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6
5 4567890123 8901234 48901234 1234 PATTERNS AND ALGEBRAMathematicians 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 that describe mathematical and scientific relationships.
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In this chapter you will: Wordbank
• build a geometric pattern, complete a table of values and describe the pattern in words and in algebraic symbols
• use the rule to calculate the corresponding value for a larger number
• apply a given rule expressed in words
• use letters to represent numbers and develop the notion that a letter is used to represent a variable • translate between words and algebraic symbols • recognise and use equivalent algebraic
abbreviations
• 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. • evaluate To find the value of an algebraic
expression after substituting.
• formula A general mathematical rule written using letters and symbols.
• pronumeral Another name for a variable. • substitute To replace a letter (variable) with a
number.
• variable A letter of the alphabet used to stand for a number.
Start up
1 If = 3, what is the value of:
a + 5? b 7 − ? c 2 × ? d 12 + ? e − 1? f × 10? g + ? h 27 ÷ ? i −6 + ? j × (−1)? k −3 ÷ ? l −4 − ? 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 ? i 12 ÷ =−3 What is ? j −4 + = 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? k If = −2, what does × 4 + 3 equal? l If = −1, what does 10 × 2 − equal?
6-01 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
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.
Exercise 6-01
Number of triangles 1 2 3 4 5 6 Number of toothpicks Worksheet 6-01 Brainstarters 6 Worksheet 6-02 Geometric patterns L 1922 Bridge builder: triangles 1TLF
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. 2
a Copy the table above.
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
a Copy the table above.
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 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
a Copy the table above.
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 Bridge builder: quadrilaterals L 1924
TLF
Bridge builder: complex squares
L 1925
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 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. 5
a Copy the table above.
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 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
a Copy the table above.
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 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.
Number of triangles 1 2 3 4 5 6
Number of toothpicks
Number of hexagons 1 2 3 4 5 6
Number of toothpicks
L 1926
Bridge builder: complex pentagons
6-02 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.
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.
Example 1
For the pattern below, 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
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?
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?
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?
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 55 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 5 shapes? c 50 shapes?
6 The number of toothpicks equals 6 times the number of shapes minus 4. How many toothpicks are needed to build:
a 3 shapes? b 12 shapes? c 80 shapes?
7 The number of toothpicks equals the number of shapes squared. How many toothpicks are needed to build:
a 3 shapes? b 50 shapes? c 100 shapes?
8 The number of toothpicks equals 4 times the number of shapes plus 2. How many toothpicks are needed to build:
a 5 shapes? b 10 shapes? c 75 shapes?
9 The number of toothpicks equals 5 times the number of shapes minus 2. How many toothpicks are needed to build:
a 3 shapes? b 10 shapes? c 50 shapes?
10 The number of toothpicks equals the number of shapes squared plus 7. How many toothpicks are needed to build:
a 12 shapes? b 50 shapes? c 80 shapes?
11 The number of toothpicks equals the number of shapes squared minus one. Select A, B,
C or D. The number of toothpicks needed to build 10 shapes is:
A 10 B 20 C99 D9
12 The number of toothpicks equals the number of shapes plus 6. How many shapes can be built with 16 toothpicks?
13 The number of toothpicks equals 4 times the number of shapes plus 1. How many shapes can be built with 33 toothpicks?
14 The number of toothpicks equals the number of shapes squared minus 1. How many shapes can be built with 80 toothpicks?
Exercise 6-02
6-03 The language of algebra
Algebra is the use of letters and symbols to write rules simply and easily.
n = 4 × s + 2 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.
Exercise 6-03
Example 2
1 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
2 Let t = number of paddocks and f = number of fence posts. Rewrite this rule using algebra: the number of fence posts equals 5 times the number of paddocks minus 1. Solution
f = 5 × t − 1
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 A restaurant owner arranges tables as shown below. If T is the number of tables, select
the rule gives the number of chairs, c. Select A, B, C or D.
