Chapter 5
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05 NCM7 2nd ed SB TXT.fm Page 150 Saturday, June 7, 2008 4:44 PM
NUMBER
Mercury, the planet closest to the Sun, has an average temperature of 350°C while Pluto, the planet furthest from the Sun, has an average temperature of 230°C. The highest recorded temperature in Australia is 53°C at Cloncurry in Queensland and the lowest is 23°C at Charlotte Pass in the Snowy Mountains. The values in these measurements are all examples of integers.
5
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05 NCM7 2nd ed SB TXT.fm Page 151 Saturday, June 7, 2008 4:44 PM
0123456789
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0123456789 567890123 3 8901234
4567890123 8901234 0123456789
Wordbank
In this chapter you will: 567890123
4567890123 8901234 0123456789
456789 012345 0123456789 678901234
56789012345678 • recognise the5678 direction and magnitude 4567890123456789012345 of an • ascending Increasing, from smallest to largest. 56789012345678 1234567890123456789 5678 integer • coordinates The two values that give the position 234567890123 45678901234 0123456789 • order directed numbers and place them on a 0123456789 56789012 456789012345678901234567890123456789012345678 56789012 of a point on a number plane. 8 7890123456789012 3456789012345678 number901234567890123456789012345678901234567890123456789012345678 line • descending Decreasing, from largest to smallest. 789012345678 567890123456789012345678901234567890123456789012345 • add and subtract integers • integer A positive or negative whole number or 0123456789 56789012345678901234567890 1234567890123456789 78901234 5678 234567890123 45678901234 zero. • multiply and divide integers 0123456789 456789016789012345678901234567890123456 7890123456789012 • apply ‘order of operations’ to expressions involving 345678901• negative number A number less than zero. 4567890129012345 6789012345678901234567890123456789012345678901234567890123456789012 integers 6789012345678901234567890123456789012345678901234567890123 • number plane A grid that is made up of two 345678 and plot points on the9012345 90123456789012345678901234567890123456 7890123456789012 89012345678 • locate number67890123456789012345678 plane123456789012345678901234 number lines, meeting at right 345678901234567890123 901234567890123 01234567890 5678 angles. 012345678901234 5678901 0123 4567890 the 1234567890123456789 012345678901234567 890123456789012345678901234567890123456789012345678901234567890123456789012345678 • identify origin and the four quadrants of the • origin The centre 678901234567890 point of a number plane, with2345678901 234567 45678901234567890123456 789012345678901234567 8901234 plane. 6789012345678901 756789012345678901234number 45678901234 34567890123456789012345678901234567 0123456789012345 coordinates (0, 0). 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05 NCM7 2nd ed SB TXT.fm Page 152 Saturday, June 7, 2008 4:44 PM
Start up Worksheet 5-01 Brainstarters 5
1 Find the answers to the following. a 7+9 b 12 − 4 e 13 + 8 f 30 − 5 i 6+7 j 14 − 6
c 7×4 g 2 × 11 k 3×5
d 20 ÷ 5 h 15 ÷ 3 l 24 ÷ 4
2 Find the answers to the following. a 12 + 37 b 652 − 48 e 79 × 3 f 384 ÷ 3 i 30 − 12 ÷ 6 j 20 × (3 + 17)
c 22 × 5 g 3+5×4 k 12 − 3 × 2 + 6
d 66 ÷ 2 h 18 ÷ 2 − 7 l 43 − [5 × 2 + 1]
3 Write true (T) or false (F) for each of the following. a 15 12 b 8 11 d 4 + 3 10 e 12 − 3 11 g 15 ÷ 3 12 ÷ 4 h 8+3+26×1 Skillsheet 5-01
c 63 f 73×2 i 18 ÷ 3 2 + 2 + 2
4 Write each of these sets of numbers in ascending order (smallest to largest). a 3, 8, 2, 1, 4 b 2, 18, 5, 7, 3, 16 c 38, 51, 49, 27, 66, 54
Ordering numbers
5 Write each of these sets of numbers in descending order (largest to smallest). a 6, 11, 18, 17, 5, 3 b 8, 7, 14, 23, 31, 5, 2 c 89, 91, 37, 69, 41, 3, 46 6 Find the answers to the following. a 32 b 23
c 52
d 33
5-01 Number lines A number line is used to show the position and size of numbers: RIGHT 0
1
2
3
4
5
6
7
8
We have shown the position of 2, 4 and 7. As we move to the right on the number line, the numbers become bigger.
Exercise 5-01 1 Write the whole numbers shown by the arrows on this number line. A
0
B
5
C
D
10
15
E
F
20
25
2 Estimate the whole numbers pointed to by the arrows on these number lines. A B C D a 0
152
4
8
NEW CENTURY MATHS 7
12
16
20
24
05 NCM7 2nd ed SB TXT.fm Page 153 Saturday, June 7, 2008 4:44 PM
A
b 0
B
10
c
A
0
20 C
10
20
e
30
B
0
20 A
C
40
B
D
30
E
40 D
E
40
50
D
50 F
60
E
60
C
0
D
30
B
A
d
C
F
80
100
E
120
F
60
140
G
90
120
3 Write the temperature shown on each of these thermometers. a 0
10
20
30
40
50
60
70
80
90
100 °C
b 10
0
30
20
40
°C
4 Draw a number line and use dots to show the numbers 3, 6, 9, 12 and 15. 5 Draw a number line and show all the whole numbers between 3 and 12. 6 Draw a number line marked from 0 to 10 and show all the whole numbers less than 8. 7 Write the next three numbers in the pattern each time: a 3, 6, 9, 12, … b 18, 19, 20, 21, … c 109, 110, 111, … d 79, 78, 77, 76, … e 33, 32, 31, … f 161, 160, 159, … g 8, 7, 6, 5, … h 21, 19, 17, … 8
S 12
14
16
18
20
22
24
26
28
On this number line, where will S be if it moves: a 4 places to the right? b 10 places to the right? d 5 places to the left? e 10 places to the left?
30
32
c 19 places to the right? f 20 places to the left?
9 The temperature was 22°C at 9:00am. If it rose 8°C during the day, what was the maximum temperature? 10 The temperature was 27°C. When a cool change hit, it dropped by 11°C. What did the temperature fall to?
5-02 Numbers above and below zero Numbers greater than zero, such as 5, 8 1--- and 14.37, are called positive numbers. 2
Numbers less than zero, such as −2, −10 3--- and −21.6, are called negative numbers. 4 The number ‘−2’ is read as ‘negative 2’, and not ‘minus 2’. CHAPTER 5 INTEGERS
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Directed numbers and integers Positive numbers and negative numbers together are called directed numbers. Directed numbers show both direction and magnitude (size). For example, ‘5’ or ‘+5’ means 5 units in the positive direction, while ‘−2’ means 2 units in the negative direction. A directed number that is a whole number (not a fraction) is called an integer. Some examples of integers are −3, 4 and −20.
!
Integers are the positive and negative whole numbers and zero.
Exercise 5-02 1 Icebergs Icebergs can be dangerous to ships because about 5--- of their volume lies 6 beneath the water.
