Math Smart - 7

Maths







SMART
f


For Cambridge Lower Secondary



» LEARNER'S BOOK

CONTENTS




Unit 1: Factors, Multiples and Primes
Chapter 1.1 Recognising Factors, Multiples and
Introduction Prime Numbers

1.2 Lowest Common Multiples and
Welcome to the Maths SMART series. This Highest Common Factors 10
series is designed to help you master all 1.3 Tests of Divisibility 16
the concepts and skills you need for the 1.4 Squares and Square Roots 23
Cambridge Lower Secondary Mathematics 32
Unit 2; Integers
Curriculum Framework for Stage 7. 33
Chapter 2.1 Integers
This book is divided into clear and 45
manageable units covering Numbers, Unit 3: Introduction to Algebra and Equations 46
Algebra, Measurement, Geometry and Data Chapter 3.1 Algebraic Expressions
Handling. The units are further divided into 3.2 Deriving and Using Formuiae 56
chapters, each dealing with a topic from the 3.3 Functions and Mapping 60
syllabus. I 3.4 Constructing and Solving Equations 64
The chapters have many useful features to I Unit 4: Decimals 73
make sense of the topics by investigating Chapter 4.1 Understanding Decimals 74
and deriving mathematical concepts. These 4.2 Operations on Decimais 76
features are explained on the next spread.
4.3 Ordering Decimals 88
4.4 Rounding Numbers 93
4.5 Estimation and Approximation of
V- . • .• . .• ' Decimals in Word Problems 98
^ 4 ■A* '. '"'5 "T" ■-
Unit 5: Measurement 105
Chapter 5.1 Metric Units 106
5.2 Suitable Units of Measurement 112
• •« ••• • . '
J •< V '
' 'J -v: • <0 5.3 Reading Scales 115
> . . •- J " Gi
. 6? K
• ••* -r
* ■ ' a*"- . - 120 is
Unit 6: Angles and Their Properties
Chapter 6.1 Types of Angies 121
6.2 Estimating. Measuring and Drawing
i _
. . Angles 123
6.3 Angle Properties 131 1
O
6.4 Parallel Lines, Perpendicular Lines
and Transversals 142 -
6.5 Solving Geometrical Problems 144
i''
,

Unit?: Data Handling 159 Unit 14: Geometrical Constructions 299
Chapter 14.1 Drawing and Measuring
Chapter 7.1 Collecting and Recording Data 160
Perpendicular and Parallel Lines 300
7.2 Frequency Tables for Grouped
14.2 Constructing Polygons 304
and Ungrouped Data 163

Unit 15: Measuring Time, Area, Perimeter and
Units: Fractions 172 Volume 324
Chapter 8.1 Understanding Fractions 173 Chapter 15.1 Time 325
8.2 Equivalent Fractions and
15.2 Area and Perimeter 333
Simplifying Fractions 176
15.3 Volume of Cubes and Cuboids 345
8.3 Converting Between Improper
15.4 Surface Area of Cubes and
Fractions and Mixed Numbers 180
Cuboids 351
8.4 Comparing and Ordering
Fractions 184
Unit 16: Transformation 361
8.5 Adding and Subtracting Fractions 189
Chapter 16.1 Transformation 362
8.6 Fraction of a Quantity 197
16.2 Translation, Reflection
and Rotation 366
Unit 9: Terms and Sequences 203
Chapter 9.1 Number Sequences 204 Unit 17: Data Handling 385
9.2 Generating Number Sequences Chapter 17.1 Finding Average: Mean, Median,
and Finding the General Term 209
Mode and Range 386
17.2 Presenting and Interpreting Data 399
Unit 10: Shapes and Symmetry 221
Unit 18: Probability 416
Chapter 10.1 Shapes and Solids 222
18.1 Introducing Probability 417
10.2 Symmetry 233
18.2 Calculating Probabilities 422
18.3 Estimating Probabilities 427
Unit 11: Coordinate Geometry and Graphs 243
Chapter 11.1 Coordinate Grid 244
Answers
11.2 Plotting Linear Graphs Using
Vocabulary List
Coordinate Pairs 251
11.3 Drawing Lines Parallel to the
v-axis or the .v-axis 254
11.4 Application of Linear Graphs 255

Unit 12: Percentages 268
Chapter 12.1 Understanding Percentages 269
12.2 Percentages in Quantities 276

Unit 13: Ratio, Rate and Proportion 286
Chapter 13.1 Ratio 287
13.2 Proportion 293

Textbook features









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investigative activities are brief introduction to the chapter.
usually found before a new
concept so that you can explore
how a concept is derived by
You will work on your own,
observing patterns and logical
in pairs or in groups, as
reaasoning. You are expected to
indicated by this icon.
make your own conclusions.
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Information relevant
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discussed.
Look out for QR codes
Examples are worked out with throughout the book. These
annotated notes to help you supply you with quizzes,
grasp the mathematical skills and games, applets and additional
procedures. information online to help revise
o what you have learnt

Check My Understanding provides immediate
feedback on your understanding and reinforces
your knowledge.



Recommended vocabulary is
Think and Share challenges
highlighted or bold in the book. The
you to think critically about
meanings of words are included in the
a concept or an observation
Vocabulary List at the end of the book.
made, then debate logically
about your viewpoints with
your partners.
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Factors, Multiples







and Primes





Numbers occur all around us in many formats and circumstances. For example, your
house number, a phone number, a televison channel number, your password or prices
on shop items. It is important to understand the properties of numbers such as factors,
multiples, prime numbers, squares and square roots to help us in everyday situations.
[This makes tasks such as working out how many movie tickets you can buy, or how
uch time until the next train arrives, easier and quicker.



You Will learn about:

In this chapter
CHAPTER 1.1 Pupils should be able to:

• recognise factors
Recognising Factors, • recognise multiples

• recognise prime
numbers
Multiples and Prime • use the sieve of

Eratosthenes to
generate prime
Numbers numbers


Following a recipe when baking can be tricky, especially if you want to make
more than what Is in the recipe. You might need to calculate multiples of the
amount of ingredients needed in the recipe.




Think and Share


You could use multiples to calculate the ingredients if
you need to bake more than 1 loaf of bread.
1 loaf of briMcl
For example, if you want to bake 2 loaves of bread, you
will need 600 g (2 x 300 g) of wholemeal flour.
~^=M
For 3 loaves of bread, you will need 900 g (3 x 300 g) ^00 ^ whole-Me-al four
of wholemeal flour. For 4 loaves of bread, you will ^00 ^ bread four
need 1200 g (4 x 300 g) of wholemeal flour. 2 f?p dr\cd
2 f6?p brown ^u^r
You are using the multiples of 300 to calculate how
1 f;p ?alf
much wholemeal flour you would need to make more
1 cAi^ rolled oaiK
loaves of bread!
2 +i>^p olive- oil
i5i9 ryil warm wafer
1 C-fffi.



^ RECALL


Recall how we represent a multiplication using dot array. These diagrams
show how we can represent the number 12 using the area of a rectangle.
/
• • ••••• • • • • • •
i 1
• • :; Ml!
• • ##•••#
•
# • • •
4 by 3 • 12by 1 « •


4 and 3 are factors of 12 6 and 2 are factors of 12 12 and 1 are factors of 12

o

^ RECALL Factors and prime numbers


d List the first 3
multiples of each of
Investigate!
the following:
PAIR WORK
a) 15 b)20
Use the numbers in the box. Find all the number pairs that give the products:
c)25 d)50 1. X __ = 24

Q List the first 10 prime 2. X __ = 24
numbers. 3. X __ = 36
8
Q Draw a dot array to 4. X __ = 8
show the factors for 5. X = 30
the number 42.


Factors

A factor is a number which divides into another number exactly without leaving
In other words, factors of a
a remainder. For example, 6 and 4 are factors of 24. However, 6 and 4 are not the
number, when multiplied
only factors of 24.
together, would give you
the number itself as the 1 X 24 = 24
product.
2 X 12 = 24
3 X 8 = 24
4x6 = 24 So, the factor list for 24 is 1, 2, 3, 4, 6, 8, 12,24




Example 1


5 is not a factor of 36 Write a factor list for 36.
because it cannot divide Step 1: Make a list of the pairs of numbers that give a product of 36.
36 exactly.
1 X 36 = 36
2 X 18 = 36
Although 6 x 6 = 36, we I 3 X 12 = 36
only write the number 6 I 4 X 9 = 36
once in the factor list. We ; 6 X 6 = 36
do not repeat them.
Step 2: Write the factors of 36 in an Increasing order.
The factor list for 36 is 1, 2, 3, 4, 6, 9, 12, 18 and 36.


















