For Aotearoa New Zealand Schools
Algebra
Unit 1 Algebraic Expressions
Exercise 1.1 Writing algebraic expressions involving addition and subtraction
Coursebook Recap
Joel has 7 fewer toy robots than Ariki.
If Ariki has x toy robots, Joel has (x – 7) toy robots. If Joel has y toy robots, Ariki has (y + 7) toy robots.
x – 7 and y + 7 are algebraic expressions in terms of x and y.
1. Write an algebraic expression for each of the following.
a) Add 9 and d.
b) Subtract h from 12.
2. Hana has 4 stickers. If her mother gives her a more stickers, how many stickers does she have? Express your answer in terms of a.
3. Mark has 5 more lollies than Ken. If Mark has s lollies, how many lollies does Ken have? Express your answer in terms of s
4. Aria has r ribbons. Tara has 6 more ribbons than Aria. Express the number of ribbons Tara has in terms of r.
5. A tailor used 3 fewer buttons on a skirt than on a shirt. If the tailor used p buttons on the skirt, how many buttons did he use on the shirt? Express your answer in terms of p.
6. Box A is 2 kilograms heavier than box B. If box A has a mass of w kilograms, what is the mass of box B? Express your answer in terms of w kilograms
Exercise 1.2 Finding the value of an algebraic expression involving addition or subtraction
Coursebook Recap
Maata had $p at first. She spent $30.
a) Express the amount of money Maata had left in terms of p Amount of money Maata had left = $(p – 30)
b) If Maata had $55 at first, how much money did she have left? p – 30 = 55 – 30 = 25 She had $25 left.
1. There were 15 oranges in a basket.
a) Kaiah used b oranges to make juice. How many oranges were left in the basket? Express your answer in terms of b.
(15 – b) oranges were left in the basket.
b) If Kaiah used 12 oranges to make juice, how many oranges were left in the basket?
15 – b = 15 – 12 = 3 3 oranges were left in the basket.
2. Find the value of each algebraic expression when x = 5.
Exercise 1.3 Writing and evaluating algebraic expressions involving multiplication
Coursebook Recap
1. There are b bags. There are 2 feijoas in each bag.
a) Express the total number of feijoas in terms of b.
Total number of feijoas = 2 × b = 2b
b) If b = 5, how many feijoas are there altogether?
2b = 2 × b = 2 × 5 = 10
There are 10 feijoas altogether.
2. Tomoe uses 1 5 metre of string to make a knot. If she makes q knots, what is the total length of string Tomoe will use?
q × 1 5 = q 5
Tomoe will use a total of q 5 metres of string.
1. Write an algebraic expression and notation for each of the following.
Multiply
a) e and 6
b) 12 and f
c) g and 1 12
Algebraic expression
× 6
Algebraic notation
2. Cindy made m necklaces. She used 10 beads for each necklace.
a) How many beads did Cindy use altogether? Express your answer in terms of m
Cindy used 10m beads altogether.
b) If Cindy made 9 necklaces, how many beads did she use altogether?
10m = 10 × 9 = 90
She used 90 beads altogether.
3. a) There were w children at a party. Each child was given 1 10 of a cake. How many cakes were given to all the children? Express your answer in terms of w.
w 10 cakes were given to all the children. w 10 = 30 10 = 3
b) If w = 30, how many cakes were given to all the children?
4. Find the value of each algebraic expression when y = 7.
y
5. Find the value of each algebraic expression when z = 20.
3 cakes were given to all the children. = 20 4 = 5 = 20 10 = 2 = 20 20 = 1 = 3 × 7 = 21 = 6 × 7 = 42 = 5 × 7 = 35 = 8 × 7 = 56
Exercise 1.4 Writing and evaluating algebraic expressions involving division
Coursebook Recap
There are 3 boxes. There are x sandwiches altogether. Alex puts an equal number of sandwiches in each box.
a) Express the number of sandwiches in each box in terms of x.
Number of sandwiches in each box = x 3
b) If x = 15, how many sandwiches are there in each box?
x 3 = 15 3 = 5
There are 5 sandwiches in each box.
1. Kelly has 27 kilograms of sugar. She packs all the sugar equally into m bags.
a) What is the mass of sugar in each bag? Express your answer in terms of m.
The mass of sugar in each bag is 27 m kilograms.
b) If m = 3, what is the mass of sugar in each bag?
27 m = 27 3 = 9
The mass of sugar in each bag is 9 kilograms.
2. Find the value of each algebraic expression when p = 6.
Exercise 1.5 Writing and evaluating algebraic expressions involving more than one operation
Coursebook Recap
Noah has some cherries. He puts y cherries each on 4 plates and has 3 cherries left.
a) Express the number of cherries Noah has in terms of y
Number of cherries on the 4 plates = y + y + y + y = 4 × y = 4y
Noah has (4y + 3) cherries.
b) If y = 10, find the number of cherries Noah has.
