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978-1-107-67883-5 – Cambridge Mathematics NSW Syllabus for the Australian Curriculum Year 9 5.1 and 5.2
Stuart Palmer David Greenwood Sara Woolley Jenny Vaughan Jenny Goodman David Robertson Georgia Sotiriou and Voula Sotiriou
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Chapter 1 Integers, decimals, fractions, ratios and rates
1
Integers, decimals,
fractions, ratios
and rates
Chapter
What you will learn
1A
1B
1C
1D
1E
1F
1G
1H
Adding and subtracting positive and negative integers Multiplying and dividing positive and negative integers Decimal places and signifi cant fi gures
Rational numbers and irrational numbers REVISION
Adding and subtracting fractions REVISION
Multiplying and dividing fractions REVISION
Ratios REVISION
Rates and direct proportion
© in this web service Cambridge University Press
REVISION
REVISION
www.cambridge.org
Number and Algebra
NSW Syllabus
for the Australian
Curriculum
Strands: Number and Algebra
Measurement and Geometry
Substrands: CoMputAtioN WitH iNtEGERS
FRACtioNS, DECiMAlS
AND pERCENtAGES
RAtioS AND RAtES
NuMBERS oF ANY MAGNituDE
Outcomes
A student compares, orders and calculates with integers, applying a range of strategies to aid computation.
(MA4–4NA)
A student operates with fractions, decimals and percentages.
(MA4–5NA)
A student operates with ratios and rates, and explores their graphical representation.
(MA4–7NA)
Negative numbers in the
ancient world
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The ancient Babylonians, Hindus and Egyptians were all known for their work with numbers. There is no mention of negative numbers in their writings.
The fi rst mention of negative numbers is dated back to only 200 BC in China. The Chinese used abacuses with black rods for negative numbers and red rods for positive numbers.
A student recognises direct and indirect proportion, and solves problems involving www.cambridge.org
direct proportion.
(MA5.2–5NA)
A student interprets very small and vary large units of measurement, uses scientifi c notation, and rounds to signifi cant fi gures.
(MA5.1–9MG)
3 Chapter 1 Integers, decimals, fractions, ratios and rates
pre-test
4
ridge University Press
1 Arrange the following mathematical terms under four headings: ‘Addition’, ‘Subtraction’,
‘Multiplication’ and ‘Division’.
a Sum
b Total
c Less than
d Lots of
e Product
f Into
g Take away
h Difference
i Add
j Times
k Minus
l More than
m Quotient
2 Without using a calculator, find an answer to each of the following.
a 16 less 12
b 24 more than 8
c The difference between 12 and 8
d Increase 45 by 7
e The total of 40, 34 and 0
f 9 into 45
g The quotient of 7 and 63
h 480 shared between 12
3 Evaluate the following.
a 9 + 47
b 135 - 35
e 9×7
f 320 ÷ 4
c 19 - 19
g 17 × 60
4 Use a number line to find:
a -5 - 7
b 12 - 15
e 16 - 17
f -4 + 3
c
g
d 56 + 89 - 12
h 200 - 47 - 100
-6 + 9
-7 + 4 + 3
d -12 + 12
h -4 - 4 - 4
5 Copy and complete each of the following statements.
a 5+5+5=
c 9 - (+10) = 9
×5
10
b
-6 - 6 - 6 =
× (- 6)
d
12 - (-2) = 12
2
6 The population of Australia in 2050 is projected to be 26 073 258. Round this
number to the nearest:
a ten
b hundred
c thousand
d million
7 Write down the place value of the 5 in each of the following numbers.
a 1256
b 345
c 5049
d 0.56
e 0.15
f 9.005
8 Arrange the numbers in each of the following sets in descending order (largest to smallest).
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a 2.645, 2.654, 2.465 and 2.564
b 0.456,
0.564, 0.0456 and 0.654
9 Evaluate each of the following.
a 4.26 + 3.73
b 3.12 + 6.99
c 10.89 - 3.78
10 Evaluate each of the following.
a 7 × 0.2 b 0.3 × 0.2 c 2.3 × 1.6
d 4.2 × 3.9
11 Evaluate each of the following.
a 0.345 × 100
b 3.74 × 100 000 c
37.54 ÷ 1000
e 14.8 ÷ 4
d 3.7754 ÷ 100 000
12 Complete these equivalent fractions.
2
5 25
1
3
=
=
=
d
b
c
=
6
2 12
9 18
4 16
13 Find the lowest common denominator for these pairs of fractions.
1
1
1
1
1
c 1 and
and
and
a
b
3
10
4
6
5
5
14 Find:
3
1
1
3 2
c 4÷
a
+
d 3×
b 24
2
7 7
4 2
a
f 12.6 ÷ 0.07
5 Number and Algebra
1A Adding and subtracting positive and
negative integers
R EVI S I ON
Stage
Integers are the set of positive and negative whole numbers, as well
as zero.
Being able to work with whole numbers is very important,
since whole numbers are used every day for counting, calculating
measuring and ordering.
5.3#
5.3
5.3§
5.2
5.2◊
5.1
4
let’s start: Naming groups
Numbers are used in marketplaces all around the world to describe prices, quantity and sometimes even quality.
