Cambridge
Lower Secondary
Complete
Mathematics
Second Edition
Deborah Barton
8
TEACHER HANDBOOK
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2 Expressions
2
•
•
3 × 5a = 15a
(multiply numbers together, then write
in front of the letter)
2 p × 3q = 2 × p × 3 × q
(multiply numbers first)
2 × 3 × p × q = 6 pq
y × y = y 2 ( we say ‘y squared’)
p × p × p = p3 ( we say ‘p cubed’)
•
a÷b=
•
Expressions
•
Objectives
In this chapter you will learn about
zz expressions, including variables and
constants
z
b
c
a
(write as a fraction)
b
zz expanding brackets
zz forming an algebraic expression
2
zz collecting like terms
2
For example:
−
2+5= 3
4 + − 7 = 4 − 7 = −3
−
6 − −1 = −6 + 1 = −5
What’s the point?
The use of symbols or letters
for numbers helps to describe
relationships among variables.
For example, the speed (v ) of a race
car is related to the time (t ) it takes to
travel a particular distance (d ) by v =d ÷t.
3
The area of a rectangle is length × width.
4
The perimeter of a shape is the distance around it.
Work out:
a
b
c
d
e
f
How to add and subtract with negative numbers
Simplify:
p+ p+ p+ p+ p
i
ii G + G
iii b + b + b − b
iv m + m − m − m
v p× p
vi m × m × m
vii t ÷ p
3 x can be written as 3 × x or in full as
x + x + x. Write in full:
ii 5 y
i 4m
−
8 + 10
4 − 12
3 + −9
−
3− 4
−
7 − −5
−
1 + −8
3
What is the area of a rectangle of length
12 cm and width 8 cm?
4
Find the perimeter of this shape.
3 cm
2 cm
4 cm
5 cm
Before you start
You should know ...
Check in
1
1
The basics of algebra:
•
•
•
•
26
a + a + a = 3 × a or 3a for short. No
need for the multiplication symbol
when letters are used. This is called
simplifying or collecting like terms.
a × 5 = 5a (write the number first)
a × b = ab for short
b × 3 × a = 3ab for short (number
first, then letters in alphabetical order)
a
Write the following in a shorter way.
i 4× p
ii t × 3
iii h × k
iv a × b × c
v 2 × 4m
vi 7 y × 5
vii a × 2 × b
viii 3n × 4u
ix 4t × 6r
2.1 Expressions
Here is some key language about, for example, 3 x + 7.
In maths we try to make things simpler by writing
as few words as possible. For example, these
two sentences can be written in a shorter way:
zz3 x + 7 is an expression, in this case involving
3 apples and 7 bananas = 3a + 7b
6 apples and 12 bananas = 6a + 12b
We have used the letter a to represent apples, the letter
b to represent bananas and the + symbol to replace the
word ‘and’. The process of using letters to represent
unknown numbers or variables is algebra.
numbers letters and a + symbol.
zzA constant is a symbol which always means the
same thing, so 7 is a constant.
zzA term is part of an expression so 3 x and 7 are the
two terms in the expression. 3 x is the algebraic
term and 7 is the number term.
zzAn unknown is part of an expression that you
don’t know the value of, so x is the unknown.
zzThe coefficient is the number part of an algebraic
term so, in 3x, the 3 is the coefficient.
