# Appendix A Appendix A

Appendixes

Appendix A

A:01 Beginnings in number

A:02 Number: Its order and structure

A:03 Fractions

A:04 Decimals

A:05 Percentages

A:06 Angles

A:07 Plane shapes

A:08 Solid shapes

A:09 Measurement

A:10 Directed numbers

A:11 The number plane

A:12 Algebra

Appendix B

B:01 Challenge: Finite differences

Appendix C

C:01 Multiplying with indices

C:02 Dividing with indices

C:03 Power of a power

C:04 Miscellaneous exercises on indices

C:05 Investigation: Large numbers on a calculator

Appendix D

D:01 Non-numerical geometrical reasoning

(extension)

Appendix E

E:01 Fun Spot: The Tower of Hanoi

Appendix F

F:01 Bisecting intervals and angles

F:02 Constructing angles of 60°, 120°, 30°

and 90°

F:03 Constructing perpendicular and parallel lines

F:04 Constructing regular polygons in a circle

Appendix G

G:01 The minimum conditions for congruent

triangles

Appendix H

H:01 Similar triangles

H:02 Fun Spot: Drawing enlargements

Appendix I

I:01 The pictogram or picture graph

I:02 The column graph

1

• Much of the work of Year 7 will form a foundation for our work this year.

• Revision exercises based on this review can be found in Chapter 1 of the text.

APPENDIX A

A:01 | Beginnings in Number

A:01A Roman Numerals

Numerals

1 I

5 V

10 X

50 L

100 C

500 D

1000 M

• LX means 50 + 10.

• XL means 50 − 10.

• XXXIV means

10 + 10 + 10 + 5 − 1.

worked examples

1 392 = CCCXCII

2 1988 = MCMLXXXVIII

3 56 000 = LVM

4 594 = DXCIV

5 LXXIII = 73

6 MDCXVI = 1616

7 DMMLX = 502 060

• Large numerals are formed by

placing a stroke above the symbol.

V = 5000

X = 10 000

M = 1 000 000

see

1:01

A:01B Place Value and Expanded Notation

314 million

1 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

1 0 0 0 0 0 0

952 thousand

1 0 0 0 0 0

1 0 0 0 0

1 0 0 0

1 0 0

867 units

1 0

1

3 1 4 9 5 2 8 6 7

In the numeral to the left:

• the value of the 3 is 300 000 000,

• the value of the 1 is 10 000 000,

• the value of the 4 is 4 000 000,

• the value of the 9 is 900 000,

• the value of the 5 is 50 000,

• the value of the 2 is 2000,

• the value of the 8 is 800,

• the value of the 6 is 60,

• the value of the 7 is 7.

worked examples

1 Write ‘six million, ninety thousand’ as a basic (simple) numeral.

2 Write (8 × 10 000) + (4 × 1000) + (7 × 100) + (0 × 10) + (5 × 1) as a basic numeral.

3 Write (7 × 10 3 ) + (5 × 10 2 ) + (3 × 10) + 9 as a basic numeral.

Solutions

1 We use zeros as place holders: 6 000 000 + 90 000 = 6 090 000

2 (8 × 10 000) + (4 × 1000) + (7 × 100) + (0 × 10) + (5 × 1) = 84 705

3 (7 × 10 3 ) + (5 × 10 2 ) + (3 × 10) + 9 = 7539

2 NEW SIGNPOST MATHEMATICS 8

A:01C Long Multiplication and Long Division

Multiplication

Division

1 Estimate first. 1 Estimate first.

2 Multiply by the ones digit. 2 Use either the contracted form method or the

3 Multiply by the tens digit. preferred multiples method.

Example

Extended form

531 ×

48

4 248 ← 8 × 531

21 240 ← 40 × 531

25 488

A:01D Powers of Numbers

five squared

5 2 = 5 × 5

= 25

five cubed

5 3 = 5 × 5 × 5

= 125

Example (long division)

Contracted form

28 r 10

31) 878

− 62↓

258

248

10

Answer: 28 remainder 10 or 28 10 -----

31

Preferred multiples

28 r 10

31) 878

− 620 20

258

− 248 8

10 28

■ The small raised number (the index) tells how

many times that number is used in the product.

worked examples

1 Write 2 5 in expanded form and write its basic numeral.

2 Write 10 × 10 × 10 × 10 as a power of ten.

3 What is the basic numeral of 8 × 10 4 ?

Solutions

1 2 5 2 10 × 10 × 10 × 10 3 8 × 10 4

= 2 × 2 × 2 × 2 × 2 = 10 4 = 8 × (10 × 10 × 10 × 10)

= 32 = 8 × 10 000

= 80 000

A:01E Expanded Notation Using Powers

Ten-thousands

10 000

Thousands

1000

Hundreds

100

Tens

10

Units

1

10 × 10 × 10 × 10 10 × 10 × 10 10 × 10 10 1

10 4 10 3 10 2 10 1 1

6 9 8 7 5

■ 6 ten-thousands

9 thousands

8 hundreds

7 tens

5 units

69 875

69 875 = (6 × 10 000) + (9 × 1000) + (8 × 100) + (7 × 10) + (5 × 1)

= (6 × 10 4 ) + (9 × 10 3 ) + (8 × 10 2 ) + (7 × 10) + (5 × 1)

APPENDIX A 3

A:01F Estimation

Estimate before you calculate.

We usually estimate an answer by rounding off each number involved to its

leading digit and then performing the operation.

worked examples

Use leading digit estimation to estimate the answer for each of these.

■ 5314

Question Rounding off Estimate

1 618 + 337 + 159 + 409

2 38 346 − 16 097

3 3250 × 11·4

4 1987 ÷ 4

600 + 300 + 200 + 400

40 000 − 20 000

3000 × 10

2000 ÷ 4

1500

20 000

30 000

500

see

A:02 | Number: Its Order and Structure

A:02A Order of Operations

Mathematicians have agreed on an order of operations so that confusion does not occur.

1:02 Order of Operations

1 Do operations within grouping symbols.

2 Do multiplications and divisions as they appear (from left to right).

3 Do additions and subtractions as they appear (from left to right).

■ Order of

Operations

1( )

2 × and ÷

3 + and −

Where grouping symbols occur inside other grouping symbols, deal with the innermost grouping

symbols first.

Parentheses Brackets Braces

( ) [ ] { }

worked examples

1 4 + 20 ÷ (4 + 1) 2 12 − 6 × 2 3 50 − (25 − [3 + 19])

= 4 + 20 ÷ 5 = 12 − 12 = 50 − (25 − 22)

= 4 + 4 = 0 = 50 − 3

= 8 = 47

1

4 80 ÷ -- of 40 5 6×

9+

25

6

2

= 80 ÷ 20 = 6 × 3 + 5

= 4 = 23

50 + 5

--------------

20 ÷ 4

55

= -----

5

= 11

4 NEW SIGNPOST MATHEMATICS 8

A:02B Using Number Properties

• Multiplying any number by 1 leaves the number unchanged,

eg 347 × 1 = 347.

• Multiplying any number by zero gives the answer zero

eg 84 × 0 = 0.

• Adding zero to any number leaves it unchanged

eg 36 + 0 = 36.

• When adding two numbers, the order does not change the answer

eg 4 + 1 = 1 + 4.

• When multiplying two numbers, the order does not change the answer

eg 7 × 9 = 9 × 7.

• When adding more than two numbers, we may add them in any order

eg 4 + 169 + 96 = (4 + 96) + 169 = 100 + 169 = 269.

• When multiplying more than two numbers, we may multiply them in any order

eg 4 × 836 × 25 = (4 × 25) × 836 = 100 × 836 = 83 600.

• Multiplication can be distributed over addition or subtraction

eg 8 × (17 + 3) = 8 × 17 + 8 × 3, 4 × (100 − 3) = 4 × 100 − 4 × 3,

7 × 99 = 7 × 100 − 7 × 1, 17 × 102 = 17 × 100 + 17 × 2.

A:02C The Number Line and Using Inequality Signs

To graph a number on the number line, we place a large dot on the point which represents that

number.

worked examples

1 Graph the odd numbers less than 8. 2 On the number line graph the set {0, 4, 7}.

–1 0 1 2 3 4 5 6 7 8

–1 0 1 2 3 4 5 6 7 8

If one number is to the left of another on the number line, then it is smaller than that number,

eg 3 < 5, −1 < 0. The number to the right is larger, eg 7 > 4, 0 > −1.

