Chapter06-Patterns and Rules

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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

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Number rules from geometric patterns

In the following exercise you will be looking at patterns and ﬁnding rules to describe them. You will need toothpicks or matches to build geometric patterns.

1 If = 3, what is the value of:

a + 5? b 7 − ? c 2 × ? d 12 + ? e − 1? f × 10? g + ? h 27 ÷ ? 2 a + 4 = 10 What is ? b − 7 = 5 What is ? c 6 × = 36 What is ? d ÷ 5 = 4 What is ? e + = 20 What is ? f × = 81 What is ? g 18 − = 11 What is ? h 33 ÷ = 3 What is ?

3 In each of these equations, which number can replace the to make one side equal to

the other?

a 6 + 2 = + 4 b 4 + = 7 + 2 c 2 × = 3 × 6 d 3 + 4 = × 1 e 6 + 9 = + 11 f + 3 = 4 + 7 g 3 × 8 = 6 × h 9 × = 6 × 6 i 18 − 2 = 8 + j 4 + = 13 − 5 k 9 × 5 = − 5 l 26 ÷ = 6 + 7

4 a If = 9, what does + 7 equal? b If = 6, what does 5 × equal? c If = 3, what does 10 − equal? d If = 12, what does ÷ 4 equal? e If = 7, what does 2 × + 3 equal? f If = 4, what does 3 × + 5 equal?

g If = 15, what does ÷ 3 − 4 equal? h If = 5, what does 4 × − 6 equal? i If = 2, what does 9 × 2 + equal? j If = 5, what does 4 × 3 − equal?

Start up

Worksheet 6-01

Brainstarters 6

1

a Copy the table above.

b Build one triangle. Write the number of toothpicks used in the table. c Build two triangles. Write the number of toothpicks used in the table. d Repeat the process for three, four, ﬁve and six triangles.

e Write the relationship between the number of triangles and the number of toothpicks

needed to build them. Start with ‘The number of toothpicks equals …’

f Compare your answer to part e with those of others in your class. Write any different answers you ﬁnd.

g Predict how many toothpicks are needed to build 100 triangles. Number of triangles 1 2 3 4 5 6 Number of toothpicks

Exercise 6-01

Worksheet 6-02 Geometric patterns

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b Build one diamond. Write the number of toothpicks used. c Repeat the process for two, three, four, ﬁve and six diamonds.

d Write the relationship between the number of diamonds and the number of toothpicks

needed to build them: The number of toothpicks equals …

e Compare your answer to part d with those of others in your class. Write any different

answers you ﬁnd.

f Predict how many toothpicks are needed to build 80 diamonds.

3

b Build one hexagon. Write the number of toothpicks used. c Repeat the process for two, three, four, ﬁve and six hexagons.

d Write the relationship between the number of hexagons and the number of toothpicks

answers you ﬁnd.

f Predict how many toothpicks are needed to build 40 hexagons.

4

b Build one square. Write the number of toothpicks used. c Repeat the process for two, three, four, ﬁve and six squares.

d Write the relationship between the number of squares and the number of toothpicks

answers you ﬁnd.

f Predict how many toothpicks are needed to build 50 squares.

Number of diamonds 1 2 3 4 5 6 Number of toothpicks Number of hexagons 1 2 3 4 5 6 Number of toothpicks Number of squares 1 2 3 4 5 6 Number of toothpicks

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5

b Build one triangle. Write the number of toothpicks used. c Repeat the process for two, three, four, ﬁve and six triangles.

d Write the relationship between the number of triangles and the number of toothpicks

e Compare your answer to part d with others in your class. Write any different answers. f Predict how many toothpicks are needed to build 100 triangles.

6

b Build one hexagon. Write the number of toothpicks used. c Repeat the process for two, three, four, ﬁve and six hexagons.

d Write the relationship between the number of hexagons and the number of toothpicks

e Compare your answer to part d with others in your class. Write any different answers. f Predict how many toothpicks are needed to build 20 hexagons.

7 Make up your own geometric pattern and draw it. Follow the steps from the previous

questions to ﬁnd how many toothpicks are needed to build 100 of your shapes.