UNIT 1 Factors, Multiples and Primes

Check My
Understanding Speed Challenge

O List the factors of each of the following,
Factor race
a) 9 b) 16 c) 48
d) 99 e) 49 f) 13 O into groups of 4.
O Write a factor list for each of the following. O Race to complete the
factor list of each
a) 12
number that the
b) 7
teacher calls out.
c) 16
O When your group has
d) 21
completed the factor
e) 50
list, raise your hand.
f) 48
O The group that
O Cross out the incorrect number(s) in each factor list, completes the factor
a) 1,2,3,4,6,9,18 b) 1,2,3,7,9,27 list first, wins the
round.
c) 1,5,5,25 d) 1,2,4,5, 10, 15,20,25,50, 100
@ Compare your factor
O State whether the statements are true or false. If false, rewrite the list to the complete
statement to make it true. factor list that your
a) 2 is a factor of every natural number teacher writes on the
b) 1 is a factor of every natural number board.
c) 6 Is a factor of 34
d) 38 is a factor of 38

o Spotlight
Prime numbers
Natural numbers are the
set of positive numbers,
mbers.
Investigate! from 1 onwards. TheyThey g
CO
are whole numbers
Refer to the numbers in the box. excluding zero.
a) Write a factor list for each of the numbers in the table.
CO
Number Factor list
100 27
1

53
30





13
24


b) List the numbers that have only two factors.




O

A number that has only two factors is called a prime number. The factors of
a prime number are always the number 1, and the number itself. The prime
numbers in Investigate! on the previous page are: 13 and 53. The number 13 has
why is the number
only two factors: 1 and 13. The number 53 has only two factors: 1 and 53.
1 neither prime nor
composite? A composite number has more than two factors. As you saw in Investigate! on
the previous page, the numbers 100, 8, 30, 24 and 18 are composite numbers.
All of these numbers have more than two factors. The number 1 is neither prime
nor composite.

Check My

Understanding

O Circle the prime numbers.




8 12 27 7 36 18 11 19 9


Q Match the terms to their descriptions.

1. Prime number The result obtained when two or more
Amazing numbers are multiplied
Mathematician 2. Factor A number that has exactly two factors:
1 and itself

3. Product A number which divides into another
number exactly without leaving a remainder

@ Put a tick in the correct column.

Number Prime number Composite number

2
The sieve of
8
Eratosthenes can be
13
used to find prime
numbers. 25
Find out more about 70
Eratosthenes.

SMH
The sieve of Eratosthenes
Eratosthenes was a Greek mathematician. He developed a method of finding
prime numbers up to 100 by eliminating or sieving out all composite numbers,
and the number 1. This is called the sieve of Eratosthenes.











Factors, Multiples and Primes

Spotlight
To discover the first 25 prime numbers, sift out all the composite numbers
between 1 and 100 using multiples. Currently, the largest
Refer to the hundred grid below. known prime number
was found in 2018 and is
O Cross out the number 1 since it is not prime.
23 249 425 digits long. It
© Circle the 2 and then cross out all the multiples of 2.
was found by the Great
© Circle the 3 and then cross out all the multiples of 3. Internet Mersenne Prime
Search (GIMPS).
O Do the same for 5 and 7 and their multiples.
© Circle the remaining numbers. These are the first 25 prime numbers. Find out more.

© List the 25 prime numbers less than 100 in ascending order.


@ 3 X 5 6 7 8 9 10
n 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100



Multiples



Multiples of a number are the results of multiplying the number by a positive
whole number.

15 is a multiple of 3 3 Is a factor of 15

\ /
15 = 3X5
15 is also a multiple of 5 ^ ^ 5 is a factor of 15



308 = 4 X 7 X 11
/ \

308 is a multiple of 4, 7 and 11 4, 7 and 11 are all factors of 308

€ Investigate!


in a car park in Singapore in 2016, the parking charges were
$3.00 for each hour of parking. For every hour that your car is parked,
you need to pay another $3.00.
This means that the price of parking will go up in multiples of 3.











s Figure 1.1 Parking 3 6 9 12 15
charges are $3.00 for
(1 X 3) (2X3) (3X3) (4X3) (5X3)
each hour of parking
s Figure 1.2 Car parking charges over a period of 5 hours at $3.00 per hour
O In 2019, the car park charges increased to $4.00 per hour.
Complete the table.

Number of hours Cost ($)
1
2
3
5

11
Q Find a carpark in your city. Calculate the cost of parking a car there from
8 a.m. to 10 p.m. Write a sentence to describe the cost using the word
'multiple'.


Example 2


o Spotlight List the first four multiples of 12.
12 X 1 = 12
12 X 2 = 24
There is an infinite
12 X 3 = 36
number of multiples for
12 X 4 = 48
any natural number.
So, the first four multiples of 12 are 12, 24, 36 and 48.
List the multiples of 24 between 23 and 97.
24 X 1 = 24
24 X 2 = 48
24 X 3 = 72
24 X 4 = 96
24 X 5 = 120
We are looking for the multiples between 23 and 97. So, we can stop at
the 5*'' multiple of 24, which is 120.
So, the multiples of 24 between 23 and 97 are 24, 48, 72 and 96.

UNIT 1 Factors, Multiples and Primes

Check My
Understanding



15 = 5x 3
a) State the factors.
b) State the multiple.

© On the hundred grid:
a) Colour the multiples of 9 in yellow.
b) Circle the factors of 9.
c) Which number is circled and coloured?
d) State whether true or false. If false, rewrite the statement to make it true.
i) The number 9 is a factor of 9, but not a multiple of 9.
ii) The number 3 is a multiple of 9, but not a factor of 9.
iii) The number 3 is not a prime number.
iv) The number 27 is a prime number because It only has factors: 1 and 9.


1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

© List

a) the first five multiples of 9.
b) the multiples of 6 between 21 and 48.
O State whether the statements are true or false. If false, rewrite the statement to make it true.
a) 30 is a multiple of 10.
b) 0 is a multiple of 6.
c) 32 is a multiple of 10.
©* Challenge! Write down a number that is a factor and a multiple of 8.










e

CHAPTER 1.2


In this chapter
Lowest Common
Pupils should be able to:
• find the Lowest
Multiples and Highest
Common Multiple (LCM)
• find the Highest
Common Factor (NCR)
Common Factors
• use factor trees
Now that you have worked through factors and multiples, you will
know that a factor is a number that divides another number exactly,
with no remainder. In this chapter we are going to look at Lowest
^ RECALL Common Multiples (LCM) and Highest Common Factors (HCF).



List all the factors of
72 and 108. Then find We can use lowest common multiples to solve
the highest common problems that involve:
factor. • purchasing multiple items in order to have
enough. For example, putting together
List the multiples
some party bags. Balloons are sold in bags
of 12 and 16. Find
of 20 and candy bars are sold in bags of 8.
the lowest common
What is the minimum number of balloons
multiple.
and candy bars you need in order to have
Find the lowest an equal number of balloons and candy
common multiple of bars in each party bag?
5, 6 and 9. • figuring out when two things will happen
again at the same time. For example, two bells are set to ring at different
time intervals. One bell rings every 7 minutes, and the other bell rings every
8 minutes. Both bells have just rung. After how many minutes will both bells
ring at the same time again?

We can use highest common factors to solve problems that involve:
• arranging objects in rows or groups. For example, you have 12 movie DVDs,
24 wildlife DVDs, and 30 musical DVDs. You need to pack these DVDs into
containers. You can only pack one type of DVD in each container. Each
container must have the same number of DVDs. What is the greatest number
of DVDs you can pack in each container?
• splitting things into parts of equal size, so that the parts are as large as
possible. For example, you can use your knowledge of highest common
factors to help you solve an everyday problem. You have two pieces of cloth.
One piece is 32 cm wide and the other piece Is 48 cm wide. You want to cut
both pieces into equal widths that are as wide as possible. How wide should
you cut the pieces into?