A:02D Special Groups of Whole Numbers

Name of group Pattern Diagram or

explanation

Cardinal numbers 0, 1, 2, 3, . . . Zero + counting

numbers

Counting

numbers

Zero is included.

Comment or rule

1, 2, 3, 4, . . . Zero is not included.

, , , , , , ...

0 1 2 3 4 5 6 7

Even numbers 2, 4, 6, 8, . . . All dots are in pairs. (2 × n)

, , , , ...

0 1 2 3 4 5 6 7

Odd numbers 1, 3, 5, 7, . . . One dot not paired. (2n − 1)

, , , ...

0 1 2 3 4 5 6 7

0 1

2

3

4

5

6 7

APPENDIX A 5

Name of group Pattern Diagram or

explanation

Comment or rule

Square numbers 1, 4, 9, 16, . . . A counting number times itself,

, , , , ... eg 1 × 1, 2 × 2, 3 × 3, etc. (n 2 )

Triangular

numbers

Hexagonal

numbers

Fibonacci

numbers

Palindromic

numbers

1, 3, 6, 10, 15,

. . .

1, 7, 19, 37,

. . .

1, 1, 2, 3, 5, 8,

. . .

Examples

929, 7337

, , , , ...

1, 1 + 2, 1 + 2 + 3,

1 + 2 + 3 + 4, . . .

The sum of the first n counting

numbers.

, , , ...

1, 1 + (6 × 1), 1 + (6 × 1) + (6 × 2),

1 + (6 × 1) + (6 × 2) + (6 × 3), . . .

+6 +12 +18 +24 +30

ie 1, 7, 19, 37, 61, 91, . . .

Except for the first two in the pattern, each term is the

sum of the two terms before it.

the same forwards

or backwards.

‘Able was I ere I saw Elba’

is a palindromic sentence.

A:02E Factors and Multiples

• A factor of a counting number divides it exactly.

• A multiple of a counting number is found when you multiply

it by another counting number.

■ 3 × 8 = 24. Here 3 and 8

are factors of 24, and 24 is

a multiple of both 3 and 8.

worked examples

1 List all factors of 102. 2 List the first nine multiples of 7.

To do this we look for numbers that 7 × 1 = 7, 7 × 2 = 14, 7 × 3 = 21,

multiply together to give 102. 7 × 4 = 28, 7 × 5 = 35, 7 × 6 = 42,

1 × 102 = 102 7 × 7 = 49, 7 × 8 = 56, 7 × 9 = 63.

2 × 51 = 102

The first nine multiples of 7 are:

3 × 34 = 102

7, 14, 21, 28, 35, 42, 49, 56, 63.

6 × 17 = 102

Note: 4, 5, 7, 8, 9, 10 and 11

do not divide 102 exactly.

The factors of 102 are:

1, 102, 2, 51, 3, 34, 6, 17.

■ You need not check numbers higher than 11.

102 ÷ 11 9·27

you have gathered all factors.

• The highest common factor of two numbers is the largest number that is a factor

of both. This is called the HCF.

• The lowest common multiple of two numbers is the smallest number that is a multiple

of both. This is called the LCM.

6 NEW SIGNPOST MATHEMATICS 8

worked examples

1 Find the highest common factor 2 Find the lowest common multiple

(HCF) of 102 and 153. (LCM) of 15 and 9.

Number Factors Number Multiples

102 1, 102, 2, 51 , 3, 34, 6, 17. 15 15, 30, 45 , 60, 75, 90, . . .

153 1, 153, 3, 51 , 9, 17. 9 9, 18, 27, 36, 45 , 54, . . .

The common factors are red.

The highest common factor is 51.

There would be other common multiples

but the lowest common multiple can be

seen in the table to be 45.

A:02F Prime and Composite Numbers

• A prime number is a counting number that has exactly two factors, itself and 1.

7 has only two factors, 7 and 1, so 7 is a prime number.

• A composite number has more than two factors.

25 has three factors, 25, 1 and 5, so 25 is a composite number.

• The number 1 has only one factor, 1, so it is neither prime nor composite.

Use a factor tree to write 252 as a product of prime numbers.

252

worked examples

4

63

2 2 3 21

252 = 2 × 2 × 3 × 3 × 7

= 2 2 × 3 2 × 7

2

2

3

3

7

APPENDIX A 7

A:02G HCF and LCM by Prime Factors

When we express numbers as products of their prime factors it is sometimes easier to find their

HCF and LCM.

For the numbers 400 and 1080 find:

1 the HCF, and 2 the LCM.

400

20 20

4 5 5 4

2 2 5 5 2 2

worked examples

1080

20 54

4 5 6 9

2 2 5 2 3 3 3

From these factor trees we find:

Number

The two numbers have 2 × 2 × 2 × 5 in common.

2 × 2 × 2 × 5 = 2 3 × 5 or 40.

∴ HCF = 40

The highest common factor of 400 and 1080 is 40.

The LCM must contain the whole of both numbers, so it will contain the whole of 400

(ie 2 × 2 × 2 × 2 × 5 × 5) as well as any other prime factors of 1080 not already written down.

LCM = (2 × 2 × 2 × 2 × 5 × 5) × (3 × 3 × 3)

= 2 4 × 3 3 × 5 2

= 10 800

The lowest common multiple of 400 and 1080 is 10 800.

■ Note:

The HCF is found by using the smallest common power of each prime factor of the numbers.

The LCM is found by using the highest power of each prime factor of the numbers.

400

1080

Product of prime factors

2 × 2 × 2 × 2 × 5 × 5

or 2 4 × 5 2

2 × 2 × 2 × 3 × 3 × 3 × 5

or 2 3 × 3 3 × 5

A:02H Divisibility Tests

If 4 divides a number and leaves no remainder, then the number is divisible by 4. Here are some

simple divisibility tests you should know.

Divisor Divisibility test Example

2 The number is even,

ie it ends in 0, 2, 4, 6 or 8.

3 The sum of the digits is

divisible by 3.

4 The number formed by the last

two digits is divisible by 4.

4136 is divisible by 2

as it is even.

30 012 is divisible by 3

as (3 + 0 + 0 + 1 + 2) is

divisible by 3.

76 112 is divisible by 4

as 12 is divisible by 4.

5 The last digit is 5 or 0. 11 225 is divisible by 5

as it ends with a 5.

Learn these!

They’re useful.

8 NEW SIGNPOST MATHEMATICS 8

Divisor Divisibility test Example

6 The number is divisible by

both 2 and 3

8 The number found by the last

three digits is divisible by 8.

9 The sum of the digits is

divisible by 9.

10 002 is even and its

digit sum is divisible by 3.

963 216 is divisible by 8

as 216 is divisible by 8.

142 128 is divisible by 9

as (1 + 4 + 2 + 1 + 2 + 8)

is divisible by 9.

10 The last digit is 0. 814 170 is divisible by 10

as it ends with 0.

11 The sum of the odd-numbered

digits is equal to the sum of the

even-numbered digits or differs

from it by a multiple of 11.

25 The last two digits are 00, 25,

50 or 75.

7 081 426 is divisible by

11 as (7 + 8 + 4 + 6) −

(0 + 1 + 2) is 22 and this

is divisible by 11.

80 925 is divisible by 25

as it ends in 25.

100 The last two digits are 00. 81 700 is divisible by 100

as it ends in 00.

By using the tests in

the table, we can

construct other tests.

■ • Divisibility

by 12 is

the same as

divisibility by

both 3 and 4.

• Divisibility by

50 is the same

as divisibility by

both 2 and 25.

A:02I Square and Cube Roots

• The number which when squared gives 225 is called the

square root of 225.

• The number which when cubed gives 512 is called the

cube root of 512.

If the square of 15 is 225, then the square root of 225 is 15.

■ If 15 2 = 225, then 225 = 15 .

If the cube of 8 is 512, then the cube root of 512 is 8.

■ If 8 3 = 512, then 3 512 = 8 .

3

means ‘the square root of’.

means ‘the cube root of’.

A:03 | Fractions

A:03A Basic Concepts

3

• ----- means 3 of 10 equal parts.

10

• 10 tenths make 1 whole.

3 ← numerator

• A mixed numeral has a ‘whole number’

10 ← denominator

3

part and a ‘fraction’ part, eg 2 ----- .

10

23

• An improper fraction has a numerator that is bigger than its denominator, eg ----- .

10

see

1:03

APPENDIX A 9

A:03B The Fraction Bar

The fraction bar is like a division sign.