Number of triangles 1 2 3 4 5 6 Number of toothpicks Number of hexagons 1 2 3 4 5 6 Number of toothpicks SkillBuilder 7-02 Number patterns with shapes

Atomic physicist

Maria Goeppert Mayer (1906–1972) shared the Nobel Prize for physics in 1963. She worked on atomic particles and found the pattern of ‘magic numbers’ in the nuclei of atoms. Mayer found that nuclei that have 2, 8, 20, 28, 50, 82, and 126 protons or neutrons are stable. The physical properties of the atoms determine these ‘magic numbers’. Mayer first studied to be a mathematician but later turned to physics and became one of the few women of the time to study atomic physics.

Find where Maria Goeppert Mayer was born and with whom she shared the Nobel prize.

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Using pattern rules

In the previous exercise you discovered a rule for each given pattern. Now you will do the reverse and ﬁnd the pattern from a rule given to you.

Number cruncher

This game is written in Excel and is in the accompanying spreadsheet.

The number cruncher changes the number in its left (green) eye socket into the number in its right (yellow) eye socket using a rule. There is a rule written in the number cruncher’s mouth, but the formula for this rule is really written behind the number in its right eye socket.

Make up a new formula and type it into the right eye socket. (Remember the input number is in D16.) Then erase the rule in the mouth.

Show the number cruncher to a friend and ask if he or she can work out the rule only by changing the numbers in the left eye socket. When your friend has worked out the rule, type it in the number cruncher’s mouth. Take turns to make up rules.

Who can work out the rule in the least number of guesses?

Using technology

INPUT OUTPUT RULE OUTPUT = 2 × INPUT + 1 6 13 Spreadsheet 6-01 Number cruncher

Example 1

For this pattern, the rule is: The number of toothpicks

equals 4 times the number of shapes plus 2.

How many toothpicks are needed to build:

a 1 shape? b 5 shapes? c 100 shapes?

Solution

a Number of toothpicks = 4 ×1+ 2

= 6

So 6 toothpicks are needed to build 1 shape.

b Number of toothpicks = 4 ×5+ 2

= 22

So 22 toothpicks are needed to build 5 shapes.

c Number of toothpicks = 4 ×100+ 2

= 402

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1 The number of toothpicks equals 5 times the number of shapes. How many toothpicks

are needed to build:

d 10 shapes? e 25 shapes? f 100 shapes?

2 The number of toothpicks equals 7 times the number of shapes. How many toothpicks

3 The number of toothpicks equals 3 times the number of shapes plus 1. How many

toothpicks are needed to build:

d 20 shapes? e 55 shape? f 108 shapes?

5 The number of toothpicks equals 10 times the number of shapes minus 5. How many

7 The number of toothpicks equals the number of shapes squared. How many toothpicks

d 20 shapes e 50 shapes? f 100 shapes?

10 The number of toothpicks equals the number of shapes squared plus 7. How many

Exercise 6-02

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The language of algebra

Algebra is the use of letters and symbols to write rules simply and easily.

Example 2

Let n stand for the number of toothpicks and s stand for the number of shapes. Use algebra to rewrite the rule which states that the number of toothpicks equals 4 times the number of

shapes plus 2.

Solution

n = 4 × s + 2

This is called an algebraic rule or formula. It has an equals sign in it. The letters n and s are called variables because their values can vary. Variables are also called pronumerals because they stand in place of numerals.

1 Use n to stand for the number of toothpicks and s to stand for the number of shapes.

Rewrite each of these rules as an algebraic sentence.

a The number of toothpicks equals ﬁve times the number of shapes. b The number of toothpicks equals seven times the number of shapes.

c The number of toothpicks equals three times the number of shapes plus one. d The number of toothpicks equals four times the number of shapes plus two. e The number of toothpicks equals two times the number of shapes plus three. f The number of toothpicks equals ten times the number of shapes minus ﬁve.

2 Choose your own letters and rewrite each of

these rules as an algebraic sentence.

a The number of tiles equals three times the

number of shapes.

b The number of apples equals four times

the number of apple trees.

c The number of toothpicks equals six

times the number of shapes.

d The number of paving stones equals three

times the number of metres plus three.

e The number of serves equals two times

the number of strawberries plus seven.

f The number of toothpicks equals ﬁve times the number of shapes minus three.

g The number of toothpicks equals six

times the number of shapes minus four.

h The number of dots equals the number of

shapes squared.

i The number of buttons equals four times the number of shirts plus two.

j The number of toothpicks equals ﬁve times the number of shapes minus two.