Height +40 m
a Using the scale, find: i the height of the iceberg above the water ii the depth of the iceberg below the water iii the total height of the iceberg. b Why are ‘+’ and ‘−’ signs useful in these measurements?
+30
m
+20
m
+10
m
0m −10 m −20 m −30 m −40 m −50 m −60 m −70 m −80 m −90 m −100 m
2 Temperature Temperature is measured in degrees Celsius. It can be above or below zero. a What is the normal body temperature? b At what temperature does water freeze? c At what temperature does dry ice evaporate? d If the highest temperature recorded last week was + 48°C at Salah, in Algeria (in Africa), and the lowest was −69°C at the South Pole in Antarctica, what is the difference between the two temperatures?
water boils
body temperature water freezes
dry ice evaporates
liquid nitrogen freezes
154
NEW CENTURY MATHS 7
Temperature 100°C 80°C 60°C 40°C 20°C 0°C −20°C −40°C −60°C −80°C −100°C −120°C −140°C −160°C −180°C −200°C
05 NCM7 2nd ed SB TXT.fm Page 155 Saturday, June 7, 2008 4:44 PM
3 Deposits or withdrawals Here is a business account showing money coming in, and money to be paid to others. (Withdrawals have a minus sign.) Calculate the final balance at the end of the day. Balance at start: $721 −$49 $61 −$1 −$101 $24 −$261 Balance at the end of the day: $ 4 Copy this timeline into your book and place each event in the correct place. BC
AD
500
A 490 BC B 400 BC C 330 BC D 240 BC E 55 BC F AD 100 G AD 190 H AD 260 I AD 370 J AD 410
0
500
Phidippides of Athens set out on his 26-mile run that inspired the marathon. The first temple dedicated to the Greek god Zeus was built. Euclid showed that an infinite number of prime numbers exist but occur in no logical pattern. Eratosthenes estimated the circumference of the Earth using two sticks. Roman forces under Julius Caesar invaded Britain. The first Chinese dictionary was compiled. The abacus was invented. The two-year war between Rome and Persia ended. Hypatia, a female mathematician, was born in Alexandria, Egypt. Roman forces left Britain.
5 Settlement in Australia Australian archeologists have found human bones which suggest that people have been living here for 40 000 years. These early inhabitants were ancestors of the Aboriginal people. Europeans settled in Australia in 1788 when Governor Phillip arrived at Botany Bay with English convicts. a Explain how −40 000 could describe the settlement of Australia. b If this year is zero, in what year did Governor Phillip arrive? c Why do historians use BC and AD (or BCE and CE) when talking about dates? d If the year you were born is zero, what year is it now? e If 1788 is zero, what year is it now? 6 Puzzle Archeologists found a coin marked 27 BC and tried to sell it to a museum for $1000. Why would you advise the museum not to buy it? CHAPTER 5 INTEGERS
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Just for the record
Oetzi iceman In 1991, a preserved body of a man was found in a glacier in the Alps between Italy and Austria. This man was probably a shepherd or a hunter who lived about 3300 BC. Oetzi, as he was named, was wearing a loin cloth and leather shoes stuffed with straw. He also wore a cloak over his tunic. An examination of his body showed that he was killed by an arrowhead that was found beneath his left shoulder. He is now kept in a refrigerated chamber in the town of Bolzano in northern Italy. How many years ago did Oetzi live?
5-03 Ordering integers Skillsheet 5-01
Every integer has an opposite. The opposite of 5 is −5. The opposite of −4 is 4. Integers can be positioned and ordered on a number line.
Ordering numbers
−4
−3
−2
−1
0
1
2
3
4
On a number line, any number is bigger than all the numbers to its left.
Example 1 Fill in the missing numbers for every mark on this number line. −7
−5
−2
0
1
0
1
4
Solution −7
!
−6
−5
−4
means ‘is greater than’.
−3
−2
−1
2
3
4
5
means ‘is less than’.
Think of the and signs as being the mouths of crocodiles. They always open towards the bigger number.
156
NEW CENTURY MATHS 7
05 NCM7 2nd ed SB TXT.fm Page 157 Saturday, June 7, 2008 4:44 PM
−2 is less than 3. This is written as −2 3.
3 is greater (or more) than 1. This can be written as 3 1.
3
−2
1
3
It follows that: • 3 is greater than − 4, or 3 − 4 • −5 is less than −3, or −5 −3 • −50 is less than 2, or −50 2 • −8 is more than −9, or −8 −9.
Exercise 5-03 1 Copy these number lines, using a ruler to mark the positions evenly. Fill in the missing number for each mark. a −6
−3
−1
0
1
2
3
6
−20
−10
0
c
10
40 −5
d
−100
0 0
e
5
10
100 −3
0
3
2 Copy each pair of numbers and use or to make true statements. a 4 1 b 0 9 c 7 −7 d −3 e 5
−11
f −1
−6
g −12
i −6
5
j −6
−2
k −2
q 17
TLF
9
L 2001
Scale matters: negatives
b
m −136
Ex 1
36 23
n −872 r 8
5
o −120
8 −6
120
0
h −10
−2
l 24
−24
p −12
−78
−47
3 What directed number is opposite of: a 6? b 1? c −10?
d 11?
4 Write a word that is the opposite of: a up b more c down f south g withdraw h decrease
e 0? d left i west
f −2? e ascending
5 Here are some numbers: 1, −2, 3, −4, 5, −6, 7, −8. a Which is the biggest number? b Which is the smallest number? c Rewrite the numbers in order, from smallest to biggest. CHAPTER 5 INTEGERS
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6 Rewrite these numbers in ascending order (smallest to largest). a −3, 2, −1, 3 b 5, −5, 2, −8, −3 c −4, −6, −3, −10, 0 d 6, −3, 4, −2, −5 e −48, 36, −24, 8, 0, −11 f 15, 12, −10, −26, 3, −2 7 Rewrite these numbers in descending order (largest to smallest). a 4, 3, −1, 5, −2 b −4, 8, −7, −2, 0 c 1, −1, 4, −5, −11, −3 d 8, − 4, −18, 3, −2 e − 6, −15, − 48, −3, 1 f 33, 1, −100, −58, −36 8 The distance between −3 and 1 on a number line is 4. −4
−3
−2
−1
0
1
2
3
4
Find the distance between these pairs of numbers. a 2 and 4 b −2 and − 4 c 0 and 4 e −3 and 3 f −10 and 10 g − 6 and 1
d 0 and −3 h −1 and 6
9 Which of the following is true? Select A, B, C and D. A −6 2 B 4 −5 C 0 −3
D 12 −2
10 In this question, POS means positive and NEG means negative. Negative −8
−7
Positive −6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
7
8
a Start at −3. Move 4 steps in the NEG direction. Move 9 steps in the POS direction. Move 6 steps in the NEG direction. Where are you now? b Start at 8. Move 12 steps in the NEG direction. Move 5 steps in the POS direction. Move 7 steps in the NEG direction. Where are you now? c Start at −5. Move 6 steps in the POS direction. Move 4 steps in the NEG direction. Move 7 steps in the POS direction. Where are you now? d Start at 3. Move 5 steps in the POS direction. Move 8 steps in the NEG direction. Move 2 steps in the NEG direction. Where are you now? e Start at −1. Move 10 steps in the NEG direction. Move 4 steps in the POS direction. Move 5 steps in the NEG direction. Where are you now?
