prim_maths_6_2ed_tr_learner_book_answers.pdf

CAMBRIDGE PRIMARY MATHEMATICS 6: TEACHER’S RESOURCE
1 Cambridge Primary Mathematics 6 – Wood, Low, Byrd & Byrd © Cambridge University Press 2021
Learner’s Book answers
Unit 1 The number system
Getting started
1 a 9 ones b 9 hundredths
c 9 tenths
2 a
two hundred and 205408.9
five thousand, four
hundred and eight
point nine
b
five hundred and 570036.01
seventy thousand
and thirty six point
zero one
3 a 0.1 b 5.55
4 10
5 a 6m b 37cm
c 12m d 11cm
6 a 99.5 b 10.4
Exercise 1.1
1 a 7 hundred b 7 thousandths
c 7 tenths d 7 hundredths
2 2.046
3 Odd one out is 12.34. All the others are
equivalent to 1.234.
4 a
two point one 2.139
three nine
b
negative nine −909.909
hundred and
nine point nine
zero nine
c
twenty-five point 25.431
four three one
d
negative three point −3.481
four eight one
5 0.8 or
8
10
and 0.004 or
4
1000
6 a 0.14 b 0.019
7 a 7200 b 85 c 42.8
d 6.7 e 0.151 f 0.55
8 a C b D c E
9 5
Think like a mathematician
98 889 petals left (100000−1111=98889)
Exercise 1.2
1 4 8 3 7
2 $15
3 4.5 5.05 4.55 5.35
4 4.5 7.8 2.4 9.1
5 7.51 7.49 7.53
6 A
False, 3.04 is 3.0 when rounded to the
nearest tenth.
B True
C False, 6.95 is 7.0 when rounded to the
nearest tenth.
7 55.6 litres 12.2 metres 35.5 kilograms
8 0.5
9 7.97 is 8 when rounded to the nearest whole
number.
7.97 is 8.0 when rounded to the nearest tenth.
The 7 in the hundredths place increases the
tenths by one so 7.9 becomes 8.0. If the number
is rounded to the nearest tenth, there must be a
digit in the tenths place, even if it is zero.
5 cm is between 4.50 and 5.49cm
6 cm is between 5.50 and 6.49cm
Smallest possible perimeter=4.50+4.50+5.50+
5.50=20.00cm
Largest possible perimeter=5.49+5.49+6.49+
6.49=23.96cm

2 Cambridge Primary Mathematics 6 – Wood, Low, Byrd Byrd © Cambridge University Press 2021
Check your progress
1 0.6+0.05+0.005
2 97.314
3 1000
4 a 3.1 b 10
5 a 10 b 100
c 1000 d 0.034
6 13.94 seconds
Unit 2 Numbers and
sequences
Getting started
1 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
2 a
4, 11, 18, 25, 32, 39
b 2, 11, 20, 29, 38, 47
3 4, 7, 10
4 a
Any three from 12, 24, 36 or 48
b Any three from 2, 4, 6 or 12
Exercise 2.1
1 a
multiply by 6 b 60
2 a
6, 9, 12 b add 3
c multiply by 3
3 a
8, 16, 24, 32, 40, 48
b multiply by 8
c 400
4 a
100 triangles and 150 triangles and
rectangles altogether
b Position Term
1 3
2 6
3 9
4 12
5 15
c multiply by 3
d 60
5 1
1
4
and 1
3
4
6 a
1, 1.01, 1.02, 1.03, 1.04
b 1.09
7 All the tenths digits are even and 5 is odd.
8 −2 −3
3
5
−4
9 30.01 (30.010)
+5, +9, +13, +17, +21 (the difference between the
terms increases by 4 each time)
Next number: 45+21=66
12=6+6, 21=15+6, 29=28+1, 30=15+15
Exercise 2.2
1 81
2 a
25 b 100 c 49
3 9 and 36
4 A: 8, B: 64, C: 125
5 a
125 b 1 c 27
6 Odd Not odd
Cube number 1 or 27 8 or 64
Not a cube
number
Learners’
own answers
Learners’
own answers
7 27 and 125
8 23
, half of 42
, 32
−1and 22
×2=8
32
and 23
+1=9
Adding two consecutive square numbers
4+9=13
9+16=25 25−13=12
16+25=41 41−25=16
25+36=61 61−41=20
36+49=85 85−61=24
The differences between the terms increase by
4 each time.

Adding odd numbers
1+3=4
The sum of the first two
odd numbers=22
1+3+5=9 The sum of the first three
odd numbers=32
1+3+5+7=16 The sum of the first four
odd numbers=42
1+3+5+7+9=25 The sum of the first five
odd numbers=52
Exercise 2.3
1
18
24
5
30
60
20 15
29
14 21
Multiples of 5
Multiples of 2 Multiples of 3
Common multiples of 2, 3 and 5: 30, 60
2 105, 120, … (any multiple of 15 bigger
than 100)
3 24 and 48
4 once more on day 12 (12 is a common multiple
of 3 and 4)
Common multiples. 12 is a common multiple
of 3 and 4 so they both play football on the
12th day which is one time in a fortnight.
5 a
1, 2, 3, 6, 9, 18
b 1, 2, 3, 4, 6, 8, 12, 24
c 1, 2, 3, 6 circled
6 1, 2, 5 and 10
7 Pierre shares the cards equally so any factor of
32 could be the number of friends. The factors
of 32 are 1, 2, 4, 8, 16 and 32.
8 a
65 b 15
One possible answer is: 5, 1, 3, 6, 2, 4
Check your progress
1 4.6
2 a
multiply by 7 b 70
3 7×7 and 7+7+7+7+7+7+7
4 42
5 8
Unit 3 Averages
Getting started
1 a red b blue c white
2 Group A: 6 years old
Group B: 7 years old
Group C: 9 years old
Group D: 11 years old
3 The mode is 2. The median is 3.
The shopkeeper should use the mode as the
average as only one size 3 was sold. Size 2 is
more typical of the data as four size 2 tops
were sold.
Exercise 3.1
1 a mode: 4cm and 5cm
median: 5cm
b mode: 51mm and 59mm
median: 52mm
c mode: 1.2m and 1.8m
median: 1.5m
d mode: 101cm and 102cm
median: 101.5cm
2 a 6 b 7 c 11 d 4
3 a
Player C has the highest mean
bowling score.
b B is better because their scores are more
consistent and all their scores are above
100 or D is better because they scored
over 150 in two of their games whereas B
never scored over 150.