3 Using your dictionary, ﬁnd the meanings of ‘number’, ‘numeral’, ‘pronumeral’ and

‘variable’.

Exercise 6-03

Example 2 SkillBuilder 8-04 Working with symbols

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Doubling and halving numbers

You can double or halve a number by splitting it up ﬁrst, then doubling or halving.

1 Examine these examples:

a 92 × 2 Think: Double 92= double 90 + double 2 = 180 + 4

= 184

b 37 × 2 Think: Double 37= double 30 + double 7 = 60 + 14

= 74

c × 86 Think: Half of 86= half of 80 + half of 6 = 40 + 3

= 43

d × 244 Think: Half of 244= half of 240 + half of 4 = 120 + 2

= 122

e × 78 If the tens number is odd, then:

Think: Half of 78= half of 60 + half of 18 = 30 + 9

= 39

f × 132 Think: Half of 132= half of 120 + half of 12 = 60 + 6

= 66

2 Now simplify these:

a 54 × 2 b 77 × 2 c 83 × 2 d 105 × 2 e 26 × 2 f 41 × 2 g 98 × 2 h 162 × 2 i × 182 j × 274 k × 92 l × 138 m × 506 n × 76 o × 48 p × 170 1 2 ---1 2 ---1 2 ---1 2 ---1 2 --- 1 2 --- 1 2 --- 1 2 ---1 2 --- 1 2 --- 1 2 --- 1 2

---Skillbank 6A

SkillTest 6-01 Doubling and halving

Jailbreak

Jailbreak is a mathematical game which involves algebra. The game is played in pairs. The aim of the game is for both you and your partner to crack the cell code (ﬁnd the correct rule) in order to escape jail.

How to play the game

Step 1: Open your spreadsheet application. Copy the headings below into the spreadsheet.

Step 2: Without letting your partner see what you are doing, enter a number in cell A3.

(Try using decimals when you are conﬁdent with how the game operates.)

A B C 1 Jail break 2 In Out Rule(s) 3 4

Using technology

Spreadsheet

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Tables of values

Algebra can be used to complete tables. Remember any letters can be used as pronumerals.

Step 3: In cell B3, enter a formula which uses cell A3. You can use up to three

mathematical operations in your formula. For example:

Step 4: Your partner uses the keyboard and must now try to work out your formula by

listing all the possible rules in column C. The time allowed for working out the rules is 2 minutes.

Step 5: After the time has expired, check all the possible formulas recorded in column C

against the formula locked in at cell B3. If your partner matches the formula to that in cell B3, he or she will receive 5 points. Your partner also receives 1 point for each other correct formula.

Step 6: The winner of the game is the ﬁrst person to record 50 points.

Be the ﬁrst player to gain 50 points and you will have mastered cracking coded cells!

A B C 1 Jail break 2 In Out Rule(s) 3 4 =2*A3+1 4 Remember: × = * ÷ = / + = + − =

-Example 3

Complete this table for the given rule.

t = s + 5

Solution

To ﬁnd t each time, add 5 to the number given for s.

s 1 2 3 0 12 7 9 6

t 17

s 1 2 3 0 12 7 9 6

t 6 7 8 5 17 12 14 11

1 Complete each table for the given rule. a y = x − 1 b q = p × 3 x 3 7 1 8 4 5 y 3 p 8 3 2 5 10 4 q 15

Exercise 6-04

Example 3

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c n = m − 2

d y = x + 7

e b = 2 × a

f y = x − 3

2 Complete each of these tables for the given rule. a r = p + 2 b w = 2 × t c b = 3 × a d k = h ÷ 2 e p = (5 × m) − 1 f n = (2 × m) + 3 m 6 4 11 5 8 3 n 2 x 2 11 7 0 13 9 y 18 a 8 1 5 0 4 2 9 b 10 x 10 4 7 5 8 9 6 11 y 7 p 3 8 2 6 0 4 1 14 r 10 t 7 11 2 0 8 12 5 9 w 16 a 4 5 7 2 0 10 8 6 b 21 h 12 8 2 6 0 4 10 16 k 5 m 3 10 1 4 8 6 2 12 9 p 49 m 1 5 0 3 2 8 4 10 n 19