10

worked examples

1 ----- = 10 ÷ 2 2 -- = 9 ÷ 4 3 ----- = 87 ÷ 10

2

4

10

1

7

= 5 = 2 --

= 8-----

9

4

8

4

1

4

Note: -- = -- + -- = 2

87

-----

10

80

10

7

10

1

--

4

= ----- + ----- = 8

7

-----

10

9

4

87

10

A:03C Mixed Numerals and Improper Fractions

To write a mixed numeral as an improper fraction, express the whole number with the same

denominator as the fraction part.

worked examples

1

1 3

3 -- = --

1

+ --

2 5 5

----- = --

3

+ ----- 3 1 -- =

2 1 × 2 2

10 1 × 10 10

4

6

=

1

=

50

--

3

+ --

----- + -----

2 2

10 10

=

7

53

= --

= -----

2

10

=

1

--

1

4

--

4

7

--

4

× 4

× 4

3

+ --

4

3

+ --

4

A:03D Simplifying Fractions

A fraction may be simplified by casting out equal factors from both the numerator and

denominator. (This process is called cancelling.)

8 8

1 ----- = ----- ÷ 2 2 20 ----- =

10 10 ÷ 2

50

8 4

= -------

10 5

=

4

= --

5

=

worked examples

20 ÷ 10

-----

50 ÷ 10

20

-------

2

50 5

2

--

5

-----

8

10

and

4

--

5

the same!

are

A:03E Equivalent Fractions

The size of a fraction is unchanged if both the numerator and denominator are multiplied by the

same number.

10 NEW SIGNPOST MATHEMATICS 8

worked examples

Complete the following equivalent fractions:

2

3

1 -- = -----

2 -- = --------

5 10

4 100

Multiply top and

Multiply top and

bottom by 2. bottom by 25.

2

--

× 2 4

= -----

3 × 25

--

75

= --------

5 × 2 10

4 × 25 100

They’re

the same!

3

4

-- = -----

1 10

Multiply top and

bottom by 10.

4 × 10

--

40

= -----

1 × 10 10

To add or subtract fractions that have the same denominator, add or subtract the numerators.

To add or subtract fractions that have different denominators, use equivalent fractions to give each

the same denominator.

A:03H Comparing Fractions

worked examples

3 4 3+

4

1 ----- + ----- = -----------

2

19 6 19 – 6

-------- – -------- = -------------- 3

3 7 3+

7

-- + -- = -----------

10 10 10

100 100 100

8 8 8

7

13

10

= -----

= --------

= -----

10

100

8

2

= 1 -- or 1

worked examples

3 3

1 2

9 3

--

3

3

+ -----

2

----- – --

-- + --

5 10

10 4

4 5

3

= --

× 2 3

+ -----

= 9 × 2 3

----- – --

× 5 =

3 × 5 2 × 4

-- + --

5 × 2 10

10 × 2 4 × 5

4 × 5 5 × 4

=

6 3

=

18 15

-----

=

15

+ -----

8

----- – -----

----- + -----

10 10

20 20

20 20

9

3

23

= -----

= -----

= ----- or 1-----

3 10

20

20 20

To compare fractions, use equivalent fractions to give them the same denominator.

8

1

--

4

APPENDIX A 11

worked examples

1 Which fraction is 2 Arrange in order, from smallest

3 6

8 17 3

smaller, -- or ----- ? to largest: { ----- , ----- , -- }.

4

10

3

--

× 5 = 15

8

-----

-----

× 2 =

16 -----

4 × 5 20

10 × 2 20

6

-----

× 2 = 12

17 17

-----

----- = -----

10 × 2 20

20 20

6

3

∴ ----- is the smaller. --

× 5 =

15 -----

10

4 × 5 20

3 8 17

In order they are -- , ----- , ----- .

10

20

4

4

10

20

■ When denominators

are the same, compare

numerators.

A:03I Addition and Subtraction of Mixed Numerals

worked examples

2

3

1 3 -- + 1--

2 10 -- − 1--

5 4

5 2

2 3

1 1

= (3 + 1) + (--

+ -- ) = (10 − 1) + (--

− -- )

5 4

5 2

2

= 4 + (-- × 4 3

+ -- × 5

1 1

) = 9 + (--

− -- )

5 × 4 4 × 5

5 2

8 15

1 1

= 4 + (-----

+ ----- ) = 8 + (1 -- − -- )

20 20

5 2

23

3

6

= 4 ----- (ie 4 + 1 ----- ) = 8 + (-- × 2 1

− -- × 5 )

20

20

5 × 2 2 × 5

3

12 5

= 5 -----

= 8 + (-----

− ----- )

20

10 10

7

= 8-----

10

3

3 5 − 2--

5

3

= (5 − 2) − --

5

3

= 3 − --

5

3

= 2 + (1 − -- )

5

2

= 2--

5

1

1

1

■ -- is bigger

2

1

than -- , so use

5

one whole.

A:03J Multiplication

To multiply fractions, we may multiply numerators and multiply denominators.

1

worked examples

3

2

■ -- has been ■ of the shaded part

4

5

-- ■ ( 2

of 3

) or ( 2

× 3

) is 6

-- -- -- -- -----

5 4 5 4 20

.

has been cut out.

12 NEW SIGNPOST MATHEMATICS 8

2

5

Conclusion: -- × -- = ----------- = ----- or

If a number will divide into a numerator and a denominator, this should be done. It is easier to

do it before multiplying, rather than doing it when you simplify the answer.

3 7

3

2 -----

2

× -----

9

3 --

15

× --

4 ----- × -----

10 10

5 3

10 16

(No number will exactly

1

3 2

9 15

(divide top and bottom.) = ---- × ----

= -------

5 3 1

2 10 × -------

3

16

7

= -----------------

10 × 10

2

3

= -----------

= --------------

1

16

21

= --------

100

2

27

= --

= -----

5

32

A:03K Multiplication of Mixed Numerals

To multiply mixed numerals, write them as improper fractions.

3

5

4

1

A:03L Division

3

4

2 × 3

5 × 4

6

20

3

-----

10

worked examples

1 4 × --

2 1 -- × 3 3 3 -- × 1

3

5

= -- × --

= ----- × --

= ------- ×

12

5

2

5

= -----

= -----

=

7

8

= 2 --

= 5 --

= 5

15

8

45

8

5

8

3

1

There are two methods for doing division involving fractions. The method on the right is more

popular.

3

4

5

15

1 4

5

--

1

1

--

3

4

----

1

2 1

When the fractions have the same

denominator, you can divide the

3 5 6 5

1

eg -- ÷ -- = -- ÷ -- = 6 ÷ 5 = 1--

4 8 8 8

5

To divide by a fraction, invert the

fraction and multiply (ie multiply

by the reciprocal).

3 2 3 5 15 7

eg -- ÷ -- = -- × -- = ----- = 1 --

4 5 4 2 8 8

Mixed numerals should first be changed to improper fractions.

9 1

10 2

9

5 10

9

5

worked examples

1 ----- ÷ --

2 4 ÷ --

3 1 -- ÷ 3

2 1

= ------- × ----

= -- × --

= -- ÷

1

= --

= -----

= ---- ×

4

5

= 1 --

= 6 --

=

4

1

20

3

3

5

2

3

5

3

1

2

3

2

1

3

1 2

2

--

5

3

--

4

15

-----

4

4

-------

2

15 5

APPENDIX A 13

A:03M A Fraction of a Quantity

When finding a fraction of a quantity, it is sometimes wise to change to a smaller unit. To find one

quantity as a fraction of another, make sure the quantities have the same units.

worked examples

1 Find -- of 2 km. 2 What fraction is 40 cm of 2 m?

1

--

5

3

--

5

3

5

1

40 cm

of 2 km = -- of 2000 m Fraction = -----------------

5

200 cm

4

= 2000 ÷ 5m = -----

20

= 400 m =

of 2 km = 3 × 400 m

= 1200 m or 1·2 km

1

--

5

see

1:04

A:04 | Decimals

A:04A Our Number System is Based on Ten

5 hundreds 500·

7 tens 70·

3 units 3·

1 tenth 0·1

9 hundredths 0·09

4 thousandths 0·004

2 ten-thousandths 0·0002

5 7 3·1 9 4 2

573·1942

■ 0·8 0·65

means

means

8 of 10 65 of 100

equal parts. equal parts.

100 10 1

1

-----

10

1

--------

100

1

-----------

1000

1

----------------

10 000

5 7 3 · 1 9 4 2

If the ‘large cube’ represents one whole, the ‘flat’ represents 0·1, the ‘long’ represents 0·01 and the

‘short’ represents 0·001, then the number represented below is 2·152.