27
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Contents
About this book
4
Active learning resources
Negative numbers jigsaw
5
Factorizing expressions wheel
11
Mindfulness fractions and decimals puzzle
18
Dining investigation
20
Equations treasure hunt
24
Angle properties pairs game
33
Statistics word search
36
Transformations puzzle
38
Probability collective memory activity
42
Resources for use with Student Book 8
The sieve of Eratosthenes
44
Percentages game
45
Revision material
Multi-choice Exercise A
49
Multi-choice Exercise B
54
Multi-choice Exercise C
59
Answers to multi-choice Exercises A, B and C
64
Topic tests
Topic test – Chapters 1–5
66
Topic test – Chapters 6–10
74
Topic test – Chapters 11–15
82
Answers to topic tests
93
Progression test-style papers
Paper 1 (non-calculator)
97
Mark scheme for Paper 1
105
Paper 2 (calculator)
108
Mark scheme for Paper 2
119
Worked solutions
Worked solutions for some of the more difficult
questions from Student Book 8
122
Answers
Answers for Student Book 8
129
Answers for Homework Book 8
169
Additional support can be found on
Kerboodle. As well as extra assessments to
accompany the Teacher Handbook, there
are resources for each chapter in the Student
Book, including interactives and worksheets
for both formative and summative
assessment.
3
About this book
This book is designed to provide teachers with useful
resources to support teaching of the Cambridge
Secondary Stage 8 Mathematics curriculum
framework in preparation for the Cambridge
Checkpoint assessments and further study at IGCSE.
The active learning resources included ensure that
there is plenty of scope for discussion and group work,
which are valuable tools for reinforcing learning.
Often students complete such activities thinking that
they have done very little work as they have written
little or nothing down. However, the work completed
is often of greater value than a written task, as there is
more scope for sharing knowledge and eliminating
common misconceptions than when students work
individually. There is less pressure than in a written
exercise as these resources can have different starting
points, meaning students can work in an order that
suits them and their level of understanding.
Often there are no limits to learning, particularly
when using the creative alternative approaches. You
will find many students work at higher levels than
expected, because of their natural competitiveness and
desire to improve. Also, in group work and discussion,
students are required to articulate sometimes quite
complex mathematical ideas that wouldn’t be covered
when working individually. Not everyone learns best
by writing things down. These activities address
alternative learning styles, particularly if you find
your classes often take the ‘teacher gives examples,
students do an exercise’ approach.
All good resources should provide you with the
opportunity to simplify or extend the resource as
necessary. Explanations are provided here for how to
differentiate the resources to cater for differing ability
levels. Sometimes an activity takes less time than
4
About this book
anticipated and this can be difficult to manage in
lessons, so there are sections suggesting alternative
ways to use the resources.
There are also resources for exercises in the Student
Book that promote active learning; this ensures better
understanding, and consequently better retention of
the subject knowledge.
Revision material is included in the book in the form
of multi-choice questions. These can be given to the
students as worksheets at the end of a chapter or
series of chapters, or they could be used in a more fun
approach, for example as a quiz.
Assessment materials are provided in the form of topic
tests and end-of-year progression tests. The assessment
materials are balanced between all the content areas in
the framework: number, algebra, geometry and
measure, statistics and probability. They are also
underpinned by thinking and working mathematically
questions, providing a structure for the application of
mathematical skills. Mental strategies are tested
throughout. Mark schemes are included for all topic
tests, and these are provided in great detail for the endof-year progression tests (all mark schemes have been
written and developed by the author).
There are worked solutions to some of the more
difficult questions in the Student Book and answers to
all of the exercises in the Student Book and the
Homework Book and of course, don’t forget
Kerboodle, where you will find yet more resources.
Finally, I hope that all the material in this book is
extremely useful for you, both in monitoring your
students’ progress and helping them increase their
understanding and enjoyment of maths.
Deborah Barton
Active learning resources
Mindfulness fractions and decimals puzzle
For use with Chapter 6 of Student Book 8
Teacher’s notes
Example 6
This mindfulness colouring activity is a fun
alternative to a written exercise. This activity requires
students to be able to:
Work out 6 2 × 4
5
•
multiply and divide by 0.1 and 0.01
= 24 + 8
•
round to given significant figures
= 24 + 1 3
•
change a fraction to a recurring decimal
= 25 3
•
subtract mixed numbers
•
divide by a proper fraction
•
multiply by a mixed number.
Introductory activity
This activity can be used as a plenary or revision
exercise supporting the work completed in Chapter 6.
