Chapter Integers - Cambridge University...

Chapter

What you will learnWhole number addition and subtractionWhole number multiplication and divisionNumber propertiesDivisibility and prime factorisationNegative numbersAddition and subtraction of negative integersMultiplication and division of integersOrder of operations and substitution

1.11.21.31.41.51.61.71.8

IntegersIntegers111111Chapter1Chapter11111111Chapter1Chapter

Cambridge University Press978-0-521-17864-8 - Essential Mathematics: For the Australian Curriculum: Year 8David Greenwood, Franca Frank, Jenny Goodman, Bryn Humberstone, Justin Robinson and Jennifer VaughanExcerptMore information

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3

Public key encryption

Today, most of the world’s electronic commercial transactions are encrypted so that important information does not get into the wrong hands. Public key encryption uses the RSA algorithm named after its inventors Rivest, Shamir and Ademan, who invented the mathematical procedure in 1977. The algorithm creates public and private number keys that are used to encrypt and decrypt information. These keys are generated using products of prime numbers. Because prime factors of large numbers are very diffi cult to fi nd (even for today’s powerful computers) it is virtually impossible to decrypt the code without a private key. The algorithm uses prime numbers, division and remainders, equations and the 2300-year-old Euclidean division algorithm to complete the task. If it wasn’t for Euclid (about 300 BCE) and the prime numbers, today’s electronic transactions would not be secure.

Australian curriculumN U M B E R A N D A L G E B R A

N u m b e r a n d p l a c e v a l u e

Carry out the four operations with integers, using effi cient mental and written strategies and appropriate digital technologies (ACMNA183)

•






Chapter 1 Integers4 Chapter 1 Integers

1 Complete these calculations mentally.

a 126 + 254 b 301 + 249 c 387 − 124 d 140 − 78

e 11 × 10 f 25f 25f × 8 g 79 × 3 h 101 × 5i 88 ÷ 8 j 150 ÷ 3 k 7 × 10 000 l 980 ÷ 20

2 Complete the following additions and subtractions.

a 12339+

b 497278+

c 7819−

d 336289−

3 Complete the following multiplications and divisions.

a 1219×

b 33814×

c )3 453 d )7 364

4 What is the remainder when 6 is divided into these numbers?

a 73 b 41 c 106 d 594

5 a List the fi rst 5 multiples of 6.

b List the fi rst 4 multiples of 9.

c What is the lowest common multiple (LCM) of 6 and 9?

6 a List all the factors of 12.

b List all the factors of 15.

c What is the highest common factor (HCF) of 12 and 15?

7 Prime numbers have 2 factors. List the fi rst 6 prime numbers starting at 2.

8 Use order of operations to complete these calculations.

a 2 + 3 × 4 b 10 − 8 ÷ 2 c (5 − 2) × 7 d 24 ÷ (3 + 9)

9 If 32 = 3 × 3 = 9 then 9 = 3. Find the answer to these squares and square roots.

a 42 b 52 c 36 d 121

10 Write the missing numbers in these patterns.

a 3, 2, 1, —–, —–, —– b 9, 4, -1, —–, —–, —–

c -21, —–, —–, -12, -9, —– d 2, -4, 8, —–, 32, —–, —–

11 Use this number line to help fi nd the answer.

10-1-2 2 3 4 5-3-4-5

a 2 – 5 b 0 – 3 c -4 + 6 d -2 + 7

Pre-

test






Number and Algebra 5

Whole number addition and subtractionThe number system that we use today is called the Hindu–Arabic

or decimal system and uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

The value of each digit depends on its place in the number, so, for

example, the 4 in 3407 has a place value of 400. Whole numbers

include 0 (zero) and the counting (natural) numbers 1, 2, 3, 4, …

Two numbers can be added to fi nd a sum or subtracted to fi nd a

difference. If 22 child tickets and 13 adult tickets were purchased

for fairground rides, the sum of the number of tickets (35) is

found using addition and the difference between the number of

child and adult tickets (9) is found using subtraction.

Let’s start: Sum and differenceUse a guess-and-check method to try to fi nd a pair of numbers

described by these sentences.

