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739613412 primary maths 2ed 4 LB answers
Material and Energy Balance (Shiv Nadar University)
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
Learner’s Book answers
1 Numbers and the number system
Getting started
1
There is no unique answer but possible
answers are:
−5, −2, 1, 4, . . . is the only one that includes
negative numbers.
9, 12, 15, 18, . . . is the only one that includes
multiples of 3.
5
a
Example: 30, 33, 36, 39, . . . (make sure
they know they can use numbers outside
the 3 × table).
a
subtract 5
c
subtract 100
a
six hundred and one
b
two hundred and ninety-nine
c
one hundred and eleven
3
a
364
b
909
4
a
562 = 500 + 60 + 2
b
305 = 300 + 5
b
Example: 1 , 4 , 7 , 10 , . . .
5
a
160
b
10
c
Not possible as the sequence must be odd,
even, odd, even, etc.
d
Example: 100, 103, 106, 109, 112, 115,
118, 121, 124, 127, . . .
2
b
4
add 10
Exercise 1.1
1
a
1046
b
948
d
8999
e
−1
2
1
2989
1
3
5
7
9
3
5
7
9
11
5
7
9
11
13
7
9
11
13
15
9
11
13
15
17
Possible answers are:
2, 4, 6, 8, . . . and 2, 5, 8, 11, . . . both have
a first term of 2 but 3, 5, 7, 9, . . . has a first
term of 3.
2, 4, 6, 8, . . . and 3, 5, 7, 9, . . . both have a
term-to-term rule of ‘add 2’ but 2, 5, 8, 11, . . .
has a term-to-term rule of ‘add 3’.
2, 5, 8, 11, . . . and 3, 5, 7, 9, . . . both have
a second term of 5 but 2, 4, 6, 8, . . . has
a second term of 4.
1
2
1
2
1
2
6
No, together with an explanation:
They could keep subtracting 3, but it would
take a very long time and they are quite likely
to make errors.
You might encourage them to think about
multiples of 3 (3, 6, 9, 12, . . .). If the sequence
ended at 0 it would have to include multiples
of 3.
397 ÷ 3 leaves a remainder, therefore Abdul
is not correct.
7
Linear sequences:
+2
+2
3
c
1
2
a
Add five – the next term is 19.
b
Subtract four – the next term is 8.
All of these sequences have a term-to-term
rule that generates successive terms with the
same difference between them.
Non-linear sequences:
The other sequences have different
differences between successive terms and are
therefore non-linear sequences:
c
Add one more each time: 2, 3, 5, . . .
Differences are 1, then 2 so the next
difference will be 3 giving 8 as the
fourth term.
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
Multiply by three: 2, 6, 18, so the next
term is 18 × 3 which is 54.
e
Subtract one less each time: 50, 41,
33, . . . Differences are 9, then 8, so the
next difference will be 7 giving 26 as the
fourth term.
f
8
Exercise 1.3
d
Divide by 2: 32, 16, 8, so the next term
is 8 ÷ 2 which is 4.
1
a
nine hundred thousands
b
fifty thousands
2
25 025
3
a
805 469 = 800 000 + 5000 + 400 + 60 + 9
b
689 567 = 600 000 + 80 000 + 9000 + 500
+ 60 + 7
c
508 208 = 500 000 + 8000 + 200 + 8
The next term in the pattern is:
4
The largest 5-digit number is 99 999.
One hundred thousand is 100 000.
100 000 – 99 999 = 1 so Bruno is correct.
5
This represents 16; the sequence shows
square numbers.
6
a
8, 9, 10
(27 ÷ 3 = 9 which is the middle
number)
b
3, 4, 5, 6, 7
(25 ÷ 5 = 5 which is the middle
number)
7
−4
2
a
A = −5
b
B
670
b
4
c
36
d
4150
e
35
f
3500
606 × 10 = 6060
Same: digits 6 and 6 remain.
Different: all digits change their place value.
b
−6
c
0
d
B = −2
C=3
D=5
−3
Think like a mathematician
a
The numbers are:
15, 24, 33, 42, 51
3
a = −8, b = −2, c = 11
114, 123, 132, 141, 213, 222, 231, 312, 321, 411
4
−6 ° C
5
−4 ° C
1113, 1122, 1131, 1212, 1221, 1311, 2112,
2121, 2211, 3111
6
ANTARCTICA
7
−5 ° C is colder than −4 ° C.
Marcus has not taken any notice of the
negative signs. He should place his numbers
on a number line to help him correct the
mistake.
b
The largest number is 111 111
c
The smallest number is 15.
a
2° C
b
−4 ° C
1
430
d
–3 ° C
e
4° C
2
520
3
Any justified answer, for example:
8
11 112, 11 121, 11 211, 12 111, 21 111
111 111
c
Think like a mathematician
1° C
Check your progress
6, 8, 10,12, . . . and 1, 3, 5, 7, . . . both have
a term-to-term rule of ‘add 2’ but 8, 11, 14,
17, . . . has a term-to-term rule of ‘add 3’.
Learners’ posters based on their own
investigations.
2
a
Answers should include:
Exercise 1.2
a
= 30
All the other missing numbers are 300.
Think like a mathematician
1
× 100 = 3000 so
4
−32 °C
5
a
335 271
b
105 050
c
120 202
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
6
a
Three hundred and seven thousand two
hundred and one.
b
Five hundred and seventy seven thousand
and six.
c
Seven hundred and ninety thousand three
hundred and twenty.
Exercise 2.2
1
45 minutes
2
10 hours
3
09:00 to 09:25 is 25 minutes but 09:25 to 10:00
is 35 minutes.