14 NEW SIGNPOST MATHEMATICS 8

A:04B Conversions: Fractions and Decimals

The following examples show the connection between fractions and decimals.

7

10

worked examples

13

100

9

1000

1 ----- = 0·7 2 2 -------- = 2·13 3 ----------- = 0·009

1 zero 1 figure 2 zeros 2 figures 3 zeros 3 figures

after the after the after the

point point point

To write a decimal as a fraction, use the method above to write the decimal as a fraction with a

denominator of 10, 100, 1000, etc, then simplify.

worked examples

Change each to a fraction or mixed number in simplest form.

1 0·85 2 0·025 3 1·8 4 9·04

85

= -------- ÷ 5

25

= ----------- ÷ 25

8

= 1 ----- ÷ 2

= 9

100 ÷ 5

17

20

1000 ÷ 25

1

40

10 ÷ 2

= -----

= -----

= 1 --

= 9

4

5

4

-------- ÷ 4

100 ÷ 4

1

-----

25

To write a fraction as a decimal, we may use either of two methods.

1 Express the denominator of the fraction as

a power of ten, then change to a decimal.

or

2 Divide the numerator by the denominator.

If a decimal keeps repeating,

it’s called a repeating

or recurring decimal.

Method 1

worked examples

Change each fraction to a decimal.

3 3 × 2

1 -- = --

5 5 × 3

6

= -----

10

= 0·6

■ 0·3333333333 … = 0·3˙

0·1717171717 … = 0·1˙ 7˙

0·1081081081 … = 0·1˙ 08˙

33

33

-------- × 5

× 5

2 -------- =

200 200

165

= -----------

1000

= 0·165

APPENDIX A 15

Method 2

5

3 -- = 5 ÷ 8

8

= 0 · 6 2 5

= 8 ) 5 · 0 2 0 4 0

= 0·625

3

4 ----- = 3 ÷ 11

11

= 0 · 2 7 2 7 …

= 11 ) 3 · 0 8 0 3 0 8 0 3 0

= 0·2˙ 7˙

When a decimal

does not repeat,

it’s called a

terminating decimal.

■ 0·6, 32·75

8·11, 0·0067

A:04C Comparing Decimals

■ To compare decimals,

give them the same number

of decimal places.

eg

0·3 0·300

0·33 0·330

0·303 0·303

‘Ascending order’ means

‘smallest to largest’.

Whole numbers may be written as decimals by placing

zeros after a decimal point

eg 7 = 7·00, 130 = 130·0, 2 = 2·000

The number line can be used to compare two decimals.

The decimal to the left will be the smaller one. The

decimals 0·5, 2 and 3·1 have been graphed on the

number line.

0

0·5

1

1·5

2

2·5

3

3·5

We place tenths under tenths, hundredths under hundredths, and thousandths under thousandths.

To add or subtract decimals, use the rule: Place Points Under Points.

worked examples

1 3·7 + 1·52 2 63·85 − 2·5 3 8 − 1·625

= 3 · 7 0 +

= 63·85 −

= 8·000 −

1 1· 5 2 2·50 1·625

5 · 2 2 61·35 6·375

16 NEW SIGNPOST MATHEMATICS 8

A:04E Multiplication with Decimals

When multiplying, we must have the same number of figures after the decimal point in

the answer as we had altogether after decimal points in the question.

1 0·006 × 0·5 2 38·2 × 0·11

= 0·0030 = 4·202

= 0·003

3 (0·05) 2 4 1·3 × 19·1

= 0·05 × 0·05 = 24·83

= 0·0025

worked examples

■ Multiply as though

there were no decimal

points, then use the

rule to place the point

It helps to estimate

A:04F Multiplying and Dividing by 10, 100, 1000

1 When we multiply a decimal by 10, 100 or 1000, we move the decimal point 1, 2 or

3 places to the right.

2 When we divide a decimal by 10, 100 or 1000, we move the decimal point 1, 2 or

3 places to the left.

worked examples

1 0·60 × 100 2 34·180 × 1000 3 0·075 × 10

= 60 = 34 180 = 0·75

4 81·6 ÷ 100 5 056· ÷ 1000 6 0·045 ÷ 10

= 0·816 = 0·056 = 0·0045

■ Zeros may have to

be put in to fill an

empty space.

A:04G Division of a Decimal by a Whole Number

To divide a decimal by a whole number use the rule: Place Points Under Points.

worked examples

1 48·9 ÷ 3 2 1·5 ÷ 5 3 0·123 ÷ 4

1 6·3 0·3 0·030 7 5

= 3 ) 4 1 8·9 = 5 ) 1·5 = 4 ) 0·123 3 0 2 0

■ In each case we

have kept POINT

UNDER POINT.

A:04H Division of a Decimal by a Decimal

To divide one decimal by another decimal we change the question so that we are

dividing by a whole number. We achieve this by moving the decimal point to the right

in both numbers.

APPENDIX A 17

1 0·8136 ÷ 0·04

0·8136

= -----------------

0·04

0·8136 × 100

= -----------------

0·04 × 100

81·36

= -------------

4

= 20·34

OR

worked examples

1 0·8136 ÷ 0·04 2 875 ÷ 0·05

Move the decimal point = 875·00 ÷ 0·05

two places to the right = 87 500 ÷ 5

in each to make the = 17 500

second number whole.

= 81·36 ÷ 4 3 3·612 ÷ 1·2

= 20·34 = 36·12 ÷ 12

= 3·01

A:04I Decimal Currency

The rules for decimals also apply to amounts of money.

worked examples

1 Dh362 + Dh3.42 2 Dh100 − Dh41.63 3 Dh8.37 × 8 4 Dh90 ÷ 8

Solutions

1 Dh362.00 + 2 Dh100.00 − 3 Dh 8.37 × 4 Dh1 1 . 2 5

Dh 3.42 Dh 41.63 8 8 ) Dh9 1 0 . 2 0 4 0

Dh365.42 Dh 58.37 Dh66.96

■ 25fils = Dh0.25

see

A:01F

A:04J Rounding Off

This is an important skill used in estimation.

To round off (or approximate) a number correct to a given place, we round up if the

next figure is 5 or more, and round down if the next figure is less than 5.

worked examples

Round off: 1 96 700 000 to the nearest million.

2 0·085 to the nearest hundredth.

3 86·249 correct to one decimal place.

Solutions

1 96 700 000 has a 6 in the millions column.

The number after the 6 is ‘5 or more’ (ie 7),

∴ 96 700 000 = 97 000 000, correct to the

nearest million.

2 0·085 has an 8 in the hundredths column.

The number after the 8 is ‘5 or more’ (ie 5),

∴ 0·085 = 0·09, correct to the nearest hundredth.

18 NEW SIGNPOST MATHEMATICS 8

3 86·249 has a 2 in the first decimal place.

The number after the 2 is ‘less than 5’ (ie 4),

∴ 86·249 = 86·2, correct to one decimal place.

A:05 | Percentages

A:05A Changing Percentages to Fractions and Mixed Numerals

A percentage is a convenient way of writing a fraction that has a denominator of 100.

‘Per cent’, written %, means ‘per 100’ or ‘for every 100’.

worked examples

see

1:05

9

100

1 9% = --------

2 64% =

(9 for every 100) =

125

64

-------- ÷ 4

100 ÷ 4

16

-----

25

3 125% = --------

4 14 -- % = --------

100

2 100 × 2

25

= 1-------- ÷ 25

100 ÷ 25

29

= --------

1

200

= 1--

4

1

14 1 2 --

× 2

Write the percentage

as a fraction and

simplify.

100% = 1

200% = 2

A:05B Changing Fractions to Percentages

100

To express a fraction as a percentage, multiply by -------- %.

1

worked examples

3 3

1 -- = --

3 11

× 1

2 1 -- = ----- × 1

4 4

8 8

3 100

11

= -- × --------%

= -----

4 1

8

25

3 100

11

= -- × --------%

= -----

14

1

28

275

= 75% = --------%

2

= 137·5%

100

× --------%

1

25

100

× --------%

1

1

20

37 37 100

3 -------- = --------

3 23 100

× --------% 4 4 -- % = ----- × --------%

300 3300

1

5 15

1

37

460

= -----%

= --------%

3

1

1

= 12 -- % = 460%

3

That is the

same as

multiplying by 1.