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Finding the rule

g q = (3 × p) − 2

h t = (4 × r) + 1

i z = (5 × y) + 2

j g = h × h

3 These exercises can also be done using a spreadsheet. Use the link to take you to the

accompanying Excel spreadsheet.

p 4 1 8 5 7 10 2 6 q 16 r 2 0 3 9 5 4 1 6 t 9 y 4 8 1 0 7 5 6 2 z 42 h 2 8 5 4 0 3 1 6 g 25 Spreadsheet 6-02 Filling in tables

Example 4

1 What is the rule for this table of values?

Solution

What has been done to the number in the top row to get the number in the bottom row? The pattern is: 1 − 1= 0

3 − 1= 2

4 − 1= 3, and so on. The rule is q = p − 1.

2 What is the rule for this table of values?

Solution

The pattern is: 3 × 8= 24 3 × 3= 9 3 × 2= 6 The rule is n = 3 × m. p 1 3 4 6 7 8 q 0 2 3 5 6 7 m 8 3 2 5 10 4 n 24 9 6 15 30 12

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1 Find the rule used for each of these tables of values: a Rule: b = ? b Rule: k = ? c Rule: y = ? d Rule: j = ? e Rule: p = ?

2 Find the rule used for each of the following tables, then complete the tables:

t 2 3 4 6 7 8 9 10 b 0 1 2 4 5 6 7 8 p 1 2 3 4 5 8 10 12 k 5 6 7 8 9 12 14 16 x 4 5 7 8 9 11 15 18 y 1 2 4 5 6 8 12 15 h 0 2 4 5 6 8 9 11 j 0 4 8 10 12 16 18 22 m 0 5 10 15 25 30 40 45 p 0 1 2 3 5 6 8 9 a a 2 8 9 5 7 10 4 1 b 6 24 27 15 21 30 b k 8 7 4 10 15 20 6 9 n 4 3 0 6 11 16 c u 18 4 8 10 20 16 14 2 w 9 2 4 5 10 8 d x 4 5 6 7 8 9 10 11 y 1 2 3 4 5 6 e t 1 2 3 4 5 6 7 8 b 3 6 9 12 15 18 f e 4 5 6 7 8 9 10 11 f 9 10 11 12 13 14

Exercise 6-05

Example 4

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Finding harder rules

These rules involve multiplication and either addition or subtraction.

3 Number patterns can also be shown as graphs. Use the link to take you to a technology

activity in which you can graph the tables in Question 2.

g u 0 1 2 3 4 5 6 7 v 0 8 16 24 32 40 h e 3 5 1 6 0 7 4 8 f 9 15 3 18 0 21 i p 7 5 10 8 2 4 1 9 r 14 12 17 15 9 11 Spreadsheet 6-03 Rules and graphs Worksheet 6-03 Finding the rule

Example 5

1 Find the rule for this table of values:

Solution

If the values in the top row are consecutive (increase by 1 each time), the bottom row helps us ﬁnd the multiplier.

The bottom row values go up by 2 each time, so the multiplier is 2. This means the formula must have 2 × r in it.

2 × 1 − 1= 1 2 × 2 − 1= 3

2 × 3 − 1= 5, and so on. The rule is t = 2 × r − 1.

2 Find the rule for this table of values:

Solution

The bottom row goes up by 4, so the multiplier is 4. 4 × 1 + 4= 8 4 × 2 + 4= 12 4 × 3+ 4= 16, and so on. The rule is e = 4 × d + 4. r 1 2 3 4 5 6 7 t 1 3 5 7 9 11 13 d 1 2 3 4 5 6 7 e 8 12 16 20 24 28 32 2 2 2 2 2 2 4 4 4 4 4 4

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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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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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b The number of tiles increases by 2 each time the arm length increases. This means the

multiplier is 2. In words, the rule is:

The number of tiles is always 2 times the arm length plus 1. c The rule is:

t = 2 × a + 1

(Check that it works.)

d i t = 2 × 50 + 1 ii t = 2 × 100 + 1

= 101 = 201

So 101 tiles are needed for So 201 tiles are needed for an arm length of 50. an arm length of 100.