A c = 2 × T + 4 B c = 10 × T Cc = 3 × T + 4 Dc = 2 × T + 2
4 Using your dictionary, find the meanings of ‘pronumeral’ and ‘variable’.
6-04 Tables of values
1 Complete each table for the given rule.
a y = x − 1 b q = p × 3 c n = m − 2 d y = x + 7
Exercise 6-04
x 3 7 1 8 4 5 p 8 3 2 5 10 4 y 3 q 15 m 6 4 11 5 8 3 x 2 11 7 0 13 9 n 2 y 18Example 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
s 1 2 3 0 12 7 9 6
t 6 7 8 5 17 12 14 11
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 q = (3 × p) − 2 g t = (4 × r) + 1
h z = (5 × y) + 2 i g = h × h
3 Which table of values follows the rule c = ?Select A, B, C or D.
6-05 Finding the rule
a 8 1 5 0 4 2 9 x 10 4 7 5 8 9 6 11 b 10 y 7 p 3 8 2 6 0 4 1 14 t 7 11 2 0 8 12 5 9 r 10 w 16 a 4 5 7 2 0 10 8 6 h 12 8 2 6 0 4 10 16 b 21 k 5 m 3 10 1 4 8 6 2 12 9 p 49 p 4 1 8 5 7 10 2 6 r 2 0 3 9 5 4 1 6 q 16 t 9 y 4 8 1 0 7 5 6 2 h 2 8 5 4 0 3 1 6 z 42 g 25 A h 0 2 4 B h 0 2 4 C h 0 2 4 D h 0 2 4 c 1 3 5 c 1 2 5 c 0 1 3 c 1 2 3 h+2 2
---Example 4
1 What is the rule for this table of values?
Solution
What has been done to the top row number 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.
p 1 3 4 6 7 8
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.
Exercise 6-05
t 2 3 4 6 7 8 9 10 p 1 2 3 4 5 8 10 12 b 0 1 2 4 5 6 7 8 k 5 6 7 8 9 12 14 16 x 4 5 7 8 9 11 15 18 h 0 2 4 5 6 8 9 11 y 1 2 4 5 6 8 12 15 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 k 8 7 4 10 15 20 6 9 b 6 24 27 15 21 30 n 4 3 0 6 11 16 c u 18 4 8 10 20 16 14 2 d x 4 5 6 7 8 9 10 11 w 9 2 4 5 10 8 y 1 2 3 4 5 6 e t 1 2 3 4 5 6 7 8 f e 4 5 6 7 8 9 10 11 b 3 6 9 12 15 18 f 9 10 11 12 13 14 g u 0 1 2 3 4 5 6 7 h e 3 5 1 6 0 7 4 8 v 0 8 16 24 32 40 f 9 15 3 18 0 21 i p 7 5 10 8 2 4 1 9 r 14 12 17 15 9 11 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. m 8 3 2 5 10 4 n 24 9 6 15 30 12 Ex 4
6-06 Finding harder rules
These rules involve multiplication and either addition or subtraction.
1 Find the rule used for each of the following tables, then complete the last two columns of each table.
Exercise 6-06
a f 1 2 3 4 5 6 7 8 b m 1 2 3 4 5 6 7 8 h 1 4 7 10 13 16 p 2 7 12 17 22 27 c m 0 1 2 3 4 5 6 7 d h 3 4 5 6 7 8 9 10 b 3 6 9 12 15 18 k 8 10 12 14 16 18 e r 0 1 2 3 4 5 6 7 f a 2 3 4 5 6 7 8 9 s 1 4 7 10 13 16 b 2 4 6 8 10 12 Worksheet 6-03 Finding the ruleExample 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 themultiplier 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 Circus towers: triangular prisms L 1937