Key ideas
Integers are the positive and negative whole numbers, including zero.
…, -3, -2, -1, 0, 1, 2, 3, …
■ To add a negative number, you subtract its opposite: a + (-b) = a - b
ridge University PressFor example: 5 + (-7) = 5 - 7
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-6 + (-2) = -6 - 2
■ To subtract a negative number, you add its opposite: a - (-b) = a + b
For example: 5 - (-7) = 5 + 7
-6 - (-2) = -6 + 2
■ Adding or subtracting zero leaves a number unchanged.
a+0=a
e.g. 5 + 0 = 5
a-0=a
e.g. 5 - 0 = 5
■
REVISION
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1 Match each of the following sentences to the correct expression on the right.
a The sum of 5 and 7
i 5 + (-7)
b The total of negative 5 and 7
ii 5 - (-7)
c The difference between negative 5 and 7
iii 5 + 7
d The sum of 5 and negative 7
iv -5 - 7
e The difference between 5 and negative 7
v -5 + 7
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Exercise 1A
LL
Here are some groups of numbers. In groups of two or
three, use the correct mathematical terms to
describe each group. (Suggestions include: ‘multiples
of’, ‘factors of’, ‘integers’, ‘squares’ and ‘cubes’.)
• 2, 4, 6, 8, …
• 1, 4, 9, 16, …
• 1, 3, 5, 7, 9, …
• 1, 2, 3, 4, 5, 6, …
• -1, -2, -3, -4, -5, …
• 1, 8, 27, 64, …
• 1, 2, 3, 4, 6 and 12
M AT I C A
1A
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-3 - 5
SolutioN
ExplANAtioN
Draw a number line showing -10
on it.
Start at -10 and count up
(for addition) 8 places to finish
at -2.
a
–8
–6
–4
–2
0
2
-10 + 8 = -2
Draw a number line showing -3
on it.
Count down (for subtraction)
5 places to -8.
b
–9 –8 –7 –6 –5 –4 –3 –2 –1
0
1
2
-3 - 5 = -8
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4 Use the number line below to help you simplify the following.
–8 –7 –6 –5 –4 –3 –2 –1
a -6 + 4
d -1 + 5
g -1 - 1
0
1
b -6 + 8
e 6 - 10
h -3 + 3
2
3
4
5
6
7
8
c 0-3
f -2 - 3
i -3 - 3
5 Use a number line to evaluate the following.
a 9-6
b 9-7
d 9-9
e 9 - 10
g -9 + 8
h -9 - 8
j -3 + 12
k -8 + 6
m -12 + 5
n -10 - 8
c
f
i
l
o
6 Evaluate the following.
a 4+8-7
d -9 + 8 + 1
g -12 - 12
c -19 - 1
f 8+3-5
i -6 - 5 - 4
b 5+6+9
e -12 - 3 + 8
h 15 + 5 - 15
9-8
-9 - 0
-6 - 4
-1 + 12
18 - 19
On the number
line, move right for
addition and left for
subtraction.
Do in order from
left to right.
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Example 1 using a number line for addition and subtraction of integers
–10
F
PS
c -6 + 6 = 0
f 4 - (-2) = 4 - 2
U
b
C
M AT I C A
WO
Use a number line to find:
a -10 + 8
R K I NG
R K I NG
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b 18 + 0 = 18
e 6 - (-4) = 6 + 4
MA
3 True or false?
a 9 - 7 = -2
d 9+8=8+9
U
LL
2 Match each expression on the left to a number on the right.
a 10 + (-7)
i -17
b -10 - (-7)
ii -3
c 10 - (-7)
iii 0
d -10 - 7
iv 3
e -10 + 7 + 3
v 17
Y
Chapter 1 Integers, decimals, fractions, ratios and rates
LL
6
M AT I C A
7 Number and Algebra
MA
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Find 17 + (-12).
SolutioN
ExplANAtioN
17 + (-12) = 17 - 12
=5
Adding a negative is the same as subtraction:
17 + (-12) = 17 - 12
7 Find the value of the following. Check your answers with a calculator.
a 9 + (-5)
b 12 + (-16)
c 3 + (-7)
d 15 + (-24)
e -9 + (-23)
f -13 + (-25)
g -100 + (-89)
h 56 + (-80)
i -9 + (-9)
j 18 + (-18)
k -245 + (-560)
l 98 + (-155)
m -89 + (-78)
n 145 + (-3)
o -567 + (-237)
R K I NG
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PS
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Example 2 Adding a negative integer
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M AT I C A
9 + (-5) = 9 – 5
-9 + (-23) = -9 – 23
Example 3 Subtracting a negative integer
Find -13 - (-9).