UNIT1 Factors, Multiples and Primies

Think and Share


Sam used the factors of 32 and 48 to work out the length of the pieces she should cut.
Factors of 32 are 1, 2, 4, 8, 32
Factors of 48 are 1, 2, 3, 4, 6, 8, 12,1£ 24, 48
Common factors are 1, 2, 4, 8, 16. The highest factor is 16.

The highest common factor of the numbers 32 and 48 is 16.
32 cm 48 cm


16 cm 16 cm 16 cm 16 cm 16 cm
► Figure 1.2 Sam cut
them into widths of
16 cm






1.2.1 Lowest Common Multiple (LCM) Multiples that are


common to two or more
Example 3 numbers are called
common multiples. The
Kit and Tina are running on a circular track. They start from the same point but numbers 90 and 180 are
Kit takes 30 seconds to run a lap while Tina takes 45 seconds to run a lap. After both common multiples
how many seconds will they meet again at the starting point? of 30 and 45. The number
90 is the lowest common
Answer
multiple of 30 and 45.
We can list the multiples of 30 and 45 to help us;
Multiples of 30 are 30, 60, 90, 120, 150, 180, 210, 240, 270...
Multiples of 45 are 45, 90, 135, 180, 225, 270....

90 is the first multiple that occurs in both lists. We say that 90 is the Lowest Check My
Common Multiple (LCM) of 30 and 45. Kit and Fandi will meet at the start line
again after 90 seconds. They will meet again after 180 seconds. Understanding


Example 4 O Find the lowest common
multiple of:
O Find the first three common muitiples of 6 and 8. a) 5 and 7
© What is the LCM of 6 and 8? b) 6and 15
c) 4 and 6
Answer
d) 7 and 14
The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72...
e) 11 and 12
The multiples of Bare: 8, 16, 24, 32, 40, 48, 56, 64, 72...
f) 20 and 36
O The first three common multiples of 6 and 8 are 24, 48 and 72. The g) 15 and 50
numbers 24, 48 and 72 occur in both lists of multiples. h) 18 and 24
O The LCM of 6 and 8 is 24. The number 24 is the smallest number that
occurs in both lists of multiples.

Think of a factor tree as a [ Using Factor Trees to find LCMs

tree with branches. •
1
You can use factor trees to find the LCM of two or more numbers.
You do not need to draw '
the tree! Just the diagram.
How to draw a factor tree
60
Any number can be written as the product of Its prime numbers. A factor tree
6^^ ^^10
breaks down a number into its prime numbers.





Draw the factor tree for the number 60.




















Write down any factor pair Repeat the process, adding When you get to a prime
that multiplies to give the new branches. As 2 x 3 = 6 number (a number whose
number 60. and 5x2 = 10, write these only factors are 1 and itself),
as new branches. draw a circle. Continue until
all branches are finished.
r
Express the number as a product of its
prime numbers in order. The exponent is 2 because
60 = 2X2X3X5
Then write the prime numbers with the number 2 appears twice as
= 2^ X 3 X 5
same base number in its exponential form. a prime factor of 60



Exponent
Exponents The number of times
the base is multiplied
The exponent of a number
tells you how many times to I
Base number
multiply the number if —8^ = 8X8X8
The number
multiplied by itself.
being multiplied




2"^ =2X2X2X2 O 3^X5^ = 3X3X5X5
= 16 = 9 X 25
= 225



Factors, Multiples and Primes

Check My

Understanding

O Draw a factor tree for each of the following numbers.

a) 36 b) 72 c) 80
Q Work out the product and write them using exponents,
a) 2x2x2x3x3x5x5 b) 2x2x3X5x5



How to use factor trees to find the LCM





Drawing the factor tree to find the LCM of 120 and 45


: 120




















Draw the factor tree Write the number of 120 Repeat the process for the Write the number of
for number 120. as the product of its prime other number, that is 45. 45 as the product of
numbers. its prime numbers.
120 = 2X2X2X3X5 45 = 3 X 3 X 5
= 23 X 3 X 5 = 32 X 5


The LCM of 120 and 45 is the product of the
largest multiple of each prime that appears 120 = 2^ x 3 X 5 Spotlight
on one of the factor trees. 45 = 3^ x 5 O
• Prime factor 2 which appears three times: 7}
Try growing a
• Prime factor 3 which appears twice: 3^
factorisation forest!
• Prime factor 5 which appears once: 5


h
LCM of 120 and 45 = 2^ x 3^ x 5
=2x2x2x3x3x5
=8x9x5
= 360
Therefore the LCM of 120 and 45 is 360.

o

Check My
Understanding

O Use factor trees to find the LCM of the following numbers:
a) 15 and 30 b) 12 and 16 c) 15 and 20 d) 28 and 96 e) 10. 16 and 40

o A business woman goes to New York every 15 days for one day. Another business woman goes
to New York every 24 days, also for one day. Today both of them are in New York.
How many days later will they be in New York on the same day again? Hint: Use a factor tree.





.22 Highest common Factor (HCF)



Example 6



O Find the common factors of 30 and 20.
O What is the highest common factor of 30 and 20?

Answer
You can use a list of the factors of 30 and 20 to find the answer:
Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30.
Factors of 20 are 1, 2, 4, 5, 10 and 20.
O The common factors of 30 and 20 are 1, 2, 5 and 10.
O The highest common factor of 30 and 20 is 10.





How to use factor trees to find the HCF


Try and Apply!

Chris is making identical flower
- -' AC?'')
arrangements for a party. She has 24
orange flowers, and 16 yellow flowers.
She wants each arrangement to have
the same number of each flower
colour. What is the greatest number
of arrangements that she can make if
every flower is used?
16 24













Factors, Multiples and Primes

\

0









Write the number of Write the number 16
24 as the product of its as the product of its
Draw the factor tree Repeat the process for the
prime numbers. prime numbers.
for number 24. other number, that is 16.
24 = 2X2X2X3 16 = 2 X 2 X 2 X 2
= 23 X 3 = 2"



The HCF of 24 and 16 is the product of the prime factors
that are common to both numbers. 24 = 2X2X2X3
16 = 2 X 2 X 2 X 2
The numbers 24 and 16 both have a prime factor of 2.
There are three prime factors of 2 in the number 24.
There are four prime factors of 2 in the number 16.
Therefore, only 2 x2 x2 is common to both 16 and 24.




h
2X2X2=8
Chris can make 8 arrangements with the
same number of each flower colour.
Each arrangement will have
24 T 8 = 3 orange flowers and
16^8 = 2 yellow flowers.




Check My

Understanding


o Use factor trees to find the HCF of the following numbers.
a) 15 and 30 b) 12 and 16 c) 15 and 20 d) 28 and 96 e) 10, 16 and 40
e Josh is making identical balloon arrangements for a party. He has 32 red balloons, 24 white
balloons, and 16 orange balloons. He wants each arrangement to have the same number of
each balloon colour. Use a factor tree to find the greatest number of arrangements that he
can make if every balloon is used.





e

CHAPTER 1.3


In this chapter
Tests of Divisibility
Pupils should be able to:
• use the tests of
divisibility for 2, 3, 4, 5, The bar code on items you buy is like a fingerprint of the item. Each
different item has its own unique barcode. The first six digits in a
6, 8, 9, 10 and 100
12-digit barcode number are the manufacturer identification number.
The next five digits are the item number. The last digit is called the
^ RECALL check digit. This digit can be used to check whether the number has

been scanned correctly.
Q Even numbers end
with a digit which
is . All even
numbers can be
• Think and Share
divided by .
Q Odd numbers end Let's check whether the bar code 639382000393
with a digit which is
is correct:
. Odd numbers
O Cross out the last digit: 63938200039^
cannot be divided by
© Add all the digits in odd positions
(1«, 3rd^ 5ih^ j jp, code; 39382 00039
6 + 9 + 8 + 0 + 0 + 9 = 32
© Multiply the answer by 3.
32 X 3 = 96
O Add all the digits in even positions (2"^, 4'^ 6'^ ...) in the code:

The calculation of the 3 + 3 + 2 + 0 + 3 = 11
check digit in a bar code © Add the answer in Step 4 to the answer in Step 3: 96 + 11 = 107
requires you to know
O Create the check digit by determining what number should be
whether one number
added to the number in step 4 to make the number divisible by 10.
is exactly divisible by
107 + 3 = 110
another number. The
tests of divisibility can be The check digit is therefore 3.
used to help you quickly The complete bar code number is therefore: 639382000393
establish that, especially
when working with
large numbers such as a
12-digit barcode.















