CAMBRIDGE PRIMARY MATHEMATICS 3: TEACHER’S RESOURCE

Learner’s Book answers

1 Numbers to 1000
Getting started
1  
36 77 80 246

45 46 47 87 90 255 256 257 145 146 147

56 97 98 99 100 266 155

 

11 772

21 22 23 42 781 782 783

32 51 52 53 792

42 62
  2 428 = 400 + 20 + 8, 913 = 900 + 10 + 3,
576 = 500 + 70 + 6; 395 = 300 + 90 + 5.
2 42 87
3 a 215
b 632
0 10 20 30 40 50 60 70 80 90 100
4 564
3 70, 30, 50, 60
5 3 tens, 30; 9 ones, 9; 9 hundreds, 900; 9 tens, 90;
Exercise 1.1 8 ones, 8; 2 hundreds, 200.

1   1, 4, 7 and 8 tens, 10, 40, 70, 80.

132 135 458 460
Think like a mathematician
142 145 468 470
The unused place value cards are: 500, 800, 10, 60,
2 and 6. All the possible numbers are: 512, 516,
152 153 154 155 478 479 480 562, 566, 812, 816, 862 and 866.

488 490 6 eight hundred and seventy-three,

eight hundred and fifty-three,
498 500
three hundred and seventy-eight,
three hundred and fifty-eight.

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Exercise 1.2 3 120, 680, 390, 910, 740, 600

1   4 100, 700, 400, 900, 700, 600

220 790
5 598

310 320 330 880 890 900

Think like a mathematician
420 990 The number should be greater than or equal to
x 95 and less than (x + 1)05, where x is any non-
negative whole number. For instance,
x = 0: 95, 96, 97, 98, 99, 100, 101, 102, 103, 104 ➝
550 all round to 100
x = 1: 195, 196, 197, 198, 199, 200, 201, 202, 203,
640 650 660 204 ➝ all round to 200
x = 2: 295, 296, 297, 298, 299, 300, 301, 302, 303,
750
304 ➝ all round to 300 …
x = 9: 995, 996, 997, 998, 999, 1000, 1001, 1002,
2 a 643 is greater than 458 and 458 is 1003, 1004 ➝ all round to 1000
less than 643. Sofia is correct, as 100, 200, 300, …, and 1000 are
b 475 is greater than 472 and 472 is in the above list.
less than 475.
c 883 is greater than 838 and 838 is Check your progress
less than 883. 1  
374 744 746
3 smallest 38, 475, 563, 621, greatest 679
4 greatest 834, 483, 438, 384, smallest 48 383 384 385 754 756
5 48 marked about halfway between 0 and 100,
384 marked approximately three-quarters of 394 764 765 766
the way between 300 and 400, 438 marked
less than halfway between 400 and 500, 483 774 776
marked about three-quarters of the way
between 400 and 500, 834 marked less than 784 786
halfway between 800 and 900.
6 Estimates from 160 to 190, 310 to 340, 830 2 7 ones, 7; 7 hundreds, 700; 7 tens, 70; 3 ones, 3;
to 870. 8 hundreds, 800; 4 tens, 40.
7 Any number less than 263, any number greater 3 Number Round to the Round to the
than 671, any number greater than 457, any nearest 10 nearest 100
number less than 346.
234 230  200
Think like a mathematician 471 470  500
Learners’ own statements and answers. 896 900  900
750 750  800
Exercise 1.3 303 300  300
1 200 to 300 spots, or 200 to 400 spots. There 987 990 1000
are 287 spots.
2 a No, 500 to 600 would be a better estimate.
b Yes, the mass of 24 grams is between
20 grams and 30 grams for 200 to 300 beans.
c 700 to 800 beans or 700 to 900 beans.

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Think like a mathematician

2 Statistics: Tally charts
Learners’ own answers.
and frequency tables Check your progress
Getting started 1 a third week b 32
c 4
1 Animal Tally 2 Fruits Votes
giraffes
mango
lions

camels apple

meerkats grapes

fish
banana
penguins
total 49
seals

3 Score Tally Frequency

Exercise 2.1 30 2
1 Learners’ own answers.
40 5
2 a Football is liked the most.
50 2
b Cricket and basketball are liked by the
same number of people. 60 3
c 50 people took part in the survey. 70 2
d For example, the tables do not say how 80 6
often the games are played; the tables do
not tell you if they surveyed boys or girls. 90 4

3 Learners’ own answers.

4 Favourite hobby Tally Frequency 3 Addition, subtraction

painting 2
and money
dancing 1
Getting started
football 4 1
54 68 56
reading 5
75 99 77
a 2 b reading c dancing
d Learners’ discussion should mention that 89 64 83
the information tells you what activities
are most popular, so you could plan these 2 Answer depends on the numbers chosen.
for the club sessions. You might also want
3 For example: $10, $5, $1, half dollar (50c),
to know what people’s favourite snacks
quarter dollar (25c), two dimes (2 × 10c), one
and drinks are, and so on.
nickel (5c); three $5 (3 × $5), $1, three quarter
dollars (3 × 25c), two dimes (2 × 10c).