5-04 Adding and subtracting integers Example 2 Worksheet 5-02 Adding and subtracting integers
1 What is −6 + 7? Solution −8
Skillsheet 5-02 Integers using diagrams
−7
−6
−5
−4
• −6 is where you start. • + direction is right. • 7 is how far you move.
158
NEW CENTURY MATHS 7
−3
−2
−1
−6 + 7 = 1
0
1
2
3
4
5
6
7
8
05 NCM7 2nd ed SB TXT.fm Page 159 Saturday, June 7, 2008 4:44 PM
2 What is 8 + (−6)? Solution −6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
7
8
9
10
7
8
9
10
• 8 is where you start. • + direction is right. • −6 is how far you move in the opposite direction (that is, left). 8 + (−6) = 2 So 8 + (−6) is the same as 8 − 6. 3 What is 2 + (−5)? Solution −6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
• 2 is where you start. • + direction is right. • −5 is how far you move in the opposite direction (that is, left). 2 + (−5) = −3 So 2 + (−5) is the same as 2 − 5.
Example 3 1 What is 3 − 4? Solution −8
−7
−6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
7
8
−3
−2
−1
0
1
2
3
4
5
6
7
8
• 3 is where you start. • − direction is left. • 4 is how far you move. 3 − 4 = −1 2 What is −3 − 4? Solution −8
−7
−6
−5
−4
• −3 is where you start • − direction is left • 4 is how far you move. −3 − 4 = −7 CHAPTER 5 INTEGERS
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05 NCM7 2nd ed SB TXT.fm Page 160 Saturday, June 7, 2008 4:44 PM
3 What is 4 − (−3)? Solution −6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
7
8
9
10
7
8
9
10
• 4 is where you start. • − direction is left. • −3 is how far you move in the opposite direction (that is, right). 4 − (−3) = 7 So 4 − (−3) is the same as 4 + 3. 4 What is −3 − (−1)? Solution −6
−5
−4
−3
−2
−1
0
1
2
3
4
5
6
• −3 is where you start. • − direction is left. • −1 is how far you move in the opposite direction (that is, right). −3 − (−1) = −2 So −3 − (−1) is the same as −3 + 1.
Note: Negative numbers can be entered into a calculator using the sign change key
TLF
+/− or (−) .
L 585
Integer cruncher
Exercise 5-04 Ex 2
Ex 3
Ex 2
1 Find the answers to these additions. a 6+9 b 8 + (−9) e −2 + 2 f −2 + 5 i −8 + (−18) j 8 + 12 m 7 + (−9) n −13 + 4
c g k o
8 + (−5) −4 + (−7) −11 + 3 −16 + 15
d h l p
8 + (−15) −2 + (−2) −10 + 10 −7 + 13
2 Find the answers to these subtractions. a 4−2 b 13 − 12 e 7 − (−5) f −3 − (−2) i −6 − 4 j −3 − (−8) m −6 − 7 n 12 − 18
c g k o
2 − 13 −5 − (−5) 5 − 11 9 − (−3)
d h l p
5 − (−1) −5 − 5 4 − (−2) −2 − 7
3 Copy each question and work out the answer. Use the number line if you are unsure. a 6 + (−6) b −2 + 2 c −5 + 12 d −11 + (−9) e 3 + (−8) f −7 + (−7) g 0 + 70 h −27 + 6 i −4 + (−15) j 9 + (−1) k −32 + 0 l −13 + 21
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4 Find the value of: a 20 − 9 e −4 − 7 i 0−8
b 9 − 20 f −4 − (−7) j 0 − (−8)
c 20 − (−9) g 13 − (−3) k 5 − (−5)
Ex 3
d −9 − 20 h −13 − (−3) l −5 − 5
5 Use your calculator to find the value of: a 6 − 13 b 4 − (−2) d −5 − (−6) e 3 + (−8) g 11 − (−15) h −83 + (−95)
c −7 + 3 f 8 − (−23) i 12 − (−108)
6 7 − 12 = ? Select A, B, C and D. A 18 B −5
C −19
D5
7 -2 + 7 = ? Select A, B, C and D. A9 B −9
C5
D −5
8 Simplify the following. a −3 + (−3) + (−4) d 5 + (−5) + (−7) + 7 g 9 + (−2) + 2 + (−8) + (−9)
b −3 + 3 + (−4) e 5 + (−5) + (−7) + (−7) h 6 − 8 + (−3) + 1
c −3 + (−3) + 4 f −5 + (−5) + (−7) + 7
9 Simplify the following. a 1 − 9 + 9 − 10 − 1 d −1 − 2 − 3 − 4 − 5 − 6 g −13 + 20 − 4 + 10 − 6
b −4 − 7 − 6 + 7 − 4 e 6 − 9 + 4 − 11 h −6 + 10 − 1 + 3
c 4−7−6−7−4 f 7 − 10 + 6 − 5
10 Find the missing number for each of the following. a 7+ =0 b −3 + =0 d −8 + 1 = e 5+ = 12 g 10 + =4 h −10 + = −4 j + −2 = −18 k + 7 = −2
c −3 + = −6 f −5 + = −12 i + −2 = 18 l + (−7) = −2
11 Copy and complete each table by adding the numbers in the left column to the numbers in the top row. a
+
−2
−7
−10
b
+
4
4
−5
−2
8
6
−3
−5
6
5
3
−4
5
−2
−11
c
+
2
3
4
d
+
−1
1
2
−3
−3
−4
6
−6
6
CHAPTER 5 INTEGERS
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Mental skills 5A
Maths without calculators
Adding 8 or 9 A quick way to mentally add 9 or 8 to a number is to add 10 and count back 1 or 2 respectively. 1 Examine these examples. a 17 + 9 = 17 + 10 − 1 b 44 + 8 = 44 + 10 − 2 = 27 − 1 = 54 − 2 = 26 = 52 Count: ‘17, 27, 26’ Count: ‘44, 54, 52’ c 128 + 19 = 128 + 20 − 1 d 256 + 38 = 256 + 40 − 2 = 148 − 1 = 296 − 2 = 147 = 294 Count: ‘128, 148, 147’ Count: ‘256, 296, 294’ 2 Now simplify these. a 146 + 9 e 29 + 19 i 714 + 8 m 386 + 8
b f j n
212 + 9 687 + 39 623 + 8 418 + 48
c g k o
308 + 9 254 + 29 207 + 8 909 + 28
d h l p
1755 + 9 933 + 19 155 + 8 277 + 18
5-05 Applying integers Example 4 The temperature on Mars is 23°C and the temperature on Jupiter is −150°C. What is the difference in these temperatures? Solution Write a number sentence to calculate the answer. 23 − (−150) = 23 + 150 = 173 The difference in temperature is 173°C.