UNITl Factors, Multiples and Primes

1.3.1 Tests of divisibility
O Spotlight
Remember:

Play the divisibility game
on this site.
dividend divisor quotient

.i-z\B
A divisibility test is a quick way of finding out whether a given number is divisible
by a divisor without doing the division calculation. This is usually done by
examining the digits of the dividend.




investigate


Tests of divisibility

Use your knowledge of factors to discover some divisibility rules.
In this investigation you will explore how some numbers are related to their factors.

O Explore 10-, as a factor.
a) Circle each number that has 10 as a factor.

148 25 010 1340 1899 6706 15 595 10 000 15 001 145 890 206 980

b) What do you notice about all the numbers you have circled?


c) Write a divisibility rule for 10. Hint: Look at the last digit of the numbers you have
circled.


d) List 6 five-digit numbers that are divisible by 10. The numbers must not appear in the list
in Question la.


Q Explore 100 as a factor.

a) Circle each number that has 100 as a factor.

200 25 010 1300 1299 8701 13 795 10 000 15 001 265 890 216 900
b) What do you notice about all the numbers you have circled?



c) Write a divisibility rule for 100.


d) List 6 six-digit numbers that are divisible by 100. The numbers must not appear in the list
in Question (2a).








o

O Explore 5 as a factor.

a) Circle each number that has 5 as a factor.
34 5258 8275 15 447 1090 5550 25 000 10 165 2222 144 555

b) What do you notice about all the numbers you have circled?



c) Write a divisibility rule for 5.


d) List 6 six-digit numbers that are divisible by 5. The numbers must not appear in the list in
Question 3a.


O Explore ||Q as a factor.


a) Circle each number that has 2 as a factor.
14 188 275 4447 1 096 8 889 1 010 10168 2 222 144986

b) What do you notice about all the numbers you have circled?


c) Write a divisibility rule for 2. Hint; Think about odd and even numbers.



d) List 6 seven-digit numbers that are divisible by 2. The numbers must not appear in the list
in Question 4a.


@ Explore 4 as a factor.

a) Circle each number that has 4 as a factor.

24 188 275 4416 1096 8889 1017 10 168 2221 134 904
b) What do you notice about all the numbers you have circled?


c) Write a divisibility rule for 4. Hint: Check the divisibility of the last two digits of the

number.

d) List 6 seven-digit numbers that are divisible by 4. The numbers must not appear in the list

in Question 5a.


O Explore 8 as a factor.
'
a) Circle each number that has 8 as a factor.
214 888 2032 4416 1011 8889 1016 10 168 2221 134 904

b) What do you notice about all the numbers you have circled?







E UNIT 1 Factors, Multiples and Primes

c) Write a divisibility rule for 8. Hint: Check the divisibility of the last three digits of the
number.


d) List 6 seven-digit numbers that are divisible by 8. The numbers must not appear in the list
in Question 6a.



O Explore 3 as a factor.
a) Circle each number that has 3 as a factor.

214 888 2032 4416 1011 8889 1016 10 168 2226 134 904
b) What do you notice about all the numbers you have circled?



c) Write a divisibility rule for 3. Hint: Check whether the sum of the digits is divisible by
three.


d) List 6 five-digit numbers that are divisible by 3. The numbers must not appear in the list in
Question 7a.



O Explore 9 as a factor.
a) Circle each number that has 9 as a factor.
214 888 2032 4446 7011 8 889 1026 19 467 2226 734 904

b) What do you notice about all the numbers you have circled?


c) Write a divisibility rule for 9. Hint: Check whether the sum of the digits is divisible by nine.



d) List 6 six-digit numbers that are divisible by 9. The numbers must not appear in the list in
Question 8a.


O Explore as a factor.

a) Circle each number that has 6 as a factor.

214 888 2032 4416 1011 8889 1026 10 168 2226 134 904
b) What do you notice about all the numbers you have circled?


c) Write a divisibility rule for 6. Hint: Check whether the number is divisible by two and also
divisible by three.



d) List 6 six-digit numbers that are divisible by 6. The numbers must not appear in the list in
Question 9a.






o

Summary of the tests for divisibility

In your investigation, you discovered some tests for divisibility. The division hints
box below summarises the divisibility tests.





Division Hints



When the last three digits are
When a number is even and ends
in 0, 2, 4, 6 or 8.
Example: 144 is divisible by 2
since the last digit is 4.
The n"'"':'®'.^°''5^®a''ndV52% 8 = 69
three digits is 552 and

When the sum of its digits is
divisible by 3.
Example: 144' 1+4 + 4 = 9
3nd9canbe divided by 3 When the sum of its digits is divisible
without a remainder. by 9.
Example: 144 is divisible by 9 since
the sum of its digits is divisible by 9.
1+4+4=9

A number is divisible by 4 iUbe
last two digits are divisible by 4.
Example; 2144 is divisible by 4
since 44 is divisible by 4.





When the number is divisible by 5 or
end in 0 or 5.
Example: 695 Is divisible by 5 since
the last digit is 5.
690 is divisible by 5 since the last
digit is 0.
A number Is divisible by 100 if its
last two digits are 00.
Example; 34 500 is divisible by 100
since the last two digits are 00.
















UNIT 1 Factors, Multiples and Primes

Investigate

lO Spotlight
Use the tests of divisibility to investigate which of the following numbers
can divide 2520 exactly.
Divislbiiity rule for
Seven
3 4 5 6
Subtract twice the
last digit from the
8 9 10 100
number formed by the
remaining digits.
Is 651 divisible by 7?
65-(1 X 2) = 63

Since 63 is divisible by 7,
Check My so is 651.
Understanding
Divisibility rule for
Eleven
O Match the number with the correct test of divisibility. Subtract the last digit

from the number
A number is divisible by this number formed by the
if the sum of its digits is a multiple remaining digits.
of 9.
Is 396 divisible by 11?
3 A number is divisible by this number
39-6 = 33
if its last two digits are 00.
Since 33 is divisible by
4 A number is divisible by this number
11, so is 396.
if the number formed by its last three
digits is a multiple of 8.

5 A number is divisible by this number
if the last digit is an even number.

6 A number is divisible by this number
if its last digit is 0.

8 A number is divisible by this number
if it is divisible by 2 and also by 3.

9 A number is divisible by this number
if the number formed by its last two Being a multiple of 3
digits is divisible by 4. means the number is also

10 A number is divisible by this number divisible by 3. Being a
multiple of 8 means it is
if the sum of its digits is a multiple
of 3. divisible by 8.
100 A number is divisible by this number
if its last digit is 0 or 5.

o Use the tests of divisibility. Circle the answers.


Number Divisible by:
Example:
© 0 © 5 © 8 10 100
28 548 ©
357 2 3 4 5 6 8 9 10 100

432 2 3 4 5 6 8 9 10 100
2 362 2 3 4 5 6 8 9 10 100

5 681 2 3 4 5 6 8 9 10 100
18 303 2 3 4 5 6 8 9 10 100

58 475 2 3 4 5 6 8 9 10 100
400 005 2 3 4 5 6 8 9 10 100

782 300 2 3 4 5 6 8 9 10 100
7 421 894 2 3 4 5 6 8 9 10 100

7 762 342 2 3 4 5 6 8 9 10 100
© Tick all the answers that apply.

