87 h 1

2 Silva 3.26 m, Dan 3.9 m, Raff 4.07 m, Lily 4.28 m, Elara 4.7 m, Nick 5.02 m, James 5.21 m

Unit 3 · Topic 1 — Money and money calculations

Guided practice

1 Rounds up to 0: 8, 9 · down to 0: 1, 2 · up to 5: 3, 4 · down to 5: 6, 7

2 $3.58 up $3.60 · $7.86 down $7.85 · $15.32 down $15.30 · $23.01 down $23.00 · $99.99 up $100.00 · $85.43 up $85.45 · $48.04 up $48.05 · $59.97 down $59.95

Independent practice

1a A $3.55 · B $1.50 · C $2.00 · D $3.00 · E $1.75
1b A $8.55 · B $6.50 · C $7.00 · D $8.00 · E $6.75

2 a–d Teacher to check.

3 a up b down c up d up

4 a B and E b A and D c B and C d D and F e C and E

5 a & b Teacher to check.

Extended practice

1 a 35c b R1.75 c R5.65 d R3.05 e R9.00 f R7.00

2 a R5 b R7.50 c 80c d R9.55 e R2.65 f R4.25

3 a 2 b 4 c 10 4 a 13 b 7 c 2

Unit 4 · Topic 1 — Number patterns

Guided practice

1 a 25, Add 2 · b 0, Subtract 11 · c 23, Subtract 3 · d 103, Add 10 · e 90, Add 9

Independent practice

1 a Multiply by 7 b Subtract 9 c Add 20 d Multiply by 10

2 a 46, 100, 109 b 8, 100, 16

3 a, b & d Teacher to check. c The multiples of 4 are both circled and shaded. e All of them.

4 a & b Teacher to check. c They all end in zero. d The numbers that are multiples of both 2 and 5 are also multiples of 10.

Extended practice

1a 1, 2, 4, 7, 11, 16, 22, 29, 37, 46 — Add 1 more each time
b 3, 5, 9, 15, 23, 33, 45, 59, 75, 93 — Add 2 more each time
c 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 — Multiply the previous number by 2

2 Teacher to check.

3 a 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 · b 14, 28, 42, 56, 70 · c 35, 70 · d 21, 42, 63

Unit 4 · Topic 2 — Problem solving

Guided practice

1a 15 + 21 = 48 – 12, answer 21 · b 42 + 16 = 31 + 27, answer 16 · c 73 – 24 = 26 + 23, answer 24

Independent practice

1 a 100 – 42 = 31 + 27 · b 56 + 31 = 108 – 21 · c 98 + 30 = 200 – 72 · d 43 + 54 = 72 + 25 · e 97 – 18 = 61 + 18

2 a 15 b 14 c 84 d 54 e 6 f 8 g 7 h 55 i 40 j 13 k 36 l 65

3 a 12 × 6 = 72 · b 8 × 9 = 72 · c 15 × 6 = 90 · d 49 ÷ 7 = 7 · e 54 ÷ 6 = 9 · f (28 + 32) × 10 = 600

4 Teacher to check.

Extended practice

1 Multiple answers — e.g. 40 green, 40 red and 26 blue; 100 green, 3 red and 3 blue; or 35 green, 35 red and 36 blue.

2 1 each for 48 people, 2 each for 24, 3 each for 16, 4 each for 12, 6 each for 8, 8 each for 6, 12 each for 4, 16 each for 3.

3 Teacher to check — multiple possible answers.

Unit 5 · Topic 1 — Length and perimeter

Guided practice

1 a 8 mm b 25 mm c 43 mm d 37 mm 2 a 17 mm b 6 mm c 68 mm 3 a 13 cm b 5 cm c 9 cm

Independent practice

1 a cm b m c cm d mm e m f mm

2 a 20 b 100 c 55 d 230 e 25 f 38 g 380 h 120 i 12 (all mm)

3 a 200 b 1000 c 550 d 125 e 350 f 475 g 3 h 3.5 i 10 (all cm)

4 a 1 m b 5 m c 2.5 m 5 Teacher to check.

6 a 20 cm b 18 cm c 20 cm d 23 cm 7 a 80 mm b 75 mm c 120 mm d 168 mm

Extended practice

1 a–k Teacher to check.

Unit 5 · Topic 2 — Area

Guided practice

1 matchbox lid 19 cm² · netball court 465 m² · smart phone 81 cm² · chopping board 600 cm² · table top 2 m²

Independent practice

1 a cm² b m² c cm² d m² 2 Teacher to check.

3 a 24 cm² b 18 cm² c 16 cm² d 12½ cm² 4 Teacher to check.

Extended practice

1 a 16 cm² b 16 cm² c 40 cm² d 15 cm² 2 Teacher to check.

Unit 5 · Topic 3 — Volume and capacity

Guided practice

1 a 6 cm³ b 12 cm³ c 16 cm³ 2 c 3 Teacher to check 4 b

Independent practice

1 a Teacher to check b 2 c 4 2 a Teacher to check b 3 c 9

3–5 Teacher to check.