■ We could also have changed

the denominator to 100,

3 30

eg ----- = --------

10 100

= 30%

APPENDIX A 19

A:05C Changing Percentages to Decimals

To change 93% to a decimal, divide 93 by 100. eg 93· ÷ 100 = 0·93

worked examples

1 47% = 47· ÷ 100 2 30% = 30· ÷ 100

= 0·47 = 0·30 or 0·3

3 325% = 325· ÷ 100 4 300% = 300· ÷ 100

= 3·25 = 3

5 12·7% = 12·7 ÷ 100 6 0·3% = 00·3 ÷ 100

= 0·127 = 0·003

A:05D Changing Decimals to Percentages

Hundredths column

13% = 0·13 13 ÷ 100 = 0·13

150% = 1·50 150 ÷ 100 = 1·50

4% = 0·04 4 ÷ 100 = 0·04

To express a decimal as a percentage, multiply by 100%.

worked examples

1 0·87 = 0·87 × 100% 2 1·30 = 1·30 × 100%

= 87% = 130%

3 5 = 5·00 × 100% 4 0·825 = 0·825 × 100%

= 500% = 82·5%

■ Add zeros when they are

needed. eg 0·8 = 0·80

A:05E Conversions You Should Know

Percentage 1% 5% 10% 12 -- % 20% 25% 33 -- % 50% 100%

Decimal 0·01 0·05 0·1 0·125 0·2 0·25 0·3˙ 0·5 1

1

100

5

100

Fraction -------- -------- ----- -- -- -- -- -- 1

A:05F Using Percentages

1

10

1

2

1

8

To find a percentage of a quantity, write the percentage as a decimal (or fraction) and

multiply by the quantity.

1

5

worked examples

1 8% of 560 L 2 70% of 680 g 3 5% of Dh800

8 ■ 24 out of 32

= 0·08 × 560 = 0·70 × 680 =

5 800

-------- × --------

is the same as

= 44·80 = 476·00

1100

1

75 out of 100.

= 44·8 L = 476 g = Dh40

1

4

1

3

1

3

1

2

20 NEW SIGNPOST MATHEMATICS 8

0

0

4 Rashid scored 24 marks out of 32. What is this as a percentage?

24

Fraction of marks obtained = -----

32

3

24 100

Percentage of marks obtained = ----- × --------%

324

1

300

= --------%

4

= 75%

75%

24

32

A:06 | Angles

A:06A Naming Angles

An angle can be named in several ways. Two ways are shown here.

B

Angle Method 1 Method 2

A

C

∠ABC

or

∠CBA

∠B

or

• In Method 1, we go from the letter on

one arm to the vertex and then to the

letter on the other arm.

• Method 2 can only be used when there

is no chance of confusion.

see

1:06

A:06B Measuring Angles with a Protractor

Place the protractor as shown below and read off the angle.

You need to choose the correct reading from the two on the arm of the angle.

20

10

30

40

50

60

70

80

180170160150140130120110100

90

100

80

110

70

120

60

130

50

140

40

150

30

160

20

10

170

180

0

This angle measures 110°, as

it is bigger than a right angle.

This angle measures 40°, as it

is smaller than a right angle.

20

10

30

40

50

60

70

80

180170160150140130120110100

90

100

80

110

70

120

60

130

50

140

40

150

30

160

20

10

170

180

0

APPENDIX A 21

A:06C Types of Angles

Angles may be classified into different types depending on their size.

Type acute right obtuse straight reflex revolution

Size

less than

90°

90° between 90°

and 180°

180° between

180° and

360°

360°

Diagram

A:06D Special Angles

Name Diagram Explanation

Vertex

Common

arm

1 have a common arm

2 have a common vertex

3 and lie on opposite sides of the

common arm.

Complementary

46°

44°

Any two angles whose sum is 90° are

complementary angles.

eg 46° + 44° = 90°

Supplementary

Any two angles whose sum is 180° are

supplementary angles.

eg 128° + 52° = 180°

128° 52°

complementary

These are formed when a right angle is

cut into two parts. The angles add up

to 90°.

supplementary

These are formed when a straight angle

is cut into two parts. The angles add

up to 180°.

22 NEW SIGNPOST MATHEMATICS 8

Angles at a

point

Name Diagram Explanation

92°

125°

143°

If angles meet at a point, then their

sum is 360° or one revolution.

eg 92° + 125° + 143° = 360°

Vertically

opposite

33° 147°

147° 33°

When two straight lines intersect, the

vertically opposite angles are equal.

A

A transversal is a line that cuts two or more other lines.

In the diagram AB is a transversal.

B

Corresponding

x

x

There are four pairs of corresponding

angles.

These are equal when the lines crossed

by the transversal are parallel.

Alternate

There are two pairs of alternate angles.

These are equal when the lines crossed

by the transversal are parallel.

Co-interior

x

x

There are two pairs of co-interior

angles.

These are supplementary when the

lines crossed by the transversal are

parallel. (They add up to 180°.)

APPENDIX A 23

see

1:07

A:07 | Plane Shapes

A:07A Naming Plane Shapes

Plane shapes are shapes that lie in one plane, or flat surface. They have area but not volume.

Below are some of the common plane shapes.

This is the family of quadrilaterals.

rectangle

trapezium

parallelogram

rhombus

square

kite

Each member of this family is a special kind of each of the shapes on its left,

eg a rhombus is a special parallelogram, trapezium, kite and quadrilateral.

Other plane shapes include:

triangle pentagon hexagon octagon circle oval (ellipse)

Shapes that have only straight sides are called polygons.

The ID cards at the front of this book (pages xv to xx) should be used often to help you remember

the language of Mathematics.

A:07B Constructing Triangles

When we construct triangles, we make use of a ruler, compasses and protractor.

Example 1

worked examples

Construct a triangle that has sides of length 6 cm, 5 cm and 4 cm.

24 NEW SIGNPOST MATHEMATICS 8

Solution 1

1 2 3 4

5 cm

4 cm

Step 1 Draw an interval 6 cm long.

Step 2 Set the compasses to 5 cm and from one end of the interval draw an arc.

Step 3 Set the compasses to 4 cm and from the other end of the interval draw an arc to cut the

arc that has already been drawn.

Step 4 Join the point where the arcs meet to both ends of the interval to obtain the triangle.

Example 2

Construct a triangle that has sides of 4 cm and 5 cm which form an angle of 60°.

Solution 2

6 cm 6 cm 6 cm 6 cm

1 2 3 4

60° 60° 60°

5 cm 5 cm

4 cm

4 cm

60°

5 cm

Step 1 Use the protractor to draw a 60° angle.

Step 2 Measure 5 cm along one arm of the angle.

Step 3 Measure 4 cm along the other arm of the angle.

Step 4 Join the ends of the arms to form the triangle.

A:07C The Angle Sum of a Triangle

The sum of the angles of any triangle is 180° (or one straight angle).

A:07D Types of Triangles and Their Properties

Equilateral

triangle

Name Diagram Properties

• All sides are equal.

• All angles are equal at 60°.

Isosceles

triangle

• Two sides are equal.

• Angles opposite equal sides are

equal. (These are called base

angles.)

APPENDIX A 25

Name Diagram Properties

Scalene

triangle

• No sides are equal.

• All angles are different.

Acute-angled

triangle

• All angles are acute (less than 90°).

Right-angled

triangle

• One angle is a right angle (90°).

Obtuse-angled

triangle

• One angle is an obtuse angle

(greater than 90°).

• The longest side is opposite the

largest angle.

Often you can construct a quadrilateral in several different ways.

worked examples

To construct this quadrilateral, use the following steps.

Step 1 Draw a 60° angle with your protractor.

Step 2 Mark off the sides AB (5 cm) and AD (3·2 cm).

Step 3 Draw an 80° angle with its vertex at B.

Step 4 Mark off the side BC (3·8 cm).

Step 5 Join CD.

A

3·2 cm

60°

D

5 cm

C

80°

3·8 cm

B

1 2 3 4 5

C

D D D

60° 60°

80°

A A 5 cm B A

B A

B A

3·2 cm

3·8 cm

D

C

B

26 NEW SIGNPOST MATHEMATICS 8

Name Diagram Properties

1 Trapezium • One pair of opposite sides are

parallel.

2 Parallelogram • Two pairs of sides are parallel.

• Opposite sides are equal.

• Opposite angles are equal.

• Diagonals bisect one another.

3 Rhombus A rhombus has all the properties of a

x x

parallelogram and …

• All sides are equal.

• Diagonals bisect each other at right

x x

angles.

• Diagonals bisect the angles through

which they pass.

4 Rectangle A rectangle has all the properties of a

parallelogram and …

• All angles are right angles.

