Integers, decimals, fractions, ratios

Cambridge University Press 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 Excerpt More information

2

Chapter 1 Integers,decimals,fractions,ratiosandrates

1Integers, decimals,

fractions, ratios

Chapter

and rates

What you will learn

1A Addingandsubtractingpositiveandnegativeintegers REVISION 1B Multiplyinganddividingpositiveandnegativeintegers REVISION 1C Decimalplacesandsignificantfigures 1D Rationalnumbersandirrationalnumbers REVISION 1E Addingandsubtractingfractions REVISION 1F Multiplyinganddividingfractions REVISION 1G Ratios REVISION 1H Ratesanddirectproportion

? in this web service Cambridge University Press



Negative numbers in the ancient world

ridge University Press

TheancientBabylonians,HindusandEgyptianswere allknownfortheirworkwithnumbers.Thereisno mentionofnegativenumbersintheirwritings. Thefirstmentionofnegativenumbersisdatedback toonly200BCinChina.TheChineseusedabacuses withblackrodsfornegativenumbersandredrodsfor positivenumbers.

Number and Algebra

3

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

Astudentcompares,ordersandcalculates withintegers,applyingarangeofstrategies toaidcomputation.

(MA4?4NA)

Astudentoperateswithfractions,decimals andpercentages.

(MA4?5NA)

Astudentoperateswithratiosandrates, andexplorestheirgraphicalrepresentation.

(MA4?7NA)

Astudentrecognisesdirectandindirect proportion,andsolvesproblemsinvolving



directproportion.

(MA5.2?5NA)

Astudentinterpretsverysmallandvary largeunitsofmeasurement,usesscientific notation,androundstosignificantfigures.

(MA5.1?9MG)

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Chapter 1 Integers,decimals,fractions,ratiosandrates

ridge University Press

pre-test

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

d 56 + 89 - 12 h 200 - 47 - 100

4 Use a number line to find:

a -5 - 7

b 12 - 15

e 16 - 17

f -4 + 3

c -6 + 9 g -7 + 4 + 3

d -12 + 12 h -4 - 4 - 4

5 Copy and complete each of the following statements.

a 5+5+5= ?5

b -6 - 6 - 6 = ? (-6)

c 9 - (+10) = 9 10

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).

a 2.645, 2.654, 2.465 and 2.564

b 0.45w6,w0w.5.c6a4m, 0b.r0id4g5e6.oarngd 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 e 14.8 ? 4 f 12.6 ? 0.07

11 Evaluate each of the following. a 0.345 ? 100 b 3.74 ? 100 000 c 37.54 ? 1000 d 3.7754 ? 100 000

12 Complete these equivalent fractions.

a

1 2

=

12

b

3 4

=

16

c

5 6

=

25

d

9

=

2 18

13 Find the lowest common denominator for these pairs of fractions.

a

1 3

and

1 5

b

1 6

and

1 4

c

1 5

and

1 10

14 Find:

a

3 7

+

2 7

b

2

-

3 4

c

4

?

1 2

d

3 4

?

1 2

Number and Algebra

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1A Adding and subtracting positive and

negative integers

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.

let's start: Naming groups

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

REVISION Stage 5.3# 5.3 5.3? 5.2 5.2 5.1 4

Numbersareusedinmarketplacesall aroundtheworldtodescribeprices, quantityandsometimesevenquality.

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 PresFs or example: 5 + (-7) = 5 - 7 -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

L LY

Exercise 1A REVISION

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

M AT

WORKING UF

C

R PS H E M ATICA

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Chapter 1 Integers,decimals,fractions,ratiosandrates

1A

L LY

L LY

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

3 True or false? a 9 - 7 = -2 d 9+8=8+9

b 18 + 0 = 18 e 6 - (-4) = 6 + 4

c -6 + 6 = 0 f 4 - (-2) = 4 - 2

Example 1 using a number line for addition and subtraction of integers

Use a number line to find: a -10 + 8

b -3 - 5

M AT

M AT

WORKING UF

C

R PS H E M ATICA

WORKING UF

C

R PS H E M ATIC A

SolutioN

ExplANAtioN

a

?10 ?8 ?6 ?4 ?2

0

2

-10 + 8 = -2

Draw a number line showing -10 on it. Start at -10 and count up (for addition) 8 places to finish at -2.

b ?9 ?8 ?7 ?6 ?5 ?4 ?3 ?2 ?1 0 1 2

-3 - 5 = -8

Draw a number line showing -3 on it. Count down (for subtraction) 5 places to -8.

ridge University Press



4 Use the number line below to help you simplify the following.

?8 ?7 ?6 ?5 ?4 ?3 ?2 ?1 0 1 2 3 4 5 6 7 8

a -6 + 4 d -1 + 5 g -1 - 1

b -6 + 8 e 6 - 10 h -3 + 3

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 9-8 f -9 - 0 i -6 - 4 l -1 + 12 o 18 - 19

On the number line, move right for addition and left for subtraction.

6 Evaluate the following. a 4+8-7 d -9 + 8 + 1 g -12 - 12

b 5+6+9 e -12 - 3 + 8 h 15 + 5 - 15

c -19 - 1 f 8+3-5 i -6 - 5 - 4

Do in order from left to right.

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