Upload
hanguyet
View
221
Download
0
Embed Size (px)
Citation preview
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
www.cambridge.org© in this web service Cambridge University Press
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)
•
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
www.cambridge.org© in this web service Cambridge University Press
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
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
www.cambridge.org© in this web service Cambridge University Press
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
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
www.cambridge.org© in this web service Cambridge University Press
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−
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
www.cambridge.org© in this web service Cambridge University Press
Number and Algebra 7
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
www.cambridge.org© in this web service Cambridge University Press
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.
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
www.cambridge.org© in this web service Cambridge University Press
Number and Algebra 9
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
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
www.cambridge.org© in this web service Cambridge University Press
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
SOLUTION EXPLANATION
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
www.cambridge.org© in this web service Cambridge University Press
Number and Algebra 11
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
Unde
rsta
ndin
gFl
uenc
y
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
www.cambridge.org© in this web service Cambridge University Press