Download Chapter06-Patterns and Rules PDF

TitleChapter06-Patterns and Rules
TagsVariable (Mathematics) Formula Atomic Nucleus Algebra Physics & Mathematics
File Size3.6 MB
Total Pages30
Table of Contents
                            Student text
	Imprint page
	Table of contents
	Preface
	How to use this book
	How to use the CD-ROM
	Acknowledgements
	Syllabus reference grid
	1 The history of numbers
		Different number systems
		The Hindu–Arabic number system
		Place value
		Expanded notation
		The four operations
		Arithmagons
		Dividing by a two-digit number
		Order of operations
		The symbols of mathematics
		Topic overview
		Chapter review
	2 Angles
		Naming angles
		Comparing angle size
		The protractor
		Drawing angles
		Angle geometry
		Naming lines
		Angles and parallel lines
		Finding parallel lines
		Topic overview
		Chapter review
	3 Exploring numbers
		Special number patterns
		Tests for divisibility
		Factors
		Prime and composite numbers
		Prime factors
		Index notation
		Squares, cubes and roots
		Topic overview
		Chapter review
	Mixed revision 1
	4 Solids
		Naming solids
		Convex and non-convex solids
		Polyhedra
		Prisms and pyramids
		Cylinders, cones and spheres
		Classifying solids
		Euler’s rule
		Edges of a solid
		The Platonic solids
		Drawing and building solids
		Different views of solids
		Topic overview
		Chapter review
	5 Integers
		Number lines
		Numbers above and below zero
		Directed numbers
		Ordering directed numbers
		Adding and subtracting integers
		Multiplying integers
		Dividing integers
		The four operations with integers
		Reading a map grid
		The number plane
		The number plane with negative numbers
		Topic overview
		Chapter review
	6 Patterns and rules
		Number rules from geometric patterns
		Using pattern rules
		The language of algebra
		Tables of values
		Finding the rule
		Finding harder rules
		Finding rules for geometric patterns
		Algebraic abbreviations
		Substitution
		Substitution with negative numbers
		Topic overview
		Chapter review
	Mixed revision 2
	7 Decimals
		Place value
		Understanding the point
		Ordering decimals
		Decimals are special fractions
		Adding and subtracting decimals
		Multiplying and dividing by powers of 10
		Multiplying decimals
		Calculating change
		Dividing decimals
		Decimals at work
		Converting common fractions to decimals
		Recurring decimals
		Rounding decimals
		More decimals at work
		Topic overview
		Chapter review
	8 Length and area
		The history of measurement
		The metric system
		Converting units of length
		Reading measurement scales
		The accuracy of measuring instruments
		Estimating and measuring length
		Perimeter
		Area
		Converting units of area
		Area of squares, rectangles and triangles
		Areas of composite shapes
		Measuring large areas
		Topic overview
		Chapter review
	9 Geometric figures
		Polygons
		Classifying triangles
		Naming geometric figures
		Constructing triangles
		Classifying quadrilaterals
		Constructing perpendicular and parallel lines
		Constructing quadrilaterals
		Topic overview
		Chapter review
	Mixed revision 3
	10 Fractions
		Highest common factor and lowest common multiple
		Naming fractions
		Equivalent fractions
		Ordering fractions
		Adding and subtracting fractions
		Adding and subtracting mixed numerals
		Fractions of quantities
		Multiplying fractions
		Dividing fractions
		Topic overview
		Chapter review
	11 Volume, mass and time
		Volume
		Volume of a rectangular prism
		Capacity and liquid measure
		Mass
		Timelines
		Converting units of time
		Time calculations
		World standard times
		Timetables
		Topic overview
		Chapter review
	12 Algebra
		Algebraic expressions
		Algebraic abbreviations
		From words to algebraic expressions
		Like terms
		Multiplying algebraic terms
		Expanding an expression
		Expanding and simplifying
		Algebraic substitution
		Topic overview
		Chapter review
	13 Interpreting graphs and tables
		Picture graphs
		Column graphs and divided bar graphs
		Sector graphs
		Line graphs
		Travel graphs and conversion graphs
		Step graphs
		Reading tables
		Topic overview
		Chapter review
	Mixed revision 4
	General revision
	Answers
	Index
Glossary
	A
	B
	C
	D
	E
	F
	G H
	I J
	K L
	M
	N
	O
	P
	Q
	R
	S
	T
	U V
	W X Y Z
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Document Text Contents
Page 1

PR_skilltest plus_L


Patterns and algebra

Patterns
and rules

6


Mathematicians and scientists try to find patterns in their investigations
and then make rules to describe the patterns. You did a similar thing
with Euler’s rule in Chapter 4.

This chapter introduces you to the language of mathematics called



algebra



. It is used to write the rules which describe mathematical and
scientific relationships.

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■ build a geometric pattern, complete a table of values and describe the
pattern in words and as a formula

■ use the rule to calculate the corresponding value for a larger number
■ translate between words and algebraic symbols
■ understand and use variables, algebraic abbreviations and formulas
■ determine a formula to describe the pattern in a table of values
■ substitute into algebraic expressions
■ generate a number pattern from an algebraic expression.

■ algebra A mathematical language for describing relationships using
letters to stand for numbers.

■ formula A general mathematical rule written using letters and symbols.
■ variable A letter of the alphabet used to stand for a number.
■ pronumeral Another name for a variable.
■ substitute To replace a letter (variable) with a number.
■ evaluate To find the value of an algebraic expression after substituting.