Exercise 5-05 Ex 4
For each question, write a number sentence to calculate the answer. 1 The heights above (and below) sea level of some well-known places are: • The Dead Sea (Jordan Valley) −397 metres • Death Valley (California, USA) −86 metres • Mt Kosciuszko (Australia) 2230 metres • Mt Everest (Nepal) 8840 metres
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a b c d
How much higher is Mt Everest than Mt Kosciuszko? How much higher is Mt Kosciuszko than Death Valley? How much lower is the Dead Sea than Death Valley? The highest point on Earth is Mt Everest and the lowest point on land is the Dead Sea. What is the difference in altitude between these two points?
2 The temperature on a day in winter reached a maximum of 11°C. It dropped to a minimum overnight of −2°C. How many degrees did it drop? 3 A man walked 16 kilometres east and then 20 kilometres west. How far is he from his starting point? 4 Anita left home and walked three kilometres west to her friend’s home. Together, they then walked west for another 4 kilometres. How far from Anita’s home were they then? 5 Mt Etna is an active volcano in Italy. It is 3200 m high. The lava starts 652 m below the Earth’s surface and shoots to a height of 750 metres above Etna. How far does the lava travel?
6 The submarine Nemesis descends from the surface to a depth of 955 metres to inspect a shipwreck. It then ascends 800 metres to send a message. How far below the surface is it when it sends the message?
$ million
7 The graph on the right shows the profit made by Moneta Pty Ltd over 6 years. a In which years did Moneta make Moneta Pty Ltd profit 200 a profit? 2002 160 b In which years did Moneta make a loss? 120 80 c What was the decrease in profit 2003 between 2003 and 2004? 40 2006 2007 d Did the company make an overall 0 Year profit or loss during the six years −40 2005 2004 shown? How much profit or loss −80 did they make? 8 A scuba diver was swimming at a depth of 18 metres. Her diving friend was at a depth of 7 metres. What was the vertical distance between them? 9 Karina had $74 in her bank account yesterday but was allowed to withdraw $121. Today, she deposited $40 into the account. What is her account balance now? 10 What is the correct number sentence for this problem? Select A, B, C or D. ‘The overnight low was −6°C. The temperature rose by 13°C during the day and dropped 5°C after the sun went down. What was the temperature then?’ A −6 + 13 − 5 B −6 − 13 − 5 C −6 + 13 + 5 D −5 − 13 − 6 CHAPTER 5 INTEGERS
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Using technology
Temperature ranges at Thredbo The following data shows the minimum and maximum temperatures for Thredbo, recorded each day for two weeks in June, 2007. 1 Enter the data, as shown below (centre data, bold headings), into a spreadsheet.
Source: www.bom.gov.au
2 In D1, enter the label ‘Difference’. To find the difference between the minimum and maximum temperatures each day, enter the formula as shown in cell D2 below.
3 To copy this formula into cells D3 to D16, click on cell D2 (with left mouse button) and position the mouse over the bottom right-hand corner of this cell. Drag down to cell D16 and let go. This is called Fill Down.
4 Using the values you obtained in column D from question 3, answer these questions. a On which day was the largest difference between the maximum and minimum temperatures recorded? Write your answer in cell E1. b On which day was the smallest difference between the maximum and minimum temperatures recorded? Write your answer in cell E2.
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5 In cell A18, enter the label ‘Lowest Minimum’. In cell A19, enter ‘Highest Minimum’. In cell D18, enter ‘Lowest Maximum’. In cell D19, enter ‘Highest Maximum’. In cell A20, enter ‘Difference’. 6 a To find the lowest minimum, in cell B18, enter =min(B2:B16). Now, enter appropriate formula into cells B19, C18 and C19 to find the values for the labels in question 5. (Note: The lowest and highest maximums go in C18 and C19.) b What do you notice about the value in cell C19, compared to the values in the other three cells from part a? 7 a Enter the formula shown in cell B20 below. From cell B20, use Fill Right to copy this formula into cell C20.
b For this fortnight in Thredbo, use your answers from question 6 a and 7 a to answer these questions. i Which column (B or C) had the greatest difference in temperature? Write your answer in cell E3. ii In cell E4, write a formula to find the difference between the lowest minimum temperature and the highest maximum temperature.
Just for the record
Brahmagupta Brahmagupta was a famous Indian mathematician who lived from AD 598 to AD 670. He wrote important works on mathematics and astronomy. His writings contained two remarkable ideas that were ahead of his time. • He defined zero as the result of subtracting a number from itself. • He gave arithmetical rules for positive and negative numbers, which he called fortunes and debts. Use the Internet to find what some of these rules were.
CHAPTER 5 INTEGERS
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Working mathematically
Reflecting and reasoning
Multiplying integers 1 Copy the table below onto a large piece of paper and, as a group: a complete the shaded section b complete the pattern for the first row, that is 25, 20, 15, 10, 5, 0, −5, … c complete the pattern for the next row, that is 20, 16, 12, 8, 4, 0, −4, … d complete the next four rows e complete the pattern for each column. ×
5
4
3
2
1
0
−1
−2
−3
−4
−5
5 4 3 2 1 0 −1 −2 −3 −4 −5
2 How do the signs (positive or negative) of the numbers in the question affect the sign of the answer? Look for a pattern. 3 Use your completed table to help you complete the following. a 4 × (−3) = b −3 × 5 = c −2 × (−2) = e 2×5= f −3 × 3 = g −1 × 4 =
Skillsheet 5-02
d −5 × (−1) = h −5 × 5 =
5-06 Multiplying integers
Integers using diagrams
!
The rules for multiplying directed numbers are: positive × positive = positive positive × negative = negative negative × positive = negative negative × negative = positive
×
+
−
+
+
−
−
−
+
When multiplying two numbers which have the same sign, the answer is positive. When multiplying two numbers which have different signs, the answer is negative.
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Example 5 Find the answers for these, taking care to get the signs correct. a −3 × 2 b −6 × (−7) c 4 × (−4) Solution a −3 × 2 = −6 b −6 × (−7) = 42 c 4 × (−4) = −16
(negative × positive = negative) (negative × negative = positive) (positive × negative = negative)
Exercise 5-06 1 Find the answer for each of these. a −3 × 6 b 6 × (−4) e −7 × (−9) f −9 × 5 i 3 × (−8) j −9 × 1 m 5 × (−2) n −7 × 6 q −12 × (−5) r 20 × (−3)
c g k o s
2 8 × (−5) = ? Select A, B, C or D. A 13 B −13
C −40
D 40
3 −7 × (−3) = ? Select A, B, C or D. A 21 B −21
C −10
D 10
4 −2 × 3 × 7 = ? Select A, B, C or D. A1 B 19
C −35
D −42
Ex 5
−3 × (−7) 7×6 −1 × (−1) −11 × (−4) −10 × (−10)
d h l p t
4 × (−8) −7 × (−6) −9 × 6 −9 × 9 −7 × (−8)
5 Complete these multiplication tables, taking care to get the signs correct. a b c 6 −4 × −5 5 × × −3 5
d
−3
5
−4
5
−5
6
×
−1
1
e
×
7
−3
f
×
−1
4
8
1
−6
2
6 Find the answer for each of these. a −6 × (-4) b 2 × 15 e 16 × (−1) f −3 × 81 i −99 × 1000 j −56 × (−100) m −300 × 100 n −4 × 4 q 8 × (−70) × (−200) r −13 × 100
c g k o s
−9 × (−9) 63 × 300 −1 × (−1) × (−1) −9 × (−4) 0×4
−3
d h l p t
−6
11 × (−3) −104 × (−40) 13 × (−4) −6 × 3 × (−5) 12 × (−3) × (−10)
CHAPTER 5 INTEGERS
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7 Find the value of: a 42 d (−8)2 × (−2) g −4 × (−1)3
b (−3)2 e (5)3 h (−2)3 × (−5)2
c (−7)2 f (−2)3 i (−3)3 × (−2)3
8 Follow the arrows along all the stepping stones and multiply by the next number as you go. Is your answer positive or negative at the finish? 1
START
−1
1 −1
1
1 1
−1 1
1
1 −1
−1 1
1 −1
−1 −1
−1
−1 1 FINISH
9 Use your calculator to find the answer for each of the following. a −8 × 6 b −11 × (−4) c −15 × (−16) d 13 × (−5) e 38 × (−11) f −25 × (−12) g (−5)2 h (−8)3 i −15 × (−7)2
5-07 Dividing integers Because division is the reverse operation to multiplication, the rules for dividing integers are the same as those for multiplying them. • 5 × (−4) = −20 so −20 ÷ (−4) = 5 • −6 × 2 = −12 so −12 ÷ 2 = −6 • −3 × (−9) = 27 so 27 ÷ (−9) = −3
!
