Chapter06-Patterns and Rules

6

Patterns and algebra

Patterns and rules

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.

In this chapter you will: ■ ■ ■ ■ ■ ■ ■

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.

Wordbank ■ ■ ■ ■ ■ ■

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.

Think! 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?

PATTERNS AND RULES

167

CHAPTER 6

Start up Worksheet 6-01 Brainstarters 6

= 3, what is the value of: + 5? b 7− ? − 1? f × 10?

1 If a e

2 a + 4 = 10 c 6 × = 36 e + = 20 g 18 − = 11

What is What is What is What is

? ? ? ?

c 2× ? g + ?

d 12 + h 27 ÷

? ?

b −7=5 d ÷5=4 f × = 81 h 33 ÷ = 3

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

If If If If

= 9, what does

+ 7 equal?

= 3, what does 10 − equal? = 7, what does 2 × + 3 equal? = 15, what does ÷ 3 − 4 equal? = 2, what does 9 × 2 + equal?

b If

= 6, what does 5 ×

d f h j

= 12, what does ÷ 4 equal? = 4, what does 3 × + 5 equal? = 5, what does 4 × − 6 equal? = 5, what does 4 × 3 − equal?

If If If If

equal?

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. Worksheet 6-02 Geometric patterns

Exercise 6-01 1

Number of triangles

1

2

3

4

5

6

Number of toothpicks

a b c d e

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

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NEW CENTURY MATHS 7

2

Number of diamonds

1

2

3

4

5

6

Number of toothpicks

a b c d

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

Number of hexagons

1

2

3

4

5

6

Number of toothpicks

a b c d

Copy the table above. Build one hexagon. Write the number of toothpicks used. Repeat the process for two, three, four, five and six hexagons. Write the relationship between the number of hexagons 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 40 hexagons.

4

Number of squares

1

2

3

4

5

6

Number of toothpicks

a b c d

Copy the table above. Build one square. Write the number of toothpicks used. Repeat the process for two, three, four, five and six squares. Write the relationship between the number of squares 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 50 squares.

PATTERNS AND RULES

169

CHAPTER 6

5

Number of triangles

1

2

3

4

5

6

Number of toothpicks

a b c d

Copy the table above. Build one triangle. Write the number of toothpicks used. Repeat the process for two, three, four, five and six triangles. Write the relationship between the number of triangles and the number of toothpicks needed to build them: The number of toothpicks equals … 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

Number of hexagons

1

2

3

4

5

6

Number of toothpicks

a b c d

SkillBuilder 7-02 Number patterns with shapes

Copy the table above. Build one hexagon. Write the number of toothpicks used. Repeat the process for two, three, four, five and six hexagons. Write the relationship between the number of hexagons and the number of toothpicks needed to build them: The number of toothpicks equals … 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.

Just for the record 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.

170

NEW CENTURY MATHS 7

Using technology 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?

INPUT

Spreadsheet 6-01 Number cruncher

OUTPUT

6

13

RULE OUTPUT = 2 × INPUT + 1

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.

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 So 402 toothpicks are needed to build 100 shapes. PATTERNS AND RULES

171

CHAPTER 6

Exercise 6-02 Example 1

1 The number of toothpicks equals 5 times the number of shapes. How many toothpicks are needed to build: a 1 shape? b 3 shapes? c 5 shapes? d 10 shapes? e 25 shapes? f 100 shapes? 2 The number of toothpicks equals 7 times the number of shapes. How many toothpicks are needed to build: a 1 shape? b 3 shapes? c 4 shapes? d 10 shapes? e 30 shapes? f 75 shapes? 3 The number of toothpicks equals 3 times the number of shapes plus 1. How many toothpicks are needed to build: a 1 shape? b 3 shapes? c 5 shapes? d 15 shapes? e 40 shapes? f 100 shapes? 4 The number of toothpicks equals 2 times the number of shapes plus 3. How many toothpicks are needed to build: a 1 shape? b 4 shapes? c 8 shapes? 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 toothpicks are needed to build: a 1 shape? b 3 shapes? c 5 shapes? d 10 shapes? e 50 shapes? f 90 shapes? 6 The number of toothpicks equals 6 times the number of shapes minus 4. How many toothpicks are needed to build: a 1 shape? b 3 shapes? c 5 shapes? d 12 shapes? e 25 shapes? f 80 shapes? 7 The number of toothpicks equals the number of shapes squared. How many toothpicks are needed to build: a 1 shape? b 3 shapes? c 5 shapes? d 20 shapes e 50 shapes? f 100 shapes? 8 The number of toothpicks equals 4 times the number of shapes plus 2. How many toothpicks are needed to build: a 1 shape? b 3 shapes? c 5 shapes? d 10 shapes? e 25 shapes? f 75 shapes? 9 The number of toothpicks equals 5 times the number of shapes minus 2. How many toothpicks are needed to build: a 1 shape? b 3 shapes? c 5 shapes? d 10 shapes? e 50 shapes? f 100 shapes? 10 The number of toothpicks equals the number of shapes squared plus 7. How many toothpicks are needed to build: a 1 shape? b 3 shapes? c 5 shapes? d 12 shapes? e 50 shapes? f 80 shapes?