SolutioN
ExplANAtioN
- 13 - (- 9) = -13 + 9
= -4
Subtracting a negative is the same as addition:
- 13 - (- 9) = -13 + 9
-8 - (-6)
12 - (-12)
-90 - (-9)
-90 - (-87)
-6 - (-100)
c
f
i
l
o
-5 – (-9) = -5 + 9
7 - (-6)
-34 - (-34)
90 - (-90)
234 www.cambridge.org
- (-6)
-230 - (-240)
WO
10 Copy and complete each of the following statements.
a
+6=8
d -9 +
= -10
b -6 +
=5
c
12 -
= 15
e 13 -
= 20
f
10 -
= -1
11 What must be added to each of the following to obtain a final result of zero?
a 8
b -7 + 3
c -8 - 5
d -124
e 19
f
0
g 12 - 8 + 18
h -98
i
12 - (-14)
Try using a
number line.
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Work from left
to right.
MA
9 Simplify each of these using the rules above. Check your answers with a calculator.
a -9 - 9 - 9
b -23 - (-8) + 12
c 50 - 46 - (-6)
d 24 - (-8) + (-6)
e -20 - (-5) - (-15)
f
-18 - (-6) + (-12)
g -90 - (-89) - 90
h 125 - 35 - (-35)
i
-90 + (-87) - 3
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b
e
h
k
n
LL
8 Evaluate the following.
a -5 - (-9)
d 2 - (-7)
g -35 - (-7)
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j -68 - (-70)
m 670 - (-85)
M AT I C A
1A
WO
12 The temperature inside a car was measured at 23°C. Fifteen minutes later it had risen by
18°C. The air conditioner was then switched on and the temperature fell by 22°C. What
was the temperature inside the car at this time?
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Enrichment: Magic squares with integers
13 a
Copy and complete this magic square. Each row, column and diagonal add to the same number.
-8
What is the sum of
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-2
-6
the diagonal?
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4
b Arrange these 9 integers into a 3 by 3 magic square.
-13, -10, - 7, -4, -1, 2, 5, 8 and 11
c Complete this 4 by 4 magic square.
5
0
10
-1
4
R
HE
6
2
8
7
-4
R K I NG
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Chapter 1 Integers, decimals, fractions, ratios and rates
LL
8
M AT I C A
9 Number and Algebra
1B Multiplying and dividing positive and
negative integers
R EVI S I ON
Stage
Multiplication is a way of writing repeated addition. It is possible to develop rules for multiplying and
dividing with negative integers.
let’s start: Repeated addition
Write each of the following as a multiplication before finding the answer.
i 4+4+4
ii (-4) + (-4) + (-4)
iii 5 + 5 + 5 + 5
5+
5+
iv (-5) + (-5) + (-5) + (-5)
v (-7) + (-7) + (-7) + (-7) + (-7)
vi (-8) + (-8)
5+
5.3#
5.3
5.3§
5.2
5.2◊
5.1
4
5
Use your results from above to answer these divisions.
i 12 ÷ 4
ii -12 ÷ (-4)
iii -12 ÷ 3
iv -20 ÷ 4
v -20 ÷ (-5)
vi 20 ÷ 5
vii -35 ÷ (-7)
viii -35 ÷ 5
ix -16 ÷ (-8)
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■
■
■
When multiplying or dividing two numbers with the same www.cambridge.org
sign, the result is a positive
number.
+×+=+
-×-=+
+÷+=+
-÷-=+
For example: -5 × (-4) = 20 and -18 ÷ (-6) = 3
When multiplying or dividing two numbers with different signs, the result is a negative
number.
+×-=-×+=+÷-=-÷+=For example: -5 × 4 = -20 and 18 ÷ (-6) = -3
On your calculator, use brackets when finding powers of negative numbers.
For example: (-13)2 = 169 (not -169)
Key ideas
What can you conclude about dividing a negative by a:
a positive number?
b negative number?
Chapter 1 Integers, decimals, fractions, ratios and rates
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1 Without actually finding the answer to any of these questions, decide whether the answer to
each would be a positive or a negative number.
a 2 × (-8)
b 78 × (-1)
c 56 ÷ (-2)
d -90 ÷ (-10)
e 8 × 12
f -18 × (-9)
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REVISION
LL
Exercise 1B
M AT I C A
2 Copy and complete this table.
×
-4
-3
-2
-1
0
1
2
3
4
-4
-3
-2
-1
0
1
2
3
4
3 Complete the following statements.
a A negative number times a negative number equals a _________ number.
b A negative number times a positive number equals a ________ number.
c A negative number divided by a negative number equals a ________ number.
d A positive number divided by a negative number equals a _________ number.
e A negative number divided by a positive number equals a __________ number.
U
Find the value of:
a -6 × (-7)
b
8 × (-12)
SolutioN
ExplANAtioN
a -6 × (-7) = 42
The product of two negative numbers gives a positive answer.
-×-=+
b 8 × (-12) = -96
The product of a positive number and a negative number results in
a negative answer.
+×-=-
4 Find these products. Use a calculator to check your answers.
a -8 × 6
b -10 × 2
c
d -9 × (-8)
e -15 × (-3)
f
g 100 × (-6)
h -9 × (-9)
i
j -13 × 11
k 11 × (-13)
l
m -24 × (-3)
n -18 × (-5)
o
-9 × 8
45 × (-2)
-12 × 12
34 × (-1)
36 × (-3)
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Example 4 Multiplying with negatives
+×+=+
–×–=+
+×–=–
–×+=–
R K I NG
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M AT I C A