Download Chapter 5 PDF

TitleChapter 5
TagsWater Multiplication Cartesian Coordinate System Elementary Mathematics Mathematical Notation
File Size1.5 MB
Total Pages32
Table of Contents
                            Cover
Imprint
Contents
	Preface
	New Century Maths user’s guide
	Acknowledgements
	Features of this book
	1 The history of numbers
		Start up
		1-01 The ancient Egyptian number system
		1-02 Australian Aboriginal number systems
		1-03 The Babylonian number system
		1-04 The Roman number system
		1-05 The modern Chinese number system
		Working mathematically: Calendar month
		1-06 The Hindu–Arabic number system
		1-07 Expanded notation
		Mental skills 1A: Multiplying by a multiple of 10
		1-08 The four operations
		Using technology: What is a spreadsheet?
		Working mathematically: Double-digit dice game
		Mental skills 1B: Dividing by a multiple of 10
		1-09 Dividing by a two-digit number
		Working mathematically: Magic squares
		Using technology: Sorting data
		1-10 Order of operations
		1-11 The symbols of mathematics
		Working mathematically: The four 4s puzzle Brain bender
		Using technology: Fruit picking
		Power plus
		Chapter 1 review
	2 Angles
		Start up
		2-01 Naming angles
		2-02 Measuring angles
		2-03 Drawing angles
		2-04 Classifying angles
		2-05 Angle relationships
		2-06 Vertically opposite angles
		2-07 Angle geometry
		2-08 Lines in geometry
		Mental skills 2: Changing the order
		2-09 Alternate angles on parallel lines
		2-10 Corresponding angles on parallel lines
		2-11 Co-interior angles on parallel lines
		2-12 Angles on parallel lines
		Using technology: Constructing angles using geometry software
		2-13 Proving lines are parallel
		Power plus
		Chapter 2 review
	Mixed revision 1
	3 Exploring numbers
		3-01 Special number patterns
		Using technology: Developing number patterns
		Start up
		Working mathematically: Figurate numbers More types of numbers
		3-02 Tests for divisibility
		3-03 Factors
		Working mathematically: Factor path puzzle
		Mental skills 3: Calculating differences and making change
		3-04 Prime and composite numbers
		Working mathematically: Goldbach's conjecture
		Working mathematically: More factor paths
		3-05 Prime factors
		3-06 Squares, cubes and roots
		Using technology: Absolute cell referencing
		Power plus
		Chapter 3 review
	4 Solids
		Start up
		4-01 Convex and non-convex solids
		4-02 Nets of solids
		4-03 Polyhedra
		Working mathematically: Stacking and packaging
		4-04 Special solids
		4-05 Classifying solids
		4-06 Euler’s rule
		Mental skills 4: Estimating answers
		4-07 Edges of a solid
		4-08 The Platonic solids
		Working mathematically: Perpendicular, parallel and skew lines
		Working mathematically: Special solids
		4-09 Drawing and building solids
		4-10 Different views of solids
		Using technology: More spreadsheet formulas
		Power plus
		Chapter 4 review
	5 Integers
		Start up
		5-01 Number lines
		5-02 Numbers above and below zero
		5-03 Ordering integers
		5-04 Adding and subtracting integers
		Mental skills 5A: Adding 8 or 9
		5-05 Applying integers
		Using technology: Temperature ranges at Thredbo
		5-06 Multiplying integers
		5-07 Dividing integers
		5-08 The four operations with integers
		Mental skills 5B: Subtracting 8 or 9
		Working mathematically: Multiplying integers
		Using technology: Spreadsheet formulas involving integers
		5-09 The number plane
		5-10 The number plane with negative numbers
		Power plus
		Chapter 5 review
	6 Patterns and rules
		Start up
		6-01 Number rules from geometric patterns
		6-02 Using pattern rules
		6-03 The language of algebra
		6-04 Tables of values
		6-05 Finding the rule
		6-06 Finding harder rules
		6-07 Algebraic abbreviations
		Mental skills 6A: Doubling and halving numbers
		Using technology: Algebra and patterns
		6-08 Finding rules for geometric patterns
		Using technology: Number patterns
		Working mathematically: Finding formulas
		6-09 Substitution
		Working mathematically: Finding more formulas
		6-10 Substitution with negative numbers
		Mental skills 6B: Multiplying and dividing by 4 or 8
		Power plus
		Chapter 6 review
	Mixed revision 2
	7 Decimals
		Start up
		7-01 Place value
		7-02 Understanding the point
		7-03 Ordering decimals
		7-04 Decimals and fractions
		Working mathematically: Decimal distances
		7-05 Adding and subtracting decimals
		Working mathematically: Comparing heights
		Working mathematically: Decimals and powers of 10
		7-06 Multiplying and dividing by powers of 10
		7-07 Using estimation to multiply decimals
		Mental skills 7A: Multiplying an even number by 5
		Working mathematically: Decimal places in multiplication answers
		7-08 Multiplying decimals
		7-09 Calculating change
		Working mathematically: Multiplication maze
		Using technology: Fruit and vegetables
		7-10 Dividing decimals by NS4.3 whole numbers
		7-11 Dividing decimals by decimals
		Mental skills 7B: Multiplying by 9, 11 or 12
		Using technology: Calculations involving decimals