The rules for dividing directed numbers are: positive ÷ positive = positive positive ÷ negative = negative negative ÷ positive = negative negative ÷ negative = positive
÷
+
−
+
+
−
−
−
+
When the two numbers involved in the division have the same signs, the answer is positive. When the two numbers involved in the division have different signs, the answer is negative.
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Example 6 Find answers for these, taking care to get the signs correct. a −15 ÷ 3 b 18 ÷ (−9) c −10 ÷ (−5) Solution a −15 ÷ 3 = −5 b 18 ÷ (−9) = −2 c −10 ÷ (−5) = 2 14 d ------ = 14 ÷ (−7) = −2 −7
14 d -----−7
(negative ÷ positive = negative) (positive ÷ negative = negative) (negative ÷ negative = positive) (positive ÷ negative = negative)
Exercise 5-07 1 Find the answers for the following divisions. a 15 ÷ (−5) b −8 ÷ 4 c e 27 ÷ (−3) f −18 ÷ (−6) g i −20 ÷ 5 j −51 ÷ (−3) k m −80 ÷ (−5) n −10 ÷ 2 o q −27 ÷ 9 r −80 ÷ (−4)
Ex 6
−12 ÷ (−3) 24 ÷ (−6) 45 ÷ (−5) 98 ÷ (−7)
d h l p
6 ÷ (−2) 100 ÷ (−2) −72 ÷ 6 48 ÷ (−8)
2 Find the answers for these divisions. -----a 12
--------b 100
-------c -18
-------d -66
e
f
g
h
-2 -126 ----------6
-50 45 ------5
3 −48 ÷ (−2) = ? Select A, B, C or D. A 24 B −24
-3 100 --------5
-2 -81 --------9
C −50
D −46
C4
D −20
------ = ? Select A, B, C or D. 4 16 -4
A 12
B −4
5 Divide the top row by the left-hand column to complete these tables. a b ÷ 8 −8 32 −32 ÷ −6 −3 3 6 −6
−2
−3
4
6 This question involves both division and multiplication. Remember to work from left to right, and find the answers. a 5 × (−4) b 6 × (−3) ÷ 9 c 7 × 6 ÷ (−2) d −3 ÷ (−1) × 1 e 24 ÷ (−6) × 50 f −56 ÷ (−8) × 4 g −100 ÷ (−20) ÷ (−5) × 23 h −27 × (−9) ÷ 9 i 3 × 8 ÷ (−4) ÷ (−2) j −14 ÷ 2 × 3 × (−2) k −16 × 5 ÷ 4 ÷ (−5) l 13 × 2 × (−5) ÷ (−10) m 500 ÷ (−10) × 4 n −14 × (−5) ÷ (−7) ÷ (−2) CHAPTER 5 INTEGERS
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7 Use your calculator to answer the following. a 45 ÷ (−9) × 2 b c −200 ÷ (−100) ÷ 2 d e −40 × (−3) ÷ (−20) f g −544 × (−15) × (−6) h
−56 × (−4) ÷ 2 32 × (−3) × (−15) 78 × (−2) × (−5) ÷ (−39) 48 ÷ 3 × (−12)
5-08 The four operations with integers Skillsheet 5-03 Order of operations
Worksheet 5-03
Now that you know how to add, subtract, multiply and divide with integers, you can evaluate mixed expressions. Remember to follow the ‘order of operations’ rules. • First: work out the value within any grouping symbols • Second: work out multiplication and division from left to right • Third: work out addition and subtraction from left to right.
Integer review
Example 7 Find the value of 3 × (−4 − 6). Solution 3 × (−4 − 6) = 3 × −10 = −30
(brackets first)
Exercise 5-08 1 Find the answers to the following. a 6−8 b −3 + 5 e −2 − 12 f 10 − 35 i 4 − (−7) j −3 − (−11) m 9 − 12 n −13 − 10 − (−1)
c g k o
−8 − 4 −8 − 13 −15 + 9 4 − 6 + 2 + (−1)
2 Find the answers to the following. a −3 × 6 b −2 × (−6) d −16 ÷ 8 e −25 × (−2) g 12 × (−4) h 48 ÷ (−6) j 21 ÷ (−3) ÷ (−7) k −2 × 5 × (−3) m −32 ÷ (−4) × (−2) n 6 ÷ (−3) × (−4)
Ex 7
c f i l o
d h l p
−7 + 9 19 − 22 −12 − (−3) 9 − 12 + 7 − 9
18 ÷ 2 −30 ÷ (−6) −100 ÷ (−10) 15 ÷ (−3) × (−2) −24 × 2 ÷ (−3) ÷ (−4)
3 6 + 4 × (−2) = ? Select A, B, C or D. A 14 B −20
C −2
D −14
4 Find the value of each of the following. a −5 × 2 + 3 b −5 × (2 + 3) e 12 − 8 ÷ (−4) f −3 × (9 − 10) i (−6 − 2) ÷ (−4) j −3 ÷ 3 + 9
c 3 × 6 × (−2) g 21 ÷ (−7) + 8 k 12 × (6 − 7) + 1
d 12 − 8 ÷ 4 h [−6 − (−4)] × (−4) l −4 × 7 ÷ (−7 − 7)
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5 Find the value of: a −5 × 6 − 8 c [18 − (−3)] × 4 e 81 − (−3 × 12) g 6 × (−2) + (−4) i −19 + 4 × 6 k (−6 − 3) ÷ (−3) m −8 + (−9) × (−10) ÷ (−5) o −15 + 19 × 2 − 8 q −3 − 10 ÷ 2 + 19
b d f h j l n p r
−5 × (6 − 8) 5 + (−8) × (−7) + 72 102 × (−5 + 3) −7 − 5 + 9 [−6 − (−9)] × (−2) 25 ÷ (−5) + 6 6 − 9 + 5 − 11 6 − 13 + 14 − 3 × (−5) 3 − 10 × (−2) − (−19)
6 A Year 7 student completed these 10 questions. Mark them, and correct the mistakes. a c e g i
−3 × 5 = −15 −6 − 8 = −2 12 ÷ (−6) = −2 −7 − 6 + 10 = 3 12 − 3 × 5 = 45
b d f h j
6−8=2 −11 + 3 = −8 (−2 + 6) × 2 = −8 (−12 − 3) ÷ (−3) = 3 2 − 9 × (−3) − (−18) = 39
7 Use your calculator to find the value of: a −2 × 15 + 11 b 6 × (13 − 27) d 25 ÷ (−5) ÷ (−5) e [12 − (−10)] × (−3) + 6 g 9 − 19 + 36 − 8 × (−6) h −12 − 30 ÷ (−2) + 58
Mental skills 5B
c −14 − 6 × (−8) f −3 × (−8) ÷ (−4) + (−11) i 6 + 4 × (−5) ÷ (−2)
Maths without calculators
Subtracting 8 or 9 A quick way of mentally subtracting 9 or 8 is to subtract 10 and add back 1 or 2 respectively. 1 Examine these examples. a 66 − 9 = 66 − 10 + 1 = 56 + 1 = 57 Count: ‘66, 56, 57’ c 72 − 49 = 72 − 50 + 1 = 22 + 1 = 23 Count: ‘72, 22, 23’