7
C: 1000 + 606 + 4 = 1610
4
a
7 hours
b
45 minutes
8
55 500
5
a
51 minutes
b
1.17 p.m. or 13:17
9
a
540 ÷ 10 = 54
b
307 × 10 = 3070
6
83 years
c
60 × 100 = 6000
d
3400 ÷ 100 = 34
7
a
20 minutes
b
15 minutes
4 °C
b
−10 °C
c
2 hours
10 a
Think like a mathematician
2 Time and timetables
a
a
years
c
minutes
b
2
9.15 and quarter past 9
3
3.05
4
a
hours
3 months
c
4 weeks
1
5.07 p.m.
2
11:45
3
15:30 is the only time that must be an
afternoon time.
15:30 is the only 24-hour digital time.
b
months
hours
4
Exercise 2.1
1
a
30
b
2
a
180
b
330
c
49
d
36
e
54
f
450
g
5
c
60
3
C
4
Missing values are:
7.15 a.m.
9.45 p.m.
d
12
60
a
5 years
c
12 weeks
b
3 days
5
40 minutes
6
a
Monday
b
3 November
c
25 November
d
1 and 15 November
7
a
10 minutes
b
8.55 a.m.
8
a
12 minutes
b
35 minutes
c
The 15:13 bus
3.20 p.m.
5
11.45 a.m.
6
17:10
7
06:00 and 18:00
8
Correct answer is 21:00. Ava added 10 to the
hours. She should have added 12 to the hours.
Think like a mathematician
3
b
Check your progress
Getting started
1
5 days
3 Addition and
subtraction of whole
numbers
Getting started
a
Ten past one or one ten.
1
78
b
Other possible times: 02:20 05:50 10:01
11:11 12:21 21:12 22:22 15:51 20:02
2
86
3
94
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
4
Less
than 10
Greater
than 10
Even
8
12
Odd
7
13 25
Think like a mathematician
4
3
Exercise 3.1
1
a
44
b
16
c
22
d
24
e
13
f
14
b
63
c
9
2
55 + 45 = 100
3
a
89
5
7
6
Think like a mathematician
Findings may include:
4
100
450
450
Answers across the two diagonals are always
the same.
Answers are always even.
300
Smallest possible answer (1 in top left-hand
corner) is 18.
200
Largest possible answer (31 in bottom right-hand
corner) is 46.
250
400
350
Exercise 3.2
5
a
6
a
b
28 + 72 = 100
55 = 70 − 15
1
25
55
5
b
25
a
216
b
595
c
278
d
336
2
Rajiv is correct. See the Teacher’s Resource
for different ways of explaining the answer.
3
86 chairs
4
340 g
5
606 stamps
6
111
7
The largest 2-digit number is 99.
99 + 99 = 198 which has 3 digits.
100
Exercise 3.3
50
7
4
Any three numbers that sum to 10, for
example, 2, 5 and 3
1
Own examples.
2
even
3
Own examples.
4
Own examples.
5
Martha is adding even numbers.
Even + even + even = even
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
6
7
Counter-example (sufficient to show that the
statement is not always true), e.g. 1 + 5 = 6 and
6 is even, or
4 Probability
General case: Salem is not correct. If you add
5 to an even number the answer is odd, but
if you add 5 to an odd number the answer is
even.
Getting started
Counter-example (sufficient to show that the
statement is not always true), e.g. 5 − 3 = 2
or
General case: Heidi is not correct because:
odd − odd = even
1
a
It will not happen
b
It will happen
c
It might happen
2
a
B
3
Spinner A.
Learners may find it helpful to use number
counters on a grid.
Odd comes from odd + odd + odd or odd +
even + even.
The various solutions must conform to this
pattern:
E
O
E
O
O
O
O
O
E
O
E
O
O
O
O
1
2
5
a
Certain
b
No chance
c
Poor chance
d
Good chance
e
Even chance
b
No chance
Learners’ own answers.
3
O
5
Check your progress
red
Exercise 4.1
4
O
c
blue
There is a greater chance of getting a red spin
on spinner A because it has fewer equally
likely outcomes than spinner B.
Or
The section for red is larger on spinner A
so there is a greater chance of the pointer
landing on it.
Think like a mathematician
Even comes from even + even + even or odd +
odd + even.
b
Total
Heads
11
Tails
9
a
Poor chance
c
Certain
Spinner with 4 approximately equal sections
coloured red, blue, yellow and purple. No
section coloured green.
1
42
2
78
Think like a mathematician
3
76 chairs
a
There is a poor chance of rolling a 3.
4
466 books
b
There is no chance of rolling a 7.
5
26 + 34 or 24 + 36
c
There is an even chance of rolling an odd
number.
6
69
d
7
Own examples showing the sum of three even
numbers is even.
The chance of rolling a number less than 10
is certain.
8
No. Counter example, e.g. 3 + 4 = 7 which
is odd.
Check your progress
1
a
False
b
False
c
False
d
True
e
False
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
2
There is an even chance of flipping a tail.
Tally
Total
Head
IIII IIII
IIII IIII
IIII III
28
Tail
32
IIII IIII
IIII IIII
IIII IIII II
5
1 box containing 50 apples
2 boxes containing 25 apples
5 boxes containing 10 apples
50 boxes containing 1 apple
25 boxes containing 2 apples
10 boxes containing 5 apples
6
18, 27, 36, 45, 90
Other answers are possible.
5 Multiplication,
multiples and factors
7
57 + 7 = 64
57 + 15 = 72
57 + 23 = 80 etc.
8
Getting started
factors
of 40
factors
of 30
1
×
1
5
10
2
5
6
10
10
50
100
20
5
5
25
50
10
2
2
10
20
4
1
1
5
10
2
8
7
Think like a mathematician
3 × 4 = 4 × 3 = 12
3 × 5 = 5 × 3 = 15
2
5, 10, 15, 20
3
9
4
39 × 3 + 39 × 7 = 39 × (3 + 7)
= 39 × 10
= 390
57 with any method shown.
