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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.
05 NCM7 2nd ed SB TXT.fm Page 150 Saturday, June 7, 2008 4:44 PM
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In this chapter you will: Wordbank
• recognise the direction and magnitude of an
integer
• order directed numbers and place them on a
number line
• add and subtract integers
• multiply and divide integers
• apply ‘order of operations’ to expressions involving
integers
• locate and plot points on the number plane
• identify the origin and the four quadrants of the
number plane.
•
ascending
Increasing, from smallest to largest.
•
coordinates
The two values that give the position
of a point on a number plane.
•
descending
Decreasing, from largest to smallest.
•
integer
A positive or negative whole number or
zero.
•
negative number
A number less than zero.
•
number plane
A grid that is made up of two
number lines, meeting at right angles.
•
origin
The centre point of a number plane, with
coordinates (0, 0).
05 NCM7 2nd ed SB TXT.fm Page 151 Saturday, June 7, 2008 4:44 PM
Page 16
165CHAPTER 5 INTEGERS
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.
05 NCM7 2nd ed SB TXT.fm Page 165 Saturday, June 7, 2008 4:44 PM
Page 17
166 NEW CENTURY MATHS 7
5-06 Multiplying integers
Working mathematically
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.
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) = d −5 × (−1) =
e 2 × 5 = f −3 × 3 = g −1 × 4 = h −5 × 5 =
× 5 4 3 2 1 0 −1 −2 −3 −4 −5
5
4
3
2
1
0
−1
−2
−3
−4
−5
Reflecting and reasoning
Skillsheet
5-02
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.
× + −
+ + −
− − +!
05 NCM7 2nd ed SB TXT.fm Page 166 Saturday, June 7, 2008 4:44 PM
Page 31
180 NEW CENTURY MATHS 7
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, , , b 1, 0, −1, , ,
c −6, −8, , , d 2, 0, −2, , ,
e 1, −2, 3, , , f −1, −3, −5, , ,
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
4 Answer true (T) or false (F) for each of these.
a −4 � 2 b −2 � −3 c −1 � 1 d 2 � −1
e −11 � −10 f −3 � −4 g −8 � 11 h −4 � −3
i 5 � −4 j −16 � −17 k 1 � −5 l −9 � −2
5 Find the opposite of each of these integers.
a −2 b −23 c 56 d −10 e 8 f 0
6 Evaluate the following.
a 3 + (−4) b 3 − (−4) c −8 + (−2)
d 10 + (−13) e −5 − (−9) f 7 + 11
g −4 − 8 h −5 − (−5) i −7 − 8
j −3 + 6 k −5 + (−4) + (−2) l −8 + 5 − 4
m 8 − 6 − (−4) n 1 − 3 + 0 o −3 − 14 − 5
p −7 + 6 + (−7) q 4 + 6 + 7 r 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) b −8 × (−6) c −2 × 5
d −100 × 4 e 7 × (−9) f 6 × 8
g −6 × (−4) × (−1) h −2 × 3 × 7 i −4 × 2 × (−5)
9 Evaluate the following.
a −12 ÷ (−4) b 15 ÷ (−5) c −6 ÷ (−6)
d 16 ÷ (−4) e −24 ÷ 12 f −1000 ÷ 10
g 24 ÷ (−6) ÷ (−2) h −36 ÷ 3 ÷ 4 i −64 ÷ (−4) ÷ 2
Exercise 5-01
Exercise 5-01
Exercise 5-02
Exercise 5-03
Exercise 5-03
Exercise 5-04
Exercise 5-05
Exercise 5-06
Exercise 5-07
Topic test 5
05 NCM7 2nd ed SB TXT.fm Page 180 Saturday, June 7, 2008 4:44 PM
Page 32
181CHAPTER 5 INTEGERS
10 Evaluate the following.
a −6 + (−3) × 2 b −4 − 3 × 6
c −12 ÷ 4 − 10 d −3 × 5 + 6 ÷ (−2)
e 30 ÷ (−6) + 5 f −2 × 7 + 15 ÷ (−3)
g −12 + (−3) × (−7) − 2 h 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 −5 + (−4) = −9 b −5 + 4 = −9
c −2 × 3 = 6 d −2 × (−3) = 6
e 5 × (−4) = −20 f 0 × (−7) = −7
g 8 + (−7) + (−2) = −7 h −4 − (−6) = 2
i [8 × (−9)] + (6 − 7) = −71 j −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) D (0, 4) E (5, 2) F (6, 5)
13 Give the coordinates of each labelled point on this number plane.
14 a In which quadrant is the point (−3, −3)?
b Write the coordinates of a point on the y-axis.
c In which quadrant is a point with a positive x-coordinate and a negative y-coordinate?
d Write the coordinates of a point in the 2nd quadrant.
e What are the coordinates of the origin?
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?
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 5-08
Exercise 5-08
Exercise 5-09
1
2
−1 1 2 3 x
y
4
−2
−3
−2−3−4
4
−4
A
−5
−5 5
C
H
−1
3
F
J
D
E
I
K
B
G
Exercise 5-10
Exercise 5-10
Exercise 5-10
Exercise 5-10
05 NCM7 2nd ed SB TXT.fm Page 181 Saturday, June 7, 2008 4:44 PM
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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.