b 83 − 8 = 83 − 10 + 2 = 73 + 2 = 75 Count: ‘83, 73, 75’ d 141 − 28 = 141 − 30 + 2 = 111 + 2 = 113 Count: ‘141, 111, 113’
2 Now simplify each of these by counting. a 26 − 9 b 44 − 9 e 161 − 29 f 187 − 59 i 82 − 8 j 131 − 8 m 44 − 8 n 577 − 28
c g k o
123 − 9 75 − 19 96 − 8 203 − 18
d h l p
270 − 9 457 − 39 120 − 8 365 − 48
CHAPTER 5 INTEGERS
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Using technology
Spreadsheet formulas involving integers Remember that a spreadsheet is like a calculator. We use special symbols to do calculations; the basic operations are shown below. Mathematical operation
Symbol used in spreadsheet
Addition (+)
+
Subtraction (−)
-
Multiplication (×)
*
Division (÷)
/
Remember: A spreadsheet formula always begins with an equals (=) sign. 1 Enter the following numbers into cells as shown below, where m represents the value in cell B1, n is the value in cell B2, p is the value in cell B3, and so on.
2 Enter the following formulas into the given cells. (Try to predict the answer before you enter each formula.) a C1, m + r (means enter the formula into cell C1 as shown above) b C2, n + q
c C3, n − q
d C4, p − q − m
e C5, p − q + m
f C6, m − n
g C7, n − r
h C8, m × n
i C9, n × r
j C10, n × q × r
k C11,
n2
l C12,
r3
m C13, q ÷ r
n C14, m ÷ q r o C15, ---m (When you have obtained the answer, right click on the cell, choose Format Cells and convert the answer to a fraction.) p C16, the minimum value from the set of cells B1 to B5 q C18,
q
r C19, m ÷ (q × r) (Format Cells as in o above to convert to a mixed numeral.) s C20, (n + q) × m t C21, n × (r − q) n–q u C22, m × n + 25 v C23, ----------r m w C24, ----------x C25, m2 + n3 r–q
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3 Use the values for m, n, p, q and r given in Question 1. a i In cell E1, enter =B2/B3 ii In cell E2, enter =B3/B2 iii Compare your answers in cells E1 and E2. In two sentences, explain the validity of each answer. b Does (m + n)2 = m2 + n2? i In cell D1, enter a formula for (m + n)2. ii In cell D2, enter a formula for m2 + n2. iii Compare your answers in cells D1 and D2 to justify your answer.
5-09 The number plane A number plane is a grid made from a horizontal number line called the x-axis, and a vertical number line called the y-axis. Note: the plural of ‘axis’ is ‘axes’.
y-axis
Worksheet 5-04
5
Plane grid 1
4
Skillsheet 5-04
3
The number plane
2 1 0 1
Points on the number plane are found by using a pair of numbers called coordinates, for example (3, 2) meaning 3 across and 2 up. A set of coordinates such as (3, 2) can also be called an ordered pair. In the number plane on the right: • A is the point (2, 1) • B is the point (0, 3). How do we locate points? Always start from 0. The first number tells how far to move across, the second tells how far to move up. The first number is called the x-coordinate. The second number is called the y-coordinate. (2, 1) means (2 across, 1 up) • start from 0 • move 2 units across • then 1 unit up
2
3
4
5 x-axis
Up y 5 E (3, 4)
4 3
B (0, 3) C (3, 2)
2 1
D (5, 2)
A (2, 1) origin
0 1
2
3
4
5
x Across
(0, 3) means (0 across, 3 up) • start from 0 • do not move across • move 3 units up
Note: The point (0, 0) is called the origin.
CHAPTER 5 INTEGERS
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Exercise 5-09 1 Write the coordinates of the points shown on this number plane.
y P
5 4
A
D E
3 B
2
F C
1 0
2 Record the coordinates of the eight points shown on this number plane.
1
2
3
4
5
6
x
y B
3
J
2 M 1 0
U P
N A 1
2
E 3
4
5
x
3 Draw a number plane with the numbers from 0 to 6 on each axis. Start at the point (1, 1) and draw a line joining it to (4, 1). Join (4, 1) to (6, 2), then continue through (3, 2) → (1, 1) → (1, 4) → (3, 5) → (6, 5) → (4, 4) and finish at (1, 4). Now join (3, 2) to (3, 5). Join (4, 1) to (4, 4). Join (6, 2) to (6, 5). What have you drawn? 4 Write the letters shown at these points on the grid on the right, to spell some words. a (3, 1), (4, 3), (2, 4), (3, 5), (5, 2), (1, 5) b (2, 3), (6, 1), (6, 4), (5, 5), (2, 1) c (6, 5), (6, 4), (6, 3), (5, 3), (5, 1) d (2, 4), (5, 2), (6, 3), (5, 1), (5, 3) e (3, 1), (4, 3), (5, 1), (2, 2), (2, 1) f (4, 3), (2, 4), (2, 3), (1, 3), (1, 1), (2, 1)
y 5
R
D
B
V
N
W
4
L
M
K
J
M
A
3
I
P
F
U
E
T
2
C
S
G
H
E
K
1
R
E
N
H
R
L
0
Worksheet 5-04 Plane grid 1 Worksheet 5-05 Bigtop employee
1 2 3 4 5 6 x 5 You may use grid paper for these questions. a Draw a number plane or use the link to print Plane grid 1. On the x-axis, mark 0 to 15. On the y-axis, mark 0 to 20. b Plot these ordered pairs and join them with lines according to these instructions: (0, 4) → (2, 1) → (13, 1) → (15, 4) → (0, 4) STOP (0, 6) → (8, 6) → (15, 8) → (8, 20) → (0, 6) STOP (8, 4) → (8, 20) c What did you draw?