• Diagonals are equal.

5 Square A square has all the properties of a

rhombus and a rectangle.

6 Kite • Two pairs of adjacent sides equal.

• Diagonals are perpendicular.

• One diagonal is an axis of symmetry.

APPENDIX A 27

A:07G Finding the Size of an Angle

Diagram Rule Example

a° c°

triangle

a° + b° + c° = 180°

50°

70°

Find the value of a.

a + 50 + 70 = 180

a + 120 = 180

∴ a = 60

a° + b° + c° + d° = 360°

55°

110° 115°

Find the value of b.

b + 55 + 115 + 110 = 360

b + 280 = 360

∴ b = 80

isosceles triangle

a° = b°

3 cm

72° 3 cm

Find the value of x.

This is an isosceles

triangle since two

sides are equal.

∴ x = 72

equilateral triangle

a° = b° = c°

Find the value of y.

This is an equilateral

triangle as all sides

are equal.

∴ All angles are equal.

3y = 180

y = 60

a° b°

straight angle

a° + b° = 180°

135°

Find the value of y.

135 + y = 180

∴ y = 45

a° b°

angles at a point

a° + b° + c° = 360°

70°

30°

40°

Find the value of m.

m + 70 + 30 + 40 = 360

m + 140 = 360

∴ m = 220

A:07H The Angle Sum of a Quadrilateral

The sum of the angles of any quadrilateral is 360° (or one revolution).

The angle sums of other polygons can be found by multiplying the number that is two less than the

number of sides by 180°:

angle sum = (n − 2) × 180°, where n is the number of sides,

eg angle sum of a hexagon = (6 − 2) × 180°

= 720°

28 NEW SIGNPOST MATHEMATICS 8

A:08 | Solid Shapes

A:08A Know the Names of Solids

The ID cards at the front of this book (pages xv to xx) should be used often to help you remember

the language of Mathematics.

A:08B Prisms

see

1:08

(1) (2) (3) (4)

Prisms

All prisms have a special pair of parallel faces. These faces are the only two faces that need not be

rectangular in shape. If the prism is ‘sliced’ parallel to these faces, the same shape always results.

This shape is called the cross-section.

A prism is named according to the shape of its cross-section.

worked example

Name this solid.

Solution

The shape of the crosssection

of this solid is a

hexagon so it is called a

hexagonal prism.

■ The prisms above are:

1 cube

2 triangular

3 hexagonal

4 pentagonal.

APPENDIX A 29

A:08C Pyramids

(1) (2) (3) (4)

Pyramids

All pyramids have one face that need not be triangular. This face is used to name the pyramid.

A right pyramid has its point above the centre of its base.

A pyramid is named according to the shape of its non-triangular face. If all of its faces

are triangles the pyramid is a triangular pyramid.

worked example

Name this solid.

Solution

The solid is a pyramid.

The non-triangular face

is a square. Hence the

solid is a square pyramid.

■ An oblique

pyramid does

not have its

point above

the centre of

its base.

■ The pyramids

above are:

1 square

2 triangular

3 hexagonal

4 pentagonal.

Euler’s theorem: For any convex polyhedra, F + V = E + 2 where

F = number of faces, V = number of vertices and E = number of edges.

A:08D Nets of Solids

Solid shapes can be made from plane shapes. This can be done by drawing the net of the solid on a

piece of paper. The net shows how the faces of the solid are joined to each other. When these faces

are folded along their edges, the solid is formed.

worked example

Draw the net of a square pyramid on a sheet of

square grid paper.

Solution

A square pyramid has four

identical triangular faces

joined to a square base, so:

Step 1 Draw a square base.

triangles.

F = 5, V = 6, E = 9

30 NEW SIGNPOST MATHEMATICS 8

A:08E Drawing Pictures of Solids

Below, a rectangular prism has been drawn using two different projections.

Rectangular prism

Use grids like these

to practise drawing solids.

1 Using square grid

2 Using triangular grid

worked example

Draw the front view, top view and right

side view of the solid pictured below.

Solution

top

front

front

right side

A:09 | Measurement

A:09A Metric Units and Equivalents

The Metric Units ID card on page xv and the Metric Equivalents table on page xiv should be used

A:09B Measuring Lengths

worked example

Write down each measurement in centimetres, giving your

answers correct to the nearest millimetre.

a b c d

An interval is

part of a line. It

has a definite length.

see

1:09

cm 1 2 3 4 5 6 7 8 9

Solution

a 1·5 cm (ie 15 mm) b 3·8 cm (ie 38 mm)

c 6·1 cm (ie 61 mm) d 8·4 cm (ie 84 mm)

APPENDIX A 31

A:09C Conversions in the Metric System

When converting a measurement from one unit to another, you need to know the meanings of

metric prefixes.

kilo … hecto … deca … deci … centi … milli …

1

1000 100 10 1 -----

10

The value of each column is ten times the value of the column on its right.

1

--------

100

1

-----------

1000

To convert a unit into a larger unit you divide.

To convert a unit into a smaller unit you multiply.

worked examples

1 3000 mm = 3000 ÷ 10 cm 2 2500 mL = 2500 ÷ 1000 L 3 630 mg = 630 ÷ 1000 g

= 300 cm = 2·5 L = 0·63 g

4 7 km = 7 × 1000 m 5 7·8 kg = 7·8 × 1000 g 6 2·5 m = 2·5 × 100 cm

= 7000 m = 7800 g = 250 cm

7 An interval is 8·4 cm long. The interval must be divided into twelve equal parts. How many

millimetres would be in each part?

The length 8·4 cm = 8·4 × 10 mm

= 84 mm

∴ the length of each part = 84 ÷ 12

= 7mm

A:09D Perimeter

The perimeter of a figure is the sum of the side lengths.

It is the distance around the figure.

worked examples

Solutions

Find the perimeter

of each figure

1 9 m

P = (5 + 10 + 8 + 9) m

= 32 m

8 m The perimeter is 32 metres.

5 m

■ Perimeter of a square:

P = 4 × s (or P = 4s)

s

10 m

32 NEW SIGNPOST MATHEMATICS 8

2 For a square, all sides are equal.

P = (5·2 + 5·2 + 5·2 + 5·2) m

5·2 m

= (4 × 5·2) m

= 20·8 m

The perimeter is 20·8 metres.

3 For a rectangle, opposite sides

are equal.

P = (2 × 1·2 + 2 × 2·6) cm

2·6 cm

= (2·4 + 5·2) cm

= 7·6 cm

The perimeter is 7·6 centimetres.

1·2 cm

■ Perimeter of a

rectangle:

P = 2 × l + 2 × b

(or P = 2l + 2b)

l

The formulae

make the working

easier.

b

A:09E Time

You should be familiar with both conventional and digital time.

worked examples

1 a b c d

12

12

12

12

9

3

9

3

9

3

3

6

6

6

25 minutes Half past 10 10 minutes to 7 A quarter to 5

past 1

2 a b c d

AM PM AM PM

15 minutes 36 minutes past 20 minutes 55 minutes

past 10 8 (after noon) past 11 past 7

(before noon) (before noon) (after noon)

APPENDIX A 33

3

A 24-hour clock uses a 4-digit number, the first two digits indicating

the hour past midnight and the second two indicating the minutes.

a b c d

20 minutes past 30 minutes past 12 noon 57 minutes past

5 (before noon) 5 (after noon) 11 (after noon)

■ The Days in a Month

To work out how many days are in each month,

we can use the knuckles on two hands.

The knuckle stands for a month of 31 days;

the lower skin joining two knuckles stands for a

month of 30 days, with the exception of February

which has 28 days, and 29 days each leap year.

Jan

Feb

31

28

or

29

Mar

Apr

May

31

31

30 30

June

July

31

Aug

Oct

Nov

Dec

31

31

31

30 30

Sep

A:09F Distance, Speed and Time

If speed does not change (or we are referring to average speed), the following formulae can be used.

• Distance = Speed × Time

• Speed = Distance ÷ Time

• Time = Distance ÷ Speed

We can write these formulae using letters instead of words:

D = S × T, S = D ÷ T, T = D ÷ S

worked examples

1 Yousif ran at a speed of 5 metres per second for 20 seconds. How far did he run?

2 Sara walked at a constant speed for 50 seconds. During this time she travelled 150 metres.

What was her speed?

3 A train travelling with a speed of 30 kilometres per hour travelled a distance of

120 kilometres. How long did it take?

Solutions

1 Speed = 5 m/s. Time = 20 s. 2 Time = 50 s. Distance = 150 m.

Find the distance.