5
3 × 6 = 6 × 3 = 18
4 × 5 = 5 × 4 = 20
4 × 6 = 6 × 4 = 24
5 × 6 = 6 × 5 = 30
Exercise 5.2
Exercise 5.1
6
1
17
2
21 and 42
3
1, 2, 4, 8, 16, 32
4
The dates for Saturdays are 6, 13, 20 and 27.
Bruno is right because:
6 + 1 = 7 which is 1 × 7
13 + 1 = 14 which is 2 × 7
20 + 1 = 21 which is 3 × 7
27 + 1 = 28 which is 4 × 7
1
2
2
Any reason as the methods are essentially
the same. They both use factors 4 = 2 × 2
and 15 = 5 × 3. Other valid methods can be
based on the worked example.
3
a
235
b
116
c
267
d
444
a
47 × 3 =
4
×
40
7
3
120
21
120 + 21 = 141
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
b
c
d
93 × 4 =
9
×
90
3
4
360 12
360 + 12 = 372
51 × 5 =
×
50
1
5
250
5
123
615
345
1725
567
2835
6 2D shapes
320 + 28 = 348
Getting started
×
80
4
320 28
1
5
280
a rectangle
hexagon
6
380 cents or $3.80
2
pentagon
7
32 × 5 = 160
3
8
a
696
b
903
Shape C is not a hexagon because it has 7 sides
and 7 vertices and a hexagon has 6 sides and
6 vertices.
c
567
d
952
4
This pentagon is regular because it has 5 equal
length sides and 5 equal angles.
5
a
yes
b
no
d
yes
e
yes
Think like a mathematician
Answer: 897 × 3 = 2691
Check your progress
7
OUT
250 + 5 = 255
87 × 4 =
7
IN
b
c
triangle
c
yes
Exercise 6.1
1
7 × 8 = 56 or 8 × 7 = 56
2
Fatima is not correct. Multiples of 5 end in 5
or 0.
3
3200
4
3 and 4, 5 and 6, 8 and 9
5
6
16 × 2 × 5
16 × 2 × 5
= 32 × 5
= 16 × 10
= 160
= 160
Igor chose the better method.
6 × 2 × 15
6 × 2 × 15
= 12 × 15
= 6 × 30
= 180
= 180
Ingrid chose the better method.
75 and 30
7
The factors of 16 are 1, 2, 4, 8 and 16.
8
The factors of 18 are 1, 2, 3, 6, 9 and 18.
The factors of 20 are 1, 2, 4, 5, 10 and 20.
16 has an odd number of factors because it
is a square number.
a 632
b 3852
c 1169
1
a
The four triangles make a square.
b
The two pentagons make a hexagon.
c
The four triangles make an irregular
quadrilateral (parallelogram).
a
Any shape with at least one right angle,
for example, a square.
b
Any shape with at least one curved side,
for example, a semicircle.
c
Any shape with at least one pair of
parallel sides, for example, a rectangle.
d
Any shape with at least seven vertices,
for example, an octagon.
e
Any shape that is not a polygon,
for example, a circle.
3
a
no
4
a
hexagons
b
squares and triangles
c
octagons and squares
d
squares, hexagons and octagons
2
5
b
yes
c
yes
d
no
Yes, all the triangles tessellate.
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
Think like a mathematician
Check your progress
a
1
2 triangles
2 quadrilaterals (trapeziums)
1 square and 1 rectangle
1 triangle and a pentagon
1 triangle and 1 quadrilateral (trapezium)
b–e Learners’ own investigations.
Exercise 6.2
1
2
3
One possible answer:
a
2
b
4
c
1
d
4
e
0
f
2
g
4
h
0
a
2
b
0
c
4
d
1
e
2
f
1
g
4
h
1
a
1
b
1
c
1
d
0
3
Drawing of hexagon tessellating.
e
4
f
5
g
0
h
4
4
a
yes
b
yes
c
no
5
a
4
b
3
c
2
6
a
6
b
2
c
1
d
0
e
1
4
B, D, F
5
The parallelograms that have diagonal lines
of symmetry all have all four sides the same
length.
a No
b C and E
c
8
2
7
d
yes
8
B
Think like a mathematician
7 Fractions
Shape Name
Sides Vertices Lines of
symmetry
Getting started
A
Regular
(equilateral)
triangle
3
3
3
Short version of answers (see teacher guide for
fuller information).
B
Regular
4
quadrilateral
(square)
4
4
C
Regular
pentagon
5
5
5
D
Regular
hexagon
6
6
6
E
Regular
heptagon
7
7
7
F
Regular
octagon
8
8
8
G
Regular
nonagon
9
9
9
H
Regular
decagon
10
10
10
1
Parts must be equal in size
2
1
4
3
1
6
<
1
3
Draw on the same diagram to compare
different fractions.
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
Exercise 7.1
5
1
1
and 1 but accept 1 , 1 , 1 , 1 , 1 or 1 instead of 1
3
6
5 7 8 9 10
11
6
2
1
8
3
B1
3
Parts are equal in area so each part is a
quarter of a whole.
5
B
6
Same: 4 parts each part is a quarter of
the whole
Different: parts are a different shape
3 3 3 3
12 8 6 4
Finding one-third is equivalent to dividing
by 3 and so on.
6
1
of $15 = $5 and 1 of $24 is $6 so I would
3
4
choose 1 of $24.
4
7
1
of 60 = 30
2
Individual answers.