4 The range of Sarah’s practice times is
18 minutes.
The range of Anita’s practice times is
4 minutes.
Anita has the smallest range of times, so she
is more consistent in the amount of time she
practises for over the 5 days.
5 a 3 b 10 c 65 d 42
e 5 f 9 g 25
6 The range of Group 1 is 25cm.
The range of Group 2 is 14cm.
Group 1 has the largest range.
The children in Group 2 are closer in height
to each other. The heights of the children
in Group 1 are more spread out and there is
a bigger difference between the tallest and
smallest child in Group 1.
7 a Range Mode Median Mean
Kali 6 6 6 7
Summer 8 0 3 3
Benji 4 0 0 1
Kyle 5 4 and 7 5 5
b Kali has been most successful at skipping
because Kali has the highest median and
mean averages.
c Learners should indicate the mean or the
median. The mode could represent two
tries that were much lower or higher than
all the others; the mean and the median
represent the middle of the data.
8 a Range Mean Mode Median
Week 1 7°C 22°C 20°C 20°C
Week 2 6°C 27°C 25°C 27°C
Week 3 8°C 25°C 27°C 27°C
Week 4 8°C 18°C 19°C 19°C
Week 5 10°C 23°C 28°C 23°C
Week 6 8°C 30°C 26°C 30°C
b Week 5
c Learners should use the information from
the table to argue which is the warmest,
for example: Week 6 is the warmest
because it has the highest mean and
median, and the mode in Week 6 is the
third highest, which also makes it warmer
than four other weeks.
d range: 12°C
mode: 20°C
median: 25°C
mean: 25°C
9 Fratania: mean (99mm)
Spanila: mode (95mm)
Brimland: mode (97mm)
Gretilli: median (92mm)
Many possible solutions. For example:
5, 5, 5, 5
4, 5, 5, 6
3, 5, 5, 7
4, 4, 6, 6
1, 1, 9, 9
The total number of puppies in each solution
is 20, because 20 divided by 4 equals 5.
Check your progress
1 a mode: 4
median: 5
mean: 6
range: 6
b mode: 7
median: 6
mean: 6
range: 12
c mode: 29
median: 29.5
mean: 30
range: 19
d mode: 6 and 11 (bimodal)
median: 11
mean: 10.5
range: 9
2 mode: $0, median: $37, mean: $61
The median best describes the average amount
of money collected.
The mode $0 is not typical of the data because
5 of the 7 buckets collected more. The mean
$61 is not typical because one amount ($263)
has pushed up the mean and 6 of the 7
buckets collected less than $61.

Unit 4 Addition and
subtraction (1)
Getting started
1 □=15 ○=9
2 a
4412 (3214+1198)
b 1016 (3154−2138)
3 a
−5 b −4 c 1
Exercise 4.1
1 a 120000 b 140000
2 a
3428km b 34530km
3 Ravi should have regrouped 800 into 700+100
and written:
30000 + 6000 + 700 + 100 + 4
30000 + 1000 + 0 + 80 + 0
5000 + 700 + 20 + 4
= 31080
–
or
–
3 6 8 0 4
3 1 0 8 0
5 7 2 4
7
1
4 137 points
5 −9°C
6 3°C
7 −21°C
8 12°C
9 a
9°C b 19°C c 33°C
10 a 4 b 55
Three digits: answer is always 1089
Four digits: answer is always 10890
Two digits: answer is always 99
Exercise 4.2
1 a
3 spaces b 4 spaces c 7 spaces
2 a
equal b equal
c not equal
3 No, Khalid is not correct. Learners use a
counter example or other explanation, for
example:
If d=2 then 2+3=3+2 but 2−3=−1 and
3−2=1
4 a x 1 2 4
y 3 4 6
b y−x=2, or equivalent
5 a
x 1 2 3 4 5 6 7 8 9
y 8 7 6 5 4 3 2 1 0
b x+y=9, or equivalent
6 a=6cm and b=9cm
7 a
20cm b 28cm c 8cm
8 a
For example, x=4, y=7; x=6, y=3;
x=7, y=1
b p=x+x+y or p=2x+y
a 1 1 1 1 1 2 2 2 2 3 3 3 4 4 5
b 1 2 3 4 5 1 2 3 4 1 2 3 1 2 1
c 5 4 3 2 1 4 3 2 1 3 2 1 2 1 1
Check your progress
1 Missing answers: 5, 6 and 5
Calculation is: 43259+8526=51785
2 a
−5°C b 9°C
3 m=3cm and n=7cm
Method:
• m+n+n=17
m+n=10
n=7
• m+7+7=17
m+14=17
m=3

Unit 5 2D shapes
Getting started
1 A and iii, B and v, C and iv, D and i, E and vi,
F and ii
2 a 1 b 0 c 3
3 a The shape has 2 right angles.
b The shape has 1 curved edge.
c The shape has 3 straight edges.
d The shape has 1 pair of parallel sides.
4
Exercise 5.1
1 a
A square is a quadrilateral.
b It has 4 equal sides.
c It has 2 pairs of parallel sides.
d The sides meet at 90°.
e The diagonals bisect each other at 90°.
f It has 4 lines of symmetry.
2 a
A parallelogram is a quadrilateral.
b It has 2 pairs of equal sides.
c It has 2 pairs of parallel sides.
d It has 2 pairs of equal angles.
e The diagonals bisect each other.
f It has 0 lines of symmetry.

Think like a mathematician 1
a Group 1: A, D, E Group 2: B, C, F
b i
An isosceles trapezium is
a quadrilateral.
It has one pair of equal sides.
It has 1 pair of parallel sides.
It has 2 pairs of equal angles.
It has 1 line of symmetry.
ii
A trapezium that is not isosceles is a
quadrilateral.
It has 1 pair of parallel sides.
3 a square
rectangle
parallelogram
trapezium
isosceles trapezium

rhombus kite
b
Quadrilateral
Square Rectangle Parallelogram Trapezium Isosceles
trapezium
Rhombus Kite
Four equal sides ✓ ✓
Two pairs of
equal sides
✓ ✓ ✓
One pair of
equal sides
✓
One pair of
parallel sides
✓ ✓
Two pairs of
parallel sides
✓ ✓ ✓ ✓
All angles 90º ✓ ✓
One pair of
equal angles
✓
Two pairs of
equal angles
✓ ✓ ✓
Diagonals bisect
each other
✓ ✓ ✓ ✓
Diagonals meet
at 90º
✓ ✓ ✓
4 Sofia is correct. An isosceles trapezium is the only quadrilateral with one pair of equal sides.
5 ii
True. Learners’ own answers, for example: Because we are told that the shape is a parallelogram
and opposite sides of a parallelogram are parallel.
iii True. Learners’ own answers, for example: Because we are told a is parallel to e and in the
parallelogram we know that c is parallel to a, so c must also be parallel to e.
iv False. Learners’ own answers, for example: Because in the trapezium there is only one pair of
parallel sides and these are the opposite sides e and g, so f cannot be parallel to g.