• The sum of two numbers is 88 and their difference is 14.

• The sum of two numbers is 317 and their difference is 3.

Describe the meaning of the words ‘sum’ and ‘difference’ and discuss how you found the pair of

numbers in each case.

1.1

� Two numbers can be added in any order. This is called the commutative law for addition.

The commutative law does not hold for subtraction.

a + b = b + a e.g. 7 + 11 = 11 + 7a − b ≠ b − a e.g. 5 − 2 ≠ 2 − 5

� Three or more numbers can be added in any order. This uses the associative law for

addition. The associative law does not hold for subtraction.

(a + b) + c = a + (b + c) e.g. (2 + 5) + 4 = 2 + (5 + 4)

(a − b) − c ≠ a − (b − c) e.g. (9 − 5) − 2 ≠ 9 − (5 − 2)

� Addition and subtraction algorithms can be used for larger numbers.1 14 31 14 31 1 91 8 2

6 2 1

+

2 12 13 13 12 13 12 12 13 12 1 41 4 2

1 7 2

−

� Strategies for mental arithmetic include:

• Partitioning e.g. 247 + 121 = (200 + 100) + (40 + 20) + (7 + 1) = 368

85 − 22 = (80 − 20) + (5 − 2) = 63

• Compensating e.g. 134 + 29 = 134 + 30 − 1 = 163

322 − 40 = 320 − 40 + 2 = 282

• Doubling or halving e.g. 35 + 37 = 2 × 35 + 2 = 72

240 − 123 = 1

2 of 240 − 3 = 117

Key

idea

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Chapter 1 Integers6

Example 1 Using mental arithmetic

Evaluate this difference and sum mentally.

a 347 − 39 b 125 + 127

SOLUTION EXPLANATION

a 347 − 39 = 308 347 − 39 = 347 − 40 + 1= 307 + 1= 308

b 125 + 127 = 252 125 + 127 = 2 × 125 + 2= 250 + 2= 252

Example 2 Using an algorithm

Use an algorithm to fi nd this sum and difference.

a 938217+

b 14186−


a 9 3 82 1 7

1 1 5 5

19 319 3+

8 + 7 = 15 (carry the 1 to the tens column)

1 + 3 + 1 = 59 + 2 = 11

b 1 11 18 68 6

5 5

1 141 11 141 131 131 11 141 131 141 111 111 1−

Borrow from the tens column then subtract 6 from 11.

Now borrow from the hundreds column and then Now borrow from the hundreds column and then

subtract 8 from 13.

1 Write the number that is:

a 26 plus 17 b 43 take away 9

c 134 minus 23 d 451 add 50

e the sum of 19 and 29 f the sum of 111 and 236f the sum of 111 and 236fg the difference between 59 and 43 h the difference between 339 and 298

i 36 more than 8 j 142 more than 421

k 32 less than 49 l 120 less than 251

Exercise 1A

Unde

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g







2 Write the digit missing from these sums and differences.

a 2 3 7

+ 4

2 7 9

b 4 9+ 3 8

8

c 4 9 3+ 2 1 4

7 7

d 1 4

+ 3 9 25 5 6

e 3 8− 1 9

1

f 1 2 8

− 8

3 9

g 3 4

− 1 6 21 4 2

h 2 5 1

− 1 4

8 7

Unde

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gFl

uenc

y

3 Evaluate these sums and differences mentally.

a 94 − 62 b 146 + 241 c 1494 − 351 d 36 + 19

e 138 + 25 f 251f 251f − 35 g 99 − 20 h 441 − 50

i 350 + 351 j 115 + 114 k 80 − 41 l 320 − 159

4 Use an algorithm to fi nd these sums and differences.

a 12846+

b 94337+

c 9014927421

++

d 8141439326

++

e 9436−

f 421204−

g 17261699−

h 14072328−

i 42831410729

+++

j 100424079116

10494

+++

k 30172942−

l 10024936−

5 A racing bike’s odometer shows 21 432 km at the start of a race and 22 110 km at the end of

the race. How far was the race?