Exercise 7.2
1
of 50 = 10
5
Think like a mathematician
27 (from 1 of 27 = 9)
Check your progress
1
A
2
5
6
3
$4
2
a
1
3
3
a
3
4
b
1
8
b
9
1
= $12
2
1
= $3
8
3
10
c
1
= $8
3
4
1
= $4
8
$32
1
= $8
4
1 1 1 1
, , , The larger the denominator the more
6 5 4 3
For unit fractions, the larger the denominator
the smaller the fraction. To find 1 of a
1
= $6
4
1
= $16
2
1
parts the fraction is divided into, making each
part smaller.
$24
1
= $4
6
4
quantity, divide the quantity by four.
6
1
5
7
a
8
12 16
6
3
= $24
4
7
10
1
2
0
10
5
9
1
of 80 = 20
4
3
Think like a mathematician
1
Finding one-half is equivalent to dividing by 2.
In each case, the fractions are acting
as operators.
4
7
Yes.
9
25
b
30
c
15
24
No, 1 of $30 = $15 and 1 of $60 = $20
2
3
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
Think like a mathematician
8 Angles
Maryam could draw a line through an obtuse
angle that gives a right angle and an acute angle
or an obtuse angle and an acute angle, so she
won’t always end up with two acute angles.
Getting started
1
Right angle
2
B and E
3
a
4
Exercise 8.3
b
c
0
d
1
3
Exercise 8.1
1
a
D
2
a
c
b
c
G
True
b
True
True
d
False
E
3
L, J, K
4
r, p, t, q, s
5
Angles A and B are the same size. The lines
for angle A are longer, but that does not mean
that the angle is greater. You could convince
Sam by tracing one of the angles and placing
it on top of the other angle to check they are
the same.
1
a
2
After four right angles you are facing in the
same direction as when you started. You have
turned a full circle.
3
a
Estimate between 20 and 40 degrees.
b
Estimate between 70 and 89 degrees.
a
Estimate between 100 and 120 degrees.
b
Estimate between 150 and 170 degrees.
4
6
No answer (talking activity).
270
c
360
Either 10 degrees or 20 degrees because using
the decision tree the angle is between 0 degrees
and 45 degrees, and using the angle diagram
we can tell it is much smaller than 45 degrees.
6
Carly could use the decision tree to work out
that the angle is between 135 and 180 degrees.
By using the angle diagram she could see that
the angle is much closer to 135 degrees than
180 degrees so a better estimate would be closer
to 135 degrees.
Think like a mathematician
Estimate between 110 and 125 degrees for the
first angle.
Estimate between 65 and 80 degrees for the
second angle.
Exercise 8.2
1
Independent learner activity.
Check your progress
2
a
acute
b
obtuse
1
B
c
obtuse
d
right angle
2
D, G, C, F, E
e
acute
3
4
a
An angle drawn less than 90 degrees.
4
b
An angle drawn between 90 and 180
degrees.
5
An obtuse angle is between 90 degrees and
180 degrees.
or
An obtuse angle is larger than a right angle,
but smaller than a straight line.
Estimate between 60 and 80 degrees.
6
Estimate between 160 and 179 degrees.
3
4
A right angle is an angle of 90 degrees.
An acute angle is smaller than 90 degrees.
An obtuse angle is greater than 90 degrees
and smaller than 180 degrees.
10
b
5
Think like a mathematician
Many possible answers. For example, 1 minute
past 12 o’clock would create one of the smallest
acute angles; 1 minute past 6 o’clock would
create one of the largest obtuse angles. (Note:
Learners are not expected to use the words ‘acute’
or ‘obtuse’.)
180
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9 Comparing, rounding
and dividing
7
Think like a mathematician
Getting started
The ones digit forms a repeating pattern in every
case, for example:
1
17 ÷ 4 and 21 ÷ 5
•
2
18
3
145
149
4
a
216 > 126
c
216 < 226
150
153
•
b
226 > 216
•
1
a
50 000
b
20 000
c
50 000
2
a
100 000
b
900 000
c
200 000
3
a
5000
b
5200
c
5210
4
335, 336, 337, 338, 339, 340, 341, 342, 343, 344
5
a
5500
b
5500
c
5000
d
5500, 6000
e
The answers are different, 5000 and 6000.
6
645 123 < 645 213
7
a
2228
5895
6194
6962
8848
b
2200
5900
6200
7000
8800
Think like a mathematician
a
A = 5500
E = 5505
Remainder 1 when divided by 4: 5, 9, 13, 17,
21, 25, 29, 33, 37, . . .
Ones digit: 5 → 9 → 3 → 7 → 1 → 5 (repeat)
Remainder of 1 when divided by 5: 6, 11, 16,
21, 26, . . .
Ones digit 6 → 1 → 6 (repeat)
Exercise 9.1
B = 5050
C = 5045
Remainder of 1 when dividing by 6: 7, 13, 19,
25, 31, 37, 43, . . .
Ones digit: 7 → 3 → 9 → 5 → 1 → 7 (repeat)
Learners may go on to investigate other
numbers and patterns.
Check your progress
1
a
16 787, 16 976, 32 622, 48 150, 150 966
b
17 000, 17 000, 33 000, 48 000, 151 000
2
43
3
6162, 6164, 6166, 6168
4
42 ÷ 6 = 7 because all the other answers are 8.
5
11 melons
10 Collecting and
recording data
D = 5455
b–c Round to 5000: 5046, 5047, 5048, 5049
Round to 5100: 5051, 5052, 5053, 5054
Getting started
Exercise 9.2
1
a
Shoe size Tally
Total
5 weeks
30
I
1
2
5 packs
31
II
2
3
a
32
II
2
33
II
2
4
Remainder 1: 25 ÷ 3,
7 ÷ 3, 1 ÷ 3
another example: 31 ÷ 3
(any correct answer)
34
IIII I
6
Remainder 2: 20 ÷ 3,
23 ÷ 3, 14 ÷ 3, 2 ÷ 3
7 boats
another example 32 ÷ 3
(any correct answer)
35
III
3
36
III
3
37
I
1
1
5
6
11
Any valid reason that matches their
chosen method.