□ 2 Last digit is divisible by 2
□ 3 Digits sum to a multiple of 3

What divides □ 4 Last two digits are a multiple of 4
2856 □ 3 Last digit is a 0 or a 5
exactly? □ 6 Is divisible by 2 and 3

□ s Last three digits are divisible by 8
□ 9 Digits sum to a multiple of 9
□ 10 Last digit is a 0




O Tick all the answers that apply.
□ 2 Last digit is divisible by 2

□ 3 Digits sum to a multiple of 3
□ 4 Last two digits are a multiple of 4
What divides
1174 □ 3 Last digit is a 0 or a 5
exactly? □ o Is divisible by 2 and 3
□ s Last three digits are divisible by 8
□ o Digits sum to a multiple of 9

□ 10 Last digit is a 0






Factors, Multiples and Primes

CHAPTER 1.4


In this chapter
Squares and Square Roots Pupils should be able to:


• recognise squares of
Squares and square roots are Important because they are needed when we
whole numbers to at
calculate areas. Suppose, one day, you have a square kitchen in your apartment
least 20 x 20
that needs tiling.
• recognise square roots
How many tiles do you need to tile the floor? of numbers up to the
square root of 400
• use square and square
root notation
Think and Share


Jackie is tiling her kitchen floor. The kitchen ^ RECALL
is square. The floor tiles are also square. Five
tiles fit along each side of the kitchen.
Q Fill in the boxes.
Help Jackie work out how many tiles she
a) 1 X 1 =
needs altogether by completing the diagram
on the right. b) 2 X 2 =
Jackie could also have calculated the number
c) 3 X 3 =
of tiles she needed like this:
• number of square tiles along one side x number of square tiles d) 5 X 5 =
along the other side

5 X 5 = 25 tiles OR 5^=25 tiles
□
We say five squared equals 25.
Q What are the square
What if Jackie used 16 square tiles to tile her
numbers from 6 to
square bathroom floor?
10?
Draw a diagram using the square here to
help Jackie work out how many tiles there
would be along each side of the bathroom.
Jackie could also have calculated the number
of tiles along each side of the bathroom
like this:
• number of square tiles along one side x number of square tiles
along the other side
50, 16 = 4x4 Vie = 4 tiles
We say the square root of 16 equals 4.

Square numbers




Investigate!


You will need 10 square sticky notes of the same size.

O Try to build squares using different numbers
of square sticky notes given in the first
column of the table. If you can build a
square using that number of square sticky
notes, place a tick in the 'Yes' column,
otherwise place a tick in the 'No' column.


Number tiles Yes No
9
5

6
4

2
10

1

Use your table from Question 1. Draw the different sized squares on a
square grid paper.
O Use what you have learned in Questions 1 and 2 to help you complete
the table.
Number of squares along one Total number of small squares
side of a square used to make up the square

1 1
2 4
3 9
4
5
6
7
8
9
10
20






Factors, Multiples and Primes

The numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and 400 that you have listed in the
second column are called square numbers.

A square number is the product obtained when a whole number is multiplied
by itself.
O Spotlight

1 X 1 can be written as 1^
in each case, the number
2x2 can be written as 2^ Most calculators have
is multiplied by Itself.
for
a key like this Q
3x3 can be written as 3^ finding the square of a
number.
Example 7




• ••••
• ••• • ••••
••
• • ••••
••
•• • • ••• • ••••
••
• •• • • ••• • ••••
1 X 1 2x2 3X3 4X4 5X5

1 4 9 16 25

Check My
Understanding



O Complete the table. 3^ has been done as an example for you. Spotlight

Square numbers Square numbers
Did you
I Did vou know that when
32 9 122
you add odd numbers in
42 132 sequence, you get square
numbers?
5^ 142
6^ 152
72 162
8^ 172

92 182 1 +3 + 5 + 7 = 16
10^ 192
112 202
The numbers 4, 9, 16
and 25 are ail square
@ On a hundred grid, shade all the square numbers. numbers.

Q Draw a dot pattern to represent each square number,

a) 72 b) 9'



©

.4.2 Square roots


Finding the square root of a number is the opposite of finding the square of a
number. The square root of a number is a value that can be multiplied by itself to
give that number.

Example 8
'O Spotlight squared



Most calculators have
a key like this® for or 4 = 16
finding the square root
VTe =4
of a number.
square root
squared


or 13 = 169

yw = 13
square root

We say that finding the square root of a number is the inverse operation of
finding the square of a number. The square of the number 4 is 16, and the
square root of 16 is 4. The square of the number 13 is 169, and the square
root of 169 is 13.



Check My
Understanding


O Complete the table without using a calculator.


1^= 1 □ Vi"=i
5^=25 V25 =5
2^ = 4 \^=2 II



V^ = 3 6^=36





4^= 16










UNIT 1 I Factors, Multiples and Primes

0 Without roots. Then use a calculator to check that



a) b)
o
o
c) V144 d)
e) V361 f) V441

g) V676 h) V196


0 Complete the crossword. Write the answers in words.


V2^

^/roo

o
o




3


Vieoo VilT















•viT












O State whether true or false. If false, rewrite the statement to make it true.
a) The square of a certain number is a value that can be multiplied by itself to give that certain
number.
b) A square root is the product obtained when a whole number is multiplied by itself.
c) The numbers 16, 25, 36 and 49 are all square numbers.
d) The square root of 4 is 16.

Investigate]

PAIR WORK
Squares and Roots Bingo

Each pair will need: Squares and Roots Bingo
one game board (supplied by your
teacher) □
4 0 16 49
How to play:
n
O Your teacher will write one 7 9 2 5
calculation on the board. You
need to calculate the answer. Free
12 54 Space 4 □
0 the answer appears on your
If
game board, use a pencil to cross
3 144 8 9
out the number. You may only
cross out one number at a time.
0 When your numbers have been 100 1 □ 121
crossed out in a row or column
or diagonal (your teacher will tell
you this), raise your hand and call out 'Bingo!'.
O The first pair to call out Bingo, is the winner of the game.

0 Another pair will check your answers. You can only be the winner if
you have answered correctly.

Revision

O Find the lowest common multiple of:
List:
a) 8 and 12 b) 7 and 11
a) factors of 12 c) 4 and 15 d) 13 and 12
b) first five multiples of 12 0 a) Is it possible to list the last three
c) factors of 18 multiples of 7?
d) multiples of 9 between 30 and 81
b) Give a reason for your answer.
e) factors of 17
0 Use the clues to find the mystery numbers.
2, 3 and 5 are all factors of 30. However, a) This is a square number smaller than 100.
they are not the only factors of 30. One of its factors is 9.
a) List the other factors of 30. b) This number Is between 40 and 85. It is
b) List the factors of 30 that are prime an even multiple of 9, and 24 is a factor
numbers. of it.
State whether the statements are true or c) This is an even number between 40 and
false. If false, rewrite the statement to make 80. The numbers 5, 4 and 3 are all factors
it true. of this number.
a) Any even natural number is a multiple d) This is an odd number less than 100.
of 4. It has an odd number of factors. The
b) All prime numbers are odd. sum of the two digits that make up the
c) 7 Is a multiple of 49. number is 9.
d) 1 is a multiple of every natural number. e) This is a prime number with two digits.
e) There are only two even numbers that It is smaller than 20. If you double it and
are prime numbers. subtract 9, you get a square number.


Factors, Multiples and Primes

Sarah is packing party bags. She has 54 Evaluate without the use of a calculator:
sweets, 32 chocolates, and 18 lollipops. She 112
a) 6^ b)
wants each party bag to have the same
number of each candy. What is the greatest c) 1^ d) 92
number of party bags that she can make if e) 15^ f) 0^
all the candies are used?
g) Vl21 h) Vm
Mark has a bag of marbles. He wants to
i) V49 j)
share his marbles with his friends.
He can share the marbles equally between k) 52 + VIOO 1) 5^-V
8, 9 and 12 people.
Write down the smallest prime number that
He has less than 100 marbles.
is also a multiple of 29.
How many marbles does he have in the
bag? <D Write down the smallest prime number that
is also a factor of 20.
Q Hotdogs come in packages of 10, while rolls
Write down the smallest prime number that
come in either 8 or 12. What is the smallest
is also a factor of 27 and a multiple of 3.
number of packages of each you have to
buy in order to have the same number of
hotdogs and rolls?

Describe the test of divisibility for:
a) 6 b) 4
Use your tests of divisibility to work out
whether the numbers 9042, 76 543 and 124
800 are divisible by 2, 3, 4, 5, 6, 8, 9, 10 and
100. Place a tick in the cells. You do not
have to calculate the answer.