TLF
Ex 5 Circus towers: rectangular prisms L 1938TLF
2 Find the rule used for each of the following tables, then complete the last two columns of each table.
3 Find the rule for the table below. Select A, B, C or D.
A m = 2p + 1 B m = 3p − 1 Cm = 2p + 4 Dm = 3p + 1
6-07 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 m × m = m2 a × a × a × a = a4
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 m y × y n a × a × a o 2 × a × a × a × a
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 a m 0 1 2 3 4 5 6 7 b c 1 2 3 4 5 6 7 8 n 5 8 11 14 17 20 d 11 21 31 41 51 61 c w 1 2 3 4 5 6 7 8 d y 3 4 5 6 7 8 9 10 x 4 9 14 19 24 29 z 0 2 4 6 8 10 e a 1 2 3 4 5 6 7 8 f z 4 5 6 7 8 9 10 11 m 5 9 13 17 21 25 t 1 3 5 7 9 11 g d 0 1 2 3 4 5 6 7 h g 3 1 5 2 10 6 7 4 a 4 7 10 13 16 19 h 4 0 8 2 18 10 i j 8 3 10 1 7 2 5 4 j w 5 1 3 6 4 2 8 10 k 22 7 28 1 19 4 t 49 9 29 59 39 19 p 2 4 8 5 m 5 11 23 14
Exercise 6-07
m 4 ----h 4 --- 6 p ---dw 4 --- 5y 7---3 Karen was asked to write m × m × 3 + 4 in abbreviated form. Which of the following is the correct answer? Select A, B, C or D.
A 3m2 + 4 B 2m3 + 4 C6m + 4 Dm2 + 7
4 For each of the following tables of values, write the rule:
a in words b using algebra.
i m 1 2 3 ii h 2 3 4 5
p 2 4 6 k 0 1 2 3
iii a 2 8 9 10 iv x 0 1 2 3
b 6 24 27 30 y 1 3 5 7
Mental skills 6A
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
---Maths without calculators
Using technology
Algebra and patterns
1 a The following spreadsheet shows a table for the expression x + 2. Copy this table
into a spreadsheet, as shown.
b To enter the rule shown in cell A2, type =B1+2 into cell B2.
c To complete the table, use Fill Right to copy the rule in cell B2 into cells C2 to
K2. Centre your answers in each of the cells.
d Use your spreadsheet to predict the following answers. i When x = 6, the answer is .
ii When x = 17, the answer is .
iii When x = 31, the answer is .
iv What is the value of x when the answer is 96? 2 a Copy this table into a new spreadsheet.
b In cell B2, enter a formula to represent x − 3. Hint:
=B1-c Use Fill Right to =B1-copy this formula into =B1-cells C2 to L2. Centre your answers. d Use your spreadsheet to predict the following answers.
i When x = 9, the answer is .
ii When x = 28, the answer is .
iii What is the value of x when the answer is 37? iv What is the value of x when the answer is 105? 3 a Copy this table into a new spreadsheet.
b In cell B2, enter a formula to represent x ÷ 2.
c Use Fill Right to copy this formula into cells C2 to L2. Centre your answers. d Use your spreadsheet to predict the following answers.
i When x = −10, the answer is .
ii When x = 6, the answer is .
6-08 Finding rules for geometric patterns
4 a Copy this table into a new spreadsheet.
b In cell B2, enter a formula to represent 10 × x − 1.
c Use Fill Right to copy this formula into cells C2 to G2. Centre your answers. d Use your spreadsheet to predict the following answers.
i When x = −7, the answer is .
ii When x = 8, the answer is .
iii What is the value of x when the answer is 199? 5 Open a new spreadsheet.
a In cell A1, enter the label ‘x’. Enter -2 in cell B1, -1 in cell C2, 0 in D1, 1 in E1.
Continue this pattern to cell H1.
b In cell A2, enter the label ‘x ^ 2’ or ‘x × x’ to show x2. In cell B2, enter a formula to represent this rule.
c Fill Right to complete the answers for cells C2 to H2.
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 Write the rule for your completed table in words. c Write the rule as a formula.
d How many tiles would be needed for an arm length of:
i50? ii 100?
Arm length, a 1 2 3 4 5 7 9
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?
Exercise 6-08
Number of cuts, c 1 2 3 4 5 7 9
Number of pieces, p 2 4 Solution
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.
a Arm length, a 1 2 3 4 5 7 9
Number of tiles, t 3 5 7 9 11 15 19
Ex 6
2 cuts
2 T-shapes
a Here are the first two T-shapes. Draw the next three T-shapes.