14
b
16
c
12
d
14
4 bags (the last bag will only have 3 apricots
in it)
b
6
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2
Check your progress
a–b Answers depend on the class.
Exercise 10.1
b
1
1
a
2
a–c Individual investigations.
3
a–c Individual investigations.
32
c
Cerys
113
a
Number of biscuits
b
3
c
3
2
4
7
8
9
10
11
12
Number of biscuits
3
1
2
3
4
5
a
Completed sentences.
b
Table drawn with labelled heading
columns, e.g. Number of names, Number
of people.
6
Number of books
5
11 Fractions and
percentages
Getting started
3
4
5
6
7
8
9
10
Scores in the spelling test
6
a
4
b
d
Completed sentence, beginning ‘The dot
plot shows that . . .’
0
c
1
1
8
2
3
>1
5
5
3
6
10
4
A and C
4
Think like a mathematician
8
12
Answers depend on learners’ choices in
collecting, representing and interpreting the
data. For example, they might find that Book
1 is easier to read than Book 2 because Book
1 has only 1 word with more than 8 letters and
Book 2 has 5 words with more than 8 letters on
the pages shown.
Individual investigations.
5
8
7
8
Exercise 11.1
1
1 3
= =6
4 12 24
2
1 5
=
2 10
3
3
and 9 are equivalent so 4 is the odd one
4
12
6
3
4
out OR and both contain 4 so 9 is the
4
6
12
Learners’ own tables/preferences.
7
3
8
3
= 30
10 100
3 9 18
= =
4 12 24
7 35
=
10 50
4 8
=
5 10
4
is left
10
odd one out
4
3
or 6
6
12
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5
1
4
3
8
1
2
5
8
3
4
7
8
6
1
< 5 <3
4 16
8
7
The larger the denominator the smaller the
parts, so eighths are smaller than quarters.
Three quarters are equivalent to six eighths
3
3
<
4
8
5
3
= 30 = 6 so 3 is the odd one out.
10 100 20
9
6
a
15%
b
30
7
a
42%
b
70%
8
2%
12 Investigating
3D shapes and nets
Think like a mathematician
There are 9 different fractions:
1 3
=
2 6
1 4
=
2 8
1 5
=
2 10
1 2
=
3 6
1 3
=
3 9
1 2
=
4 8
1 2
=
5 10
2 4
=
3 6
2 6
=
3 9
3 6
=
4 8
Getting started
Exercise 11.2
1
a
1
4
2
a
75%
3
50%
4
a
25%
b
75%
5
a
30%
b
70%
6
a
50%
b
1
2
7
Smaller
Bigger than
than
b
75%
b
25%
c
2
a
3
A – cuboid
b
4
6
C – sphere
50%
D – square-based pyramid
E – triangular prism
1
4
c
but smaller than
3
4
than
35%
80%
14%
50%
85%
45%
9
2 triangles
3
4
2
1
2
<3<5
3
4
6
2
1
3
3
Sets are: 3 and not 3
4
4
A and B have 3 of the shape shaded.
4
1 2
= =3=4
5 10 15 20
Exercise 12.1
Bigger
1
4
8%
8
25%
1
Check your progress
13
6
B – cylinder
74%
4
1
a
A triangular prism has 3 rectangular faces
and 2 triangular faces.
b
A cuboid has 6 rectangular faces and
0 triangular faces.
c
A square-based pyramid has 1 rectangular
faces and 4 triangular faces.
d
A cone has 0 rectangular faces and
0 triangular faces.
a
7
b
2 pentagons and 5 rectangles
3
1 hexagon and 6 triangles
4
a
8
5
a
Two of the possible common properties
are: A and C have the same number of
vertices, or B and E are both pyramids.
b
B and E c
b
5
c
d
12
d
A, C and D
8
B and E
6
Descriptions of hexagonal prism. May include
that it has 8 faces, 18 edges and 12 vertices.
7
a
6
d
For example: Cubes and cuboids both
have 6 rectangular faces. Cubes need to
have 6 square faces, but cuboids do not.
b
0
c
2
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
Think like a mathematician
5
Hexagon-based pyramid
No 3D shapes with 1 to 5 straws.
6
a
Triangular prism
b
Rectangle
6 straws – tetrahedron
7 straws – no shape
13 Addition and
subtraction
8 straws – square-based pyramid
9 straws – triangular prism
10 straws – pentagon-based pyramid
11 straws – no shapes
12 straws – cuboid (including cube), hexagonbased pyramid, octahedron
Exercise 12.2
1
a
Cuboid
2
a
Hexagonal prism
b
8
c
2 hexagons and 6 rectangles
3
C
4
A
5
a
Cone
b
Tetrahedron
c
d
b
6
c
6 rectangles
1
a
743
b
107
2
a
207
b
225
3
a
395
b
684
4
a
3
5
b
2
or 1
8
4
5
a
4
5
b
5
8
Exercise 13.1
1
a
1245
b
1632
c
1134
2
a
333
b
245
c
48
Cylinder
3
889
Square-based pyramid
4
256
5
a
882 − 435 = 447. The student has always
subtracted the smallest digit from the
largest digit.
b
531 + 278 = 809. The student has forgotten
to add the carrying digit.
a
1173
Think like a mathematician
a
Getting started
A hexagon-based pyramid has 7 faces.
A heptagon-based pyramid has 8 faces.
An octagon-based pyramid has 9 faces.
b
The number of faces is one more than the
number of sides of the base shape of the
pyramid.
c–d For prisms: The number of faces is two more
than the number of sides of the base shape of
the prism.