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Exercise 3.1 Exercise 3.3

1 Learners’ own calculations. 1 $4.50, $8.70, $24.05, $10, $0.99
2 Learners’ own calculations. 2 $20 and 45c, $9 and 75c, $15 and no cents,
3 estimate: 130 + 50 = 180; learners’ own choice $2 and 9c, $0 and 30c
of method, 134 + 53 = 187. 3 a $30 and 76c b $80
4 estimate: 220 + 70 = 290; learners’ own choice c 95c d 8c
of method, 215 + 67 = 282.
5 estimate: 150 + 140 = 290; learners’ own choice
Think like a mathematician
of method, 148 + 136 = 284. Learners recognise that they need to use the two
lowest value coins and banknotes ($1 + $2 + 1c +
6 estimate: 440 + 330 = 770; learners’ own choice
5c = $3.06) to find the smallest possible value, and
of method, 439 + 326 = 765.
use the two highest value banknotes and coins
($100 + $50 + 50c + 25c = $150.75) to find the
Think like a mathematician greatest value of Zara’s money.
All the possibilities: 4 a 18 + 26 = 44, Sumi spends 44c, 50 − 44 = 6,
242 + 139 = 381 Sumi will have 6c change.
243 + 138 = 381 b 37 + 37 = 74, Virun spends 74c,
244 + 137 = 381 90 − 74 = 16, Virun will have 16c change.
245 + 136 = 381 c 75 − 12 = 63
246 + 135 = 381 Highlighter + eraser = 37c + 26c = 63c
247 + 134 = 381 Thick felt pen + pencil = 45c + 18c = 63c.
248 + 133 = 381 5 a $3 + $1 and 10c + $3 and 25c = $7 and 35c,
$10 − $7 and 35c = $2 and 65c
249 + 132 = 381
b Any two drinks that total less than
Exercise 3.2 $5; for example, tea and orange
juice $2 + $2 and 20c = $4 and 20c.
1 46 − 8 = 38, 48 − 6 = 42, 68 − 4 = 64, 64 − 8 = 56, Change $5 − $4 and 20c = 80c.
86 − 4 = 82, 84 − 6 = 78
6  +   = $8. 8 ÷ 2 = 4, or 4 + 4 = 8,
2 For example: 573 − 9 = 564, 975 − 3 = 972.
 = $4. One ice cream costs $4.
3 estimate: 180 − 30 = 150, 178 − 25 = 153
4 estimate: 260 − 40 = 220, 262 − 37 = 225 7 $50 −   = $17. Inverse: $17 + $33 = $50.

5 estimate: 470 − 270 = 200, 472 − 267 = 205  = $33. The trainers cost $33.

6 estimate: 680 − 550 = 130, 683 − 548 = 135 8 $6 and 50c −   = $1 and 20c. Inverse:

$6 and 50c − $1 and 20c = $5 and 30c.
Think like a mathematician
 = $5 and 30c. The comic costs $5 and 30c.
All the possibilities:
490 – 145 = 345 9 Learners’ own problem, such as paying for
something with $50 and getting $21 change.
491 – 146 = 345
492 – 147 = 345 Check your progress
493 – 148 = 345
1 estimate: 150 + 230 = 380; learners’ own choice
494 – 149 = 345 of method, 147 + 225 = 372.
2 estimate: 380 − 160 = 220; learners’ own choice
of method, 384 − 158 = 226.

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3 $50 −   = $24. Inverse: $24 + $26 = $50. 5

 = $26. The jacket costs $26.

4 3D shapes
Getting started
1 one or more curved surfaces: sphere, cylinder;
all faces rectangular: cube, yellow cuboid, blue
cuboid; more than five vertices: cube, yellow
cuboid, blue cuboid
2 Fewer than six vertices: sphere, cylinder,
triangle-based pyramid, square-based
pyramid.

Exercise 4.1
1 Curved edges Straight edges A prism has two ends that are the same shape
and size. The faces are flat. The remaining
tinned
shapes don’t have those properties.
tomatoes
6 Name of Prism, Properties
shape pyramid
or neither
cube prism 12 edges
biscuits cereal
6 faces
8 vertices
cylinder neither 0 edges
Learners’ own labels, lines and shapes. 2 faces
2 Learners’ own answers. and 1
curved
3 Learners’ own answers. surface
4 What can it be? It is a … 0 vertices
A shape that has faces that square- triangular prism 9 edges
are triangles and a square based prism 5 faces
pyramid 6 vertices
A shape that has no vertices sphere hexagonal prism 18 edges
A shape that has 8 faces hexagonal prism 8 faces
prism
12 vertices
A shape that has 6 faces cuboid
square- pyramid 8 edges
A shape that has a curved cone based 5 faces
surface and a circular face pyramid
5 vertices

7 Results will depend on the throw of the dice.

8 Learners’ sketch of a cuboid, two cubes joined
together and another 3D shape with the
correct name.

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Think like a mathematician numbers as they have 5 ones. The numbers in

the overlap are all even numbers because they
a 18 b 8 have 0 ones.
c 18 d 16 Multiples of 10 circle: all numbers are
multiples of 10. The numbers in the overlap are
Number of different rectangular prisms even numbers as they have 0 ones. There are
that can be made depends on the number of no numbers in the right-hand side of the circle
bricks chosen. because all multiples of 10 are also multiples
For rules and patterns, learners must look at of 5 and so they belong in the overlap.
factors, as well as odd and even numbers. Outside the circles: numbers are not multiples
9 Learners’ own answers. of 5 or 10. All numbers have a ones digit that
is not 5 or 0.
Check your progress numbers to 1000
132
1 Learners’ own answers.
multiple multiple
2 cylinder, sphere; learners’ own explanations. of 5 120 of 10
45 350
3 A prism is a three-dimensional (3D) shape 675 490
with flat faces. It has two ends that are the 805 740
same shape and size. A prism has the same 215
387
cross-section all along the shape from end to 401
96
end. If you cut through it, you would see the
same 2D shape at either end.
A pyramid is also a 3D shape. It has a polygon
Think like a mathematician
base and flat triangular faces that join at a Sofia is right. Any multiples of 10 are also
point called the apex. multiples of 2 and 5, as 10 = 2 × 5.
4 5 × 6 = 30, 6 × 5 = 30, 30 = 5 × 6, 30 = 6 × 5,
30 ÷ 5 = 6, 30 ÷ 6 = 5, 6 = 30 ÷ 5, 5 = 30 ÷ 6
5 Multiplication and 5 No, there are two mistakes. 30 = 10 ÷ 3 should
division be 3 = 30 ÷ 10 and 30 = 3 ÷ 10 should be
10 = 30 ÷ 3.
Getting started
Think like a mathematician
1 1 × 10 = 10, 2 × 5 = 10, 5 × 2 = 10, 10 × 1 = 10
Learners’ own answers.
There are two pairs of facts: 1 × 10 = 10 and
6 Learners’ own completed multiplications.
10 × 1 = 10, 2 × 5 = 10 and 5 × 2 = 10. Learners
may suggest that, just like addition, they can 7 Each one becomes a ten and each ten becomes
multiply in any order. a hundred so the whole number is ten times
bigger. For example, 28 × 10 = 280; the 20
2 50 ÷ 10 = 5, 25 ÷ 5 = 5, 10 ÷ 2 = 5, 5 ÷ 1 = 5
becomes 200 and the 8 becomes 80. When you
3 204, 214, 224, 234 multiply a 1-digit and 2-digit number by 10,
the answer is always an even number because
Exercise 5.1 there is always a 0 in the ones place.