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 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 The first two rockets are shown below. Draw the next three rockets. 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 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
arm length = 1 arm length = 2
flight shape 1 flight shape 2
Using technology
Number patterns
This activity uses a spreadsheet to record and extend matchsticks patterns. The rules that describe and predict the pattern are entered as formulas.
1 Consider this geometric pattern formed using matchsticks.
a The following spreadsheet shows a table. Copy it into a spreadsheet.
b Using the diagrams above, complete the spreadsheet for cells B2, C2 and D2. c In cell E2, enter a formula to represent the rule for this pattern. Fill Right to
cell F2.
d Extend your spreadsheet (i.e. Fill Right) to answer the following questions. i How many matchsticks are needed to make 9 squares?
ii Find the number of matches needed to make 17 squares.
iii If 52 matchsticks were used, how many squares were made? iv Is it possible to make this square pattern with 66 matchsticks?
2 Consider this geometric pattern formed using matchsticks.
a Open a new spreadsheet and copy the table shown below.
b Using the diagrams above, complete the spreadsheet for cells B2, C2 and D2. c In cell E2, enter a formula to represent the rule for this pattern. Fill Right to
cell F2.
d Extend your spreadsheet (i.e. Fill Right) to answer the following questions. i Find the number of matchsticks needed to make 8 shapes.
ii How many matchsticks are needed to make 20 shapes? iii If 51 matchsticks were used, how many shapes were made?
iv Is it possible to make this pattern with 91 matchsticks?
6-09 Substitution
To complete a table of values, we replace the variable in the formula with a number. Replacing a variable with a number is called algebraicsubstitution. 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’.
Working mathematically
Finding formulas
1 Garden pathsTeresa 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
Reasoning and communicating
Circus towers: square stacks L 1935
TLF
Circus towers: triangular towers L 1936TLF
Circus towers: square towers L 1939TLF
1 a If a = 3, b = 2 and c = 5, find: i 6a ii 4b iii a + c iv 2abc b If a = 4, b = 6 and c = 10, find: i a + b − c ii 2b + c iii c If x = 8, y = 4 and z = 5, find: i xy ii yz − x iii
Exercise 6-09
Example 7
If a = 6, b = 3 and c = 4, find: a a + b b ab − c Solution a a + b = 6 + 3 b ab − c = 6 × 3 − 4 = 9 = 14Example 8
Complete the table shown on the right.
Solution d = 2 × 5 − 5 = 5 d = 2 × 8 − 5 = 11 d = 2 × 4 − 5 = 3, and so on. d = 2a − 5 a 5 8 4 10 12 9 d a 5 8 4 10 12 9 d 5 11 3 15 19 13
Example 9
If r = 6m + 3, evaluate r when m = 4. Solution Substituting 4 for m: r = 6m + 3 = 6 × 4 + 3 = 27 Ex 7 a+b c ---xz y---d If m = 4, n = −2 and p = 3, find:
i m2 ii 3n + p iii mnp
e If a = 6, b = 8 and c = -4, find:
i 2a2 ii ac2 iii
2 Copy and complete each of the following tables using the given rule.
a y = 4x b h = c y = 2x − 3 d c = 3b 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 3 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. x 0 3 10 4 7 6 2 5 d 4 16 10 2 8 14 12 11 y 12 24 h 8 5.5 x 2 7 8 5 10 4 3 6 b 1 4 9 2 7 5 10 3 y 11 c 27 e 12 16 8 4 10 2 6 20 r 0 1 2 3 4 5 6 7 f 3 2.5 s 8 j 3 4 6 8 2 10 5 9 d 8 2 1 4 3 7 0 6 k 8 f 7 m 7 2 6 5 4 10 1 3 v 27 6 12 0 30 15 9 18 n 37 w 10 m 5 2 1 10 4 7 8 11 t 3 1 7 2 5 8 10 6 p 47 v 24 ab c ---Ex 8 d 2 ---e 4 ---v 3 ---Ex 9
6-10 Substitution with negative numbers
Working mathematically
Finding more formulas
Arranging tables
Students set up a coffee shop by arranging square tables and chairs as shown 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
Reasoning and communicating
Example 10
a If a = −3 and b = 4, find: i 5a ii a + b iii 2a − b Solution i 5a = 5 × a ii a + b = −3 + 4 iii 2a − b = 2 × (−3) − 4 = 5 × (−3) = 1 = -6 − 4 = −15 = −10 b If m = 6 and n = −2, find: i m2n ii iii (m − n) ÷ n Solution i m2n = 62× (−2) ii = iii (m − n) ÷ n = [6 − (−2)] ÷ (−2) = 36 × (-2) = −3 = 8 ÷ (−2) = −72 = −4 m n ----m n ---- 6 -2---1 If a = −5, b = −2, evaluate: i 3a ii 4b iii ab
2 If b = 4a − 5, evaluate b when a = −2. 3 If w = 2f + 8, evaluate w when f = −10.
4 If p = 2x + 2y, evaluate p when x = −3 and y = −4.
5 If w = mn − n, evaluate w when m = −1 and n = −5. Select A, B, C or D.
A w = 0 B w = 10 Cw = −10 Dw = −20
6 If a = c2 ÷ b, evaluate a when c = 4 and b = −2. 7 If m = , evaluate m when n = 100 and p = 5.
8 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
9 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 =
Exercise 6-10
x 7 2 5 0 −2 −5 12 10 m 1 −1 −3 0 4 −6 −4 −10 y p a 3 0 1 −4 −2 2 −6 −1 t 4 8 −4 −1 6 −3 −5 1 b 4 u −12 r −3 −2 −1 0 1 2 3 4 n 3 5 0 −2 −4 −1 −5 6 s −4 p 10 12 x 2 5 −1 4 −3 0 1 6 j 1 −1 −5 5 0 3 −2 −7 y -2 4 k 25 v 3 0 8 6 10 −3 7 −1 n 1 −1 −3 3 0 −5 −7 7 w 7 10 mExample 11
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 Worksheet 6-06 Tables of values 2 Worksheet 6-05 Tables of values 1 Ex 10 n p ---Ex 11 n+3 2
---Mental skills 6B
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 = 260b
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 m624 ÷ 8 n 312 ÷ 8 o 256 ÷ 8 p 152 ÷ 8 1 2 ---1 2 ---1 2 ---1 2 ---1 2 ---1 2 ---1 2
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.
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 B = A2 + 1
c y = 3x + 2
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 A 0 1 3 f 9 19 24 99 B 5 26 101 10 001 x 0 1 3 4 y 8 32 62 92 101 3002 a x 3 4 5 6 7 8 b x 4 5 6 7 8 9 y 22 21 20 19 18 17 y 42 40 38 36 34 32 c v 1 2 4 7 10 11 w 55 50 40 25 10 5
Chapter 6 review
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’.
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.
PATTERNS
AND
RULES
Language of algebra Substitution Variable Completing tables Abbreviations Given a rule Rule Table Finding a rule Building patterns Rules Finding a pattern Words Rule or formula Worksheet 6-07 Algebra crosswordChapter revision
1 Here are the first three bridge shapes. Copy and complete the results table below.
2 Suppose that the number of toothpicks equals 2 times the number of shapes minus 1. How many toothpicks are needed to build:
a 4 shapes? b 6 shapes? c 19 shapes?
3 Rewrite these rules as algebraic sentences.
a The number of glasses (g) equals three times the number of chairs (c). b The number of dots (e) equals six times the number of squares (s) plus three. 4 Copy and complete each of these tables.
a n = 5m
b q = p − 7
c d = 3c − 4
5 Find the rule used for each of these tables.
6 Find the rule for this table of values.
7 Write each of the following in abbreviated form.
a 4 × p b a × d c m ÷ n 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 m 1 2 3 4 5 p 2 5 8 11 14 Exercise 6-01
1-lane bridge 2-lane bridge 3-lane bridge
Exercise 6-02 Exercise 6-03 Exercise 6-04 Exercise 6-05 Exercise 6-06 Exercise 6-07 Topic test 6
8 Each new paddock pattern is made by adding fence sections.
a Make a table of values to show the pattern for the number of paddocks and the number of fence sections.