		Working mathematically: Back-to-front problems
		7-12 Fractions and decimals
		7-13 Rounding decimals
		Working mathematically: Rounding prices
		7-14 Applying decimals
		Using technology: Rainfall figures
		Power plus
		Chapter 7 review
	8 Length and area
		Start up
		8-01 The metric system
		Working mathematically: Ancient measures The British Imperial system History of the metric system
		8-02 Converting units of length
		8-03 Reading measurement scales
		8-04 The accuracy of measuring instruments
		Working mathematically: ‘Guess the length’ game
		Mental skills 8: The metric system
		Working mathematically: Measuring long distances
		8-05 Estimating and measuring length
		8-06 Perimeter
		8-07 Area
		Working mathematically: The human lagoon
		8-08 Converting units of area
		Working mathematically: Standing room only! Student space in a classroom
		Working mathematically: Areas of rectangles and triangles
		8-09 Areas of squares, rectangles and triangles
		Using technology: Calculating area
		Using technology: Perimeter and areas
		8-10 Areas of composite shapes
		Working mathematically: Area puzzles
		8-11 Measuring large areas
		Working mathematically: Local areas
		Power plus
		Chapter 8 review
	9 Geometrical figures
		Start up
		9-01 Naming geometrical figures
		9-02 Classifying polygons
		Working mathematically: Logos and designs
		9-03 Classifying triangles
		Mental skills 9: Simplifying multiplication by factorising
		Working mathematically: Building with shapes
		9-04 Constructing triangles
		9-05 Classifying quadrilaterals
		Working mathematically: Shape puzzles
		9-06 Constructing perpendicular and parallel lines
		9-07 Constructing quadrilaterals
		Using technology: More spreadsheet formulas involving integers
		Power plus
		Chapter 9 review
	Mixed revision 3
	10 Fractions and percentages
		Start up
			10-01 Naming fractions
			10-02 Highest common factor and lowest common multiple
			10-03 Equivalent fractions
			10-04 Simplifying fractions
			10-05 Ordering fractions
		Working mathematically: Shaded fractions
			10-06 Adding and subtracting fractions
			10-07 Adding and subtracting mixed numerals
			10-08 Finding a fraction of a quantity
		Mental skills 10: Commonly used fractions and decimals
		Working mathematically: Netball
			10-09 Multiplying fractions
			10-10 Dividing fractions
		Working mathematically: Than’s gorgeous garden
		Working mathematically: Fraction puzzles
			10-11 Percentages and fractions
			10-12 Percentages and decimals
		Using technology: Sorting students’ results
			10-13 Converting percentages
			10-14 Finding a percentage of a quantity
		Using technology: Calculating test marks as percentages
		Power plus
		Chapter 10 review
	11 Volume, mass and time
		Start up
		Working mathematically: Comparing volumes
		11-01 Volume
		Working mathematically: Building a cubic metre Volumes of rectangular prisms
		11-02 Volume of a rectangular prism
		Working mathematically: What is your volume?
		11-03 Capacity
		Working mathematically: Volume by displacement
		Using technology: Comparing volume and capacity of two dams
		11-04 Mass
		Working mathematically: Mass of household objects
		Working mathematically: Investigating mass
		Mental skills 11: 24-hour time
		11-05 Timelines
		Working mathematically: Timeline display
		11-06 Converting units of time
		11-07 Time calculations
		11-08 Standard time
		11-09 Australian standard times
		11-10 Timetables
		Working mathematically: Round trip
		Using technology: Graphing winning times
		Working mathematically: Time puzzlers
		Power plus
		Chapter 11 review
	12 Algebra
		Start up
		12-01 Variables and expressions
		12-02 Algebraic abbreviations
		12-03 From words to algebraic expressions
		12-04 Like terms
		Mental skills 12A: Multiplying by 5, 15, 25 or 50
		12-05 Adding and subtracting like terms
		12-06 Multiplying algebraic terms
		12-07 Expanding expressions
		Working mathematically: Numerical expansions
		12-08 Expanding and simplifying expressions
		Mental skills 12B: Dividing by 5, 15, 25 or 50
		12-09 Substitution
		Using technology: Substitution
		Working mathematically: Generalised arithmetic
		Power plus
		Chapter 12 review
	13 Interpreting graphs and tables
		Start up
		Working mathematically: Collecting graphs and tables
		13-01 Picture graphs
		Column graphs and divided bar graphs
		Using technology: Creating graphs
		13-03 Sector graphs
		Mental skills 13: Reading linear scales
		13-04 Line graphs
		Working mathematically: Temperature graphs
		13-05 Travel graphs and conversion graphs
		13-06 Step graphs
		Working mathematically: Investigating graphs
		13-07 Reading tables
		Power plus
		Chapter 13 review
	Mixed revision 4
	General revision
	Appendix 1: Calculator skills
	Appendix 2: Glossary
	Answers
	Index
                        
Document Text Contents
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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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