1 rings around 76, 532, 210, 1000, 784, 38, 670 100s 10s 1s
2 A multiple of 2 is made up of groups of two. 2 8
Even numbers of objects can always be put 2 8 0
into groups of two (pairs).
8 23 × 10 = 230, so 230 pencils.
3 Multiples of 5 circle: all numbers are multiples
of 5. The numbers on the left are all odd 9 16, 21, 26, 31

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Exercise 5.2 3 3 × 5 = 15 double → 6 × 5 = 30

3 × 7 = 21 double → 6 × 7 = 42
1 4 × 6 = 24 3 × 8 = 24 ← halve 6 × 8 = 48
2 24, 28, 32, 36, 40. All these numbers are 3 × 9 = 27 ← halve 6 × 9 = 54
multiples of 4 and are even because they have
an even number in the ones place. 4 Coloured 9, 18, 27, 36, 45, 54, 63, 72, 81,
90, 99. The number that is coloured moves
3 2 × 7 = 14 double → 4 × 7 = 28 one place back to the left on the next row.
2×3=6 double → 4 × 3 = 12 To find the next multiple of 9, the ones
2 × 5 = 10 ← halve 4 × 5 = 20 digit decreases by 1 and the tens digit increases
2×4=8 ← halve 4 × 4 = 16 by 1. The numbers coloured are odd, even,
odd, even and so on. Learners may have other
4 8 × 5 = 40
ideas.
5 coloured: 72, 80, 88, 96, 104, 112, 120. All
5 For example, add the multiplication tables for
these numbers are multiples of 8 and are
3 and 6 (3 × 2 = 6, 6 × 2 =12, 9 × 2 = 18) or add
even because they have an even number in the
the multiplication tables for 4 and 5 (4 × 2 = 8,
ones place.
5 × 2 = 10, 9 × 2 = 18).
6 2 × 9 = 18 double → 4 × 9 = 36 double → 8 × 9 = 72
6 wall 3, 3, 3, middle row 9, 9, top 81;
2 × 5 = 10 double → 4 × 5 = 20 double → 8 × 5 = 40
wall 1, 9, 1, middle row 9, 9, top 81.
2 × 6 = 12 ← halve 4 × 6 = 24 ← halve 8 × 6 = 48
2×3=6 ← halve 4 × 3 = 12 ← halve 8 × 3 = 24 7 a Learners’ own sequence with a term-to-
term rule of add 9.
7 9, 13, 17, 21, 25. All these numbers are odd b The numbers will be multiples of the
because adding an even number to an odd start number, so they will follow the
number gives an odd number. They all have an same pattern as the multiplication table
odd number in the ones place. products for that number.
8 8, 27 and 43. Both numbers are odd because 8 Dominos (or ten frames or something else to
adding an even number to an odd number show that 5 + 3 = 8) with five spots and three
gives an odd number. They both have an odd spots, drawn four times.
number of ones in the ones place. 8 × 4 = 5 × 4 + 3 × 4 = 20 + 12 = 32.
9 There are many possible solutions including 5,
13, 21, 29, 37; term-to-term rule add 8 9 9 × 6 = 54, 54 ÷ 6 = 9, 54 ÷ 9 = 6, 9 × 3 = 27,
12 × 9 = 108. Learners may have other ideas.
37, 29, 21, 13, 5; term-to term rule subtract 8
13, 17, 21, 25, 29; term-to-term rule add 4 Think like a mathematician
The digit sums of multiples of 3 and 6 are always
Think like a mathematician either 3, 6 or 9. The digit sums of multiples of 9
are always 9.
Zara’s conjecture is right. The sum of any two
even numbers is an even number and the difference
between any two even numbers is an even number. Check your progress
Learners’ own discussion.
1 6 × 9 = 54, 54 = 6 × 9, 9 × 6 = 54, 54 = 9 × 6,
54 ÷ 6 = 9, 9 = 54 ÷ 6, 54 ÷ 9 = 6, 6 = 54 ÷ 9
Exercise 5.3
2 rings around 50, 340, 580, 700 and 10
1 1 2 3 4 5 6 7 8 9 10
3× 3 6 9 12 15 18 21 24 27 30 3 80, 30, 270, 45, 32, 42, 20, 48, 27
6× 6 12 18 24 30 36 42 48 54 60 4 7, 13, 19, 25, 31, 37