Find the speed.

Distance = Speed × Time

Speed = Distance ÷ Time

D = S × T

S = D ÷ T

∴ D = 5 × 20 m

∴ S = 150 ÷ 50 m/s

= 100 m = 3 m/s

∴ Yousif ran 100 metres.

∴ Sara’s speed was 3 m/s.

34 NEW SIGNPOST MATHEMATICS 8

3 Speed = 30 km/h. Distance = 120 km.

Find the time.

Time = Distance ÷ Speed

T = D ÷ S

= 120 ÷ 30

= 4h

∴ the train took 4 hours.

km/h means

kilometres

per hour.

A:09G Area

When we measure the area of a shape, we are measuring

the amount of space there is inside that shape. We try to

work out how many square units would cover the shape.

Area of a rectangle

1 cm

One square

centimetre

(1 cm 2 ).

• Area = length × breadth

A = l × b

l

b

100 cm

One square

100 cm 1 m 2

metre

(1 m 2 ).

1 m 2 = 10 000 cm 2

1 Find the area of the rectangle below.

3 m

7·5 m

worked examples

Solution

A = l × b

= 7·5 × 3

= 22·5 m 2

The area is 22·5 m 2 .

100 m

100 m 1 ha

1 ha = 10 000 cm 2

One

hectare.

2 Find the area of the figure below.

30 cm

1 m (100 cm) Note: i We must make all units the same,

so 1 m = 100 cm.

1

ii We divide the figure into two rectangles.

50 cm

2

30 cm

Solution

A = Area 1 + Area 2

= (100 × 30) + (50 × 30)

= 3000 + 1500

= 4500 cm 2

The area of the figure is 4500 cm 2 .

APPENDIX A 35

Area of a square

A square is, of course, a rectangle

with its length equal to its breadth,

so if a square has sides of s cm, then

the area is equal to s × s = s 2 cm 2 .

• Area of a square

s

A = s 2

s

worked example

Find the area of a square that has sides of length 7 mm.

7 mm A = s 2

= 7 × 7

= 49 mm 2

The area of the square is 49 mm 2 .

Area of a triangle

1

Area of a triangle = -- (base × height)

2

h

1

A = ----------- or A = -- bh

2

2

b

h

h

b

b

h

worked examples

Find the area of each triangle.

1 2 3

5 cm

8 m

5 cm

12 cm

6 m

7 cm

h

h

h

A = -----------

A = -----------

A = -----------

2

2

2

12 × 5

8

5

= --------------

= -----------

= -----------

2

2

2

= 30 cm 2 = 24 m 2 = 17·5 cm 2

36 NEW SIGNPOST MATHEMATICS 8

A:09H Volume

When we measure the space an object occupies, we are

measuring its volume. We try to work out how many

cubic units it would take to fill the object.

Irregular solids

We may be able to find the volume by counting cubes.

One cubic

centimetre (1 cm 3 ).

1 cm

1 cm 1 cm

worked example

Find the volume of the solid below.

3 cm

3 cm

3 cm There are 5 cm 3 in each layer.

There are 3 layers.

∴ volume = 5 × 3

= 15 cm 3

∴ the volume of the solid is

15 cm 3 .

One cubic

metre

100 cm 100 cm

1 m 3 = 100 100 100

= 1 000 000 cm 3

100 cm

Rectangular prisms

To find the volume of a rectangular prism, use the following rule:

V = l × b × h or simply V = lbh

h

l

b

Find the volume of each rectangular prism.

1 2

7 cm

2 cm

3 cm

worked examples

6 cm

V = lbh

V = lbh

= 7 × 3 × 2 = 6 × 2 × 3

= 42 cm 3 = 36 cm 3

∴ volume is 42 cm 3 . ∴ volume is 36 cm 3 .

3 cm

2 cm

■ Notice:

The number of cubes

in each layer is the

same as the area of

the base rectangle in

square units.

APPENDIX A 37

3

9 m

11 m

20 m

V = lbh

= 11 × 9 × 20

= 1980 cm 3

∴ volume is 1980 cm 3 .

Summary: Perimeter, area and volume

1 Perimeter: distance around 8

P = 24 units

4

4

2 Area: number of squares inside 4

A = 12 units 2

8

3

3 Volume: number of cubes inside V = 48 units 3

A:09I Capacity

3

When we measure the amount of a liquid that a container can hold, we are measuring its capacity.

We might say that an amount of a liquid

is so many cupfuls or spoonfuls, but as

with the volume of solids, we need some

standard units.

4

4

The basic unit in the metric system for capacity is the litre (L).

This means that:

1000 millilitres (mL) = 1 litre (L)

1000 litres (L) = 1 kilolitre (kL)

There is a link between the capacity of a liquid and the volume of a solid.

A container with a volume of 1 cube centimetre would hold 1 millilitre of liquid.

∴ 1mL = 1cm 3 1L = 1dm 3 1L = 1000 cm 3

38 NEW SIGNPOST MATHEMATICS 8

A:09J Mass

Mass is the amount of matter that makes up an object. We can measure the mass of an object by

measuring its weight at sea level or by comparing it with a known mass.

To get a measure of an object’s mass, we need some standard units. The metric units for mass are:

100 milligrams (mg) = 1 gram (g)

1000 grams (g) = 1 kilogram (kg)

1000 kilograms (kg) = 1 tonne (t)

The base unit

is the kilogram.

The combined mass of a container and its contents is its gross mass.

The mass of the contents only is its net mass.

worked examples

Calculate the missing mass.

1 Gross mass = 500 g 2 Gross mass = 1·2 kg 3 Net mass = 800 g

Net mass = 400 g Container’s mass = 250 g Container’s mass = 170 g

Container’s mass = g Net mass = g Gross mass = g

Container’s mass Net mass Gross mass

= Gross − Net = Gross − Container = Net + Container

= 500 g − 400 g = 1200 g − 250 g = 800 g + 170 g

= 100 g = 950 g = 970 g

A:10 | Directed Numbers

A:10A Directed Numbers

Directed numbers have both size and direction.

−20 is read as ‘negative 20’ or ‘minus 20’.

+20 is read as ‘positive 20’ or just ‘20’.

Note: +20 can be written more simply as 20.

... –4, –3, –2, –1, 0,

1, 2, 3, 4, ...

see

1:10

■ If a directed number is a whole

number, it is called an integer.

APPENDIX A 39

When adding or subtracting, a negative can be thought of as a loss and a positive can be thought

of as a gain.

worked examples

1 −7 + 11 A loss of 7 and 2 −9 + 2 A loss of 9 and

= 4 a gain of 11. = −7 a gain of 2.

3 2 − 13 A gain of 2 and 4 −7 − 5 A loss of 7 and

= −11 a loss of 13. = −12 a loss of 5.

A:10C Signs Side by Side

When two signs occur side by side (with no numeral between them):

• two like signs give a plus

• two unlike signs give a minus.

worked examples

1 6 − −10 2 3 + (+7) 3 10 − (3 − 9) 4 3 + (−7 + 11)

= 6 + 10 = 3 + 7 = 10 − −6 = 3 + (+4)

= 16 = 10 = 10 + 6 = 3 + 4

= 16 = 7

A:10D Multiplication and Division

For multiplication and division:

• two like signs give a plus

• two unlike signs give a minus.

worked examples

1 −4 × −3 2 −0·2 × −3 3 −4 × 14 4 7 × (−1·1)

= 12 = 0·6 = −56 = −7·7

5 −35 ÷ (−5) 6 60 ÷ −6 7 –21

8 –1·8

--------

----------

= 7 = −10

–3

2

= 7 = −0·9

9 −8 + 6 × −3 10 (2 − 20) ÷ 3

= −8 + −18 = −18 ÷ 3

= −8 − 18 = −6

= −26

40 NEW SIGNPOST MATHEMATICS 8

A:11 | The Number Plane

A:11A The Number Line Extended

To graph a number on the number line, place a large dot at that position on the number line.

see

worked examples

1 Graph the set of numbers {−2, 0, 1·5, 4}.

–3 –2 –1 0 1 2 3 4 5

2 List the set of numbers graphed below.

■ A set of numbers is a

collection of numbers.

1:11

0 0·5 1 1·5

The set is {−0·3, 0·2, 0·6, 1}.

A:11B The Number Plane

(2, 5) is called an ordered pair or the coordinates of a point.

The first number is the reading on the x-axis (how far right or left).

The second number is the reading on the y-axis (how far up or down).