6 Learners’ own answers, for example:
a
b
c
d
Learners’ own posters showing how you can
decompose each of the seven special quadrilaterals
into other shapes.
Exercise 5.2
1 Radius
Diameter
Centre
Circumference
2 Learners’ accurate drawings of a circle with a
radius of:
a 5cm b 60mm
a Arun is correct. Learners’ own
explanations, for example: Sofia’s circle
has a diameter of 60mm, which is the
same as 6cm. The diameter is the distance
all the way across the circle, so halfway
across is 3cm, which is the radius.
b radius=half of diameter or
diameter=twice the radius
3 Group 1 (radius of 2 cm): A, F, H
Group 2 (radius of 4 cm): C, E, G
Group 3 (radius of 10 cm): B, D, I
4 a
Gethin’s diameter does not go through the
centre of the circle.
b
Diameter
5 a, b
Learners’ accurate drawings of a circle
with a radius of 3.5cm
6 a radius b circumference
c diameter
Learners’ own answers
a,b
Learners’ accurate drawings of the circles A
and B
c 8.5cm
d
Learners’ own answers, for example: Distance
between the centres is the same as the sum of
the two radii.
e
Learners’ own answers, for example: Distance
between the centres is the same as the sum of
the two radii.
f
The distance between the centres of two
touching circles is the same as the sum of the
two radii.

Exercise 5.3
1 a rotational symmetry order 4
b rotational symmetry order 2
c rotational symmetry order 2
d rotational symmetry order 1
e rotational symmetry order 4
f rotational symmetry order 2
2 A, b and ii B, c and i C, a and iii
a No. Learners’ own answers, for example:
A trapezium has no lines of symmetry but
rotational symmetry order 1.
b No. Learners’ own answers, for example:
A parallelogram has no lines of symmetry
but rotational symmetry order 2.
4 a
A: rotational symmetry order 4
B: rotational symmetry order 2
C: rotational symmetry order 1
D: rotational symmetry order 3
Check your progress
1 a A square has 4 equal sides.
b A parallelogram has 2 pairs of
parallel sides.
c The diagonals of a kite meet at 90°.
d An isosceles trapezium has 1 pair of
equal sides.
e A rectangle has 2 lines of symmetry.
f A rhombus has 2 pairs of equal angles.
2 a
Learners’ accurate drawings of a circle
with a radius 2.5cm
b Correct labelling of the centre,
circumference, a radius and a diameter
on their circle. For example:
Centre
Diameter
Radius
Circumference
Unit 6 Fractions and
percentages
Getting started
1 150g
2 a
44%,
44
100 b 75%,
75
100
c 30%,
30
100
3 4.84.5
4.85.5
4 0.2
1
4
0.3 70%
3
4
5
1
5
=0.2
Exercise 6.1
1 a
5
6
b
6
5
c
10
4
d
4
10
2
3
5
3 operator
4 a
$12 b $15
c 10 metres

5 Halima swims further than Bella.
1
2
of 500=250m and
3
10
of 800=240m
6 Bricks left:
2
5
of 90=36 and
1
6
of 90=15
36+15=51
7
Fraction
1
4
3
4
5
4
7
4
9
4
11
4
Amount 6 18 30 42 54 66
8 a
36 b 49 c 16 d 20
6
6
6
5
6
4
6
3
6
2
6
1
3
3
3
2
3
1
5
5
5
4
5
3
5
2
5
1
2
2
2
1
4
4
4
3
4
2
4
1
1
1
9 4
6
10
4.6 4
3
5

23
5
10 9
3
5
48÷5 9.6
Exercise 6.2
1 a
4 b 7cm
c $2 d 12kg
2 48kg
3 10%=
1
10
and to find
1
10
of a quantity you
divide it by 10
50%=
50
100
=
5
10
=
1
2
so you divide by 2, not by 5
4 75%
5 a
34% b 20% c 20%
6 a
18 b 108 c 117
7 $160
8 25%
9 a
$44 b $6
10 80 children
32 children is 40% of the class so 8 children
are 10% of the class. 100% or the whole class
is 8×10 or use the diagram to show that each
division represents 8 children or any other
suitable method
10% of 600 is 60 20% of 500 is 100
50% of 150 is 75 25% of 160 is 40
Exercise 6.3
1
1
2
2 a
1
4
b
3
5
c
3
4
3
2
8
=
4
16
1
4
is the simplest form
4 a
3
2
b
5
4
c 1
2
5
5 A: true
B: false (
4
5
=80%)
C: false (
7
10
=70%)
6
60
100
and
6
10
7 Fraction Decimal Percentage
57
100
0.57 57%
8
100
=
2
25
0.08 8%
1
1
4
1.25 125%
8 a
1
8
,
1
4
,
3
8
,
5
8
,
3
4
b
1
2
,
7
12
,
2
3
,
3
4
,
5
6
9 7.77, 7.71, 7.7, 7.17, 7.07
10
7
10
0.07
23%=0.23
75%
4
5
11 Anil is correct.
2
5
=0.4 and 0.40.25
12 50%
3
5
0.65
7
10

1
1
=1,
1
2
=50%,
1
4
=25%,
1
5
=20%,
2
1
=2,
2
2
=1,
2
4
=50%,
2
5
=40%,
4
1
=4,
4
2
=2,
4
4
=1,
4
5
=80%,
5
1
=5,
5
2
=2.5,
5
4
=1.25,
5
5
=1
5, 4, 2.5, 2, 1.25, 1; 80%, 50%, 40%, 25%, 20%
Check your progress
1
4
3
of 24 because
3
4
of 40=30 and
4
3
of 24=32
2
3
4
3 0.42
9
20
55%
3
5
66%
4 a
A: true; B: false; C: true
b 30% c 8
Unit 7 Exploring
measures
Getting started
1 a area: 12m2
, perimeter: 16m
b area: 25mm2
, perimeter: 26mm
c area: 115km2
, perimeter: 48km
2 a 22:03 b 11:44
Exercise 7.1
1 a 18cm2
b 9cm2
2 a 81cm2
b 40.5cm2
3 a 630cm2
b 315cm2
c 315cm2
4 A: 2cm2
, B: 5cm2
, C: 6cm2
, D: 4.5cm2
,
E: 7, 7.5 or 8cm2
5 a
Triangle 1 2 3 4 5 6
Area (cm2
) 0.5 2 4.5 8 12.5 18
b The area of the triangle would be 24.5cm2
.
c The area of the triangle would be 50cm2
.
d Learners should notice that the area
increases with each triangle. They might
notice that the pattern of the increase
between the areas is 1.5, 2.5, 3.5, 4.5, etc.
so they could use this to predict the area
of the next triangle. Learners might notice
that the area of the triangle is the number
of the triangle multiplied by itself and
divided by 2, or link this to the area of the
rectangle that the triangle sits inside.
The area of each rectangle is 9cm2
. The triangles
all have approximately the same area. (The
triangles have exactly the same area, but this
cannot be seen from the investigation.)
6 a rectangle: 16cm2
triangle: 8cm2
b rectangle: 15cm2
triangle: 7.5cm2
c rectangle: 20cm2
triangle: 10cm2
d rectangle: 12cm2
triangle: 6cm2
e rectangle: 21cm2
triangle: 10.5cm2
f rectangle: 12cm2
triangle: 6cm2
7 18cm2
8 27 biscuits