6
Learners’ spoken explanations.
6
14
381
Think like a mathematician
The answer is 1110 for all lines that pass through 5
in the middle of the array.
Two of the other lines give palindromic numbers
(they read the same forwards or backwards):
741 + 147 = 888
Check your progress
b
123 + 321 = 444
Exercise 13.2
1
8
2
1 pentagon and 5 triangles
3
A tetrahedron has 6 edges, 4 vertices and 4
faces.
4
A
1
5
6
2
a
1
4
b
7
8
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
3
a
1 1
+ =1
2 2
2 1
+ =1
3 3
1 3
+ =1
4 4
6 2
+ =1
8 8
5 4
+ =1
9 9
5 2
+ =1
7 7
3 3
+ =1
6 6
4 1
+ =1
5 5
3
712 − 486 = 226. The student has always
subtracted the smallest digit from the
largest digit.
Then write each addition as two
subtractions.
b
1 3
+ =1
4 4
1 7
+ =1
8 8
1 1
+ =1
2 2
6 3
+ =1
9 9
2 1
+ =1
3 3
2 4
+ =1
6 6
2 5
+ =1
7 7
3 2
+ =1
5 5
456 + 352 = 808. The student has forgotten to
add the carrying digit.
4
506 is 500 to the nearest hundred. 789 is 800 to
the nearest hundred. 800 + 500 is 1300. Leroy’s
answer of 1295 is close to 1300 so his answer
is reasonable.
5
a
6
A and C are correct 3 = 6
Then write each addition as two
subtractions.
4
10
9
a
b
5
3
7
c
d
4
12
10
9
4
9
b
5
2
8
10
3 6
=
5 10
5
9
So
1 2 6
+ =
5 5 10
Yuri should add the numerators but not the
denominators to give 5 .
1 2
+ does not equal 3 . It is not correct to add
5 5
10
Fatima is correct. Parveen has incorrectly
added denominators.
the denominators.
b is incorrect as the denominators have been
added.
9
7
12
8
1 2 3
+ =
5 5 5
6
9
1
9
3
9
6
3
5
The student did not work out an
approximation for either calculation.
Think like a mathematician
1
2
1 1
2 +2=1
1
2
1
2
1 +1=1
2 2
1
2
2 7 9
+ =
8 8 8
3 6 9
+ =
8 8 8
4 5 9
+ =
8 8 8
14 Area and perimeter
Getting started
1
4
3
4
7
3 1
4 +4=1
Check your progress
1
345
2
455 biscuits
7
8
7 +1=1
8 8
1
8
1
The perimeter is 16 cm.
2
Rectangle drawn 4 cm by 6 cm.
3
The area of the blue shape is 5 square units.
4
The area of the red shape is 8 square units.
Exercise 14.1
1
a
A Between 14 cm and 22 cm
B Between 12 cm and 24 cm
C Between 6 cm and 12 cm
b
A 18 cm
B 18 cm
C 9 cm
15
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2
3
a
5000 m
b
60 mm
c
i 12 m
ii
1200 cm
a
m2
b
cm2
c
km2
d
mm2
c
Answer dependent on learners’ investigations.
The smallest possible area with whole square
centimetres is 9 cm2.
9
a
c
16 cm2
29 cm2
4
15 km2
Check your progress
5
14 cm2
1
20 m
6
Jo will get the best estimate. Lee’s estimate will
be too low. Zaid’s estimate will be too high.
2
90 mm
3
18 km2 or 18 1 km2
4
5
Rectangle with area of 10 cm2 drawn,
e.g. 5 cm by 2 cm.
Perimeter matches the rectangle drawn,
e.g. 14 cm.
a 18 cm2
b 16 cm2
c 18 cm2
6
66 m2
Exercise 14.2
1
2
a
Rectangle accurately drawn with sides
2 cm and 5 cm.
b
14 cm
a
Rectangle accurately drawn with sides
6 cm and 3 cm.
c
10 cm2
b
18 cm
c
18 cm2
3
a
15 cm2
b
24 cm2
4
a
16 cm2
b
21 cm2
5
a
6
7
8
2
15 Special numbers
Getting started
length
width
area
3a
5 cm
3 cm
15 cm2
1
6 and 12
b
6 cm
4 cm
24 cm2
2
16
4a
4 cm
4 cm
16 cm2
3
6, 12, 18 and 24
b
7 cm
3 cm
21 cm2
4
a
−5 ° C, −2 ° C, −1 ° C, 0 ° C, 1 ° C, 3 ° C
b
−5 ° C, −2 ° C, −1 ° C, 0 ° C
b
We have found out that the area is equal
to the length multiplied by the width.
a
14 cm2
b
24 cm2
c
36 cm2
d
35 cm2
There are three other rectangles that can be
drawn using whole numbers of centimetres:
1 cm by 24 cm; 2 cm by 12 cm; and 4 cm
by 6 cm
Learners should discover that Lila’s method
works, as long as the two sides measured meet
at a vertex.
5
−3
6
A, C, D
Exercise 15.1
1
−7, −6, −5, −2, −1
2
a
−4 ° C
b
−4 ° C, −2 ° C, −1 ° C, 3 ° C
a
−18, −12, −6, 0, 6, 12
b
The numbers count on in 6s; they are
multiples of 6.
c
121 will not be in the pattern because it
is odd, and 6, which is even, is repeatedly
added onto the even numbers in the
pattern so the terms will always be even.
4
a
<
5
−4 or −3
3
Think like a mathematician
16
b
16 cm2
a
Possible rectangles include: 4 cm by 4 cm, 3 cm
by 5 cm, 2 cm by 6 cm, 1 cm by 7 cm.
b
Answer dependent on learners’ investigations.