6 a A & C b 4 litres (4 L) c 3 litres 700 millilitres (3.7 L)

Extended practice

1 C, D, A, E, B

2 a 1 litre 400 mL b 2 litres 500 mL c 3 litres 859 mL d 7 litres 643 mL

3 a 3025 mL b 5340 mL c 7654 mL d 19 999 mL

Unit 5 · Topic 4 — Mass

Guided practice

1 a 1.3 kg / 1 kg 300 g · b 3.2 kg / 3 kg 200 g · c 2.5 kg / 2 kg 500 g · d 5.5 kg / 5 kg 500 g · e 4.2 kg / 4 kg 200 g · f 26.7 kg / 26 kg 700 g

Independent practice

1 & 2 Teacher to check.

3 a 1.1 kg b 150 g c 160 g d 600 g e 1.85 kg f 150 g

Accept equivalents — e.g. 1100 g for 1.1 kg.

Extended practice

1 1.7 kg = 1 kg 700 g = 1700 g · 4.5 kg = 4 kg 500 g = 4500 g · 3¼ kg = 3 kg 250 g = 3250 g · 0.62 kg = 0 kg 620 g = 620 g · 7.75 kg = 7 kg 750 g = 7750 g · 5.03 kg = 5 kg 30 g = 5030 g

2a 125 g · 840 g · 2000 g · 1500 g · 1650 g · 250 g b 715 g c 4715 g (4 kg 715 g, 4.715 kg) d 140 g (0.14 kg)

Unit 5 · Topic 5 — Temperature

Guided practice

1 a 30 °C b 60 °C c 0 °C d 44 °C e 89 °C f 100 °C

Independent practice

1 Teacher to check.

2 a 74 °C b 7 °C c 67 °C d 10 °C and 35 °C e 35 °C and 36 °C · f–g vary with location; likely f 7 °C and 10 °C, g 35 °C, 36 °C and 49 °C

3 Circle: a Snow scene b Glass of water c Cupcake d Person in shade

4 a hot b freezing c warm or hot d cold or cool

Extended practice

1 Boiling kettle 100 °C · Hot bath 42 °C · Cup of tea 65 °C · Cold winter morning 12 °C · Inside a fridge 2 °C

2 a–f Teacher to check.

Unit 5 · Topic 6 — Time

Guided practice

1 a 60 b 60 c 24 d 7 e 365 (or 366) f 52

2 a 120 b 360 c 180 d 300 e 90 f 150 g 2 h 72 i 7 j 35

Independent practice

1 Todd 75 s = 1 min 15 s, rank 2 · Harper 140 s = 2 mins 20 s, rank 6 · Jessica 100 s = 1 min 40 s, rank 4 · Mario 90 s = 1 min 30 s, rank 3 · Stirling 120 s = 2 mins, rank 5 · Anthony 70 s = 1 min 10 s, rank 1

2 a 27 days b 2 hours c 2 years d 660 minutes e 3 days f 4000 days g 3½ hours h 1 hour

3 a 35 b 300 c 300 d 60 e 730 (or 731) f 48

4 a am b pm c pm d am e pm f am

5 2 am, 9 am, 11 am, 1 pm, 3:15 pm, 9 pm

6 a 6:59 am b 8:26 pm c 12:10 am d 12:47 pm

Extended practice

1 a 11:30 am b 16 minutes c 9 hours 35 minutes d My Mother the Plumber e 3:01 pm f Cop Capers g Cakes on a Train h 7:20 pm to 8:52 pm

Unit 5 · Topic 7 — Timelines

Guided practice

1 a after b 6 months c 2 years

2 a “I broke my arm” at three and a half years · b “I started school” just before 5 years · c “I learned to swim” just after two and a half years

Independent practice

1 Teacher to check. Ben’s birthday is 7 July; the school play is 25 October; the fireworks are on 31 December.

2 a 9:30 am b 30 minutes c Lunch at 12:30 pm d Any time around 2:45 pm e Wombats then koalas f Gift shop before 2 pm

3 D, I, A, G, B, E, C, F, H

Extended practice

1a The arrows are spread out evenly but the time gaps are not all the same. b It is easier to tell the length of time between each event. c Without a scale we could not tell the length of time between each event.

2 Teacher to check.

Unit 6 · Topic 1 — 2D shapes

Guided practice

1 square 4/4 · octagon 8/8 · pentagon 5/5 · trapezium 4/4 · kite 4/4 · hexagon 6/6

Independent practice

1 a trapezium b 1 rectangle and 2 triangles c & d Teacher to check.

2 b 1 rectangle and 2 triangles (examples only).

3–5 Teacher to check.

6a A Triangle 3 angles 8 cm² · B Rectangle 4 angles 15 cm² · C Hexagon 6 angles 20 cm² · D Parallelogram 4 angles 8 cm² · E Hexagon 6b The two hexagons.