Exercise 7.2
1 Hours Hours and minutes
0.1 hours 0 hours and 6 minutes
1 hour 1 hour and 0 minutes
1.1 hours 1 hour and 6 minutes
2 a 1.25 hours
b Tom has converted 1.25 hours to 1 hour
and 25 minutes. This is wrong because
0.25 hours equals one quarter of an hour
which is 15 minutes. Each child can have
the console for 1 hour and 15 minutes.
3 Runner Hours Minutes Seconds
Emmanuel 2 8 39
Paul 2 9 15
Kazuyoshi 2 15 27
Florence 2 21 33
Yared 2 21 42
Susan 2 38 54
Gianmarco 2 39 6
Mai 2 54 18
Emily 3 2 48
Maria 3 3 3
Times where the decimal part of the hours are
0 have the same digits as the time in hours and
minutes, for example, 12.0 hours is equal to 12
hours and 0 minutes.
Check your progress
1 A 18cm2
B 3cm2
C 10cm2
2 a 4cm2
b 4.5cm2
c 6cm2
3 C
4 a 3 hours 30 minutes
b 14 hours and 6 minutes
c 9 hours and 15 minutes
d 5 hours and 42 minutes
e 11 hours and 24 minutes
f 1 hour and 3 minutes
5 2.4 hours
2 hours and 24 minutes
Unit 8 Addition and
subtraction (2)
Getting started
1 5 tenths
2 13.13
3 0.2 metres
4 a
98.73 b 7.55
5 a
7
5
b
4
8
=
1
2
c
5
10
=
1
2
6
5
6
Exercise 8.1
1 D
2 0.8 or
8
10
and 0.004 or
4
1000
3 0.88 and 0.12
4 5.05+5.115=10.165
5 a
20.478 b 30.864 c 76.934
d 29.46 e 15.853 f 20.614
6 Ahmed should make sure the decimals have
the same number of decimal places, then write
down the calculation in columns.
0 7 0
0 4 1
1 1 1
+
7 0.171

8 18.95kg
9 $191.27
10 0.066kg
1.604+2.375=3.979 and 3.476−2.501=0.975 or 3.501−2.476=1.025
Both fractions are almost a half. Both are a half of a fraction away from a half. Since sevenths are bigger
than ninths, the fraction with ninths is closer to a half, so
4
9
must be bigger.
Exercise 8.2
1 Calculation Common denominator Equivalent calculation Answer
1
3
+
1
6
6
2
6
+
1
6
3
6
=
1
2
7
10
−
1
2
10
7
10
−
5
10
2
10
=
1
5
6
5
+
1
2
10
12
10
+
5
10
17
10
=1
7
10
2 a
23
20
=1
3
20
b
7
24
c
59
40
=1
19
40
3
19
20
and
31
12
=2
7
12
4 They are both correct because
19
15
=1
4
15
5 a
23
10
=2
3
10
b
53
12
=4
5
12
c
43
24
=1
19
24
6 a
19
10
=1
9
10
b
13
12
=1
1
12
c
28
15
=1
13
15
7
7
12
8
1
36
9
1
12
1
5
+
1
2
=
7
10
1
5
+
1
3
=
8
15
1
5
+
1
4
=
9
20
1
5
+
1
5
=
10
25
1
5
+
1
6
=
11
30
1
5
+
1
7
=
12
35
1
7
+
1
2
=
9
14
1
7
+
1
3
=
10
21
1
7
+
1
4
=
11
28
1
7
+
1
5
=
12
35
1
7
+
1
6
=
13
42
1
7
+
1
7
=
14
49
1
9
+
1
2
=
11
18
1
9
+
1
3
=
12
27
1
9
+
1
4
=
13
36
1
9
+
1
5
=
14
45
1
9
+
1
6
=
15
54
1
9
+
1
7
=
16
63
General case:
1
m
+
1
n
=
m n
mn
+

Check your progress
1
67
12
or 5
7
12
2 a 10.096 b 2.638
c
13
24
3
13
30
4
2 1
5 7
9
6
6 1
9
2
9
3
–
Unit 9 Probability
Getting started
1 Learners’ own answers. C and D should point
to ‘impossible’. F should point to ‘certain’.
2 a true b false
c true d false
3 Gabriela is not correct. The next flip has an
even chance of being a head or a tail.
Exercise 9.1
1 a 1 out of 4 b 1 out of 4
c 2 out of 4 or 50%
2 a 50% b 0%
c 50% d 100%
3 a
Learners’ own answers. Set of numbers
where less than 50% of the numbers are 8
b Learners’ own answers. Set of numbers
where 5 out of 6 of them are less than 5
c Learners’ own answers. Set of numbers
where there are more 3s than 1s
d Learners’ own answers. Set of numbers
where more than 50% of them are 4
e Learners’ own answers. Set of numbers
where 2 out of 5 of them are 3
4 a 40 b heads up
c 16 out of 40, or 2 out of 5
d Learners’ own answers
e Learners’ own answers
f Learners’ own answers
g Learners’ own answers
5 a – c
odd
numbers
multiples
of 10
10
1 3
9 11
15
24
22
18
16
14
12
8 2
4 6
26 28
17
21 23
27 29
25
19
13
5 7
20
30
d The chance of winning a big prize is 0
because there are no numbers that are odd
and multiples of 10.
e yes
6 a mutually exclusive
b not mutually exclusive
c mutually exclusive
7 a i yes
ii no
iii yes
iv yes
v yes
vi no
b A and C, C and D
c Learners’ own answers, for example:
Event 1: You roll a 3 on the red dice.
Event 2: You roll an even number on the
blue dice.
d Learners’ own answers, for example:
Event 1: You roll a 1 on the red dice.
Event 2: You roll a number on the blue
dice that is less than the number on the
red dice.
8 a 8 red (H), 6 blue ( ), 2 yellow ( )
b 20 red (H), 15 blue ( ), 5 yellow ( )
c 100 red (H), 75 blue ( ), 25 yellow ( )
9 Learner’s own answers.

Learners’ own answers. Learners should compare
experimental probability with actual probability,
and show that they know that the more trials that
are carried out, the more likely that the results
would be closer to the experimental probability.
Check your progress
1 a 1 out of 8
b 2 out of 8 (or 1 out of 4)
c 1 out of 8
d 2 out of 8 (or 1 out of 4)
e 50% f 75%
2 Events 1 and 3
Unit 10 Multiplication
and division (1)
Getting started
1 a
4224 b 918 c 67
2 180×8=1440 metres
3 2
4 40
5 12×30 or 30×12
Exercise 10.1
1 No, there are too many zeros. A good estimate
would be 1500×60=90000
2 a
29568 b 37044 c 29984
3 8638
4 Pierre thinks that 0 hundreds multiplied by
7=7 hundreds. The correct answer is 42168.
5 400×60 8000×3 20×1200
6 79×60×24=113760 beats
7 a
164670 b 163950 c 533470
8 a
25764 b 67553 c 434625
9 7×18×25=3150
10 6, 7 and 4 (3627×42)
25 and 26
Exercise 10.2
1 No. The estimate of 564÷14 is not 30 because
600÷15=40 (or 600÷10=60)
2 a
109 r2 or 109
2
7
b 27
c 28
3 Both girls are correct because a remainder of
3 in this case is equivalent to
3
7
4 8 (588÷14=42 and 374÷11=34)
5 a
77
b 64
c 49
6 57 boxes
7 Mandy is wrong. She should find factors of 15
not decompose it.
She should divide by 5 and 3.
825÷5÷3=165÷3=55
8 576÷72=8 306÷34=9
9 a
2 and 1 (246÷4=61 r2)
b 61
2
4
=61
1
2
10 $27
11 a
3 b 4
Answers will depend on the numbers chosen
by learners, but the largest answer is always
found by dividing the largest dividend by the
smallest divisor and the smallest answer is always
found by dividing the smallest dividend by the
largest divisor.
Exercise 10.3
1 43719 because the sum of the digits is divisible
by 3 (4+3+7+1+9=24)
2 a
19×3=57 b 17×3=51
c 14×3=42 or 15×3=45 or 16×3=48