1 Cutting the cake.

You have the job of cutting up a birthday cake by making cuts straight through the middle of the cake.

a Draw six circles to represent the cake. Make one cut in one cake, two cuts in the next,

and so on.

b Copy and complete the results table below:

c Write the rule for the pattern in words. d Write the rule as a formula.

e How many pieces would there be if there were:

i 50 cuts? ii 100 cuts?

f How realistic is the rule? Is it possible to make 100 cuts?

2 T-shapes.

a Here are the ﬁrst two T-shapes. Draw the next three T-shapes.

Number of cuts, c 1 2 3 4 5 7 9

Number of pieces, p 2 4

2 cuts

1 cut 3 cuts 4 cuts …

arm length 1 arm length 2

Exercise 6-07

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b Copy and complete the results table below.

c

Write the rule for the pattern in words.

d Write the rule as a formula.

e How many tiles are needed to build a T-shape with arm length:

i 40? ii 70?

3 Flight shapes.

a Here are the ﬁrst two ﬂight shapes, made from toothpicks. Draw the next three.

b Copy and complete the results table below.

c

e How many toothpicks are needed to build:

i ﬂight shape 40? ii ﬂight shape 100? 4 Rockets.

a Here are the ﬁrst two rockets. Draw the next three rockets.

b Copy and complete the results table.

c

e How many toothpicks are needed to build:

i stage 60? ii stage 100?

Arm length, a 1 2 3 4 5 7 9

Number of tiles, t 4

Number of ﬂight shape, f 1 2 3 4 5 7 9

Number of toothpicks, t 7

Number of stage, s 1 2 3 4 5 7 9

Number of toothpicks, t

flight shape 1 flight shape 2

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Multiplying and dividing by 4 or 8

Multiplying and dividing by 4 or 8 involves repeated doubling or halving.

1 Examine these examples:

a

65 × 4 Think: Double twice. Double 65 = 130 Double 130 = 260

b

27 × 4 Think: Double twice. Double 27 = 54 Double 54 = 108

c

14 × 8 Think: Double three times. Double 14 = 28 Double 28 = 56 Double 56 = 112

d

236 ÷ 4 Think: Halve twice. × 236 = 118 × 118 = 59

e

564 ÷ 4 Think: Halve twice. × 564 = 282 × 282 = 141

f

392 ÷ 8 Think: Halve three times. × 392 = 196 × 196 = 98 × 98 = 49

2 Now simplify these:

a 14 × 4 b 27 × 4 c 16 × 4 d 105 × 4 e 43 × 8 f 16 × 8 g 28 × 8 h 33 × 8 i 184 ÷ 4 j 272 ÷ 4 k 560 ÷ 4 l 432 ÷ 4 m 624 ÷ 8 n 312 ÷ 8 o 256 ÷ 8 p 152 ÷ 8 1 2 ---1 2 ---1 2 ---1 2 ---1 2 ---1 2 ---1 2

---Skillbank 6B

SkillTest 6-02 Multiplying and dividing by 2, 4 or 8

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Reasoning and communicating: Finding formulas

1 Garden paths.

Teresa wanted to build a garden path. She found that she needed 11 large paving stones and 26 medium stones for each square metre of pathway.

a What is the rule linking the number of square metres of pathway and: i the number of large paving stones?

ii the number of medium paving stones?

b Help Teresa calculate her order for a pathway of 17 square metres. 2 Tree planting.

A farmer decides to plant trees along the fence line of her property. The trees cost $6.50 each and are planted 2.5 metres apart.

a Find the cost of buying enough trees for each of these lengths of fence line:

i 10 m ii 20 m iii 30 m

b i Make a table linking the number of 10-metre sections of fence line and the

cost (in dollars) of the trees.

ii What is the rule for predicting the cost of the trees?

c How much will it cost to plant trees along each of these lengths of fence line?

i 130 m ii 200 m iii 360 m iv 165 m

3 Arranging tables.

Students set up a coffee shop by arranging square tables and chairs as shown in the diagram below.

a Copy and complete this table:

b What are the next three terms of the pattern? Extend your table. c How many chairs can be placed around 10 tables?

d Write a formula to describe the number of chairs that can be placed around

various numbers of tables.

e Find the rule for arranging rectangular tables like those in the diagram below.

f What would happen if the same rectangular tables were used, but arranged so that the long sides were together? Find the rule and write it as a formula.