The largest possible area with whole square
centimetres is 25 cm2.
b
>
c
<
d
>
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6
−11 or −10 or −9 or −8 or −7 or −6 or −5; −3
or −2; any number greater than −1.
2
6
7
−4 ° C < −2 ° C < 0 ° C < 5 ° C
3
The units digit is 0, 2, 4, 6 or 8.
Anton is not correct. Any number ending in
5 is divisible by 5 but numbers ending in 0 are
also divisible by 5.
4
a
1250
1050
6700
b
525
1250
1050
a
any number divisible by 10
Exercise 15.2
b
any number divisible by 10
1
c
any number divisible by 100
6
a
6105
7
48 − 23 = 25
Think like a mathematician
−1, −3, −5, −7, −9
or
−9, −7, −5, −3, −1
The pattern is that the numbers have a difference
of 2.
5
564
2
multiples of 2
14
11
42
48
284
b
6775
1065
89 − 64 = 25
10
multiples of 4
91 − 66 = 25
16
Think like a mathematician
12
13
Hexagon maze
15
Maze 1:
4
36 and 64
5
Every number with a factor of 6 must also
have factors of 1, 2 and 3 in any order.
2 → 5 → 60; 2 → 5 → 80; 14 → 15 → 20;
14 → 15 → 70; 18 → 15 → 20;
6
133
7
Ingrid is not correct. Number 14 ends in 4 but
it is not a multiple of 4.
18 → 15 → 70; 18 → 20 → 50;
18 → 15 → 20; 18 → 15 → 70;
18 → 20 → 50;
32
c
35
d
18 → 20;
a
33
b
2 → 5; 14 → 15; 18 → 15;
10 → 25; 10 → 5
3
30
Maze 2:
18 → 20 → 90; 10 → 25 → 40;
10 → 25 → 60; 10 → 5 → 60; 10 → 5 → 80
Think like a mathematician
Multiples
Check your progress
You may need to suggest that learners start by
making lists of multiples.
1
There are various solutions, so learners could
compete to see who can use the most cards.
It is possible to use all ten cards to make multiples
of 3: 3, 9, 12, 45, 60 and 78.
a
100 and 700
b
100, 350 and 700
c
10, 60,100, 350, 530 and 700
d
5, 10, 60, 100, 125, 305, 350, 530 and 700
2
7536
Exercise 15.3
3
−8 or −9
1
a
100, 300, 700
4
a
b
10, 40, 100, 300, 530, 650, 700
5
c
all of them: 5, 10, 25, 40, 100, 300, 530,
650, 700
No. Square numbers have an odd number of
factors, for example, the factors of 9 are 1, 3
and 9.
26
b
27
c
24
d
25
Multiples of 100 are also multiples of 5
and multiples of 10.
17
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
2
6
factors
of 24
8
multiples
of 3
1 7 23
3
21
11
13
27
9
17 19
24 6
15
29
12
26
18
25
30
8 22
14 2
16
5
10
28 4
multiples
multiples 20
of 5
of 2
multiples
of 4
36
12
6
24
40
16 Data display and
interpretation
3
a
even
Getting started
1
Red
Not red
Triangle 2 red scalene 2 triangles that
triangles
are not red
Not
triangle
2
3
2 red shapes
that are not
triangles
2 shapes that
are neither red
nor triangles
a
Any three even numbers that are also
multiples of 3, e.g. 6, 12, 18.
b
Any three odd numbers that are not
multiples of 3, e.g. 5, 7, 11.
a
3
b
worm
c
ant and beetle
d
5
Exercise 16.1
4
5
1
curly hair
Norman
Filip
glasses
18
Petra
earrings
not a
multiple of
10
2, 4, 6, 8, 1, 3, 5, 7,
12, 14,
9, 11, 13,
16, 18
15, 17, 19
a
24
c
2.00 p.m. to 2.15 p.m.
d
2
f
Between 9.00 a.m. and 9.15 a.m. was
busiest. 46 vehicles passed the school
between 9.00 a.m. and 9.15 a.m. 43
vehicles passed the school between 2.00
p.m. and 2.15 p.m. 46 is greater than 43.
a
Singer 1
b
0
c
3
d
26
e
Answers include: Both the adults and
children voted singer 1 and singer 4 as
their top two favourites.
f
Answers include: The children voted
singer 2 as their least favourite, but
the adults voted singer 5 as their least
favourite.
g
Answers include: I think that the data for
the children’s vote and the data for the
adult’s vote is different because people of
different ages like different music.
Tapu
Antonella
10, 20
There cannot be any number in the
section for multiples of 10 that are not
even because all multiples of 10 are even.
Adith
Yutu
multiple of
10
b
Sophie
Sun
not even
b
1
e
46
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
6
Check your progress
a
8
b
No. It is possible that all of the
households in the group 4 to 6 had
4 people, and then there would be
17 households of 4 people, but we cannot
tell from the data.
7
Graph 2 shows the data most clearly because
the vertical scale best matches the data.
8
a
b
pictogram
For example:
1
odd
multiple 7, 21
of 7
2
a
Number of children
10
Shape C is a green shape that is not
a quadrilateral and is not a
regular polygon.
8
6
4
Shape D is not a quadrilateral, it is not
green and it is not a regular polygon.
2
0
0
1
2
3
4
5
Score
3
Number of children
Class 2 maths scores
12
10
8
b
Shape H
a
17
b
1
c
47 words
d
51 words
e
Possible answer: Both books have no
words that are 1 letter long.
f
Possible answer: Book 1 does not have
words that are 9 or 10 letters long,
Book 2 does.
g
Possible answer: I think that Book 2
might be written for older children who
can read more words and read some
longer words, and Book 1 might be
written for younger children who are not
experienced readers.