3 84
4 99, 108, 117, 126, 135
5 96
6
divisible
by 6
divisible
by 9 divisible
by 3
36
24 27
21
16
They are multiples of 3, 6 and 9.
7 Divisible
by 3
Divisible
by 6
Divisible
by 9
987 ✓
495 ✓ ✓
3594 ✓ ✓
8 210
9 a
231 or 234 or 237
b 315 or 345 or 375
c 83049 or 83349 or 83649 or 83949
2334, 1335, 2337, 1338
2367, 1368
2634, 1635, 2637, 1638
2667, 1668
2934, 1935, 2937, 1938
2967, 1968
Check your progress
1 a
27672 b 235380 c 256428
d 77 e 54 f 19
2 Always true because the number is a multiple
of 2 and a multiple of 3
3 a
92÷4=23 so the calculation is correct
b 14×9=126 so the calculation is incorrect
4 Kofi is correct.
Vijay has forgotten to add in the 1 hundred
that has been carried.
5 80
Unit 11 3D shapes
Getting started
1 A and iii, B and i, C and iv, D and ii
2 a ii, iv b i, vi c iii, v
3 G, A, D, B, F, E, C, H
Exercise 11.1
1 a cube and a (square-based) pyramid
b cuboid and a triangular prism
c cylinder and a cone
2 Group 1: A, D, E, G Group 2: B, C, F, H
3 Learners’ sketches of a compound shape made
from these simple shapes
a two different cuboids, for example:
b a cuboid and a square-based pyramid,
for example:
c two different cylinders, for example:

4 Learners’ own answers. For example:
a A cube has 6 faces.
All the faces are squares.
b A square-based pyramid has one square
face and four triangular faces.
c A cylinder has two circular faces and a
curved face which is a rectangle when flat.
d A triangular-based pyramid has four
triangular faces.
Learners’ own answers. For example:
a The surface area is the total area of all
the faces. You work it out by finding
the area of each face then adding them
all together.
b i
Find the area of one square face then
multiply by six because all six faces
are the same size.
ii
Work out the area of the square base.
Calculate the area of one triangular
face. Add the area of the square to
four times the area of the triangle.
c The surface area of a 3D shape is the total
area of all its faces.
5 a
A triangular prism has a total of five
faces. Two of the faces are triangles and
three of the faces are rectangles.
b Any correct net for the triangular prism.
For example:
6 A and iii, B and i, C and ii
a 8
b Learners’ own answers. For example, count
how many you need to complete each layer
then add these together to get the total.
7 a 7 b 6 c 6
Exercise 11.2
1 a i 500ml ii 300ml
b i 100ml ii 80ml
c i 5000ml ii 2000ml
2 a Learners’ own answers. For example:
1000ml=1 litre, so 2000ml=2 litres
2500ml=2000ml+500ml, so
2500ml=2 litres 500ml
Also 500ml=0.5 of a litre, so
2500ml=2.5 litres

b
millilitres
litres and
millilitres
litres
2500ml 2l 500ml 2.5l
3200ml 3l 200ml 3.2l
4300ml 4l 300ml 4.3l
3700ml 3l 700ml 3.7l
800ml 0l 800ml 0.8l
12100ml 12l 100ml 12.1l
a Arun is correct. Learners’ own answers.
For example: There are five increments on
the scale for 1 litre, so each increment on
the scale is worth 200ml. The water is one
increment above the 6 litres mark, so there
is 6 litres 200ml of water. This is the same
as 6.2 litres.
b Learners’ own answers. For example:
Marcus has assumed that each increment
is worth 100ml not 200ml.
c 3.8 litres or 3 litres 800ml
b i 2l ii 1.6l
c i 1000ml ii 650ml
4 a 80ml b 34ml c 2.6l
5 Learners’ own answers. For example: Fill
the jug to capacity (500ml) four times then
measure an extra 300ml.
6 90 litres
a One solution for each is given.
There may be other alternatives.
i 400ml cup A+cup B
ii 360ml cup A+cup C
iii 420ml cup A+cup C+cup D
iv 320ml 2×cup B
v 180ml cup C+cup D
vi 600ml 5×cup C
7 Estimates should be approximately:
A: 500ml/0.5 litres
B: 200ml/0.2 litres
C: 1000ml/1 litre
D: 600ml/0.6 litres
E: 750ml/0.75 litres
F: 1500ml/1.5 litres
Order: B, A, D, E, C, F
8
a 1 litre: Fill B, fill A from B, empty A. There is
1 litre in B.
2 litres: Fill A, pour contents of A into B, fill A,
fill B from A, empty B. There are 2 litres in A.
3 litres: Fill A.
4 litres: Fill B.
5 litres: Make 1 litre as above. Pour the 1 litre
into A. Fill B.
6 litres: Fill A, pour A into B, fill A. There are
3 litres in each jug which equals 6 litres.
7 litres: Fill both A and B.
b 1 litre: Fill A, pour A into B, fill A, fill B from
A, pour away B. There is 1 litre in A.
2 litres: Fill B, fill A from B, pour away A.
There are 2 litres in B.
3 litres: Fill A.
4 litres: Make 2 litres as above. Pour B into A,
fill B, fill A from B, pour away A. There are 4
litres in B.
5 litres: Fill B.
6 litres: Fill A, pour A into B, fill A. There are
3 litres in each jug which equals 6 litres.
7 litres: Make 2 litres as above. Pour B into A,
fill B. There are 2 litres in A and 5 litres in B
which equals 7 litres.
8 litres: Fill both A and B.
c Only 3, 6 and 9 litres can be made. This could
be because 6 is a multiple of 3.
d Learners’ own answers
Volume of
500ml or less
Volume of more
than 500ml
Capacity of
1 litre or less
E C F
Capacity of
more than
1 litre
B A D