4y + 3 = 4 × 10 + 3 = 40 + 3 = 43
Noah has 43 cherries.
1. Claire had some money. She spent $d each on 8 similar pens and had $100 left.
a) Express the amount of money Claire had at first in terms of d
Total amount of money spent on 8 pens = 8 × $d = $8d
Amount of money Claire had at first = $(8d + 100)
Claire had $(8d + 100) at first.
b) If d = 2, how much money did Claire have at first?
$(8d + 100) = $(8 × 2 + 100) = $(16 + 100) = $116
Claire had $116 at first.
2. Find the value of each algebraic expression when m = 8.
=
– 4 × 8
Exercise 1.6 Finding the value of an algebraic expression given in exponent notation
Coursebook Recap
If k = 6, find the value of the expression 2k2 .
2k2 = 2 × 62 = 2 × 36 = 72
1. Use the value of the variable to find the value of each algebraic expression. a) p = 4 p2 b) m = 3 m3
= 42 = 16 = 33 = 27
x = 3 5x3 + 2
4 × 52 = 4 × 25 = 100 = 5 × 33 + 2 = 5 × 27 + 2 = 135 + 2 = 137 = 3 × 24 + 5 = 3 × 16 + 5 = 48 + 5 = 53 = 72 – 4 = 49 – 4 = 45
2. A rectangle has a length of 2p centimetres and a width of p centimetres.
a) Express the area of the rectangle in terms of p.
Area of the rectangle = 2p × p = 2p2 square centimetres
The area of the rectangle is 2p2 square centimetres.
b) If p = 4, what is the area of the rectangle? 2p cm p cm
2p2 = 2 × 42 = 2 × 16 = 32
The area of the rectangle is 32 square centimetres.
Exercise 1.7 Simplifying algebraic expressions with like terms
Coursebook Recap
Simplify 3y + 5y – y + 6y.
3y + 5y – y + 6y = 8y – y + 6y = 7y + 6y = 13y
When there are only addition and subtraction operations, we work from left to right.
1. Simplify each expression.
Unit 2 Algebraic Equations
Exercise 2.1 Understanding equations
Coursebook Recap
a) 5 + 3x and 4p − 6 are algebraic expressions.
5 + 3x = 23 and 4p − 6 = 2 are algebraic equations.
The letters x and p in both the algebraic expressions and algebraic equations above are known as variables.
b) A linear equation is when the exponent of the variable is 1.
2p + 7 = 5 and 4 – 3p = 10 are linear equations because the exponent of the variable p is 1.
1. Identify each of the following as algebraic expression or algebraic equation. Tick (✓) the correct answer. Then, write the variable.
Algebraic equation Algebraic expression Variable
a) 6e + 12
b) 5b = 10
c) 9c + 46 19
d) 6x2 = 24
2. For each of the following algebraic equations, tick (✓) if it is a linear equation.
Algebraic equation Linear equation
a) 24 = 3r3
b) 23 + 7n = 9
c) g2 – 1 = 8
d) 4k + 30 = –14
Exercise 2.2 True and false equations
Coursebook Recap
Given the equation x + 4 = 7, is the equation true or false when x = 3?
When x = 3, x + 4 = 3 + 4 = 7
7 is equal to 7, so the equation is true when x = 3. x = 3 is a solution of the equation x + 4 = 7.
1. For each task, show your work clearly. Then, write true or false in the blank.
a) Is the equation y + 11 = 20 true or false when y = 8?
When y = 8, y + 11 = 8 + 11 = 19
19 is not equal to 20.
The equation y + 11 = 20 is when y = 8.
b) Is the equation 3p – 5 = 10 true or false when p = 5?
When p = 5, 3p – 5 = 3 × 5 – 5 = 15 – 5 = 10
10 is equal to 10. false true
The equation 3p – 5 = 10 is when p = 5.
2. Show that e = 7 is a solution of the equation 9e – 54 = 9.
When e = 7, 9e – 54 = 9 × 7 – 54 = 63 – 54 = 9
The equation 9e – 54 = 9 is true when e = 7. So, e = 7 is a solution of the equation 9e – 54 = 9.
Chapter 11: Practice 2.2
Exercise 2.3 Solving algebraic equations using the guess and check method
Coursebook Recap
Solve 4a + 9 = 29.
Guess Check
a = 3 4a + 9 = 4 × 3 + 9 = 21 ✗
a = 5 4a + 9 = 4 × 5 + 9 = 29 ✓ a = 5 is a solution of the equation 4a + 9 = 29.
1. Answer the questions with Yes or No
a) Is x = 8 a solution of 4x – 19 = 13?
b) Is y = 4 a solution of 3 y + 3 = 30?