05 NCM7 2nd ed SB TXT.fm Page 150 Saturday, June 7, 2008 4:44 PM
Page 2
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In this chapter you will: Wordbank
• recognise the direction and magnitude of an
integer
• order directed numbers and place them on a
number line
• add and subtract integers
• multiply and divide integers
• apply ‘order of operations’ to expressions involving
integers
• locate and plot points on the number plane
• identify the origin and the four quadrants of the
number plane.
•
ascending
Increasing, from smallest to largest.
•
coordinates
The two values that give the position
of a point on a number plane.
•
descending
Decreasing, from largest to smallest.
•
integer
A positive or negative whole number or
zero.
•
negative number
A number less than zero.
•
number plane
A grid that is made up of two
number lines, meeting at right angles.
•
origin
The centre point of a number plane, with
coordinates (0, 0).
05 NCM7 2nd ed SB TXT.fm Page 151 Saturday, June 7, 2008 4:44 PM
Page 16
165CHAPTER 5 INTEGERS
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.
05 NCM7 2nd ed SB TXT.fm Page 165 Saturday, June 7, 2008 4:44 PM
Page 17
166 NEW CENTURY MATHS 7
5-06 Multiplying integers
Working mathematically
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.
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) = d −5 × (−1) =
e 2 × 5 = f −3 × 3 = g −1 × 4 = h −5 × 5 =
× 5 4 3 2 1 0 −1 −2 −3 −4 −5
5
4
3
2
1
0
−1
−2
−3
−4
−5
Reflecting and reasoning
Skillsheet
5-02
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.
× + −
+ + −
− − +!
05 NCM7 2nd ed SB TXT.fm Page 166 Saturday, June 7, 2008 4:44 PM
Page 31
180 NEW CENTURY MATHS 7
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, , , b 1, 0, −1, , ,
c −6, −8, , , d 2, 0, −2, , ,
e 1, −2, 3, , , f −1, −3, −5, , ,
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
4 Answer true (T) or false (F) for each of these.
a −4 � 2 b −2 � −3 c −1 � 1 d 2 � −1
e −11 � −10 f −3 � −4 g −8 � 11 h −4 � −3
i 5 � −4 j −16 � −17 k 1 � −5 l −9 � −2
5 Find the opposite of each of these integers.
a −2 b −23 c 56 d −10 e 8 f 0
6 Evaluate the following.
a 3 + (−4) b 3 − (−4) c −8 + (−2)
d 10 + (−13) e −5 − (−9) f 7 + 11
g −4 − 8 h −5 − (−5) i −7 − 8
j −3 + 6 k −5 + (−4) + (−2) l −8 + 5 − 4
m 8 − 6 − (−4) n 1 − 3 + 0 o −3 − 14 − 5
p −7 + 6 + (−7) q 4 + 6 + 7 r 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) b −8 × (−6) c −2 × 5
d −100 × 4 e 7 × (−9) f 6 × 8
g −6 × (−4) × (−1) h −2 × 3 × 7 i −4 × 2 × (−5)
9 Evaluate the following.
a −12 ÷ (−4) b 15 ÷ (−5) c −6 ÷ (−6)
d 16 ÷ (−4) e −24 ÷ 12 f −1000 ÷ 10
g 24 ÷ (−6) ÷ (−2) h −36 ÷ 3 ÷ 4 i −64 ÷ (−4) ÷ 2
Exercise 5-01
Exercise 5-01
Exercise 5-02
Exercise 5-03
Exercise 5-03
Exercise 5-04
Exercise 5-05
Exercise 5-06
Exercise 5-07
Topic test 5
05 NCM7 2nd ed SB TXT.fm Page 180 Saturday, June 7, 2008 4:44 PM
Page 32
181CHAPTER 5 INTEGERS
10 Evaluate the following.
a −6 + (−3) × 2 b −4 − 3 × 6
c −12 ÷ 4 − 10 d −3 × 5 + 6 ÷ (−2)
e 30 ÷ (−6) + 5 f −2 × 7 + 15 ÷ (−3)
g −12 + (−3) × (−7) − 2 h 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 −5 + (−4) = −9 b −5 + 4 = −9
c −2 × 3 = 6 d −2 × (−3) = 6
e 5 × (−4) = −20 f 0 × (−7) = −7
g 8 + (−7) + (−2) = −7 h −4 − (−6) = 2
i [8 × (−9)] + (6 − 7) = −71 j −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) D (0, 4) E (5, 2) F (6, 5)
13 Give the coordinates of each labelled point on this number plane.
14 a In which quadrant is the point (−3, −3)?
b Write the coordinates of a point on the y-axis.
c In which quadrant is a point with a positive x-coordinate and a negative y-coordinate?
d Write the coordinates of a point in the 2nd quadrant.
e What are the coordinates of the origin?
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?
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 5-08
Exercise 5-08
Exercise 5-09
1
2
−1 1 2 3 x
y
4
−2
−3
−2−3−4
4
−4
A
−5
−5 5
C
H
−1
3
F
J
D
E
I
K
B
G
Exercise 5-10
Exercise 5-10
Exercise 5-10
Exercise 5-10
05 NCM7 2nd ed SB TXT.fm Page 181 Saturday, June 7, 2008 4:44 PM