Check your progress
1 a cylinder and a cone
b cuboid and (rectangular-based) pyramid
c three cuboids
2 Learners’ sketches of a compound shape
made from a cuboid and a triangular prism.
For example:
3 a
A square-based pyramid has a total of
five faces. Four of the faces are triangles
and one of the faces is a square. The
surface area of a square-based pyramid is
the total area of all its faces.
b Learners’ sketches of a net for the square-
based pyramid.
For example:
4 4
b 220ml
12 Ratio and
proportion
Getting started
1 a
4:5 b
4
10
or 40%
c 1:5:4
2 a
false (it should be 3 : 2)
b true c true
d false (it should be 2 in every 5 parts)
3 Odette has confused ratio and proportion.
She saw one triangle and three circles.
The ratio of triangles to circles is 1 : 3. She
should have said that 1 out of every 4 shapes is
a triangle (or 1 in every 4 shapes is a triangle).
Exercise 12.1
1 a
1 : 2 b 2 : 3 c 3 : 2
2 a
1 : 4 b 3 : 2 c 4 : 3
d 3 : 1 e 6 : 1 f 1 : 20
3 a
18 b 15 c 25
4
Equivalent to
2 : 3
Equivalent to
3 : 4
Equivalent to
4 : 5
18:27 4:6
14:21 8:12
12:16 24:32
18:24 21:28
28:35 16:20
36:45 32:40
36 : 27 cannot be placed in the table
5 a
36 onions
b 84 carrots and onions
6 70 boys and girls
7 a
16 b 6 c 9
Example method using trial and improvement:
Number
of white
beads
Cost
of
white
beads
Number
of
coloured
beads
Cost of
coloured
beads
Total
cost
3 30c 1 20c 50c
24 240c 8 160c 400c
or
$4
24 white beads and 8 coloured beads
Exercise 12.2
1 $10
2 4 pizzas
3 200ml cream, 250ml milk,
1
2
kg (or 500g)
raspberries and 125g sugar
4 $42

5 a
400cm or 4m
b 15cm
c 42.5cm
6 Scale 1 : 18 Scale 1 : 24 Scale 1 : 32
Puma Beetle, Embla Delta, Modi
7 Ratio of
rectangle
sizes
Length
in cm
Width
in cm
Perimeter
in cm
A 5 2 14
B A to B=1 : 2 10 4 28
C A to C=1 : 4 20 8 56
D A to D=1 : 6 30 12 84
The width on one row becomes the height on the
row below.
A6 paper will be 148mm high.
1189÷841=1.41
841÷594=1.41
And so on. They are all 1.41.
Check your progress
1 a
3 : 4 b 1 : 3
c 1 : 5 d 4 : 1
2 a
8 boys b 18 girls
3 4 mangoes
1 litre of apple juice
500ml yogurt
2 bananas
4 100cm
Unit 13 Angles
Getting started
1 A and iii, B and iv, C and i , D and ii
2 a right b obtuse
c acute d reflex
3 a 0° and 90° b 180° and 360°
c 90° and 180°
4 a a=120° b b=38°
c c=70°
Exercise 13.1
1 Learners’ own estimates
a=20° b=45° c=72°
d=100° e=125° f =168°
g=200° h=255° i=302°
x=40° y=320°
4 a
Learners’ own answers – accurate
drawings of the following angles:
i 30° ii 145°
iii 245° iv 350°
b, c Learners’ own answers
5 a
Yes. Learners’ own answers, for example:
If she measures one of the angles, she can
work out the other one by subtracting the
measured one from 180°.
b x=62° and y=118°, including learners’
own answers.
6 a v=303° and w=57°
b Calculation to check that v and w add up
to 360°
7 a No. Angle r is 23° which is greater than
20°, not less than.
b Learners’ drawings of a wheelchair ramp
with an angle, r, of between 7° and 15°
8 a x=42°, y=75° and z=63°
b Correct check, for example,
42°+75°+63°=180°
c x=42°, y=75° and z=63°
d Learners’ own answers. For example: The
angles in the triangle are the same as the
angles on the straight line. The angles in
the triangle add up to 180°.

Exercise 13.2
Learners’ own answers. For example: The three
angles fit together exactly on the straight line. The
sum of the angles in a triangle is the same as the
sum of the angles on a straight line, which is 180°.
1 a x=60° b x=50° c x=20°
2 a y=50° b y=35°
3 a
In an isosceles triangle, two of the angles
are equal. a and 50° are the same size,
so a=50°.
Yes, because he has shown every step in
his working.
c Learners’ own answers. For example:
50°+50°=100°, 180°−100°=80°, or
180°−2×50°=80°
4 a z=124° b z=66°
5 Learners’ own answers. For example:
180°−126°=54° and 54°÷2=27°
6 Yes. Learners’ own answers. For example:
180°−146°=34° and 34°÷2=17°.
Angle p=17°, which is greater than 15°
7 a Marcus
Marcus is correct because 180°÷3=60°
(or 60°×3=180°)
Arun is incorrect because 80°×3=240°,
which is not 180° (or 180°÷3=60°
not 80°)
8 a Group 1 (scalene triangles): A, E, F, H
Group 2 (isosceles triangles): B, C and G
Group 3 (equilateral triangles): D, I
b G
9 a Angles on a straight line add to 180°.
b a=53°
c b=43°; rule used: Angles in a triangle add
to 180°.
Check your progress
1 a=25° b=130° c=275°
2 Learners’ own answers. Accurate drawings of
the following angles:
a 80° b 175° c 315°
3 a x=75° b x=25°
4 a y=110° b y=30°
Unit 14 Multiplication
and division (2)
Getting started
1 a
1
3
of 21=7 b
1
4
of 24=6
c
1
5
of 40=8
2 a
3 b 4 c 6
3 36536
4 $35
Exercise 14.1
1
4
6
+
4
6
+
4
6
=
12
6
=2
3×
4
6
=
12
6
=2 or
4
6
×3=
12
6
=2
2 5×
2
8
=
10
8
=1
2
8
=1
1
4
or
2
8
×5=
10
8
=1
2
8
=1
1
4
3 a
6
5
=1
1
5
b
35
8
=4
3
8
c
20
6
=3
2
6
=3
1
3
4 Answer=1 Answer=2 Answer=3
A, D, F B, E, H C, G
5 C

6 Diagram showing the correct fractions of the
pie, for example:
Oscar
Bruno
÷ 3 =
2
3
=
1
3
3
9
2
9
Ali
7 a
5
16
b
2
15
8 a
2
5
b
5
16
c
3
25
9 a
6 b 9
5
12
÷2
5
8
÷3
5
6
÷4
5
4
÷6
5
3
÷8
5
2
÷12
5
1
÷24
Exercise 14.2
1 a
225.5 b 266.84 c 360.81
2 207.36
3 105.24
4 a
1115.4 b 4481.92 c 1915.64
5 Not correct. The answer should be:
1 7 0 5
× 1 5
8 5 2 5
1 7 0 5 0
2 5
1
5 7 5
17.05 × 5 should be 85.25 and not 85.75
because 5 × 0 = 0, not 5
To avoid similar errors, Parveen should
remember that if any number is multiplied by
zero, the answer is zero.
6 6.47×5
7 $183
8 $0.78 or 78 cents (35.28−34.50=0.78)
3.25×64=208
Exercise 14.3
1 a
9.3 b 0.4
c 0.09 d 28.01
2 a
3.07 b 5.05
3 8
4 6.2
5 12.32÷8=1.54, all the other answers are
1.55
6 $9.84
7 Shop A, as shop A charges $1.72 for a pot
and shop B charges $1.74 for a pot. Learners
will need to think about how many pots of
paint they need. The cost of 1 pot in shop A
is cheaper, but they must buy 4 pots at a time.
If you only needed 3 pots, you would go to
shop B.
8 5.25m
329.68÷52=6.34
Check your progress
1 a
12
5
=2
2
5
b
35
6
=5
5
6
c
18
3
=6
2 Both equal to
2
15
3 $5.25
4 8.6÷5=1.72 is the odd one out. All the other
answers are 1.75.
5 6×
7
8
=5
1
4
6×
3
4
=4
1
2
6×
5
8
=3
3
4
8×
7
8
=7 8×
3
4
=6 8×
5
8
=5
10×
7
8
=8
3
4
10×
3
4
=7
1
2
10×
5
8
=6
1
4