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NEW CENTURY MATHS 7

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.

Exercise 6-03 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.

Example 2

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. SkillBuilder 8-04 Working with symbols

3 Using your dictionary, find the meanings of ‘number’, ‘numeral’, ‘pronumeral’ and ‘variable’. PATTERNS AND RULES

173

CHAPTER 6

Skillbank 6A Doubling and halving numbers SkillTest 6-01 Doubling and halving

You can double or halve a number by splitting it up first, then doubling or halving. 1 Examine these examples: Think: Double 92 = double 90 + double 2 a 92 × 2 = 180 + 4 = 184 b 37 × 2 Think: Double 37 = double 30 + double 7 = 60 + 14 = 74 1 c --- × 86 Think: Half of 86 = half of 80 + half of 6 2 = 40 + 3 = 43 1 d --- × 244 Think: Half of 244 = half of 240 + half of 4 2 = 120 + 2 = 122 1 e --- × 78 If the tens number is odd, then: 2 Think: Half of 78 = half of 60 + half of 18 = 30 + 9 = 39 1 f --- × 132 Think: Half of 132 = half of 120 + half of 12 2 = 60 + 6 = 66 2 Now simplify these: a 54 × 2 b 77 × 2 c 83 × 2 d 105 × 2 f 41 × 2 g 98 × 2 h 162 × 2 e 26 × 2 i m

1 --2 1 --2

× 182

j

× 506

n

1 --2 1 --2

× 274

k

× 76

o

1 --2 1 --2

× 92

l

× 48

p

1 --2 1 --2

× 138 × 170

Using technology 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. Spreadsheet

A 1

Jail break

2

In

B

C

Out

Rule(s)

3 4

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.)

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NEW CENTURY MATHS 7

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: A

B

C Rule(s)

1

Jail break

2

In

Out

3

4

=2*A3+1

4

Remember: ×=* ÷=/ +=+ −=-

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!

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

Example 3 Complete this table for the given rule. t=s+5 s

1

2

3

0

t

12

7

9

6

17

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

1

2

3

0

12

7

9

6

t

6

7

8

5

17

12

14

11

Exercise 6-04 1 Complete each table for the given rule. a y=x−1 x

3

7

1

8

y

4

Example 3

5

3

b q=p×3 p q

8

3

2

5

10

4

15 PATTERNS AND RULES

175

CHAPTER 6

c n=m−2 m

6

4

n

11

5

8

3

7

0

13

9

5

0

4

2

9

5

8

9

6

11

2

d y=x+7 x

2

y

11 18

e b=2×a a

8

1

b

f

10

y=x−3 x

10

y

7

4

7

2 Complete each of these tables for the given rule. a r=p+2 p

3

r

8

2

6

0

4

1

14

2

0

8

12

5

9

10

b w=2×t t

7

11

w

16

c b=3×a a

4

5

b

7

2

0

10

8

6

6

0

4

10

16

21

d k=h÷2 h

12

8

2

k

5

e p = (5 × m) − 1 m

3

p

f

10

4

8

6

2

12

0

3

2

8

4

10

49

n = (2 × m) + 3 m

1

5

n

176

1

19

NEW CENTURY MATHS 7

9

g q = (3 × p) − 2 p

4

1

8

5

7

10

2

q

6 16

h t = (4 × r) + 1

i

r

2

t

9

3

9

5

4

1

6

8

1

0

7

5

6

2

5

4

0

3

1

6

z = (5 × y) + 2 y

4

z

j

0

42

g=h×h h

2

8

g

25

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

Spreadsheet 6-02 Filling in tables

Finding the rule Example 4 1 What is the rule for this table of values? p

1

3

4

6

7

8

q

0

2

3

5

6

7

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? m

8

3

2

5

10

4

n

24

9

6

15

30

12

Solution

3 × 8 = 24 3×3=9 3×2=6 The rule is n = 3 × m.

