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.
4 Add the two answers. 3 Write the answer using a remainder or fraction.
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
↑
leading digit
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 number reads
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
When your answer (9·27) is smaller than your divisor (11)
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
A:03F Addition and Subtraction (1)
To add or subtract fractions that have the same denominator, add or subtract the numerators.
A:03G Addition and Subtraction (2)
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
.
shaded.
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
3×
7
= -----------------
10 × 10
1×
2
9×
3
= -----------
= --------------
5×
1
2×
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
numerators to get the answer.
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
A:04D Addition and Subtraction
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
in your answer.
It helps to estimate
your answer first.
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
Bˆ
• 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
Adjacent
Name Diagram Explanation
Vertex
Common
arm
Adjacent angles:
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°
Adjacent
complementary
These are formed when a right angle is
cut into two parts. The angles add up
to 90°.
Adjacent
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
quadrilateral
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.
A:07E Constructing Quadrilaterals
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

A:07F Properties of Quadrilaterals
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
b°
a° c°
triangle
a° + b° + c° = 180°
50°
70°
a°
Find the value of a.
a + 50 + 70 = 180
a + 120 = 180
∴ a = 60
b°
a°
c°
d°
quadrilateral
a° + b° + c° + d° = 360°
55°
b°
110° 115°
Find the value of b.
b + 55 + 115 + 110 = 360
b + 280 = 360
∴ b = 80
a°
b°
isosceles triangle
a° = b°
x°
3 cm
72° 3 cm
Find the value of x.
This is an isosceles
triangle since two
sides are equal.
∴ x = 72
a°
b°
c°
equilateral triangle
a° = b° = c°
y°
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°
y°
Find the value of y.
135 + y = 180
∴ y = 45
a° b°
c°
angles at a point
a° + b° + c° = 360°
m°
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.
Step 2 Add the four adjoining
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
to help you remember this essential information.
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
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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
b×
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
b×
h
b×
h
b×
h
A = -----------
A = -----------
A = -----------
2
2
2
12 × 5
6×
8
7×
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

A:10B Addition and Subtraction
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)?
Answers
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
answers when
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
Ask: ‘What number Ask: ‘12 minus what Ask: ‘6 times what
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
Ask: ‘What number Ask: ‘What number ‘What plus 5 gives 13?’
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
follow one after the other,
eg 29, 30, 31, ...
46 NEW SIGNPOST MATHEMATICS 8