(–3, 1) is 3 to

the left and 1 up.

y

4

3

2

1

(4, 3) is above

‘4’ on the x-axis

and across from

‘3’ on the y-axis.

–5 –4 –3 –2 –1 0 1 2 3 4

x

(–5, –4) is

5 to the

left and 4

down.

(0, 0) is called

the origin. It is

0 to the right and

0 up.

–1

–2

–3

–4

(3, 0) is

3 to the right

and 0 up.

(2, –3) is

2 to the right

and 3 down.

APPENDIX A 41

A:11C Mastery Test

Do this test again and again until you get all the questions right.

C

E

8

F

How long

is CD?

D

y

4

3

2

2

4

Name this axis .

Give coordinates

9

What letter

is at (4, –4)?

8 3 units

9 A

10 F

5 (−5, −2)

4 (1, 3)

3 origin,

(0, 0)

B

1

–4

1

A

Name this axis .

–5 –4 –3 –2 –1 1 2 3 4 5 x

6

Give name

Give coords. –1 3

and coords.

–2 7 Give coords.

5 Give coords.

10 What letter

–3

is at (–2, 4)?

7 (0, −2)

6 (−4, 0)

Check your

you finish the

test.

2 y-axis

1 x-axis

see

1:12

A:12 | Algebra

A:12A Describing Number Patterns

A symbol can be used to describe the rule used to obtain a number pattern. Symbols that represent

numbers are called pronumerals. The most commonly used pronumerals are the letters of the

alphabet.

A pronumeral takes the place of a numeral.

worked examples

1 In each example below, t represents the number of triangles, and m the number of matches

used. Find the rule to describe each pattern and use it to complete the table of values given.

a

3

, , , ...

6 9

42 NEW SIGNPOST MATHEMATICS 8

The number of matches used is always three times the number of triangles.

m = 3 × t

t 1 2 3 4 5 6 7 8

m 3 6 9 12 15 18 21 24

b

, , , , ...

3 5 7 9

The number of matches used is always two times the number of triangles plus one.

m = (2 × t) + 1

t 1 2 3 4 5 6 7 8

m 3 5 7 9 11 13 15 17

2 At Ali’s fruit shop, apples cost 25 fils each. If C stands for the cost in fils and n stands for the

number of apples, write down a rule to give the cost of each lot of apples bought, and use it

to complete the table of values.

Cost of apples = (number of apples) × (25 fils)

C = n × 25

n 1 2 3 4 5 10 30 100

C 25 50 75 100 125 250 750 2500

A:12B Algebraic Expressions

If an expression contains one or more pronumerals, it is an algebraic expression.

eg 4 + a, 9 × y, p ÷ q, 5 + x × y

Note:

• 1 × m = 1m = m

The numeral 1 does not have to be written in front of a pronumeral.

• 4 × y = 4y

This means ‘4 times y’ or ‘4 lots of y’ or ‘y + y + y + y’.

• f × 5 is written as 5f, not f5.

• a × b = ab = ba

Since 5 × 11 = 11 × 5, a × b = b × a

x a

• x ÷ 5 = -- , a ÷ 12 = ----- .

5 12

The fraction is usually used instead of the division sign.

APPENDIX A 43

worked examples

Simplify:

1 5 × k + 2 2 a + 3 × y 3 6 × (a + 7) 4 5 × a ÷ 7

= 5k + 2 = a + 3y = 6(a + 7)

5a

= -----

7

Rewrite these expressions showing all multiplication and division signs.

a + 7

5 3a + 8 6 5p − 6q 7 4(x + 2) 8 -----------

3

= 3 × a + 8 = 5 × p − 6 × q = 4 × (x + 2)

= (a + 7) ÷ 3

A:12C Substitution

You will often be asked to evaluate an expression using values given for the pronumerals.

worked examples

1 Given x = 3:

a 6x = 6 × 3 b 2(x + 5) = 2 × (3 + 5) c 5x 2 = 5 × 3 2

= 18 = 2 × 8 = 5 × 9

= 16 = 45

Note: The multiplication signs must be written in after the values have been substituted for

pronumerals.

2 If a = 2 and b = 5 then:

10a 10 × 2

a 3a + 7b = 3 × 2 + 7 × 5 b -------- = -------------- c 4a(b − a) = 4 × 2 × (5 − 2)

b 5

= 6 + 35 = 4 × 2 × 3

= 41

= 20 ÷ 5

= 24

= 4

3 If the number of matches, m, needed to make t triangles is given by m = 2t + 1, how many

matches would be needed to make 100 triangles?

m = 2t + 1

= 2 × 100 + 1 (when t = 100)

= 200 + 1

= 201

So 201 matches would be needed to make 100 triangles.

A:12D Finding the Rule

Rules like y = 3x + 4 have a difference of 3 between

successive values of y as x increases by 1 each time.

Rules like y = 7x + 4 have a difference of 7 between

successive values of y as x increases by 1 each time.

x

y

1

7

y 3x 4

2 3

10 13

4

16

5

19

3 3 3 3

44 NEW SIGNPOST MATHEMATICS 8

worked example

Discover the rule connecting x and y. If the differences

( ) are the same, the rule will be like: y = x +

As the differences are all 4,

y = 4x +

From the table we see that when x = 1, y = 11.

So 11 = 4 × 1 +

∴ = 7

So the rule is y = 4x + 7.

x

y

1 2 3 4 5

11 15 19 23 27

4 4 4 4

A:12E Simplifying Algebraic Expressions

If terms have identical pronumeral parts, then they are called like terms.

Like terms may be added or subtracted.

worked examples

1 7a + 5a = 12a 2 3pq − 10pq = −7pq 3 8x − x = 7x

4 5m + 7m − 10m = 2m 5 5x + 2y + 7y = 5x + 9y 6 7p + 2q + 3p + q

= 10p + 3q

7 6a + 7b − 2a + 5b 8 7a 2 − 4a + 2a

■ The plus or minus

2

= 9a 2 − 4a

= 6a −2a +7b +5b sign always belongs to

= 4a + 12b

the term after it!

When multiplying algebraic terms, multiply numbers first and then pronumerals. When dividing,

write the division as a fraction and reduce the fraction to its lowest terms.

worked examples

1 3 × 5a 2 7m × 3p 3 a × 4b 4 −3k × −5

= 15a = 21mp = 4ab = 15k

5 12t ÷ 3 6 6m ÷ 2a 7 15r ÷ 10 8 3ab × 7a

4

3

12t

6m

15r

= 21aba

= -------

= -------

= --------

= 21aab

1 3

12a

210

= 21a 2 b

3m

3r

= 4t = -------

= -----

a

2

A:12F Grouping Symbols

To ‘expand’ an expression, we write it without grouping symbols.

• a(b + c) = ab + ac

■ • Parentheses: ( )

• a(b − c) = ab − ac

• Brackets: [ ]

APPENDIX A 45

A:12G Index Notation

In a 4 , ‘a’ is called the base, ‘4’ is called the index.

A:12H Algebraic Sentences or Equations

worked examples

1 5(x + 2) 2 9(2a − 3) 3 m(m + 7)

= 5 × x + 5 × 2 = 9 × 2a − 9 × 3 = m × m + m × 7

= 5x + 10 = 18a − 27 = m 2 + 7m

worked examples

4

1 a × a × a × a = a 2 5 3 = 5 × 5 × 5

3 p × p × p × p × p × p × p = p 7 (since the p is used as a factor 7 times).

4 m × m × m × b × b × b × b × b = m 3 b 5 5 5 × t × t × 7 × t = 35t 3

■ The plural of ‘index’ is

‘indices’.

To ‘solve’ an equation, we must find the value of the pronumeral that will make the equation true.

worked examples

1 x + 5 = 9 2 12 − a = 5 3 6m = 42

plus 5 is equal to 4?’ number is equal to 5?’ number is equal to 42?’

x = 4 a = 7 m = 7

x

4 -- = 21

5 m + 7 = 2 6 2x + 5 = 13

3

divided by 3 is equal plus 7 is equal to 2?’ 2x = 8

to 21?’ m = −5 ‘What times 2 gives 8?’

x = 63 x = 4

A:12I Problem Solving Using Algebra

We can use equations to solve problems.

The sum of two consecutive numbers is 91.

What are the numbers?

Solution

Let the numbers be x and x + 1.

Then x + (x + 1) = 91

2x + 1 = 91

2x = 90

x = 45

∴ x + 1 = 46

worked example

■ Using algebra is a

problem-solving strategy.

Consecutive numbers