Casey Stoner racing at the Malaysian Grand Prix

Example 1

Example 2






Chapter 1 Integers8

Prob

lem

-sol

ving

6 The sum of two numbers is 39 and their difference is 5. What is the larger number?

7 Find the missing digits in these sums and differences.

a 2 3

+ 9 4

6 1

b 3

+ 2

2 1 9

c 3 7

+ 4 9

7 2

d 3− 2 9

6

e 3 2

− 3

1 0 4

f 2 5

− 6 8

3 1 8

8 Wally has two more marbles than Ashan and together they have 88 marbles. How many marbles

does Ashan have?

9 Evaluate the following without the use of a calculator.

a 1 + 2 + 3 + 4 + … + 99 + 100 − 99 − 98 − … − 2 − 1b 1 − 2 + 3 − 4 + 5 − 6 + … − 98 + 99

10 Explain why the answers given in these problems are impossible.

a 2

+ 3 6

2 3 4 9

b 3 6

− 3 2

8 211 x, y and z represent any three numbers. Complete these statements.

a x + x + x y + z = ___ + x + x + x y b x − x − x y + z = z − ___ + ___ = x + x + x ___ − ___

12 How many different combinations of numbers make the following true? List the combinations

and explain your reasoning.

a 1 4

+ 2

4 2 7

b 3

− 1 4

1 4 3

Reas

onin

g

Enrichment: Magic triangles

13 The sides of a magic triangle all sum to the same total.

a Show how it is possible to arrange all the digits from 1 to 9 so that

each side adds to 17.

b Show how it is possible to arrange the same digits to a different

total. How many different totals can you fi nd?

c Extension. In how many different ways can you obtain each total?

Switching the two middle numbers on each side does not count as

a new combination.







Whole number multiplication and divisionIt is useful to know how to multiply

and divide numbers without the use of

technology. Mental strategies can be used

in some problems, and algorithms can be

used for more diffi cult cases. Calculating

the cost of 9 tickets at $51 each, for

example, can be done mentally, but the

short division algorithm might be useful

when calculating the number of trips a

dump truck with a capacity of 140 tonnes

will need to shift 1000 tonnes of coal.

Let’s start: Multiplication or division?In solving many problems it is important to know whether multiplication or division should be used.

Decide if the following situations require the use of multiplication or division. Discuss them in a group or

with a partner.

• The number of cookies 4 people get if a packet of 32 cookies is shared equally between them

• The cost of paving 30 square metres of courtyard at a cost of $41 per square metre

• The number of sheets of paper in a shipment of 4000 boxes of 5 reams each (1 ream is 500 sheets)

• The number of hours I can afford a plumber at $75 per hour if I have a fi xed budget of $1650.

Make up your own situation that requires the use of multiplication and another for division.

1.2

A typical large mining truck has a capacity of 140 tonnes.

� Finding a product is the result of using multiplication.

We say the product of 11 and 9 is 99.

� The multiplication algorithm can be used when products cannot

be found mentally.

� Using division results in fi nding a quotient and a remainder.

e.g. 38 ÷ 11 = 3 and 5 remainder

� The short division algorithm can be used when quotients cannot be found mentally.

� The commutative law holds for multiplication but not division.

7 × 5 = 5 × 7 but 21 ÷ 3 ≠ 3 ÷ 21

� The associative law holds for multiplication but not division.

(5 × 4) × 2 = 5 × (4 × 2) but (5 ÷ 4) ÷ 2 = 5 ÷ (4 ÷ 2)

� Mental strategies for multiplication

• Using the commutative and associative laws

e.g. 5 × 17 × 4 = 5 × 4 × 17 = 20 × 17 = 340

21726

13024340

5642

×←←←

217 × 6

217 × 20

1302 + 4340

dividend divisor quotient )7 5)7 5) 1 2 47 3 2

2 11 22 11 2

Key

idea

s Ke

y id

eas

Key

idea

s






Chapter 1 Integers10

• Using the distributive lawe.g. 4 × 87 = (4 × 80) + (4 × 7) = 320 + 28 = 348

or 4 × 87 = (4 × 90) − (4 × 3) = 360 − 12 = 348

• Doubling and halving

e.g. 4 × 74 = 2 × 148 = 296

� Mental strategies for division

• Halving both numbers

e.g. 132 ÷ 4 = 66 ÷ 2 = 33

• Using the distributive lawe.g. 96 ÷ 3 = (90 ÷ 3) + (6 ÷ 3) = 30 + 2 = 32

or 147 ÷ 3 = (150 ÷ 3) − (3 ÷ 3) = 50 − 1 = 49

Key

idea

s

Example 3 Using mental strategies

Use a mental strategy to evaluate the following.