2 3 × 4 = 6 × 2, 3 × 6 = 6 × 3, 3 × 8 = 6 × 4,

3 × 10 = 6 × 5

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10 table:
6 Measurement, area a 110 cm b 100 cm
and perimeter bed:
a 280 cm b 3 m
Getting started road sign:
1 a–d Learners’ own answers. a 20 km b 15 km
2 a–d Learners’ own answers.
Think like a mathematician
Exercise 6.1 The shortest route Silas can take while keeping to
the edges of the bricks is to travel one and a half
1 Learners’ own answers. 1
lengths and two widths. 30 × 1 + 15 × 2 = 75 cm
2
2 Learners’ own answers. Learners’ own answers on other routes.
3 Distance with 2 continents: km
Exercise 6.2
Length of a seal: m
Size of a saucepan: cm 1 These shapes all have 4 sides, straight sides
and lines of symmetry.
An Olympic marathon: km
2 It has 3 sides and 3 vertices. All sides are
Length of your foot: cm straight.
Length of a rowing boat: m 3 5 × 5 square with a perimeter of 20 cm
Width of a mobile phone: cm 6 × 6 square with a perimeter of 24 cm
Length of a golf course: km The perimeter of each square in the sequence
Height of your bedroom door: m is 4 cm longer than the perimeter of the
previous square.
Width of a glove: cm
4 a regular:
4 Learners’ own answers.
5 Learners’ own answers.
6 Learners’ own answers.
7 a learners’ own estimates; 4 cm b irregular:
b learners’ own estimates; 2 cm
8 a 7 m = 700 cm
1
b 250 cm = 2 and  m
2

1
c 3 and  m = 350 cm
2

1 c Learners’ own answers.

d  km = 500 m
2
5 For example: ten sticks can be a rectangle with
3 3, 3, 2, 2 sticks on four sides; or a rectangle
e 750 m =   km
4 with 4, 4, 1, 1 sticks on four sides.
1 For example: 12 sticks can be a rectangle with
f  km = 250 m
4
4, 4, 2, 2 sticks on four sides; or a rectangle
9 the distance between two continents with 5, 5, 1, 1 sticks on four sides; or a square
with 3 sticks on each side.
the length of a long journey
the distance of a marathon race

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6 a 12 cm b 14 cm c 18 m 4 a For example: a rectangle with sides 10 cm,

d 20 cm e 28 km f 20 cm 10 cm, 20 cm, 20 cm has area 200 square
cm; a rectangle with sides 15 cm, 15 cm,
15 cm, 15 cm has area 225 square cm.
Think like a mathematician
b For example: a square with sides 6, 6, 6, 6
a 24 cm has area 36 square units; a rectangle with
sides 10, 2, 10, 2 has area 20 square units.
b The next square will be 5 cm by 5 cm,
perimeter = 20 cm. c For example: a rectangle with sides 4, 4, 1,
1 has area 4 square units; a rectangle with
The next square will be 4 cm by 4 cm,
sides 3, 3, 2, 2, has area 6 square units.
perimeter = 16 cm.
c The next square will be 3 cm by 3 cm,
perimeter = 12 cm. 7 Fractions of shapes
The next square will be 2  cm by 2 cm,
perimeter = 8 cm. Getting started
Final square will be 1 cm by 1 cm, 1 For example:
perimeter = 4 cm.
d, e Perimeter measurements are 24 cm, 20 cm,
16 cm, 12 cm, 8 cm and 4 cm. All are multiples
of four, all are even numbers, descending by 4
each time.

Exercise 6.3
  
1 a 20 square units b 16 square units
c 3 square units d 6 square units
2 Yes (6 square units). Learners’ own answers.
3 a each side 9 units
b perimeter 36 units (9 + 9 + 9 + 9)

Think like a mathematician

Thandiwe is incorrect. For example, a rectangle
with side lengths 3 m, 3 m, 10 m, 10 m and another
rectangle with side lengths 4 m, 4 m, 9 m, 9 m
both have a perimeter of 26 metres. But the first
rectangle has an area of 30 square metres and the
second rectangle has an area of 36 square metres. Exercise 7.1
Check your progress 1 a
3
is green.
10
1 a 2.5 m, 7 m, 9 m 5
1 1 3 b or a half is yellow.
10
b 5   km, 3   km, 4   km, 8 km
4 4 4
2
2 a total length 19 cm c is not coloured.
10
b For example: 10 cm, 3 cm, 3 cm; 6 cm, 4 cm
2 a one slice b two slices
6 cm; 1 cm, 12 cm, 3 cm. There are many
possible answers. 1 2
c ,
10 10
3 a 8 cm, 30 m
b Learners’ own answers.

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1 2 1 2 e 17 minutes past 4
3 a  =  ,  = 
2 4 5 10
f 34 minutes past 8 or 26 minutes to 9
b Learners’ own answers.
3 a and e, b and d, c and f.
4 Learners’ own answers.
4 Learners’ own answers.
2 1
5  =  , so the learner would need to draw
4 2 5 a 18 minutes past 6
another two triangles. Where the learner b 14 minutes to 3
positions the triangles is their choice.
c 24 minutes past 6
Think like a mathematician d 42 minutes past 9 or 18 minutes to 10
1 1
a
2
is shaded, is not shaded.
2 Think like a mathematician
b Learners’ own designs and answers.
a Five light bars can make 2, 3 or 5.
Check your progress b 4, 5 and 6 light bars can match the number
3 shown on the display (that is, 4 light bars can
1 a 1
10 b make the number 4; 5 light bars can make
2
the number 5; and 6 light bars can make the
2   number 6).

Check your progress

1 a 16 minutes past 8
b 10 minutes to 4
c 24 minutes past 11
    3 sides  10 sides
d 40 minutes past 9 or 20 minutes to 10
2 2
   coloured  coloured 2 a twenty-two minutes to three or
3 10
thirty-eight minutes past two

8 Time b nine minutes past five

c thirty-seven minutes past seven or
Getting started twenty-three minutes to eight
1 a quarter past 7 b half past 2 d quarter past six or fifteen minutes past six
c quarter to 3 d quarter past 10 3 a 5:07 b 2:36 c 10:48
e 9 o’clock 4 clock c; learners’ own answers.