You may find that little introduction is necessary.
Alternatively, you may want to go through the
following examples.
Example 1
5
5
5
Mindfulness activity
Give each student a copy of the puzzle page. Ask
them to colour in all the correct answers to the
questions until they find a shape.
Differentiation
To make it more difficult, students could be asked to
write questions that give the ‘solutions’ around the
outside edges of cat shape.
Alternative approaches
17 × 0.1 = 17 ÷ 10 = 1.7
You could ask students to make an entirely new
puzzle of their own.
4.2 ÷ 0.01 = 4.2 × 100 = 420
Answers
.
0. 4
0.12
4
0.004 231 to 2 s.f. is 0.0042
4500
0.073
7
9
Example 3
18
39
.
0. 6
0.03
1300
38
45
1000
0.08
25.5
Example 2
30 524 to 2 s.f. is 31 000
5
Change the fraction to a recurring decimal
9
0. 5 5 5…
9 5.505050…
.
5
= 0.5
9
Example 4
Work out 4 1 − 1 2
5
3
3
= 4 − 1 10
15
15
18
10
= 3 −1
15
15
8
= 2
15
Example 5
Work out 5 ÷ 2
3
= 5× 3
2
5
= ×3
1 2
= 15
2
= 7
18
5
= 6×4+ 2 ×4
1
2
Active learning resources
17
1
2
3
11
30
Active learning resources
Mindfulness fractions and decimals puzzle
300
17.5
13
1
18
0.3
35
2
17 12
0.03
7
9
4.9
18
0.6
5
25 12
0.6
3 67
38
45
0.12
35 45
25. 5
8000
40
0.08
0.07
.
0.3
0
7
3 30
4500
14 29
1000
0.073
1
32
2 45
3 11
30
2.3
7 20
5
0.09
0.13
1200
8
4.5
101
30
999
7
10
2
0.4
0.0 1300
3.25
7.9
4
12
51
2
0.1
130
13 000
3000
30
1 14
14
5
4520 4.4
10 000
80%
1.2
0.085
45
3
0.04
39
1
34
900
1200
Shade in the regions with the answers to these:
The fraction
4
as a recurring decimal
9
4524 rounded to 2 s.f.
12 × 0.01
0.4 ÷ 0.1
0.07251 rounded to 2 s.f.
.
0. 7 as a fraction its simplest form
8
9
7 ×5
0.3 × 0.1
13 ÷ 0.01
999 rounded to 1 s.f.
0.08499 rounded to 1 s.f.
16 ÷
7÷
4
5
2
as a mixed number in its simplest form
5
The fraction
2
as a recurring decimal
3
4
9
2 −1
8×3
5
3
5
3 as a decimal
16
7
1
− 2 as a mixed number in its simplest form
10
3
Mindfulness fractions and decimals puzzle
19
Cambridge Lower Secondary
Complete
Mathematics
Teacher Handbook
Second Edition
8
Cambridge Lower Secondary Complete Mathematics embeds an excellent
understanding of the Cambridge Lower Secondary Mathematics curriculum.
The stretching approach helps learners to develop the skills required to
®
progress to Cambridge IGCSE with confidence.
●
Fully prepare for exams – comprehensive coverage of the course
●
Develop advanced skills – thinking and working mathematically
●
Progress to the next stage – differentiated extension material eases
the transition to 14–16 study
Student Books, Homework Books and Kerboodle online support
are also available as part of this series.
www.oxfordsecondary.com/cambridge-lowersecondary-maths
Empowering every learner to succeed and progress
Full Cambridge curriculum coverage
Reviewed by subject specialists
Stretching extension activities
Embedded critical thinking skills
Progression to the next educational stage
For additional practice
can be used flexibly across the whole school and alongside
any other maths resources.
www.oup.com
How to get in touch:
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www.oxfordsecondary.co.uk
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+44 (0)1536 452620
ISBN 978-1-382-01883-8
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