Number of tables 1 2 3 4 5 6

Number of chairs 4 6 8

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Matchstick patterns

Your task is to investigate matchstick patterns and ﬁnd the rules which describe and predict the pattern.

Triangle patterns

1 Consider the matchstick pattern below. Can you see a pattern relating the number of

matchsticks to the number of triangles produced?

2 Set up a spreadsheet similar to the one below to help you with your investigation.

3 Investigate a spreadsheet formula that will let you complete column B and hence

generalise a formula for the number of matches used for each number of triangles.

Square patterns

1 Consider the number of matches required to make this pattern of squares.

2 Set up a spreadsheet with headings ‘Number of squares’ and ‘Number of matches’. 3 Investigate a spreadsheet formula to ﬁnd the pattern and hence generalise a formula for

the number of matches used for each number of squares.

4 Try working out the rules for the two patterns that follow.

A B C

1 Number of triangles Number of matches Formula

2 1 3 2 4 3 5 4 6 5 7 6 3 matches 1 triangle 7 matches 3 triangles 5 matches 2 triangles 4 matches 1 square 7 matches 2 squares 10 matches 3 squares 7 matches 2 squares 10 matches 3 squares 13 matches 4 squares a

Using technology

Spreadsheet

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Algebraic abbreviations

When writing rules and formulas there is no need to write ‘×’ and ‘÷’. Instead of 3 × k, we write 3k. Instead of m ÷ 4, we write .

There are other abbreviations commonly used in algebra.

1 × h = h p × 4 = 4p m × w = mw

Substitution

We will now use algebraic abbreviations in our formulas for tables of values. To complete a table of values, we replace the variable in the formula with a number.

Replacing a variable with a number is called algebraic substitution. To ‘substitute’ means to ‘swap’ or to ‘serve in place of’.

After substitution the value of the formula can be worked out. This is called evaluation. To ‘evaluate’ means to ‘ﬁnd the value of’.

4 matches 1 square 12 matches 4 squares 24 matches 9 squares b m 4

----1 Write each of the following in abbreviated form:

a 6 × m b k ÷ 7 c 1 × p

d y × 3 e s ÷ 9 f a × b

g n × 1 h 1 × d i p × q

j m × y ÷ 6 k 2 × d + 7 l 9 − 2 × k

2 Insert multiplication or division signs in the correct places to write each of these in

expanded form: a 9m b c 14y d e 3k + 2 f f g ef h i y j 5am k l 16 − 3g h 4 ---6 p ---dw 4 ---5y 7

---Exercise 6-08

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Example 7

Complete this table:

d = 2a − 5

Solution

d =2 × 5 − 5= 5 d =2 × 8 − 5= 11 d =2 × 4 − 5= 3, and so on. If r = 6m + 3, evaluate r when m = 4.

Solution

Substituting 4 for m: r = 6m + 3 =6 × 4 + 3 = 24 + 3 = 27 So r = 27. a 5 8 4 10 12 9 d a 5 8 4 10 12 9 d 5 11 3 15 19 13

Example 8

1 Copy and complete each of the following tables using the given rule: a y = 4x b h = c y = 2x − 3 d c = 3b x 0 3 10 4 7 6 2 5 y 12 24 d 4 16 10 2 8 14 12 11 h 8 5.5 x 2 7 8 5 10 4 3 6 y 11 b 1 4 9 2 7 5 10 3 c 27 d 2

---Exercise 6-09

Example 7

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e f = f s = 12 − r g k = 3j − 1 h f = 2d + 5 i n = 10m − 3 j w = k p = 6m − 1 l v = t2_{− 1} Remember: t2_{= t × t}

2 Find the required values for each of the following. a If d = 4c − 10, find d when: i c = 5 ii c = 8 iii c = 10 b If b = 3t − 1, find b when: i t = 5 ii t = 12 iii t = 20 c If z = x − 7, evaluate z when: i x = 9 ii x = 15 iii x = 22 d If u = 4a + 1, evaluate u when a = 2. e If p = 2h − 5, find p when h = 3. f If k = 9j, find k when j = 4. e 12 16 8 4 10 2 6 20 f 6 r 0 1 2 3 4 5 6 7 s 8 j 3 4 6 8 2 10 5 9 k 8 d 8 2 1 4 3 7 0 6 f 7 m 7 2 6 5 4 10 1 3 n 37 v 27 6 12 0 30 15 9 18 w 10 m 5 2 1 10 4 7 8 11 p 47 t 3 1 7 2 5 8 10 6 v 24 e 2 ---v 3 ---Example 8