2. Solve these equations using the guess and check method.
a) p + 31 = 39 b) 43 + 3m = 58
Guess Check p = 8 p + 31 = 8 + 31 = 39 ✓ p = 8 is a solution.
c) 2n 3 = 19 d) 8y + 63 = 79
Guess Check n = 11 2n – 3 = 2 × 11 – 3 = 22 – 3 = 19 ✓ n = 11 is a solution.
Guess Check y = 2 8y + 63 = 8 × 2 + 63 = 16 + 63 = 79 ✓ y = 2 is a solution. Yes No
Guess Check m = 5 43 + 3m = 43 + 3 × 5 = 43 + 15 = 58 ✓ m = 5 is a solution.
Exercise 2.4 Solving algebraic equations using the balance method
Coursebook Recap
Solve 4a + 10 = 46.
We can carry out the same operation on both sides until only a is left on one side.
4a + 10 − 10 = 46 − 10
4a = 36
4a ÷ 4 = 36 ÷ 4 a = 9
1. Solve these equations using the balance method.
b – 14 + 14 = 99 + 14 b = 113 3q – 98 + 98 = 28 + 98 3q = 126 3q ÷ 3 = 126 ÷ 3 q = 42
17 + 3x = 50
Practice 2.4
Exercise 2.5 Reading and plotting points in all four quadrants
Coursebook Recap
1. Look at the coordinate plane below.
The coordinates of a point are always written in the form (x, y).
(x, y) is called an ordered pair of coordinates.
Point S is in the 1st quadrant. The coordinates of point S are (2, 3).
Point T is in the 4th quadrant. The coordinates of point T are (2, –3).
Point T is the reflection of point S across the x-axis.
For Tasks 1(a) to 1(c), refer to the coordinate plane on page 53.
a) Write the coordinates of each point.
A
(3, 5)
(–2, –4)
B C
(–4, 2)
(0, –5) H G D
b) Locate the points of the given pairs of coordinates on the coordinate plane. Then, write the letters representing the given points.
(–1, 0)
c) State the quadrant in which each point lies.
2nd quadrant
3rd quadrant
1st quadrant
4th quadrant
2. Write the coordinates of each point.
(2, –3)
a) Point A is located at (4, 2) on the coordinate plane. After a reflection across the x-axis, point A' is now located at .
(4, –2) (–6, 0)
b) Point B is located at (6, 0) on the coordinate plane. After a reflection across the y-axis, point B' is now located at .
c) Point C is located at (–3, 5) on the coordinate plane. After a reflection across the x-axis and then across the y-axis, point C' is now located at .
(3, –5)
d) Point D is located at (–2, –4) on the coordinate plane. After a reflection across the y-axis and then across the x-axis, point D' is now located at .
(2, 4)
3. a) Plot and label these points on the coordinate plane below.
b) State the quadrant in which each point lies.
2nd quadrant
3rd quadrant
4th quadrant
1st quadrant
c) Join the points P to T in alphabetical order. What letter do you get?
Exercise 2.6 Linear patterns and equations
Coursebook Recap
Look at the pattern. What is the 10th term?
6, 12, 18, 24, 30
+ 6 + 6 + 6 + 6
This is a linear pattern. The constant increase in the pattern is 6.
The nth term is 6 multiplied by the position.
Let x be the position and y be the number.
The equation representing the relationship between y and x is y = 6x.
10th term = 6 × 10 = 60
The 10th term is 60.
1. Look at each linear pattern below. Write increase or decrease in each of the first blank. Then, find the constant increase or decrease.
a) 30, 31, 32, 33, 34
The constant in the pattern is .
b) 20, 18, 16, 14, 12
The constant in the pattern is .
c) 36, 33, 30, 27, 24
The constant in the pattern is .
d) 5, 10, 15, 20, 25
The constant in the pattern is .
2. Look at the pattern below.
5, 6, 7, 8, 9
Then, circle the correct equation for the pattern.
= 4x y = x + 4
3. Look at the pattern in the table below.
a) Complete the table.
b) Write increase or decrease in the first blank. Then, find the constant increase or decrease.
The constant in the pattern is .
c) Write the equation to show the relationship between the number (y) and the position (x).
d) What is the 48th term in the pattern?
4. Look at the pattern below.
a) Draw the figure that comes next in the pattern.
b) Complete the table.
c) Write increase or decrease in the first blank. Then, find the constant increase or decrease.
The constant in the pattern is .
d) Write the equation to show the relationship between the number of squares (y) and the figure number (x).
e) How many squares are there in figure 49? There are squares in figure 49.
f) How many squares are there in figure 78? There are squares in figure 78.