a 5 × 160 b 7 × 89 c 464 ÷ 8


a 5 × 160 = 800 Double one and halve the other so 5 × 160 becomes

10 × 80

b 7 × 89 = 623 Use the distributive law so 7 × 89 becomes

(7 × 90) − (7 × 1) = 630 − 7

c 464 ÷ 8 = 58 Halve both numbers repeatedly so 464 ÷ 8 becomes

232 ÷ 4 = 116 ÷ 2

Example 4 Using multiplication and division algorithms

Use an algorithm to evaluate the following.

a 41225×

b 938 ÷ 13


a 41225

20608240

10300

×412 × 5 = 2060 and 412 × 20 = 8240

Add these two products to get the fi nal answer.

b b

)13 93 87 27 2

2 Rem 2

So 938 ÷ 13 = 72 and 2 remainder.

9393 ÷ ÷ 1313 = = 7 and 2 remainder7 and 2 remainder

28 ÷ 13 = 2 and 2 remainder







1 Find the result for the following.

a The product of 7 and 8

b The product of 13 and 100

c The remainder when 2 is divided into 19

d The remainder when 9 is divided into 104

e The quotient of 13 divided by 4

f The quotient of 120 divided by 59

2 Use your knowledge of the multiplication table to write down the answer to the following.

a 5 × 8 b 11 × 9 c 6 × 7 d 9 × 8e 11 × 6 f 12 × 11 g 8 × 4 h 7 × 9i 100 ÷ 10 j 88 ÷ 8 k 121 ÷ 11 l 144 ÷ 12

m 56 ÷ 7 n 33 ÷ 3 o 65 ÷ 5 p 78 ÷ 6

3 Decide if these simple equations are true or false.

a 4 × 13 = 13 × 4 b 2 × 7 × 9 = 7 × 9 × 2c 6 ÷ 3 = 3 ÷ 6 d 60 ÷ 20 = 30 ÷ 10

e 14 ÷ 2 ÷ 7 = 7 ÷ 2 ÷ 14 f 51 × 7 = (50 × 7) + (1 × 7)

g 79 × 13 = (80 × 13) − (1 × 13) h 93 ÷ 3 = (90 ÷ 3) + (3 ÷ 3)

i 133 ÷ 7 = (140 ÷ 7) − (7 ÷ 7) j 33 × 4 = 66 × 8

Exercise 1B

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4 Use a mental strategy to evaluate the following.

a 5 × 13 × 2 b 2 × 26 × 5 c 4 × 35 d 17 × 4e 17 × 1000 f 136 × 100 g 59 × 7 h 119 × 6i 9 × 51 j 6 × 61 k 4 × 252 l 998 × 6m 128 ÷ 8 n 252 ÷ 4 o 123 ÷ 3 p 508 ÷ 4q 96 ÷ 6 r 1016 ÷ 8 s 5 × 12 × 7 t 570t 570t ÷ 5 ÷ 3

5 Use a multiplication algorithm to evaluate the following.

a 679×

b 1294×

c 29413×

d 100490×

e 69014×

f 4090101×

g 246139×

h 1647209×

6 Use the short division algorithm to evaluate the following.

a )3 8)3 8) 5 b )7 214 c )10 4167 d )11 143

e )15 207 f )19 3162 g )28 196 h )31 32690

7 A university student earns $550 for 22 hours work. What is the student’s pay rate per hour?

8 Packets of biscuits are purchased by a supermarket in boxes of 18. The supermarket orders 220

boxes and sells 89 boxes in one day. How many packets of biscuits remain in the supermarket?

Example 3

Example 4a

Example 4b






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