Exercise 8.1
1 a minute hand pointing to 3; time around 9 More addition and
quarter past 3
b minute hand pointing close to 7; time subtraction
around 25 minutes to 7
c minute hand pointing to 12; time is Getting started
12 o’clock
1 estimate 520 + 350 = 870; learners’ own choice
2 b 13 minutes past 3 of method, 519 + 348 = 867
c 27 minutes past 9 2 estimate 380 − 170 = 210; learners’ own choice
d 52 minutes past 1 or 8 minutes to 2 of method, 375 − 168 = 207

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3 Xiang has forgotten to regroup 73 into 60 and 2 estimate 250 − 80 = 170, 246 − 84 = 162

13 so that he can subtract 5 ones. Instead, he
has subtracted 3 ones from 5 ones, which is 100s 10s 1s
incorrect.
473 − 245
= 400 + 70 + 3 − 200 − 40 − 5
= 400 + 60 + 13 − 200 − 40 − 5
= 400 − 200 + 60 − 40 + 13 − 5
= 200 + 20 + 8 = 228 3 a estimate 150 − 60 = 90, 148 − 60 = 88
b estimate 250 − 80 = 170, 245 − 82 = 163
Exercise 9.1
c estimate 330 − 70 = 260, 326 − 71 = 255
1 a estimate 40 + 60 = 100, 43 + 56 = 99 d estimate 530 − 90 = 440, 534 − 93 = 441
b estimate 70 + 30 = 100, 67 + 29 = 96
4 a estimate 340 − 160 = 180, 339 − 163 = 176
2 estimate 60 + 50 = 110, 64 + 53 = 117 b estimate 360 − 170 = 190, 355 − 172 = 183
100s 10s 1s c estimate 650 − 390 = 260, 647 − 386 = 261
d estimate 520 − 250 = 270, 518 − 248 = 270

5 a  = 92

3 a estimate 80 + 60 = 140, 84 + 62 = 146 b  = 17

b estimate 70 + 90 = 160, 71 + 87 = 158
c estimate 60 + 70 = 130, 64 + 72 = 136
d estimate 80 + 30 = 110, 75 + 34 = 109 Think like a mathematician
307 – 147 = 160
Think like a mathematician 317 – 157 = 160
The numbers could be 96 + 41, 46 + 91, 86 + 51,
327 – 167 = 160
56 + 81, 76 + 61, 66 + 71.
337 – 177 = 160
4 a estimate 230 + 50 = 280, 233 + 50 = 283
347 – 187 = 160
b estimate 180 + 60 = 240, 178 + 60 = 238
357 – 197 = 160
c estimate 150 + 70 = 220, 154 + 65 = 219
Triangle and square must both be odd or both
d estimate 190 + 60 = 250, 191 + 56 = 247 even to give the even result of 6 tens required.
e estimate 290 + 350 = 640, 286 + 352 = 638
f estimate 470 + 170 = 640, 473 + 166 = 639 Exercise 9.3
5 Learners’ own answers. 1 a 35 + 65 = 100 b 53 + 47 = 100
c 77 + 23 = 100 d 81 + 19 = 100
6 Learners’ own answers.
e 8 + 92 = 100
7 Learners’ own answers.
2 520 + 480, 530 + 470, 540 + 460, 550 + 450,
Exercise 9.2 560 + 440, 570 + 430, 580 + 420, 590 + 410

1 b 300 + 150 + 2 c 600 + 160 + 3
d 200 + 100 + 9

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3 a 35 + 65 = 100, 350 + 650 = 1000 3 a 370 + 290 = 660 b 550 + 180 = 730

b 53 + 47 = 100, 530 + 470 = 1000 c 670 − 290 = 380 d 740 − 460 = 280
c 77 + 23 = 100, 770 + 230 = 1000 e 370 + 360 = 730 f 820 − 350 = 470
d 81 + 19 = 100, 810 + 190 = 1000
e 8 + 92 = 100, 80 + 920 = 1000
10 Graphs
4 a 880 + 120 = 1000 b 470 + 530 = 1000
c 240 + 760 = 1000 d 510 + 490 = 1000 Getting started
e 340 + 230 = 570 f 750 + 150 = 900 1 a six ducks
g 560 + 290 = 850 h 370 + 350 = 720 b three elephants
i 670 + 140 = 810 j 390 + 180 = 570 c two more turtles
5 a 980 − 260 = 720 b 740 − 340 = 400 d 18 animals all together
c 670 − 380 = 290 d 810 − 520 = 290 e two fewer lions
e 760 − 490 = 270 f 520 − 370 = 150
Exercise 10.1
g 850 − 480 = 370 i 630 − 470 = 160
1 a 5 and a half hours
Think like a mathematician b 1 and a half hours more
There are 18 different pairs of 3-digit multiples of c Sunday has 5 and a half hours of
10, with a total of 540: (100, 440), (110, 430), sunshine,
(120, 420), (130, 410), (140, 400), (150, 390), Friday has 13 hours of sunshine.
(160, 380), (170, 370), (180, 360), (190, 350),
(200, 340), (210, 330), (220, 320), (230, 310), Title: Number of hours of sunshine
(240, 300), (250, 290), (260, 280) and (270, 270). Key
Friday
= 1 hour
Check your progress Saturday of sunshine
1 Learners’ own method. Sunday
a estimate 260 + 170 = 430, 263 + 174 = 437
b estimate 480 + 350 = 830, 475 + 353 = 828 2 a
c estimate 360 − 190 = 170, 358 − 187 = 171 Animals in the pet shop
d estimate 740 − 470 = 270, 736 − 472 = 264 y
9
2 Number Complement 8
to 100
7
54 46
6
19 91 81 5
Number of animals

4
77 33 23
3

Complement 2
Number
to 1000 1

440 660 560 0

turtle rabbit fish cat guinea x
pig
753 247
Animals
288 722 712
b Learners’ own answers.

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3 a–e Learners’ own answers. 2 Does not have

Has vertices
4 a–d Learners’ own answers. vertices

5 Learners’ own answers. 2D

Think like a mathematician

Not 2D
Learners’ own answers.