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Substitution with negative numbers

Worksheet 6-05

Tables

of values

Example 9

Complete this table for y = 3x − 1:

Solution

y = 3 × 1 − 1= 2 y = 3 × (−2) − 1= −7 y = 3 × (−5) − 1= −16, and so on. x 1 −2 −5 0 −1 7 y x 1 −2 −5 0 −1 7 y 2 −7 −16 −1 −4 20

1 Copy and complete each of these tables for the given rule: a y = x − 10 b p = 3m + 1 c b = a + 4 d u = t − 9 e s = 2r − 6 f p = n + 7 x 7 2 5 0 −2 −5 12 10 y m 1 −1 −3 0 4 −6 −4 −10 p a 3 0 1 −4 −2 2 −6 −1 b 4 t 4 8 −4 −1 6 −3 −5 1 u −12 r −3 −2 −1 0 1 2 3 4 s −4 n 3 5 0 −2 −4 −1 −5 6 p 10 12

Exercise 6-10

Example 9

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2 Copy and complete each of these tables for the given rule: a y = 2x − 6 b k = j × j c w = 10 − v d m = 3 If b = 4a − 5, evaluate b if a = −2. 4 If w = 2f + 8, evaluate w if f = −10. x 2 5 −1 4 −3 0 1 6 y -2 4 j 1 −1 −5 5 0 3 −2 −7 k 25 v 3 0 8 6 10 −3 7 −1 w 7 10 n 1 −1 −3 3 0 −5 −7 7 m n+3 2

---1 Arranging classroom tables.

Classrooms often have tables shaped like trapeziums.

Here are some examples of the different ways the tables can be arranged:

Power plus

SkillBuilder 10-02

Completing tables of values

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a Trace six copies of the table and carefully cut them out. b Draw at least three other possible arrangements for the tables. c One teacher set up the tables in straight lines like this:

i Draw a table of results showing the link between the number of tables in a row and

the number of chairs needed.

ii Write the rule in words and as a formula.

d Use your rule from part c to ﬁnd the maximum number of students that could be

seated at:

i 8 tables ii 15 tables iii 19 tables 2 Copy and complete each of the following tables for the given rule:

a f = 5h − 1

b y = 3x + 2

c B = 2A + 5

d B = A2_{+ 1}

3 Find the rule for each of these tables:

4 a Find 10 different formulas from your other subjects. b Write each formula and explain what it is used for.

c Prepare a short talk on three or four of the formulas you found.

h 2 3 10 1000 f 9 19 24 99 x 0 1 3 4 y 8 32 62 92 101 3002 A 0 2 10 B 7 45 105 205 A 0 1 3 B 5 26 101 10 001 a x 3 4 5 6 7 8 y 22 21 20 19 18 17 b x 4 5 6 7 8 9 y 42 40 38 36 34 32 c v 1 2 4 7 10 11 w 55 50 40 25 10 5

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Topic overview

• Write in your own words what you have learnt about patterns and rules, and algebra. • What parts of this topic did you like?

• What parts of this topic didn’t you understand? Discuss them with a friend or with the teacher.

• Give examples of occupations where algebra is used.

• Copy the summary of this topic given below into your workbook. Use bright colours to mark key words.

Language of maths

abbreviation algebra evaluate formula

operation pattern pronumeral rule

substitute substitution table of values variable 1 Which of the listed words means:

a the mathematical notation for writing a rule?

b to ﬁnd the value of an algebraic expression after substitution? c a shorter way of writing something?

2 Give an example of where the word ‘substitute’ is used outside of mathematics. 3 What is the meaning of ‘pronumeral’?

4 Why does the word ‘variable’ have that name?

5 Find two non-mathematical meanings of the word ‘formula’.

Worksheet 6-06 Algebra crossword Substitution Variable Completing tables Abbreviations Given a rule

PATTERNS

AND

RULES

Rule Table Finding a rule Building patterns Rules Finding a pattern Words Rule or formula Language of algebra

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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

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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