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. 9 A pattern of terrace houses made of sticks is shown below.
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.
Exercise 6-08
1 paddock 2 paddocks 3 paddocks 4 paddocks
Exercise 6-08
1-house
10 a Here are the first four diagrams of a pattern. Draw the next one.
b Copy and complete the table of values.
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?
11 Copy and complete each of the following tables.
a f = g + 5 b B = 3A − 2
c B = 5A + 10 d y = 200 − 5x
12 Copy and complete each of these tables. a j = + 3 b 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 Exercise 6-08 Exercise 6-09 Exercise 6-10 k 2
---Mixed revision 2
1 Which of the following shapes are:a prisms? b pyramids? c solids with curved faces? d polyhedra?
2 a What is the name of the figure on the right? b How many faces does it have?
c How many edges does it have? d What shape is its cross-section? 3 Draw a neat sketch of:
a a regular hexahedron b a trapezoidal prism c an oblique square pyramid d a triangular prism 4 a For the figure on the right, count the number of:
i faces ii vertices iii edges b Does Euler’s rule hold true?
5 For each solid below, name a pair of:
i intersecting edges ii parallel edges iii skew edges
6 Draw what would remain of this large shape if the smaller shape was removed.
7 Show these numbers on a number line: 3, −4, −2, 0, 1, 2, −5 8 Write these numbers in ascending order: 3, −3, −4, 2, 0, −1, 1
Exercise 4-03 F D C A B E G H Exercise 4-04 Exercise 4-05 Exercise 4-06 Exercise 4-07 S P Q W T V U R A E C D B J M N O L K a b c Exercise 4-09 Exercise 5-01 Exercise 5-03
9 For each of these solids, draw the views requested.
a b
i front view ii left view iii top view i top view ii right view iii front view 10 Find the distance between these pairs of numbers on a number line.
a −1 and −3 b −5 and 3 c 0 and −4
11 Simplify the following.
a 3 − 5 b −2 + 8 c −1 − 6 d −3 + 5 − 8
e −5 × (−3) f 6 ÷ (−2) g −27 ÷ (−3) h −4 × 4
i −3 × 2 − 5 j −2 − 3 × 8 k 12 − 14 ÷ 2 + 5 l 5 − [2 + (−2 × 6)] 12 Draw a number plane and mark the following points.
A(1, 3) B(−2, 2) C(3, −4) D(−2, −3) E(0, 7) F(−3, 0)
13
a For the pattern above, how many toothpicks would be needed to build:
i shape 1? ii shape 3? iii shape 5?
b Copy and complete this table.
c Find the rule for the number of toothpicks (n) used to the shape number (s). d Use this rule to find the number of toothpicks needed to build shape 40.
14 If the number of buttons needed is the number of shirts times 8, how many buttons are needed for:
a 5 shirts? b 50 shirts? c 300 shirts?
15 Copy and complete each of these tables for the given rule.
a m = k + 3 b q = p ÷ 3
c y = 2x − 4 d b = 30 − 4a
16 Find the rule used for each of these tables.
a g = ? b n = ? Shape number, s 1 2 3 4 5 Number of toothpicks, n k 2 4 −1 0 −3 1 p 12 9 6 3 0 −3 m q x 5 1 3 −2 0 −4 a 3 5 10 −1 6 0 y b f 3 5 6 8 10 11 m 1 2 3 4 5 6 g 1 3 4 6 8 9 n 5 8 11 14 17 20 Exercise 4-10 right front front Exercise 5-03 Exercise 5-08 Exercise 5-10 Exercise 6-01
shape 1 shape 2 shape 3
Exercise 6-03
Exercise 6-04