Exercise 10.2 3 Numbers

1 a odd multiples of 5
Numbers 1 to 20 31 25
1 17
5 40
19 19
multiples of 2 multiples of 3
29 35
2 4 multiples 10
of 2 and 3 3
8 10 8 16
6 12 9
14 16
18 15
20
5 17 4 a three times
7 11 13 b Learners’ own answers.

b Learners’ own answers.

2 a 11 More multiplication
Even Not even

Multiple of 3 18 12 24 15 21
and division
Not a multiple of 3 10 16 20 22 11 17 23 Getting started
1 8 × 4 = 32, 32 = 8 × 4, 4 × 8 = 32, 32 = 4 × 8,
b Learners’ own answers. 32 ÷ 8 = 4, 4 = 32 ÷ 8, 32 ÷ 4 = 8, 8 = 32 ÷ 4
3 a–e Learners’ own answers. 2 Add the multiplication tables for 5 and 3
4 Learners’ own answers but labels can be two or for 6 and 2; double and double again
arms/not two arms, four arms/not four arms, the multiplication table for 2; double the
odd number of legs/not odd number of legs multiplication table for 4.
and even number of legs/not even number 3 a 24 b 4 c 42
of legs.
d 50 e 10 f 130
Think like a mathematician g 72 h 2 i 0
Learners’ own answers.
Exercise 11.1
Check your progress 1 24, 8, 18, 63, 48, 36
1 a Three monsters have two legs, two 2 Each product is the result of a number
monsters have three legs, two monsters multiplied by itself.
have four legs.
3 24 ÷ 8 = 3 or 24 ÷ 3 = 8; 8 ÷ 4 = 2 or 8 ÷ 2 = 4;
b Learners’ own answers. 18 ÷ 6 = 3 or 18 ÷ 3 = 6; 63 ÷ 9 = 7 or 63 ÷ 7 = 9;
c For example, sort them by the number 48 ÷ 6 = 8 or 48 ÷ 8 = 6; 36 ÷ 4 = 9 or 36 ÷ 9 = 4
of arms.

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Think like a mathematician 2 Learners may have other estimates or use

other methods.
The numbers on the diagonal line are shown only
once because the diagonal line is a mirror line. For a estimate: 40 × 4 = 160, 35 × 4 = 140;
numbers that are not on the diagonal lines, we can b estimate: 60 × 3 = 180, 58 × 3 = 174;
find their reflection in the other half. c estimate: 90 × 2 = 180, 94 × 2 = 188;
4 28, 4 and 7: 4 × 7 = 28, 28 = 4 × 7, 7 × 4 = 28, d estimate: 80 × 5 = 400, 76 × 5 = 380.
28 = 7 × 4, 28 ÷ 4 = 7, 7 = 28 ÷ 4, 28 ÷ 7 = 4,
4 = 28 ÷ 7. 3 Learners’ own answers.
72, 8 and 9: 8 × 9 = 72, 72 = 8 × 9, 9 × 8 = 72, Think like a mathematician
72 = 9 × 8, 72 ÷ 8 = 9, 9 = 72 ÷ 8, 72 ÷ 9 = 8,
8 = 72 ÷ 9. The number is 60.

5 a 54, 63, 72; rule is + 9 4 46 ÷ 4 = 11 r2, so 12 tubes are needed for 46
b 38, 43, 48; rule is + 5 balls.
c 115, 123, 131; rule is + 8 5 36 ÷ 5 = 7 r1, so each child gets seven marbles
d 99, 90, 81; rule is – 9. with one marble left over.

6 100, 92, 84, 76, 68 6 37 ÷ 10 = 3 r7, so four benches are needed for
37 children.
Exercise 11.2 7 Learners’ own answers.
1 Learners’ own order of multiplication. Check your progress
a 5 × 4 × 3 = 60 b 6 × 5 × 3 = 90
c 6 × 4 × 2 = 48 d 8 × 3 × 2 = 48 1 Learners’ own answers.

2 Learners’ own order of multiplication. 2 Learners’ own answers.

3 a 13 × 4 = 10 × 4 + 3 × 4 = 40 + 12 = 52 3 Learners’ own answers.

b 18 × 5 = 10 × 5 + 8 × 5 = 50 + 40 = 90 4 72 ÷ 3 because there will be more groups of 3
c 12 × 2 = 10 × 2 + 2 × 2 = 20 + 4 = 24 in 72 than there will be groups of 4.
d 15 × 3 = 10 × 3 + 5 × 3 = 30 + 15 = 45
Learners may use other methods; for
example, for 18 × 5 use 18 × 10 = 180, halve
12 More fractions
it, 18 × 5 = 90. Getting started
4 a 28 ÷ 4 = 7 b 25 ÷ 2 = 12 r1 1 1 1
1 a b c
c 53 ÷ 5 = 10 r3 d 32 ÷ 4 = 8 5 3 4
e 10 ÷ 4 = 2 r2 f 46 ÷ 3 = 15 r1 1 1
2 No, of the same whole is greater than of
5 10
5 Each child gets seven sweets, with two left
the same whole. If the whole is split into five
over: 30 ÷ 4 = 7 r2.
equal pieces, each piece will be larger than if it
was split into ten equal pieces.
Think like a mathematician
3 Dad eats four slices, Mum eats four slices,
The number can be 11, 13, 17 or 19.
1 2
Hinata eats two slices; that is, or .
5 10
Exercise 11.3
1 Exercise 12.1
× 20 3
1 a 5 cm b 10 cm
4 80 12
2 50 cm
estimate: 20 × 4 = 80
80 + 12 = 92, 23 × 4 = 92

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3 a Ring drawn around any two marbles. 4

1 1 1 1 1
is less than ,  <  ; is greater than ,  >  .
1 1 1
9 5 4 5 4 4 5 4 5
b
10
2 1 2 1 1 2
4 12 cars 5 a  =  b  >  c  < 
4 2 5 5 3 3

5 7 9 1 1 1 1
1 1 1 1 3 d  <  e  >  f  >  .
20 10 10 5 10 2 3
2 3 4 10 4
1 1
6 of something is more than of the same
10 X 5 2 15 3 4
1 1
something, so of $100 is more than
3 4
6 27
of $100.