-r2 ■t3 ~4 -r5 v6 -i-8 ■t9 •f 10 ■r 100
a) 9042
b) 76 543

c) 124 800





V



number? Numbers can a
small stones as
ent ways to communicate our Ideas how many sco(
he three most common forms of
"How many" i:
count, like swe
rds — when you actually say "one" with things the
one" or "two hundred and fifty-six."
or sand or air.
rds — numbers made of letters, like
twenty-one or two hundred fifty-six. There are som(
money. Money
- numbers made of digits, for "how much" v\
21.256. "Nadia. how rr

Help
Sheet

Multiple
A multiple is a number that is the product of a Highest Common Factor (HCF)
given number and some other number.
The biggest number that is the factor of each
number.
6x3 = 13
\ 12:1,2, 3, 4, 6,12 These are the factor
multiple 9:1,3,9
lists of 12 and 9
Factor
HCF = 3
A factor is a number that is multiplied with
another number to get a product. Lowest Common Multiple (LCM)

The smallest number that is a common
6x 3 = 13
multiple of two or more numbers.
factor 6 and 8
1x6=6 1x8=8
Prime numbers
2x6 = 12 2x8 = 16
Prime numbers are numbers which have only
3X6 = 18 3X8 = 24
two factors: 1 and the number itself. The
4 X 6 = 24
prime numbers below 100 are:
CO
LCM = 24

1 2 3 4 5 6 7 8 9 10 Factor tree
11 12 13 14 15 16 17 18 19 20 A factor tree can be used to find HCFs and
LCMs.
21 22 23 24 25 26 27 28 29 30
This is an example of a factor tree for the
31 32 33 34 35 36 37 38 39 40 number 24:
41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 69 70
71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100



Sieve of Eratosthenes
So, 24 = 2 x2 x2 x3
The sieve of Eratosthenes is a method of
finding prime numbers. The result is a
hundred grid as shown above.











UNIT 1 I Factors, Multiples and Primes

Divisibility test Square numbers
A divisibility test is a quick way of finding out A square number is the product obtained when
whether a given integer is divisible by a divisor a whole number is multiplied by itself.
without doing the division calculation.
Square Numbers
Number Test for divisibility
1^=1 1l2= 121
2 All even numbers are divisible by
2^ = 4 122= 144
2. A number is divisible by 2 if the
last digit is 0, 2, 4, 6 or 8. 32 = 9 132 = 169
3 A number is divisible by 3 if the 42 = 16 II 142 = 196
sum of its digits is divisible by 3. 52 = 25 1 152 = 225
4 A number is divisible by 4 if the 62 = 36 162 = 256
number formed by its last two II MC
II
72 = 49 172 = 289
II
digits is divisible by 4.
182 = 324
5 A number is divisible by 5 if its
00
last digit is 0 or 5. 92 = 81 192 = 361
II
»st.
6 A number is divisible by 6 if it is 102= 100 202 ^ 400
divisible by 2 and also divisible by
3. You must use both tests. Square roots
8 A number is divisible by 8 if the The square root of a certain number is a value
number formed by its last three that can be multiplied by itself to give that
digits is divisible by 8. certain number.

9 A number is divisible by 9 if the
Square roots
sum of its digits is divisible by 9.
10 A number Is divisible by 10 if its ^/f=1 V121 =11 V256 = 16
last digit is 0.
V49 = 7 V144 = 12 V289 = 17
100 A number is divisible by 100 if its
last two digits are 00.
^^ = 3 V^ = 8 Vl69 = 13 V324 = 18

9
V^ = Vl96 = 14 V361 =19
VToo = loi V225 = 15 V400 = 20

UNIT 2













Historians believe that the Chinese were one of the first
people to use negative integers. In 200 B.C.E, the Chinese
used a system called the rod system. They used this system to
perform calculations involving negative integers and currency.


Tir iTT nrr
1 2 3 4 5 6 7 8
The Chinese rod system


10 20 30 40 50 60 70 80 90

They used the colour red to show positive integers and the
colour black to show negative integers.


T -6
-3 21
Positive


Chinese rod system Base-10 system
4 -
What do you think the numbers 4, -7 and 31 would look like
'• using the Chinese rod system?




You will learn about:
V '
Integers

Number lines

Adding and subtracting integers

rrr










\



1
UNIT 2

CHAPTER
In this chapter

Pupils should be able to:
Integers • recognise negative
numbers as positions
Integers include all positive whole numbers, negative whole numbers and 0. on a number line
• order positive and
We use numbers to record temperature. Look at the temperatures on the
thermometer. 0°C is the temperature at which pure water freezes. 57°C is negative integers
one of the highest air temperatures recorded on Earth. The number 57 is a • add and subtract
positive number. It is also an Integer. positive and negative
integers
We sometimes need to use numbers that are smaller than 0. For example,
-68°C is one of the lowest air temperatures recorded on Earth. All numbers
that are smaller than 0 are negative numbers. The number -68 is a negative
number. It is also an integer.


c
57°C: one of =- 70
•• Think and Share the highest air |- 60
temperatures -
recorded on |- 50
Dion wants to change the air temperature in a room. The table Earth |_ 40
below shows the air temperature before the change and the
temperature change he desires. * 1- 30
O What is the temperature in the room after the change? I- 20
Complete the table. Dion has done one example. 0®C: freezing |- 10
' >iuro
[■•III point of fresh I- 0
Before Change After 10 00 water
1—10
17°C Temperature rises by 8®C 2S°C
1—20
70c Temperature falls by 10°C
'■/Off
1—30
-4°C Temperature decreases by 5°C
•woe
|—40
-2°C Temperature increases by 4°C
© |—50
Work with a partner and explain how you got your -68°C: one of
the lowest air 1—60
answers.
temperatures
=—70
recorded on
Earth

Integers on a number line


Integers can be represented on a number line. Integers are ordered from the
smallest numbers on the left, to the greatest numbers on the right.
Integers only Include
the number 0 and Integer Number Line
positive and negative
Negative Integers Positive Integers
whole numbers.
Integers do not include
decimals or fractions.
-10 -9 -8 -7 -6 -5 ^ -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10


Zero is neither positive
^ RECALL nor negative


As you move from right to left, numbers get smaller on the
Draw a number line
from -8 to 8. Use number line.
it to compare the
As you move from left to right, numbers get greater on the
following numbers.
number line.
(a)-3 3

(b)or -1 Investigate

(c)-8
Aim: Make your own integer ruler.
Draw a number line
You will need: cardboard, a ruler, a pair of scissors, a pencil and a marker.
to help you find the
difference between Instructions:
-3 and 6.
o Cut a strip of cardboard 30 cm x 3 cm.
e Find the centre of the cardboard ruler. Mark this point as 0.
o Use your ruler. Make markings to the right and left of 0. The markings
should be at intervals of 1 cm.
o Label the integers 1 to 14 on the right of 0.
0 Label the integers -1 to -14 on the left of 0.
o Label the positive integers.

o Label the negative integers.
© Punch a hole on either end of your number line. Put a string and a
bead through it. Now you can move the bead as you count!




-10 -9-8-7-6-5-4 -3 -2-1 0 1 2 3 4 5 6 7 8 9 10
Negative Integers Positive Integers


O Store your integer ruler carefully. You will use it for future
activities.



UNIT 2 I Integers

Example 1


Which number is smaller? Use the number line to help you.
a) 2 or 4 b)-2or0 c)-3or-1


The number 2 lies to the left of the number 4 on the number line.
The number 2 is smaller than the number 4.

< l l l l l ! l l l l i l » l » l l l l i h
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
2<4
The number -2 lies to the left of the number 0 on the number line.
The number -2 is smaller than the number 0.



-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-2<0
The number -1 lies to the right of the number -3 on the number line.
The number -1 Is bigger than the number -3.