The pattern is:

PATTERNS AND RULES

177

CHAPTER 6

Exercise 6-05 Example 4

1 Find the rule used for each of these tables of values: a Rule: b = ? t

2

3

4

6

7

8

9

10

b

0

1

2

4

5

6

7

8

b Rule: k = ? p

1

2

3

4

5

8

10

12

k

5

6

7

8

9

12

14

16

c Rule: y = ? x

4

5

7

8

9

11

15

18

y

1

2

4

5

6

8

12

15

d Rule: j = ? h

0

2

4

5

6

8

9

11

j

0

4

8

10

12

16

18

22

e Rule: p = ? m

0

5

10

15

25

30

40

45

p

0

1

2

3

5

6

8

9

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

b

c

d

e

f

178

a

2

8

9

5

7

10

b

6

24

27

15

21

30

k

8

7

4

10

15

20

n

4

3

0

6

11

16

u

18

4

8

10

20

16

w

9

2

4

5

10

8

x

4

5

6

7

8

9

y

1

2

3

4

5

6

t

1

2

3

4

5

6

b

3

6

9

12

15

18

e

4

5

6

7

8

9

f

9

10

11

12

13

14

NEW CENTURY MATHS 7

4

1

6

9

14

2

10

11

7

8

10

11

g

h

i

u

0

1

2

3

4

5

v

0

8

16

24

32

40

e

3

5

1

6

0

7

f

9

15

3

18

0

21

p

7

5

10

8

2

4

r

14

12

17

15

9

11

6

7

4

8

1

9

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.

Finding harder rules

Spreadsheet 6-03 Rules and graphs Worksheet 6-03 Finding the rule

These rules involve multiplication and either addition or subtraction.

Example 5 1 Find the rule for this table of values: r

1

2

3

4

5

6

7

t

1

3

5

7

9

11

13

2

Solution

2

2

2

2

2

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: d

1

2

3

4

5

6

7

e

8

12

16

20

24

28

32

Solution

4

4

4

4

4

4

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. PATTERNS AND RULES

179

CHAPTER 6

Exercise 6-06 Example 5

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

b

c

d

e

f

f

1

2

3

4

5

6

h

1

4

7

10

13

16

m

1

2

3

4

5

6

p

2

7

12

17

22

27

m

0

1

2

3

4

5

b

3

6

9

12

15

18

h

3

4

5

6

7

8

k

8

10

12

14

16

18

r

0

1

2

3

4

5

s

1

4

7

10

13

16

a

2

3

4

5

6

7

b

2

4

6

8

10

12

7

8

7

8

6

7

9

10

6

7

8

9

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

b

c

d

e

180

m

0

1

2

3

4

5

n

5

8

11

14

17

20

c

1

2

3

4

5

6

d

11

21

31

41

51

61

w

1

2

3

4

5

6

x

4

9

14

19

24

29

y

3

4

5

6

7

8

z

0

2

4

6

8

10

a

1

2

3

4

5

6

m

5

9

13

17

21

25

NEW CENTURY MATHS 7

6

7

7

8

7

8

9

10

7

8

f

g

h

i

j

z

4

5

6

7

8

9

t

1

3

5

7

9

11

d

0

1

2

3

4

5

a

4

7

10

13

16

19

g

3

1

5

2

10

6

h

4

0

8

2

18

10

j

8

3

10

1

7

2

k

22

7

28

1

19

4

w

5

1

3

6

4

2

t

49

9

29

59

39

19

10

11

6

7

7

4

5

4

8

10

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

Finding rules for geometric patterns

Worksheet 6-04 Patterns and rules

Example 6

arm length 1

arm length 2

Spreadsheet 6-04 Tables and rules

arm length 3

a Copy and complete this table about the pattern above. Arm length, a

1

2

3

4

5

7

9

Number of tiles, t

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 a

Arm length, a

1

2

3

4

5

7

9

Number of tiles, t

3

5

7

9

11

15

19

PATTERNS AND RULES

181

CHAPTER 6

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.

Exercise 6-07 Example 6

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.