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6 You may use grid paper for this question. a Draw a number plane or use the link to print Plane grid 1. On the x-axis, mark 0 to 12. On the y-axis, mark 0 to 16. b Join the following points in order, from left to right: (0, 6) → (2, 1) → (9, 1) → (8, 3) → (9, 5) → (10, 1) → (12, 1) → (12, 3) → (10, 9) → (12, 11) → (12, 13) → (7, 16) → (4, 11) → (5, 10) → (2, 3) → (0, 6) STOP. (5, 10) → (6, 9) → (8, 11) → (8, 12) → (9, 11) → (10, 9)
Worksheet 5-04 Plane grid 1
c What did you draw? 7 Look at this plan of a country town. 5 4
Golf course
N
Secondary school Church
3
Council offices Library Shops
Station
Car park
2
Swamp
Shops Post office
Tennis courts
1 B
What features appear at: a F2? b D5?
Swimming pool
Pond
Car yard
A
Sports oval
Park
Primary school
C
D
c D3?
E
F
d C2?
G
e B3?
f C1?
8 Use the plan in Question 7 to write the map reference (letter first, number second) for: a the swamp b the library c the church d the secondary school e the sports oval f the council offices.
5-10 The number plane with negative numbers Now that we have used number lines involving both positive and negative numbers, we can extend the number plane as shown below. y 3 2nd quadrant
Worksheet 5-06 The number plane Worksheet 5-07 Plane grid 2
1st quadrant
2
Worksheet 5-08
1
Number plane review
origin (0, 0) −4
−3
−2
−1
3rd quadrant
O −1 −2
1
2
3
4
Worksheet 5-09
x
Girt by sea
4th quadrant
−3
CHAPTER 5 INTEGERS
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The number plane is divided into four regions called quadrants. The centre of the number plane is called the origin. The number plane is also called a Cartesian plane, named after René Descartes (pronounced ‘Ren-ay Day-cart’), a French mathematician who developed the idea of the number plane. A few points are labelled on this number plane as examples. y 4 C (−2, 3)
A (1, 3)
3 2
B (3, 1)
1 F (2, 0) −5
−4
−3
−2
−1
O
1
2
3
5
4
−1
D(−5, −1)
−2 −3 −4
E (4, −3)
G (0, −4)
Exercise 5-10 1 Write the coordinates of these 26 points. y 4 A B
U
T L −4
X
2
C E
1 Q
P −3
I
−2
−1
O
NEW CENTURY MATHS 7
1 F
J
−1
S
−2
M
−3
H
Z
−4
176
D
3 N
V K
2
3
4
x
R G Y
W
x
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2 a Draw a number plane with both axes (lines) extending from −6 to 6. b Mark these points. A(3, 1) B(−4, 3) F(1, −5) E(0, −2) J(−1, −6) I(−6, 0) M(6, 2) N(−3, 5)
C(−3, 4) G(4, −4) K(−3, −4) P(5, −5)
D(−2, −2) H(−3, 0) L(0, 5) Q(−1, 6)
3 List all of the points from Question 2 that are: a in the 1st quadrant b in the 2nd quadrant c in the 3rd quadrant d in the 4th quadrant e on the border of two quadrants. 4 In which quadrant would you find each of the following points? a (3, −5) b (−2, −4) c (−8, 1) 5 Copy this table and complete it by placing a + or a − sign in the blank spaces. Points in:
x-coordinate
1st quadrant
+
2nd quadrant
−
y-coordinate
3rd quadrant 4th quadrant
6 a Draw a number plane, with both axes extending from −3 to 3, or use the link to print Plane grid 2. b Mark these points: A(0, 1) B(2, −1)
Worksheet 5-07 Plane grid 2
C(−1, 2)
D(−2, 3)
E(3, −2)
F(1, 0)
c What do you notice about the points? Check this with your ruler. 7 a Draw a number plane, on graph paper, extending from −10 to 10 on both axes, or use the link to print Plane grid 2. b Start at the first point and join the points in the order given: i (5, 6) → (6, 5) → (8, 1) → (8, −1) → (6, −5) → (4, −6) → (1, −9) → (−1, −9) → (−4, −6) → (−6, −5) → (−8, −1) → (−8, 1) → (−6, 5) → (−5, 6) → (−7, 4) → (−8, 4) → (−10, 6) → (−10, 8) → (−8, 10) → (−6, 10) → (−3, 9) → (3, 9) → (6, 10) → (8, 10) → (10, 8) → (10, 6) → (8, 4) → (7, 4) → (6, 5). Stop and lift your pencil from the paper. ii (−3, 0) → (−1, 2) → (1, 2) → (3, 0) → (3, −1) → (2, −2) → (2, −1) → (1, −1) → (1, −2) → (−1, −2) → (−1, −1) → (−2, −1) → (−2, −2) → (−3, −1) → (−3, 0). Stop and lift your pencil from the paper. iii (−2, 1) → (−3, 2) → (−3, 3) → (−1, 5) → (−1, 2). Stop and lift your pencil from the paper. iv (2, 1) → (3, 2) → (3, 3) → (1, 5) → (1, 2). Stop and lift your pencil from the paper. v (−5, −4) → (−3, −6) → (0, −5) → (3, −6) → (5, −4). Stop and lift your pencil from the paper. vi (−2, −6) → (−2, −7) → (−1, −8) → (1, −8) → (2, −7) → (2, −6). c What did you draw? CHAPTER 5 INTEGERS
177
Worksheet 5-07 Plane grid 2
Worksheet 5-10 Quadrilateral man Worksheet 5-11 Coordinates code puzzle
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Power plus 1 Find the answers to the following. b (−4)3 a (−2)3
c (−2)5
d (−1)5
e (−1)9
f (−1)32
g − 16 × (−3)4
h 3 8
i 3 -8
j 5 32
k 5 -32
l 7 -1
2 Find the number (or numbers) which make each of these expressions true. 2 = 16 2 = 100 3 = −8 b c a 3 a b c d e f
Find two integers which, when added together, give −5. Find three integers which, when added together, give 7. Find two integers which, when subtracted, give −1. Find two integers which, when multiplied together, give −6. Find three integers which, when multiplied together, give 36. Find three integers which, when multiplied together, give −24.