- l i l l l l l t l t i l l l l l l l l l h
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-3<-1




Check My
Understanding
Trisha compared the
minimum temperatures in
GROUP WORK;
O Your teacher will give each pupil an integer card. five different places on a
© The class must decide among themselves how to peg their integers on the particular day. She recorded
the temperatures in the table
washing line.
below.
© Pupils have to place their integer cards in order, so that the number
cards are arranged from the smallest integers on the left, to the greatest Place Temperature
integers on the right.
Russia -30«C
O Once everyone has pegged their cards on the line, stand back and check
United -rc
the positions of the integers. Are they in the correct order? Suggest Kingdom
corrections if not.
Iceland -23<'C
South Africa 15°C
India 0°C

List the countries in order
from coldest to warmest.

Check My
Understanding
o Spotlight O Fill in the boxes with <, > or = .


-14
The symbol Z is used a) -2 |4 b) 8| 3-8 c) 9t ]
to represent the set of d) 3| Z1 e) "2 in ^1 f) -7| |8
integers.
g) -2| |-6 h) -3|_Jio i) -1 1 1-6
j) -3| □ o k) -10|^□ -11 1) -141 114
m) o| ^-4 n) in ^2 o) 6| ]6





Investigate

PAIR WORK
Aim: Compare integers.
You will need: integer cards from -13 to 10.
Instructions:
O Mix up the cards and place them face down in a stack.
© Each player takes turns to draw 4 cards from the stack.
© Both players will flip up their cards and arrange them from the smallest
number on the left, to the greatest number on the right.
o Once done, each learner will check the order of the other learner's
arrangement. The learner with the correct arrangement gets one point.
If both arrangements are correct, the fastest learner gets one point.
0 Continue doing this until all the cards have been picked.



Check My
Understanding


O which number line shows the Integers In the correct order?

a) < I I I 1 I I I I > b) < 1 1 1 1 1 1
0 1 2 3-4-3-2-1 3 2 1 0-1 -2-3-4


0 d)
-1 -2 -3 -4 0 1 2 3 -4 -3 -2 -1 0 1 2 3
© Arrange the integers in ascending order.

a) 2,-3,-4 b) -1,-2,-4
Ascending order:
c) 0,-1.^,4 d) -8,0,-3,6
arrange from f
e) 0, -6, -9, -5, 1 f) 10, -12, 13, -2, 12
smallest to greatest. [
Descending order: j © Arrange the integers in descending order.
arrange from ' a) -1;2;4;-^ b) 1;-3;^;5
greatest to smallest. c) -2; 6;-4;-6 d) -8; 9;-1; 6;-3

e) 6; -6; -20; -12; 10 f) 0; -4; -3; 2; 7; 5




UNIT 2 Integers

o Spotlighir
O Write down the next four terms in each of the following sequences:
a) 8,6,4.2 , , b) 15,5,-5 , , The Dead Sea has a
c) 2, 3, 5, 8 , , d) -10, -6, -2, , , , depth of about 380 m.
e) 4, 3, 1. -2, -6 , , f) -10, -9. -l.-A , , The surface and shores of
the Dead Sea are about
O Three divers are diving in the Red Sea. 430 m below sea level.
Diver A is 20 m below sea level. This makes the Dead
Diver B is 13 m below sea level. Sea the lowest point on
Diver C remained on the boat at 1 metre above sea level. land and the deepest
a) How can we record their positions below the sea level? Write the hypersaline lake on
divers' positions using integers. Earth.
b) List the divers from highest position to lowest position.
@* Challenge! During winter, the temperature in Dudinka, a town in
Russia, can go as low as 40°C below zero. How would you record this
temperature as an integer? If the temperature drops by another 5°C,
what would the new temperature be?






Adding and subtracting integers Integers are also known

as directed numbers
In primary school, we learnt how to add and subtract positive numbers. In this because they have either
positive or negative sign.
section, we will learn how to add and subtract with negative numbers.
/Size of the
Addition with negative integers number
-3



Direction of
X
the number
Use the integer track that you have made earlier to work out: (-3) + 6.
Place your counter at -3 and move 6 units to the right.



" 1 1 1 I I I 1 To add a positive
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
integer, you move
Did you land on the number 3? to the right on a
So, -3 + 6 = 3. horizontal number line.

We can also use number discs to help us carry out addition and subtraction that
involves negative numbers.
A mining engineer works
A number disc has two sides, like a coin. One side shows the positive number 1.
50 m underground. To
When we flip the disc, the other side shows the negative number 1.
attend a meeting, she
needs to go to an office
© ^ They are the inverse
that is 97 m above her
of each other.
mining site. What is the
front back
height of the office above
Since 1-1=0,
ground level?
when we put a pair of them together, we get a zero pair. 0
zero pair
Example 2

How can we work out (-3) + 6?

Let's put three discs on the table and six discs on the table.






Group the discs to see how many zero pairs we have.







same
We are left with 3. So, (-3) + 6 = 3
number
number


Work out 4- 6.
We can write it as 4 + (-5). (4) + (-6)
This is the ^ \

operation. This is the sign to indicate
the negative value.
To add a negative
integer, you move to We start from the number 4 and move 6 steps to the left. We land on the
the left on a horizontal number (-2).
number line.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Therefore, (4) + (-6) = -2.

We can also use number discs to help us.



00
©0


So, 4-6 = -2.

UNIT 2 I Integers

jflkj Amazing
IWIathom:
Mathematician
What about -4 + (-5)?
The Chinese number
rod system of 200 BCE
represented positive
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 numbers in red and
negative numbers in
black to work with
We start from (-4) and move 5 units to the left. So, -4 + (-5) = -9.
transactions. But it was
We can also use number discs to help us. only in the 7'" century
that negative numbers
were dealt with by the
0© ©0 Indian mathematician
Brahmagupta (598).
©0 Brahmagupta established
the basic mathematical
© rules for dealing with zero:
(1+0 = 1; 1 - 0 = 1; and
1x0 = 0).
Check My
Understanding 1
+
CO
Let's practise adding positive and negative integers.
O Work out the values.
a) (-3) + (-20) b)
c) (-5) + (-6) d) (-3) + (-8)
e) (+5)+ (-12) f) (-6) + (-3)

g) (-10)+ (+18) h) (+15)+ (-3)
i) ^ + (-1) j) -8 + (-5)
k) -7 +(-2) 1) -14 + (-^)
m)-4 + (-7) + 3 n) 4 + 2 + (-3) + (-1)
o) -4 + (+5) + 1 P) ^ 3 + (-2) +
+
6
q) (-18) + (-2) r) -4+ (-17)
s) (-14)+ (14) t) 12+ (-12)

e A submarine is 111 m below sea level. It rises by 7 m every minute. What
is the depth of the submarine after 10 minutes?
o The temperature went up from -5°C to Find the difference between
the temperatures.
o Bangkok is a city that is 150 cm below sea level. Another city, Kolkata,
is 914 cm above sea level. Represent the altitudes of the cities using
integers. Then, find the difference between the cities.









Q

Subtraction with negative integers


Example 5


Work out -1-2.
On the number line, we start at -1 and move 2 units to the left.
To subtract a negative
integer, you move to
the left on a horizontal
number line.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
So, -1 -2 = -3.
We can also use number discs to help us.


© ©0

So, -1 - 2 is the same as -1 + (-2) = -3.



Example 6


Work out 3-6.
On the number line, we start at 3 and move 6 steps to the left.




-10 -9 -8 -7 -6 -S -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10


We land on -3. So, 3-6 = -3.
We can also use the number discs to help us.

^ (^^) (-1


After arranging to find the zero pairs, we have:



We are left with
©0©


So, 3 - 6 is -3.
















UNIT 2 I Integers

The negative of a
Work out-1 -{-3).
negative number is a
Let us use the number discs to help us.
positive number. -(-3)
To find the negative of -3, flip the number disc as shown: makes (+3).
00© flip



So, -1 - (-3) is the same as -1 + 3:
-1)
© i7
same
We get 2.
Explain
On the number line, you would start at -1 and move to the right by 3 units. number
number


-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

So, -1 -(-3) = 2.