a Smallest answer is 6×
5
8
=3
3
4
b Largest answer is 10×
7
8
=8
3
4
Unit 15 Data
Getting started
1 a Weather Tally Total
Cloudy IIII IIII IIII 14
Rainy IIII I 6
Sunny IIII IIII 10
b Learners’ own waffle diagrams showing
the correct tally results, for example:
Key
Cloudy
Rainy
Sunny
Waffle diagram
showing the weather
over 30 days
2 a
10 minutes or more, but less than
20 minutes
b 3
c 32
d Learners’ own answers. For example: The
children in Group 1 might be younger and
attend a small local school. The children
in Group 2 might be older and attend a
regional school.
Exercise 15.1
1 a 9 b 34
c 43 d 10
2 a 11 b 12
c They both played 44 matches.
d Ken scores more goals and scores goals
more often. Ken might be in a goalscoring
position and Ben might be a defender.
3 a 4–6 roses
b Both 1–3 and 10–12 roses
c 12
d 10
e 9
f 4
g 49
4 a 20
b,c Answer depends on group size chosen.
For example: 1–4 peas=6,
5–8 peas=20, 9–12 peas=23,
13–16 peas=19, 17–20 peas=4
Number
of peas
Tally Total
1–4 IIII I 6
5–8 IIII IIII IIII IIII 20
9–12 IIII IIII IIII IIII III 23
13–16 IIII IIII IIII IIII 19
17–20 IIII 4
1‒4
0
5
10
15
20
25
5‒8 9‒12
Number of peas
Bar chart showing how many peas
grew in different pea pods
Frequency
13‒16 17‒20
d Answer depends on group size chosen.
For example: The farmer would have
found out that the most common number
of peas found in a pod is between 9
and 12.
5 a football b volleyball
c 25% d 5
e 70% f 14
6 a Sport Frequency Percentage
Football 5 25%
Basketball 9 45%
Table tennis 3 15%
Volleyball 1 5%
Cricket 2 10%

b
Key
football
Waffle diagram
showing the sports
chosen by 20 children
basketball
table tennis
volleyball
cricket
I would choose basketball because
12 children chose basketball altogether,
which makes it the most popular out of
the 40 children.
d Learners’ own answers.
7 a i
1
5
ii
2
5
iii
3
10
b i 10% ii 20% iii 40%
c Learners’ own answers. For example: The
groups are similar because 10% of each
group prefer chalk.
d Learners’ own answers. For example:
The groups are different because in
group B pastel is the most popular, but
in group A pastel, pencil and ink are all
equally popular.
8
walk
car
bus
cycle
Pie chart showing how Asif’s
friends travelled to school
9 a
Use a pie chart as it will be clear if a
candidate has 50% or more votes by
whether more than half of the pie chart
is shaded.
b Use a bar chart as it will be clearer who
has the most votes.
10 Learners’ own answers
11 More than one possible answer, for example:
a We need to choose a sport for our class to
demonstrate at an open day. What sport
should we choose?
b The parents’ association are buying some
dance shoes for our class. What sizes
should they buy?
c We want to find out if the planting around
our school is good for encouraging bees.
What plants do we have, and how many
of them?
d The school is buying some scissors for
right-handed and left-handed learners.
How many should they buy of each?
Learners’ own answers.
Exercise 15.2
1 a 11 b 8 c 21
3 a
Learners’ own answers, for example:
15–19, 20–24, 25–29, 30–34, 35–39. All
groups should be the same size
b Learners’ own answers, for example:
Vertical jumps (cm) Tally Total
15–19 II 2
20–24 IIII I 6
25–29 IIII IIII IIII I 16
30–34 IIII I 6
35–39 I 1
c Learners’ own answers. Frequency
diagram depends on groupings chosen,
for example:
Vertical jump (cm)
Frequency
Frequency diagram showing how
high a class could jump

The most common height to jump was
between 25cm and 29cm. One child
jumped 35cm or higher.
4 a 19°C b 10:30 a.m.
c Thermometer 1: 18.5°C
Thermometer 2: 30°C
d Thermometer 1: The line goes up a little
then stays flat.
Thermometer 2: The line goes up more
steeply, then comes back down.
e Learners’ own answers. For example:
Thermometer 2 was put in or near a
source of heat that was removed at about
12 o’clock, such as a radiator that was
turned off or a sunny window that went
into shadow.
5 Line graph of pulse rate
Pulse
rate
0
10
0 20 30 40 50 60 70 80
20
10
30
50
70
90
110
130
40
60
80
100
120
140
Time (minutes)
a 30 minutes
b Dee’s pulse rate increased by 24 beats per
minute from 106 to 130.
c The line increases from 66 to 118,
decreases to 106, increases to 130 and
decreases to 68.
d i 102 beats per minute
ii 112 beats per minute
iii 78 beats per minute
6 a 21cm b 21cm
c 18cm d 24cm
7 a, c
0 5 10 15 20
0
1
2
3
4
5
6
7
8
9
10
Height (cm)
Scatter graph showing the height and
number of leaves of 11 plants
Number
of
leaves
b The more leaves the plant has, the taller
it is.
d A plant that was 14cm tall would have
approximately 7 leaves.
e Learners’ own answers
8 a line graph
b frequency diagram
c scatter graph
Check your progress
1 a comedy
b 20%
c 5 (25% of 20)
2 a, b
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
Scatter graph showing the temperature and the number
of people who visited a garden each day
c approximately 18 visitors