1 cut

2 cuts

3 cuts

4 cuts

b Copy and complete the results table below: Number of cuts, c

1

2

Number of pieces, p

2

4

3

4

5

7

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.

arm length 1

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NEW CENTURY MATHS 7

arm length 2

9

…

b Copy and complete the results table below. Arm length, a

1

Number of tiles, t

4

2

3

4

5

7

9

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.

flight shape 1

flight shape 2

b Copy and complete the results table below. Number of flight shape, f

1

Number of toothpicks, t

7

2

3

4

5

7

9

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

stage 1

stage 2

b Copy and complete the results table. Number of stage, s

1

2

3

4

5

7

9

Number of toothpicks, t

c Write the rule for the pattern in words. d Write the rule as a formula. e How many toothpicks are needed to build: i stage 60? ii stage 100?

PATTERNS AND RULES

183

CHAPTER 6

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

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:

b 27 × 4

Think:

c 14 × 8

Think:

d 236 ÷ 4

Think:

e 564 ÷ 4

Think:

f 392 ÷ 8

Think:

2 Now simplify these: a 14 × 4 c 16 × 4 e 43 × 8 g 28 × 8 i 184 ÷ 4 k 560 ÷ 4 m 624 ÷ 8 o 256 ÷ 8

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NEW CENTURY MATHS 7

Double twice. Double 65 = 130 Double 130 = 260 Double twice. Double 27 = 54 Double 54 = 108 Double three times. Double 14 = 28 Double 28 = 56 Double 56 = 112 Halve twice. 1 --- × 236 = 118 2 1 --- × 118 = 59 2 Halve twice. 1 --- × 564 = 282 2 1 --- × 282 = 141 2 Halve three times. 1 --- × 392 = 196 2 1 --- × 196 = 98 2 1 --- × 98 = 49 2 b d f h j l n p

27 × 4 105 × 4 16 × 8 33 × 8 272 ÷ 4 432 ÷ 4 312 ÷ 8 152 ÷ 8

Working mathematically 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: Number of tables

1

2

3

Number of chairs

4

6

8

4

5

6

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. PATTERNS AND RULES

185

CHAPTER 6

Using technology 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?

3 matches 1 triangle

Spreadsheet

5 matches 2 triangles

7 matches 3 triangles

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

B

C

1

Number of triangles

Number of matches

Formula

2

1

3

2

4

3

5

4

6

5

7

6

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.

4 matches 1 square

7 matches 2 squares

10 matches 3 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

7 matches 2 squares

186

NEW CENTURY MATHS 7

10 matches 3 squares

13 matches 4 squares

b

4 matches 1 square

12 matches 4 squares

24 matches 9 squares

Algebraic abbreviations When writing rules and formulas there is no need to write ‘×’ and ‘÷’. Instead of 3 × k, we m write 3k. Instead of m ÷ 4, we write ---- . 4 There are other abbreviations commonly used in algebra. p × 4 = 4p m × w = mw 1×h=h

Exercise 6-08 1 Write each of the following in abbreviated form: a 6×m b k÷7 d y×3 e s÷9 g n×1 h 1×d j m×y÷6 k 2×d+7

c f i l

1×p a×b p×q 9−2×k

2 Insert multiplication or division signs in the correct places to write each of these in expanded form: h a 9m b --c 14y 4 6 f f d --e 3k + 2 p dw g ef h ------i y 4 5y j 5am k -----l 16 − 3g 7

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’. PATTERNS AND RULES

187

CHAPTER 6

Example 7 Complete this table: d = 2a − 5 a

5

8

4

10

12

9

d

Solution d=2×5−5=5 d = 2 × 8 − 5 = 11 d = 2 × 4 − 5 = 3, and so on. a

5

8

4

10

12

9

d

5

11

3

15

19

13

Example 8 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.

Exercise 6-09 Example 7

1 Copy and complete each of the following tables using the given rule: a y = 4x x

0

y

3

10

4

7

12

6

2

5

12

11

24

d b h = --2 d

4

h

c

16

10

2

8

14

8

5.5

y = 2x − 3 x

2

y

7

8

5

10

4

3

6

9

2

7

5

10

3

11

d c = 3b b

1

4

c

188

27 NEW CENTURY MATHS 7

e e f = --2

f

e

12

f

6

16

8

4

10

2

6

20

1

2

3

4

5

6

7

s = 12 − r r

0

s

8

g k = 3j − 1 j

3

k

8

4

6

8

2

10

5

9

2

1

4

3

7

0

6

5

4

10

1

3

15

9

18

7

8

11

h f = 2d + 5 d

8

f

i

7

n = 10m − 3 m

7

2

6

n

j

37

v w = --3 v

27

6

12

0

w

k

30 10

p = 6m − 1 m

5

2

1

10

4

p

l

47

v = t2 − 1 t

3

1

v

7

2

5

8

10

6

24

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.