Think like a mathematician

Think like a mathematician
The number is 60, or any multiple of 60. 1 1 1 1
Let of A = of B = of C = of D. We can say
1 2 3 4 10
7 a of 8 = 2, 8 ÷ 4 = 2 that D > C > B > A.
4
1 We can also say that D is five times bigger than A.
b of 24 = 12, 24 ÷ 2 = 12
2 C is twice as big as A.
1 Example: A = 4, B = 6, C = 8 and D = 20.
c of 40 = 10, 40 ÷ 4 = 10.
4
3 4 5 6
1 1 7 a  =   =   = 
8 a of 12 = 3, 12 ÷ 4 = 3; of 16 = 4, 6 8 10 12
4 4
3 4 5 6
1 b , , ,
16 ÷ 4 = 4; of 20 = 5, 20 ÷ 4 = 5; 30 40 50 60
4
2 3 4 5
1 1 c , , ,
of 24 = 6, 24 ÷ 4 = 6; of 28 = 7, 8 12 16 20
4 4
1
1 8 For example, is shown.
28 ÷ 4 = 7; of 32 = 8, 32 ÷ 4 = 8; 2
4
1 1
1 1 2 2
of 36 = 9, 36 ÷ 4 = 9; of 40 = 10,
4 4 1 1
40 ÷ 4 = 10. 4 4
1 1 1
b The fraction list uses multiples of 4, the 6 6 6
divisions are the division table for 4. 1 1 1 1
8 8 8 8
1 1 1 1 1
9 Each person gets two quarters of a sandwich, 10 10 10 10 10
2 1
which is   or each.
4 2 1
9
3
1
10 1 out of 4, .
4
Exercise 12.3
Exercise 12.2 4 3 7
1  +   = 
10 10 10
1 2
1 For a 12 cm line, marked at 3 cm, at 6 cm,
4 4 2 Learners’ own diagrams.
3
at 9 cm. 1 2
4 a estimate: >  ,
2 3
1 2
2 For a 15 cm line, at 5 cm, at 10 cm, 1 4
3 3 b estimate: >  ,
2 2 5
at 3 cm.
10 1 5 1
c estimate: =  ,  = 
4 8 9 2 10 2
3 , and .
5 10 10 1 3
d estimate: >  ,
2 4

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3
0 4 1 3 2 2
 +   = 1,  +   = 1,  +   = 1 2 a The chocolate bar must have the same
4 4 4 4 4 4 mass as two muffins, so one muffin must
4 3 1 have a mass of 50 g.
4  −   = 
5 5 5 b Each side of the scales has a mass of 150 g.
5 Learners’ own diagrams. 3 a 500 g < 5 kg b 10 g < 1 kg
1 1 1
a estimate: <  , 0 b estimate: <  , c 1000 g = 1 kg
2 2 5
1 7 1 1 4 a 80 g b 70 g
c estimate: >  , d estimate: <  ,
2 10 2 4
c 500 g d 4 g
5 Learners’ own answers.

0 5 1 4 2 3 3 2
6 Learners’ own answers.
6 1 −   =  , 1 −   =  , 1 −   =  , 1 −   =  ,
5 5 5 5 5 5 5 5
4 1 5 0 Think like a mathematician
1 −   =  , 1 −   =   = 0
5 5 5 5 For example rows could be:

Think like a mathematician 200 g 500 g 300 g

Learners’ own answers. 700 g 250 g 50 g
7 a $6 b $3 400 g 450 g 150 g

c $24 d $15 Columns can use the same amounts as above but
set vertically.
Check your progress Challenge answer, for example:
1 1 1 1 1
1 , , , , 500 g 50 g 450 g
10 5 4 3 2
250 g 350 g 400 g
1 1 4 2 1 1
2 a  <  b  >  c  <  250 g 600 g 150 g
4 3 5 5 10 5
5 1 4 10 2 3
d  =  e  =  f  <  Within the Challenge square, there are eight ways
10 2 4 10 3 3
to make a total of 1 kg: three rows, three columns
3 Learners’ own diagrams. and two diagonals.
1 2
a estimate: <  ,
2 10 Exercise 13.2
1 5
b estimate: >  ,  = 1 1 a A: 250 ml; B: 750 ml; C: 500 ml
2 5
1 2 b container B, container C, container A
c estimate: >  ,
2 3
c The same because 1000 ml is the same as
1 3 1 litre.
d estimate: >  ,
2 4
d Learners should have drawn a line at the
500 ml mark.
13 Measure 2 a
1
250 ml =   litre b
3
750 ml =   litre
4 4
Getting started c 1000 ml = 1 litre d 8 cans = 2 litres
1 Learners’ own answers. e 12 cans = 3 litres f 16 cans = 4 litres
3 Marcus is incorrect. His method will only
Exercise 13.1 work if he has two jugs that are the same size
1 One pear has a mass of 40 grams, so 4 pears and with the same capacity. Otherwise, a tall
have a mass of 160 grams. narrow jug could have a higher level of water
than a wider jug but still contain less water.