Evan the taxi driver charges different fares for different journeys. For a 5 km
trip, he charges $6.80. For a 10 km trip, it’s $10.30. For a 14 km trip. it’s
$13.10. How much will Evan charge for a 20 km trip? Can you find a
mathematical rule?

In this chapter you will:

Wordbank

Think!

PATTERNS AND RULES 167



CHAPTER 6

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180 NEW CENTURY MATHS 7

1 Find the rule used for each of the following tables, then complete the last two columns
of each table:

2 Find the rule used for each of the following tables, then complete the last two columns
of each table:

a f 1 2 3 4 5 6 7 8

h 1 4 7 10 13 16

b m 1 2 3 4 5 6 7 8

p 2 7 12 17 22 27

c m 0 1 2 3 4 5 6 7

b 3 6 9 12 15 18

d h 3 4 5 6 7 8 9 10

k 8 10 12 14 16 18

e r 0 1 2 3 4 5 6 7

s 1 4 7 10 13 16

f a 2 3 4 5 6 7 8 9

b 2 4 6 8 10 12

a m 0 1 2 3 4 5 6 7

n 5 8 11 14 17 20

b c 1 2 3 4 5 6 7 8

d 11 21 31 41 51 61

c w 1 2 3 4 5 6 7 8

x 4 9 14 19 24 29

d y 3 4 5 6 7 8 9 10

z 0 2 4 6 8 10

e a 1 2 3 4 5 6 7 8

m 5 9 13 17 21 25

Exercise 6-06
Example 5

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PATTERNS AND RULES 181 CHAPTER 6

Finding rules for geometric patterns

3 For practice, use the link to take you to an Excel file that has more of these exercises.

f z 4 5 6 7 8 9 10 11

t 1 3 5 7 9 11

g d 0 1 2 3 4 5 6 7

a 4 7 10 13 16 19

h g 3 1 5 2 10 6 7 4

h 4 0 8 2 18 10

i j 8 3 10 1 7 2 5 4

k 22 7 28 1 19 4

j w 5 1 3 6 4 2 8 10

t 49 9 29 59 39 19

Spreadsheet
6-04

Tables and
rules

Worksheet
6-04

Patterns and
rules

Example 6

arm length 1 arm length 2 arm length 3

a Copy and complete this table about the pattern above.

b In words, write the rule for your completed table.
c Write the rule as a formula.
d How many tiles would be needed for an arm length of

i 50? ii 100?

Solution

Arm length, a 1 2 3 4 5 7 9

Number of tiles, t

a Arm length, a 1 2 3 4 5 7 9

Number of tiles, t 3 5 7 9 11 15 19

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194 NEW CENTURY MATHS 7

Chapter 6 Review Topic test
Chapter 6

1 Bridges.
Here are the first three bridge shapes. Copy and complete the results table below.

2 Copy and complete each of these tables:
a n = 5m

b q = p − 7

c d = 3c − 4

3 Find the rule used for each of these tables:

4 Paddocks.
Each new paddock pattern is made by adding fence sections.

a Make a results table to show the pattern for the number of paddocks and the number
of fence sections.

Lanes on bridge 1 2 3 4 5 7 9 50 100

Number of tiles 5 6 16

m 1 3 12 0 7 4 2 11

n

p 10 7 11 15 20 18 14 30

q

c 2 12 5 3 7 20 9 6

d

a x 12 4 8 7 2 9 b a 3 1 4 6 11 5

y 10 2 6 5 0 7 b 21 7 28 42 77 35

c m 16 8 40 56 36 28 d r 7 2 0 9 4 6

n 4 2 10 14 9 7 t 11 6 4 13 8 10

e g 1 2 3 4 5 6 f p 7 8 9 10 11 12

k 5 8 11 14 17 20 q 4 6 8 10 12 14

Ex 6-01

1-lane bridge 2-lane bridge 3-lane bridge

Ex 6-04

Ex 6-05

Ex 6-01

1 paddock 2 paddocks 3 paddocks 4 paddocks

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PATTERNS AND RULES 195 CHAPTER 6

b In words, write the rule linking the number of paddocks with the number of fence
sections needed to make them.

c Write the rule as a formula.
d Calculate the number of fence sections required to make 100 paddocks.

5 Terrace houses.
A pattern of terrace houses made of sticks is shown:

a Make a table and find the rule connecting the number of houses in the terrace and
the number of sticks.

b Calculate the number of sticks needed to make a terrace of 21 houses.

6 a Here are the first four diagrams of a pattern. Draw the next one.

b Copy and complete the results table:

c Write the rule for the pattern, in words.
d Write the rule as a formula.
e How many triangles would there be if there were:

i 14 dots? ii 50 dots?

7 Copy and complete each of the following tables:
a f = g + 5 b B = 3A − 2

c B = 5A + 10 d y = 200 − 5x

e j = + 3

f q = 4p − 7

Number of dots 2 4 6 8 10

Number of triangles 0 4

g 1 3 5 7 120 A 5 6 7 10 30

f B

A 1 2 3 78 89 x 1 5 10 12 20

B y

k 2 −2 6 −4 −8 12 −10 −20

j

p 3 −1 −3 6 0 −2 1 −5

q

Ex 6-07

1-house
terrace

2-house
terrace

3-house
terrace

4-house
terrace

Ex 6-07

Ex 6-09

k
2
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