6
4
2
0
0
1
2
3
4
5
Score
e
Shape A is a regular, green quadrilateral
(square).
Shape B is a regular quadrilateral
(square), but is not green.
12
d
14, 28
not a
1, 3, 5, 9, 11,
2, 4, 6, 8, 10,
multiple 13, 15, 17, 19, 12, 16, 18, 20,
of 7
23, 25, 27, 29 22, 24, 26, 30
Class 1 maths scores
c
not odd
Possible answer: The data for Class 1 and
Class 2 show that 2 children in each class
scored 2 marks.
4
Possible answer: The data for Class 1 and
Class 2 show that no children scored 0 in
Class 2, but 2 children scored 0 in Class 1.
Kevin
Possible answer: I think that maybe the
children in Class 2 are older than the
children in Class 1 so they have learnt
more maths, which means they could
answer more questions correctly.
Think like a mathematician
Pictogram showing the number of
badges each child has
Todd
Tia
Raquel
Amanda
Individual posters.
Key:
19
= 4 badges
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
17 Multiplication and
division
Exercise 17.2
Getting started
1
380
2
2916
3
19
4
12 r3
5
4 packs
6
128 ÷ 8 = 16 flowers
16 × 7 = 112 hexagons
Exercise 17.1
6 packs
2
14 teams
3
9 cartons
4
12 friends
5
86 ÷ 3 = 28 r2
80 ÷ 3 = has a remainder
which must be added to
the ones before dividing
the ones by 3.
57 ÷ 3 = 19
Learners have worked
from R to L and not L
to R.
6
7
7
a
2
b
3
c
The divisor = remainder + 1
1
96 months
2
A = 135
3
Think like a mathematician
4
Estimate 400 × 6 = 2400
Calculated answer 2448 g
2304
10 ÷ 2 = 5
12 ÷ 3 = 4
12 ÷ 4 = 3
5
3852
10 ÷ 5 = 2
18 ÷ 6 = 3
14 ÷ 7 = 2
6
IN
OUT
14 ÷ 2 = 7
18 ÷ 3 = 6
20 ÷ 4 = 5
123
615
20 ÷ 5 = 4
30 ÷ 6 = 5
21 ÷ 7 = 3
345
1725
16 ÷ 2 = 8
21 ÷ 3 = 7
28 ÷ 4 = 7
567
2835
30 ÷ 5 = 6
42 ÷ 6 = 7
28 ÷ 7 = 4
18 ÷ 2 = 9
24 ÷ 3 = 8
32 ÷ 4 = 8
D = 297
8
28 cm
7
4
40 ÷ 5 = 8
54 ÷ 6 = 9
42 ÷ 7 = 6
8
$1672
27 ÷ 3 = 9
56 ÷ 7 = 8
63 ÷ 7 = 9
9
No. Sometimes a 3-digit number multiplied
by a 1-digit number gives a 4-digit answer,
for example, 124 × 9 = 1116. But sometimes
the answer has only 3 digits, for example
124 × 4 = 496.
16 ÷ 8 = 2
18 ÷ 9 = 2
36 ÷ 4 = 9
24 ÷ 8 = 3
27 ÷ 9 = 3
32 ÷ 8 = 4
36 ÷ 9 = 4
40 ÷ 8 = 5
54 ÷ 9 = 6
56 ÷ 8 = 7
63 ÷ 9 = 7
72 ÷ 8 = 9
72 ÷ 9 = 8
Think like a mathematician
363
121
4338
3
482
Check your progress
2905
9
415
7
Multiply the two numbers on the bottom row to
give the number on the top row.
20
1
1
Yes, because 400 × 6 = 2400
2
105 plants (15 × 7 = 105)
3
a
4
7 (all the numbers are multiples of 7)
15 r3
b
558
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
5
2905 and 12 r1
6
a
7
12 bags
8
Missing numbers are: 8, 72, 84, 28
b
9
One square South-West, one square SouthEast, one square South-East, one square
South-West
15
One square South-East, one square SouthWest, one square South-West, one square
South-East
One square South-East, one square SouthWest, one square South-East, one square
South-West
18 Position, direction
and movement
Getting started
5
North
1
West
East
One square South-East, one square SouthEast, one square South-West, one square
South-West
A (0, 2), B (2, 1), C (3, 3), D (5, 6)
6
The coordinate marked on the grid is (1, 4).
Safiya has used the wrong numbers for the
horizontal axis and the vertical axis.
7
y-axis
6
South
2
D
3
B
5
4
3
4
2
1
0
0
North-East
2
SW / South-West
3
a
Burd
b
i
South-East
ii
North-East
2
3
4
5
6
x-axis
Think like a mathematician
Exercise 18.1
1
1
1–3, 5, 6, 8 and 9 Individual answers.
4
The second number is the same for all the
coordinates on the line.
7
The first number is the same for all of the
coordinates on the line.
iii South-West
4
One square South-West, one square SouthWest, one square South-East, one square
South-East
One square South-West, one square SouthEast, one square South-West, one square
South-East
21
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CAMBRIDGE PRIMARY MATHEMATICS 4: TEACHER’S RESOURCE
Exercise 18.2
Check your progress
1
1
East, North-East, South, South-West
2
y-axis
6
5
4
3
2
a
2
1
0
0
3
1
2
3
4
5
6
x-axis
Octagon
4
b
3
a
A rectangle
B irregular four-sided polygon (rhombus)
C hexagon
b
Learners’ sketches to check predictions.
4
The lines are horizontal. The lines are parallel.
5
The lines are vertical. The lines are parallel.
6
(6, 2), (6, 5), (5, 2), (5, 5)
5
(1, 4), (4, 4), (5, 2), (2, 2)
Think like a mathematician
A reflected shape has the same area as the
original shape.
22
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