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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
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
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)
•
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
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
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
• Doubling or halving e.g. 35 + 37 = 2 × 35 + 2 = 72
240 − 123 = 1
2 of 240 − 3 = 117
Key
idea
s
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
Use an algorithm to fi nd this sum and difference.
a 938217+
b 14186−
SOLUTION EXPLANATION
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
rsta
ndin
g
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
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
rsta
ndin
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
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
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.
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
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
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
SOLUTION EXPLANATION
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
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
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
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