Unit 16 The laws of
arithmetic
Getting started
1 Learners’ own answers, showing that the
order of multiplication can be changed to give
the products of 7×8 and 5×2, for example:
7×8=56 and the other numbers are 5×2
which is 10.
2
7 × 20 28
4
168
= 140 +
7 × 24
7 × =
=
3 a
17 b 5 c 50
4 No. She is incorrect as the multiplication
should be done first. The answer should be 51.
5 a 3× 7 +9=30
b 6×8− 18 =30
Exercise 16.1
1 a
1×10−7=3. Tariq is thinking of
the number 3.
+7
×10
÷10
1
10
–7
3
b 6×2−5=7. Sonja is thinking of
the number 7.
+5
×2
÷2
6
12
–5
7
c (4+2)÷3=2. Pierre is thinking of
the number 2.
×3
+2
–2
4
6
÷3
2
d (14−11)×3=9. Lan is thinking of
the number 9.
÷3
–11
+11
14
3
×3
9
2 a
21 b 22 c 9
d 8 e 1 f 6
Calculation b is the same with the brackets
removed.
3 a
true
b true
c false (6+3)×4=36 or 6+3×4=36
4 a
(6+2)×5=40
b (3+4)×(2+4)=42
c 3×(4+2)=18
d (4+3+2)×2=18
5 Example answers:
a
(2+5)×5=35
b (7−5)×10=20
c 2×(14−5)=18
6 Learners’ own answers. Any correct way to
calculate 42×24 using two factors of 24.
7 a
(5×70)+(5×1)=350+5=355
b (6×60)−(6×3)=360−18=342
c (7×90)+(7×2)=630+14=644
d (8×40)−(8×3)=320−24=296
8 Working should be shown for all parts (see
example in part a).

a 3×(60+7)
= (3×60)+(3×7)
= 180+21
= 201
or
3×(70−3)
= (3×70)−(3×3)
= 210−9
= 201
b 744 c 336 d 711
9 a
False, the=sign should bebecause
8+5−7=6 and 8+7−5=10
b False, the=sign should bebecause
2×(3+4)=14 and 2×3+4=10
c True, (10×5)÷2=25 and 10×(5÷2)=25
Learners’ own answers. There are several possible
answers including:
11=(3×4)−1
12=3×4
13=(3×4)+1
14=(3×4)+2
15=(4+1)×3
16=(3+1)×4
17=3×(4+1)+2
18=(4+2)×3
19=4×(3+2)−1
20=(3+2)×4
Check your progress
1 2000
2 a
18 b 12 c 15
d 2 e 32 f 12
3 a
b
4 Example answers:
a
3×(4−2)=6 b 4×(3+7)=40
c 4×(15−12)=12 d (18−3)÷5=3
5 (4+5+1)×5=50
Unit 17 Transformations
Getting started
1 (4, 1)
2 (5, 1), (2, 2), (0, 3), (4, 5)
3 All points of the triangle move one square left
and two squares up.
4
5 (5, 6)
Exercise 17.1
1 A and ii, B and iv, C and i, D and iii
2 a (3, 0) b (−3, 1)
c (−2, −1) d (4, −3)
3 a, b
1 x
y
2 3 4 5
P
P′
Q′ R′
S′
P′′
Q′′ R′′
S′′
Q R
S
–4
–5 –3 –2 –1
–1
1
3
4
5
2
–2
–3
–4
–5
a P'(0, 2), Q'(1, 4), R'(4, 4), S'(5, 2)
b P''(−4, −4), Q''(−3, −2), R''(0, −2),
S''(1, −4)

a Yes. Learners’ explanations. For example:
The x-coordinate is −2. The y-coordinate
is 2
1
2
, which is the same as
5
2
, which can
also be written as 2.5.
b B is at 1
1
2
, −2 or
3
2
, −2 or (1.5, −2)
C is at 3
1
2
, 1
1
2
or
7
2
,
3
2
or (3.5, 1.5)
4 a (1, 2)
1
J
K L
x
y
2 3 4 5 6
–4
–5
–6 –3 –2 –1
–1
1
3
4
5
6
2
–2
–3
–4
–5
–6
b (−1, 1), (0, 0) or any point using decimals
or fractions between J and L such as
(0.5, −0.5) or (−0.5, 0.5)
c (−5, 2), (1, −4)
d Two from: (2, 3), (3, 4), (4, 5), (5, 6), etc.
e Parts b and d. Learners’ own answers.
For example:
b
There are two coordinates with whole
numbers, but you could also use
fractions, e.g. (0.5, −0.5).
d
J, K and L are the fixed vertices of
the ‘top’ of the kite, but the 4th vertex
can be any point along the diagonal
KM after the point (1, 2).
K
M M M
M
K K K
a Learners’ own answers. For example:
Group 1: Translation three squares right and
two squares up. E to G, C to D, I to H
Group 2: Translation six squares left. F to G,
A to C
Group 3: Translation one square left and six
squares down. C to E, B to H
Group 4: Translation 11 squares right and two
squares up. E to H, C to B
Group 5: Translation eight squares right.
D to B, G to H
a 2 squares right and 1 square down
b
7
6
1st
2nd
3rd
4th
x
y
8 9 10 11
2
1 3 4 5
4
6
5
8
9
7
3
2
1
–1
–2
–3
c
Kite 1st 2nd 3rd 4th
Coordinates (1, 7) (3, 6) (5, 5) (7, 4)
It is going up by two every time. It is the
odd numbers starting from 1.
e Learners’ own answers. For example:
It is going down by one every time.
Start at 7, and subtract 1 each time.
f 5th is (7+2, 4−1)=(9, 3)
6th is (9+2, 3−1)=(11, 2)
g Learners’ own answers

























Exercise 17.2
1 b and c
2 a
b
c
d
a No. He has turned shape A around, not
reflected it.
It is a good idea to use tracing paper,
but instead of turning the tracing paper
around, he needs to flip it over.
3 a
b
c
d
4 a i horizontal ii vertical
iii diagonal
b i
ii
iii

Exercise 17.3
1 a
C
b
C
a
C
B A
Learners’ own answers. For example: A 90°
anticlockwise turn is in the direction of this
arrow so A must be on the
right and B must be on the left.
b Learners’ own answers. For example: A
clockwise turn is in the same direction as the
hands of a clock like this:
An anticlockwise turn is in the opposite
direction to the hands of a clock like this:
2 a
C
b
C
3 a Example of groups:
Group 1: A to B 90° rotation clockwise,
i, iii, vi
Group 2: A to B 90° rotation
anticlockwise, ii, iv, v
a
b
C
i The lines that join the corresponding
vertices of the triangles after a
translation are parallel.
ii The lines that join the corresponding
reflection are parallel.
iii The lines that join the corresponding
rotation are not parallel.
d When you translate or reflect a triangle,
the lines joining the corresponding
vertices will always be parallel, but when
you rotate a triangle the lines joining
the corresponding vertices will never
be parallel.

e The rule works for any 2D shape.
In a rotation the shape is turned, but
in a translation or reflection it isn’t.
4 a
C
b
C
Check your progress
1 a A (−3, 1) b B (−4, −3)
c C (2.5, 3) or 2
1
2
, 3 or
5
2
, 3
d D (2, −2.5) or 2, −2
1
2
or 2, −
5
2
2 a (4, −2)
b Any point on the line segment EF,
including fractions and decimals.
Examples: (−1, 0), (−1, −1)
c (4, 4), (−6, −2)
3
1 x
y
2 3 4 5
–4
–5 –3 –2 –1
–1
1
3
4
5
2
–2
–3
–4
–5
4
5 a
C
b
C

























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