Check My
Understanding


Let's practise adding and subtracting integers.
Work out the values,
a) 3 + b) 3 + (-1)
1
c) -3+1 d) -3 + H)
e) 3-(-1) f) -3-(-1)
g) -11-18 h) 15-(-15)
i) ^0+11 j) -18-14

Work out the values,
a) 7-8 + 6 b) 10-(-6)-2
c) -3-9-5 d) 4 + (-5) +
1
e) 6 + (-9)-(-1) f) -3 + (-2) + (-1)
g) -8 + 9- 9 + 8 h) -4-2-7
Think about some
Maggie parked her car on one of the many shopping mail levels. examples of how
She forgot which level she had parked her car on. However, she integers are used in the
remembered that she had taken the elevator up five floors, to the 4^^ world we live in.
floor. On which floor level did she park her car? Discuss how integers
o Work out the following. help us to determine
a car speeding over
a) -4-(-1) b) 8-(-5)
the limit, indicating an
0 -7-(+2) d) (-14)-(-4)
overpaid bill, or going
e) -7-(-7) f) (10)-(+9) to the underground
g) (4)-(-12) h) (20) - (-30) carparks in a mall.
j) _7_(_8)-(+1)
j) -8-(+1)-(-2)
k) -2-(-1)-(+8) I) 4_(_3)_(_9)

o

Spotlight m) -8 - (-4) - (+4) = n) -16-(17) =
o) 8-(-22) = p) _ii_(_36) =
The additive inverse is
q) 0-(+9) = r) -4-(+7)-(+4) =
the number you must
s) 3 + 2-(+5) = t) -6 + 5-(-2) =
either add to, or subtract
from, another number to Find the value of each of the following,
get an answer of zero.
a) 1 + 5 + 3 = b) 3 + 12 + 8 =
A number and its
c) -3-2-1 = d) -13-7 + 2 =
additive inverse are
e) -8 + 2 - 5 = f) -11+6 + 10 =
always the same distance
from zero on a number g) 7-5 + 3- 9= h) -7 + 7 - 2 =
line. i) -3 + 5 + 3= j) 1+6-1-6=

<1*1 I I ! I I I I i"!*" O Kim is interested in cryogenics, which is the science of very low
-5-4-3-2-1 0 1 2 3 4 5
temperatures. He is doing an experiment on how low temperatures
The integers -5 and 5 are affect the growth of bacteria. He cools one sample of bacteria to
the same distance from a temperature of -58''C and another sample to -82°C. What is the
zero. temperature difference in the two experiments?
The integer-5 is the
We have learnt to calculate 13 + (-27).
additive inverse of 5.
a) Explain in words or diagrams how you would work this out.
The integer 5 is the
b) Describe a situation where this calculation would be used.
additive inverse of -5.



Speed Challenge


Let's do a timed practice for adding and subtracting integers. Your teacher
will set and stop the timer on a stopwatch. Answer the questions as fast and as
accurately as you can.
Ready, steady, go!


Simplify the following.

1. (-3) +(-20)= . 11. (-6) + (-3) =
2. (-5) +(-6)= . 12. (+15) +(-3) =
3. (+5) +(-12)= . 13. -8 +(-5) =
=
4. (-10)+ (+18) = 14. -l4 + (^)
5. -4 + (-1) = 15. 4 + 2 + (-3)+(-1) =
+
6. -7 +(-2) = 16. ^ 3 + (-2) + 6 =
7. -4 + (-7) + 3 = 17. -4-6 + 3 =
8. -A + (+5) + 1 = 18. 9 + 2-3-6 =
9. (-8) +(-2)= , 19. -1-15 + 4+7 + 11 =
10. (-3) + (-8) = 20. -14 + 23-12 + 5 =



Exchange books with your classmate for peer marking.
Write the score out of 20 at the top of the challenge. What went right for
you? Discuss your answers with your partner. Where did you go wrong?


Integers

Iievision


O The temperatures in four cities are recorded at Evaluate,
the same point in time. a) (-14)-47 b) (-139)-139
1
City o Temperature c) 13 + (-43) d) 37 + 42
Albany 4°C e) (-52)-41 f) (_i4) + (_ii)

Anchorage -7°C g) 12+ 15 + (^) h) 12-(-10)-8
i) 20 +13-(-8) ]) (-21)+ (-13)-11
Buffalo -6°C
k) 16+ (-12+ 5) I) (32-21)+ (-21)
Reno ire
O A window cleaner started on the 25^^ floor. He
a) List the cities in order from the lowest to
climbs up 3 floors, then descended 17 floors.
the highest temperature.
How many more floors must he descend to get
b) How much warmer is Reno than
to the ground floor?
Anchorage?
c) What is the temperature difference
between Albany and Buffalo?
d) If the temperature of Buffalo increases
by 11 °C, what is the temperature after
the change?

A bank manager is looking at some bank
accounts. Help her complete the table.


Before Change After
a) $26 Withdrawal of $60
b) $40 Deposit of $25

c) $30 -$60
Challenge! A magic square is a square in
d) $20
which all the numbers in each row, column
and diagonal add up to the same number.
O Calculate. This number is called the magic number.
a)(-12)+ 8 b)(-11)+ (-7) a) What is the magic number of this magic
square?
0 12+ (-8) d) 18+ (-27)
b) Use the magic number to find the
e) 33 + 504 f) (-47)+ 19
missing values.
g) (-1) + (-49) h)(-31)+ 100
i) (-39) + 50 j) (38)+ (-15)
k) 0 + (^) 1) 2 + 0 + (-6)
O Work out each of the following.

a) 4-(-4) b) (-11)-10
c) 84-(-7) d) (-18)-(-6)
e) 121 -(+20) f) 148-(-39)
-2
g) (-18)-(22) h) (-58) - 30
i) 0-(-8) j) (-10)-(-13)
k) 0-(+12) 1) (-1)-(-5)-(+21)

Integers
Integers are both positive and negative whole numbers.
They can be represented on a number line.


Negative integers Positive integers



-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10


Zero is neither positive
or negative


Adding integers Subtracting integers






5 + 8 = 13 5+ -8 5 - (-8) = 5 + 8 5 - (-8) = -5 + 8
= 13 = 3



8-(-5) = 8 + 5 = 13 -8)-(-5) = -8 + 5 = -3
5+ -8










Mathematics Connect



Up and down, left and right
The earliest known elevator dates back to the middle of the
19^^ century. But we can trace this form of mechanism right
back to the Greek and Roman times. They were used to lift and
swing things. For example, a shadufvjas used to raise water
from a river. In the same way, we use elevators to reach the top
and bottom of buildings. Your knowledge of integers helps you
to understand how many floors you need to go up or down to
reach your destination. The tallest building, The Burj Khalifa, has
an elevator that reaches 153 floors and covers 504 metres. But
that's just up and down! What about horizontally? In the future
you may be able to travel, vertically, horizontal
and possibly diagonally. How would you use your
knowledge of Integers in this type of elevator?
Use this link to find out more about
Thyssenkrupp's Rope-Free MULTI Elevator System.




UNIT 2 Integers

Introduction





to Algebra and















You will learn about:
Writing algebraic expressions
Simplifying algebraic expressions

Expanding algebraic expressions
Deriving and using formulae

Functions and mapping
Constructing and solving equations





4




1

T










>

CHAPTER 3.1

In this chapter

Pupiis should be able to:
• construct simple
algebraic expressions by
using letters to represent
numbers
• know the meanings
of the words term, In Stage 1, we used symbols such as A or □ to stand for a number.
expression and equation
• simplify linear expressions
by collecting like terms Investigate!
\
• expand linear expressions
by multiplying a constant What are the possible pairs of numbers that could make 10? In other
over a bracket words, what number can each symbol stand for?
• know that algebraic
operations follow the = 10
same order as arithmetic □•o-
operations
We are looking for possible answers to satisfy the sentence. We can list the
number pairs to 10. We can also use a table to help us.

□ •

^ RECALL 1 9
2 8
What number does each 3 7
symbol stand for?
4 6

O; There are many possible solutions to the mathematical sentence. Instead of
using symbols in the sentence, we can use letters x and y.
We can write x + y = 10. We see that the values of x and the values for y
can vary.
X can be 1 and y can be 9, or x = 3 and y = 7.
We say that x and y are solutions to the problem.

x + y is an algebraic expression.x + y = 10 is an equation.
10
An expression is used to represent a value in algebraic form. It is different
from an equation. An equation contains an equal sign {=) which shows that
the expressions on either side of it are equal to each other.





-• Think and Share


Can you list all the pairs of solutions to the problem in above?







Introduction to Algebra and Equations