4 a Find the integer midway between −8 and 12. b Find the integer midway between −1 and −17. c Find the integer midway between −21 and 19. Worksheet 5-07 Plane grid 2
5 a Draw a number plane extending from −7 to 7 on the x-axis, and from 12 to −8 on the y-axis, or use the link to print one out. b Start at the first point and join the points in the order given. i (3 1--- , 0) → (4, 2) → (4, 4) → (3 1--- , 7) → (4 1--- , 8) → (4 1--- , 9) → (5, 10) → 2 (4 1--- , 2
(2,
2 2 2 1 1 1 → (3 --- , 10) → (2, 10) → ( --- , 11 --- ) → (1, 10) → (1, 8) → (1 1--- , 7) → 2 2 2 2 1 1 1 1 1 (1 --- , 4 --- ) → (0, 2 --- ) → (−2 --- , 1) → (−3, 0) → (−3 --- , −4) → (−4 1--- , 6) → 2 2 2 2 2 2 −7) → (0, −8) → (2, −7 1--- ) → (2, −6 1--- ) → (0, −7) STOP 2 2 1 1 −7 --- ) → (1, −6) → (3, −6) → (4 --- , −6 1--- ) → (4 1--- , −5 1--- ) → (3, −4 1--- ) → (0, −5) → 2 2 2 2 2 2 1 −6 --- ) STOP 2
11 1--- ) 2 1 6 --- ) → 2
(−4, ii (−2, (−3,
iii ( 1--- , −2) → (1, −3) → (0, −5) STOP 2
iv (1 1--- , 0) → (1, −2) → (1 1--- , −4 1--- ) STOP 2
v (2, -6) → vi vii
(2 1--- , 2 1 (1 --- , 2
-4 1--- ) 2
2 2 1 1 -7 --- ) → (3 --- , -7 1--- ) 2 2 2 1 1 (2 --- , -2) → (3 --- ) 2 2
(2 1--- , 2 →
-3) → (1, -5) STOP
c Complete the picture as you wish. d What have you drawn?
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NEW CENTURY MATHS 7
→ (3 1--- , -6 1--- ) → (3, -6) STOP 2
2
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Chapter 5 review Language of maths ascending directed numbers negative ordered pair positive
axis/axes integer number line origin quadrant
Worksheet 5-12
coordinates magnitude number plane plus x-axis
descending minus opposite point y-axis
Integers find-a-word
1 What is the difference between a ‘directed number’ and an ‘integer’? 2 ‘Every integer has an opposite.’ True or false? 3 What is the difference between −4 (‘negative 4’) and − 4 (‘minus 4’)? 4 Look up the word ‘coordinate’ in the dictionary. Find both its mathematical and non-mathematical meanings. 5 Why do you think ‘ordered pair’ is another name for coordinates? 6 What is the origin and why does it have that name?
Topic overview • Write in your own words what integers are. • Write any rules you have learnt about integers. • What parts of this topic did you find difficult? What parts didn’t you understand? Discuss them with a friend or your teacher. • Give some examples of where negative numbers are used. • Copy this overview into your workbook and complete it, making any changes you find necessary. Ask your teacher to check your chart. Number line
Directed numbers
Addition
Multiplication
INTEGERS + 0 – Division
Subtraction
Ordered pair Origin Number plane
CHAPTER 5 INTEGERS
179
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Topic test 5
Exercise 5-01
Exercise 5-01
Chapter revision 1 Draw a separate number line to show each of the following sets of numbers. a 2, 3, 6, 1 b whole numbers between 2 and 5 c whole numbers greater than 1 and less than 7 2 Copy and complete these number patterns. a −2, −1, 0, c −6, −8, e 1, −2, 3,
, ,
, ,
,
b 1, 0, −1,
,
,
d 2, 0, −2,
,
,
f −1, −3, −5,
,
,
,
Exercise 5-02
3 Find the amount left in this bank account after each of these day’s transactions. a start $37.85 b starting position $65.30 deposit $18.20 withdraw $100.00 deposit $120.00 deposit $230.00 withdraw $45.00 withdraw $150.00 deposit $8.15 deposit $3.75 withdraw $248.20 withdraw $100.00
Exercise 5-03
4 Answer true (T) or false (F) for each of these. a −4 2 b −2 −3 c −1 1 e −11 −10 f −3 −4 g −8 11 i 5 −4 j −16 −17 k 1 −5
Exercise 5-03
Exercise 5-04
Exercise 5-05
Exercise 5-06
Exercise 5-07
d 2 −1 h −4 −3 l −9 −2
5 Find the opposite of each of these integers. a −2 b −23 c 56 d −10
e 8
6 Evaluate the following. a 3 + (−4) d 10 + (−13) g −4 − 8 j −3 + 6 m 8 − 6 − (−4) p −7 + 6 + (−7)
c f i l o r
b e h k n q
3 − (−4) −5 − (−9) −5 − (−5) −5 + (−4) + (−2) 1−3+0 4+6+7
f 0
−8 + (−2) 7 + 11 −7 − 8 −8 + 5 − 4 −3 − 14 − 5 3 − 8 + (−6)
7 The minimum overnight temperature in Goulburn was −5°C. During the day, it rose to 11°C. By how much did the temperature rise? 8 Evaluate the following. a 6 × (−3) d −100 × 4 g −6 × (−4) × (−1)
b −8 × (−6) e 7 × (−9) h −2 × 3 × 7
c −2 × 5 f 6×8 i −4 × 2 × (−5)
9 Evaluate the following. a −12 ÷ (−4) d 16 ÷ (−4) g 24 ÷ (−6) ÷ (−2)
b 15 ÷ (−5) e −24 ÷ 12 h −36 ÷ 3 ÷ 4
c −6 ÷ (−6) f −1000 ÷ 10 i −64 ÷ (−4) ÷ 2
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NEW CENTURY MATHS 7
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10 Evaluate the following. a −6 + (−3) × 2 c −12 ÷ 4 − 10 e 30 ÷ (−6) + 5 g −12 + (−3) × (−7) − 2
Exercise 5-08
b d f h
−4 − 3 × 6 −3 × 5 + 6 ÷ (−2) −2 × 7 + 15 ÷ (−3) 14 + 7 × (−3) + 5
11 Here is the test paper done by a Year 7 student. Mark the test out of 10 and correct any incorrect answers. a c e g i
−5 + (−4) = −9 −2 × 3 = 6 5 × (−4) = −20 8 + (−7) + (−2) = −7 [8 × (−9)] + (6 − 7) = −71
b d f h j
−5 + 4 = −9 −2 × (−3) = 6 0 × (−7) = −7 −4 − (−6) = 2 −1 × 2 × (−3) × 4 = 24
12 Draw a number plane with the numbers from 0 to 6 on each axis. Plot these points. A (2, 3)
B (1, 6)
C (3, 0)
Exercise 5-08
D (0, 4)
E (5, 2)
Exercise 5-09
F (6, 5)
13 Give the coordinates of each labelled point on this number plane.
Exercise 5-10
y 4 C H
3 B
2 F
G
1
J −5
−4
−3
A −2 D
−1
−1
1
2
3
4
5 x
−2 −3 −4
E K
−5 I
14 a b c d e
In which quadrant is the point (−3, −3)? Write the coordinates of a point on the y-axis. In which quadrant is a point with a positive x-coordinate and a negative y-coordinate? Write the coordinates of a point in the 2nd quadrant. What are the coordinates of the origin?
Exercise 5-10
15 a Plot the following points on a number plane, joining them with straight lines. (5, 3) → (3, 5) → (−1, 5) → (−3, 3) → (−3, −1) → (−1, −3) → (3, −3) → (5, −1) → (5, 3) b What shape have you drawn?
Exercise 5-10
16 a Plot these points on a number plane, joining them with straight lines. (2, −3) → (−2, −3) → (−2, −2) → (−1, −1) → (−1, 5) → (0, 7) → (1, 5) → (1, −1) → (2, −2) → (2, −3) b What shape have you drawn?
Exercise 5-10
CHAPTER 5 INTEGERS
181
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