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4 a 625 ml, rounded to 600 ml Think like a mathematician

b 490 ml, rounded to 500 ml Learners’ own posters.
c 250 ml, rounded to 300 ml
5 Learners’ own answers. Check your progress
6 Learners’ own answers. 1 a 200 grams b 7 kilograms
c 900 grams d 3 kilograms
Think like a mathematician 2 a 400 ml b 200 ml
a 1000 millilitres = 500 millilitres + 500 millilitres c Learners’ own drawings. The container
(cup A + cup A) holding 350 ml has more water.
b 700 millilitres = fill cup A, pour into cup B, 3 a 400 grams b 11 days
200 ml left in cup A, pour this into a container,
c 24 cm
fill cup A and pour into the container with
the 200 ml. 4 16 °C
c 100 millilitre = fill cup B and pour into cup A,
fill cup B again and fill cup A, 100 ml is left
in cup B. 14 Time (2)
d Yes. They can be made as follows. Different Getting started
answers are possible.
• 100 ml: see c 1 a 8:05 b 11:17
• 200 ml: fill cup A, pour into cup B, 200 ml c 2:35 d 1:34
left in cup A 2 a
• 300 ml: cup B
• 400 ml: twice 200 ml (see above)
• 500 ml: cup A b
• 600 ml: twice cup B
• 700 ml: see b
• 800 ml: cup A + cup B c
• 900 ml: cup A + 400 ml (see above)
• 1000 ml: see a
d
Exercise 13.3
1 a 30 °C b 30 °C c 10 °C
2 a No, Marianna is not right. The reading is 3 a
nearer to 15 °C than to 20 °C. 11 12 1
b The reading is nearest to 35 °C. 10 2
c marked a little over half way between 25 9 3
and 30
8 4
3 a thermometers B, C, D and E 7 6 5
b 18 °C c 15 °C
d thermometer C e 20 °C
f Learners’ own answers.
4 Learners’ own answers.
5 Learners’ own answers.

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b Exercise 14.2
11 12 1
1 a 8:50
10 2
b 8:45
9 3
c Bus 1 and Bus 2
8 4
7 5 d Razaan is correct. This timetable shows
6 the bus travelling in one direction only. It
does not give information about the buses
c travelling in the opposite direction, from
school to Razaan’s home.
11 12 1
10 2 2 a 11:23
9 3 b 2:58
c Train 2 and train 6
8 4
7 6 5 d 11:25
e 8:01
d
Think like a mathematician
11 12 1
10 2 Learners’ own answers.
9 3
Check your progress
8 4
7 5 1 a two months
6
b 31 days
c two weeks
Exercise 14.1 2 a Wednesday
1 a–g Learners’ own answers. b Thursday
c 1:00
2 a 26 days b 22nd March
d Tuesday
c 3rd February d 9th February
3 a 7:30
3 a 6th June
b 8:42
b 11th July
c the second train
c seven days
d Stefania was born in 2010, so she will be
20 years old in 2030.
15 Angles and
4 a 174 days
b 15 years old movement
c Song’s holiday is longer by 1 day. Getting started
5 a Saturday 19th April
1 a
b Monday 28th April
6 two years
7 Learners’ own answers.

Think like a mathematician

Learners’ own answers.

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b Think like a mathematician

a 3 steps b 3 steps
c 3 steps d 20 routes
e no f no
g,h Learners’ own answers.
5 Learners’ own answers.
c
6 Learners’ own answers. The last drawing
should be the same as the first drawing.

Check your progress

1

  

Other shapes have no right angles.

2 For example: The two half turns end in the
2 north
same place.

Exercise 15.1 west east

1 a, b Learners’ own answers. south

2 Learners’ own answers. start

3 a school b pond c north N

d east e east f south

g south
finish
h Learners’ own answers.
4 a Move south one square. Turn one right
angle clockwise. Move west four squares.
Turn one right angle anticlockwise. Move
south three squares. Turn one right angle
16 Chance
anticlockwise. Move east one square. Turn Getting started
one right angle clockwise. Move four
squares south. 1 Learners’ own answers.
b Learners’ own answers. 2 Learners’ own answers.

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3 Possible Impossible Exercise 17.1

A horse will grow 1
✓
wings and fly.
A carrot will walk. ✓
A baby will cry. ✓
It will rain
somewhere in the ✓ 2 a
world today.

Exercise 16.1
1 It might happen. One domino has more than
eight spots.
b
2 Learners’ own answers.
3 a Spinner A could land on 1, 2 or 3.
b Spinner B could land on 1, 2, 3, 4, 5, 6, 7
or 8.
c Spinner A is more likely to land on 3,
because there are fewer options. Spinner B
has more options.
d Learners’ own results. c
e Learners’ own answers.

Think like a mathematician

Learners’ own answers.

Check your progress

1 Learners’ own answers. d

2 Might happen. A total of 10 can be made

by 7 + 3 or 6 + 4.
3 Will not happen. You cannot make a total of
20 from any pair of these cards.
4 Learners’ own answers.

17 Pattern and 3 a

symmetry
Getting started
1 Learners’ own answers.

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b 7 These shapes have symmetry:

A circle has an infinite number of lines
of symmetry; only four are marked in the
diagram above.
8 Learners’ own answers. Marcus is correct: a
circle has infinite lines of symmetry.
c Learners’ own answers.
9 Learners’ own answers. Only the second and
4 Diagram c shows symmetry. third logos are symmetrical.
Symmetry means that when an object is split
along a line of symmetry, it will produce two Think like a mathematician
mirror images. Learners’ own answers.
5 Learners’ own answers.
Exercise 17.2
6
1 a 9 + 3 = 12, 12 + 3 = 15, 15 + 3 = 18
b 21

For example: c The constant is that three circles are

added each time.
2 a 4, 6, 8 b 10
c The constant is that two squares are
For example: added each time.
3 a,b,c 16 to 12 to 8 to 4.
d The constant is that four squares are
subtracted each time.
The pentagon can be regular or irregular, as 4 a 2, 3, 4
long as it is symmetrical. b 5
c The constant is that one cloud is added
each time.
5 a

b,c,d Learners’ own answers.

6 For example: Add a square to the centre of
the top block, but there are other ways.
7 Learners’ own answers.

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Think like a mathematician Check your progress

For example: 1 a

2 a, b Learners’ own answers.

3 a 2 to 4 to 6 to 8.
b There will be eight circles in the
final pattern.
4 Learners’ own answers.
5 Learners’ own answers.

There can also be other solutions where the

squares are not joined.

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