Extended practice

1 a & b Teacher to check — most likely a 3 cm by 3 cm square. c & d The hexagon is irregular. f 4 cm²

2 a 2 smaller rectangles and 4 larger rectangles · b 1 pentagon and 5 triangles · c 2 circles and 1 rectangle

Unit 6 · Topic 2 — 3D shapes

Guided practice

1 a rectangular prism b pentagonal prism 2 a triangular pyramid b pentagonal pyramid

Independent practice

1 & 2 Teacher to check.

3 a front view, side view, top view · b side view, top view, front view · c front view, side view, top view

4 Teacher to check.

Extended practice

1 rectangular prism · triangular pyramid · hexagonal prism

2 Teacher to check.

Unit 7 · Topic 1 — Angles

Guided practice

1 a smaller than a right angle → acute angle · b greater than a straight angle → reflex angle · c greater than a right angle → obtuse angle · d greater than a right angle → straight angle · e greater than a straight angle → revolution · f smaller than a straight angle → right angle

Independent practice

1 Matched in order of size: acute, right, obtuse, straight, reflex.

2 Teacher to check. 3 B, C, E, F, A, D

4a 1 right, 2 acute, 3 acute · b 1 obtuse, 2 acute · c 1 right, 2 acute, 3 reflex, 4 acute · d 1 acute, 2 acute · e 1 reflex, 2 acute, 3 acute · f 1 obtuse, 2 acute, 3 right, 4 obtuse, 5 acute, 6 obtuse

Extended practice

1 & 2 Teacher to check.

Unit 8 · Topic 1 — Symmetry

Guided practice

1 a–c Teacher to check.

Independent practice

1 & 2 Teacher to check.

3 a Teacher to check. b The trapezium should be circled.

4 & 5 Teacher to check.

Extended practice

1 a Tessellates (regular triangles) · b Doesn’t tessellate (regular octagons) · c Tessellates (regular hexagons)

2 Teacher to check.

Unit 8 · Topic 2 — Scales and maps

Guided practice

1 a 24 metres b 4 c In between the horse pavilion and the animal nursery d 115 metres

2 a–e Teacher to check.

Independent practice

1 Teacher to check — e.g. the field should be 6 cm long and 4 cm wide.

2 Teacher to check. 3 The width of the farm is 65 metres.

4 a 3 cm long and 2 cm wide · b 1 cm long and 0.5 cm wide · c 1.5 cm wide and 2 cm long

5 a 2000 m or 2 km · b & c Teacher to check.

6 a water station b Bow River c Start and/or Information

7 a I2 b L4 c C5

Extended practice

1 a Shark Alley b Coconut Island c Castaway Island d Shipwreck Cliffs

2 a–e Teacher to check.

Unit 9 · Topic 1 — Collecting data

Guided practice

1 & 2 Teacher to check.

Independent practice

1 a Do you like chocolate? · b What is your favourite ice-cream flavour?

2 a–e Teacher to check.

3a Tally for 0–7 pens: 1, 2, 4, 3, 1, 0, 0, 1
3b Bars of height 1, 2, 4, 3, 1, 0, 0, 1.

4 a & b Teacher to check.

Extended practice

1–3 Teacher to check. Students should have exactly 15 responses recorded in question 2.

Unit 9 · Topic 2 — Displaying and interpreting data

Guided practice

1a Bars of 8, 24, 1, 10, 12 b Dislike a little c Don’t know d Dislike e 14

Independent practice

1 a–c Teacher to check.

2 a Like a lot b 27 c 3 d Teacher to check.

3 Teacher to check. 4 a No b Yes c No d Yes e Yes f No

Extended practice

1a Because the scale is different. b a bit more popular. c a lot more popular. d & e Teacher to check. f 48

Unit 10 · Topic 1 — Chance events

Guided practice

1 & 2 Teacher to check.

Independent practice

1 impossible · very unlikely · unlikely · possible · equally likely · likely · probable · most likely

2 & 3 Teacher to check.

4 a equally likely b less likely c equally likely d equally likely e more likely

5 heads ↔ tails · has a cold ↔ is well · school starting ↔ school ending · on a train ↔ at home · likes vegetables ↔ dislikes beans and carrots

6 Teacher to check.

Extended practice

1 & 2 Teacher to check.

Unit 10 · Topic 2 — Chance experiments

Guided practice

1 a False b True c True d False e False f False

2 Teacher to check — e.g. 6 red, 2 each of green and pink, 1 blue and 1 yellow segment.

Independent practice

1a red & green · red & yellow · red & blue · green & yellow · green & blue · yellow & blue

1b The 12 ordered pairs of those four colours.

1c Teacher to check (pink and blue is impossible — there is no pink ice-cream).

2 All 36 ordered outcomes from 1 and 1 to 6 and 6.

3 a green b yellow c red and blue 4 a–e Teacher to check.

Extended practice

1a 2 green · 2 red · 2 blue · 1 green & 1 yellow · 1 green & 1 blue · 1 green & 1 red · 1 red & 1 blue · 1 red & 1 yellow · 1 blue & 1 yellow

1b–f Teacher to check.

A	B	C	D	E	F
The Kite	Bear Why?	Alien Dancer	Little Train	My Pony	Book F
$9.53	$5.68	$12.82	$9.39	$14.31	$17.63

Odd	Even