Figure 3
Figure 2
Figure 1
Figure 4
Unit 3 Problem Solving
Exercise 3.1 Word problems involving algebraic expressions
Solve the word problems. Show your work clearly.
1. Understand 2. Plan 3. Answer 4. Check 5. Plus
1. Victoria had b bananas in her basket. She had 3 times as many bananas as Connor had in his basket. Then, Connor ate 2 bananas from his basket. Find the number of bananas Connor had left when b = 27.
Number of bananas Connor had at first = b ÷ 3 = b 3
Number of bananas Connor had left = b 3 – 2
Connor had b 3 – 2 bananas left.
If b = 27, b 3 – 2 = 27 3 – 2 = 9 – 2 = 7
Connor had 7 bananas left.
2. Mrs Brown baked 500 biscuits. She gave x biscuits to her daughter. Then, she put the remaining biscuits equally in 4 jars. Find the number of biscuits in each jar when x = 60.
Total number of biscuits put in the jars = 500 – x
Number of biscuits in each jar = 500 – x 4
There were 500 – x 4 biscuits in each jar.
If x = 60, 500 – x 4 = 500 – 60 4 = 440 4 = 110
There were 110 biscuits in each jar.
3. The lengths of 5 similar wooden planks are each h metres. Another wooden plank is 3 metres long. Find the total length of the 6 wooden planks when h = 2.
Total length of the 5 similar wooden planks = 5 × h = 5h metres
Total length of the 6 wooden planks = (5h + 3) metres
The total length of the 6 wooden planks is (5h + 3) metres.
If h = 2, 5h + 3 = 5 × 2 + 3
= 10 + 3 = 13
The total length of the 6 wooden planks is 13 metres.
4. Rajit has 4 buckets with v litres of water in each bucket. He has another pot with 8 litres of water. He wants the volume of water in the 4 buckets and the pot to be the same. Find the volume of water that should be present in each container if v = 3.
Total volume of water in the 4 buckets and the pot = 4 × v + 8 = (4v + 8)
The total volume of water in the 4 buckets and the pot is (4v + 8) litres.
(4v + 8) ÷ 5 = 4v + 8 5
4v + 8 5 litres of water should be present in each container.
If v = 3, 4v + 8 5 = 4 × 3 + 8 5 = 12 + 8 5 = 20 5 = 4
4 litres of water should be present in each container.
5. Sosaia packed 7 kilograms less than s kilograms of rice equally in 4 bags. Find the mass of rice in each bag when s = 23.
Total amount of rice Sosaia packed = (s – 7)
Amount of rice in 1 bag = s – 7 4
Sosaia packed s – 7 4 kilograms of rice in each bag.
If s = 23, s – 7 4 = 23 – 7 4 = 16 4 = 4
The mass of rice in each bag was 4 kilograms.
6. The total length of a red ribbon and a pink ribbon is 7r metres. The length of the red ribbon is 2 metres less than the length of the pink ribbon. Find the length of the pink ribbon when r = 4. Chapter 11: Practice 3.1
Total length of 2 pink ribbons = Total length of the red ribbon and the pink ribbon + 2 = (7r + 2) metres
Length of the pink ribbon = 7r + 2 2
The length of the pink ribbon is 7r + 2 4 metres.
If r = 4, 7r + 2 2 = 7 × 4 + 2 2 = 28 + 2 2 = 30 2 = 15
The length of the pink ribbon is 15 metres.
Exercise 3.2 Word problems involving algebraic equations
Form an algebraic equation to solve each word problem. Show your work clearly.
1. Understand 2. Plan 3. Answer 4. Check 5. Plus
1. Hemi invited 104 people to his birthday hāngī. p people attended the hāngī and 16 people were not able to attend the hāngī. How many people attended the hāngī?
p + 16 = 104
p + 16 – 16 = 104 – 16
p = 88
88 people attended the hāngī.
2. Carter has two sticks. Stick A is d centimetres long. Stick B is 17 centimetres shorter than stick A. If the length of stick B is 28 centimetres, what is the length of stick A?
d – 17 = 28
d – 17 + 17 = 28 + 17
d = 45
The length of stick A is 45 centimetres.
3. Bella bought 5 boxes of apples. Each box had a apples. 3 of the apples were rotten. If 32 apples were not rotten, how many apples were there in each box?
5a – 3 = 32
5a – 3 + 3 = 32 + 3
5a = 35
5a ÷ 5 = 35 ÷ 5
a = 7
There were 7 apples in each box.
4. Whaea Leah gave y stickers to each student in her class. She had 2 stickers left. There were 25 students in her class. If she had 52 stickers at first, how many stickers did she give to each student?
25y + 2 = 52
25y + 2 – 2 = 52 – 2
25y = 50
25y ÷ 25 = 50 ÷ 25 y = 2
She gave 2 stickers to each student.