Example 8

PATTERNS AND RULES

189

CHAPTER 6

Worksheet 6-05 Tables of values

Substitution with negative numbers Example 9 Complete this table for y = 3x − 1: x

−2

1

−5

0

−1

7

y

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

2

−7

−16

−1

−4

20

Exercise 6-10 Example 9

1 Copy and complete each of these tables for the given rule: a y = x − 10 x

7

2

5

0

−2

−5

12

10

−1

−3

0

4

−6

−4

−10

0

1

−4

−2

2

−6

−1

−4

−1

6

−3

−5

1

2

3

4

−1

−5

6

y

b p = 3m + 1 m

1

p

c

b=a+4 a

3

b

4

d u=t−9 t

4

8

−12

u

e s = 2r − 6 r

−3

−2

−1

0

−4

s

f

190

1

p=n+7 n

3

5

p

10

12

NEW CENTURY MATHS 7

0

−2

−4

2 Copy and complete each of these tables for the given rule: a y = 2x − 6 x

2

5

y

-2

4

1

−1

−1

4

−3

0

1

6

−5

5

0

3

−2

−7

b k=j×j j k

c

25

w = 10 − v v

3

0

w

7

10

8

6

10

−3

7

−1

−3

3

0

−5

−7

7

n+3 d m = -----------2 n

1

−1

m SkillBuilder 10-02 Completing tables of values

3 If b = 4a − 5, evaluate b if a = −2. 4 If w = 2f + 8, evaluate w if f = −10.

Power plus 1 Arranging classroom tables. Classrooms often have tables shaped like trapeziums.

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

PATTERNS AND RULES

191

CHAPTER 6

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 h

2

f

9

3

10 19

24

1000 99

b y = 3x + 2 x

0

1

y

3

4

8

32

62

45

105

205

26

101

10 001

92

101

c B = 2A + 5 A

0

B

2

10

7

d B = A2 + 1 A

0

1

B

3 5

3 Find the rule for each of these tables: a

b

c

x

3

4

5

6

7

8

y

22

21

20

19

18

17

x

4

5

6

7

8

9

y

42

40

38

36

34

32

v

1

2

4

7

10

11

w

55

50

40

25

10

5

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.

192

NEW CENTURY MATHS 7

3002

Worksheet 6-06 Algebra crossword

Language of maths abbreviation operation substitute

algebra pattern substitution

evaluate pronumeral table of values

formula rule 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’.

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. Substitution

Variable

Completing tables

Abbreviations

Rule Table

PATTERNS AND RULES

Building patterns

Rules Words Finding a rule

Given a rule

Language of algebra

Rule or formula

Finding a pattern

PATTERNS AND RULES

193

CHAPTER 6

Chapter 6 Ex 6-01

Review

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

1-lane bridge

Ex 6-04

Topic test Chapter 6

2-lane bridge

Lanes on bridge

1

2

Number of tiles

5

6

3

4

3-lane bridge

5

7

9

50

100

16

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

1

3

12

0

7

4

2

11

7

11

15

20

18

14

30

12

5

3

7

20

9

6

n

b q=p−7 p

10

q

c

d = 3c − 4 c

2

d Ex 6-05

3 Find the rule used for each of these tables: a

c

e

Ex 6-01

x

12

4

8

7

2

9

y

10

2

6

5

0

7

m

16

8

40

56

36

28

n

4

2

10

14

9

7

g

1

2

3

4

5

6

k

5

8

11

14

17

20

b

d

f

a

3

1

4

6

11

5

b

21

7

28

42

77

35

r

7

2

0

9

4

6

t

11

6

4

13

8

10

p

7

8

9

10

11

12

q

4

6

8

10

12

14

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

1 paddock

2 paddocks

3 paddocks

4 paddocks

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

194

NEW CENTURY MATHS 7

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:

1-house terrace

2-house terrace

Ex 6-07

3-house terrace

4-house terrace

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.

Ex 6-07

b Copy and complete the results table: Number of dots

2

4

Number of triangles

0

4

6

8

10

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: b B = 3A − 2 a f=g+5 g

1

3

5

7

120

A

f

6

7

10

30

5

10

12

20

B

c B = 5A + 10 A

5

Ex 6-09

1

d y = 200 − 5x 2

3

78

89

x

B

1

y

k e j = --- + 3 2 k

2

−2

6

−4

−8

12

−10

−20

−1

−3

6

0

−2

1

−5

j

f

q = 4p − 7 p

3

q PATTERNS AND RULES

195

CHAPTER 6