100% coverage of New Zealand Mathematics and Statistics Curriculum for Phases 1-3
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1. Find the value of each of the following.
a) 3 + 9 – 7
b) 4 × 5 ÷ 2
c) 3 + 6 × 4 d) (20 – 4) ÷ 4
2. Find the value of each of the following. a) 52 b) −17 + 21
c)−7 – 3 d) 16 – 30
3. Look at the number patterns in the table below. Then, fill in the blanks.
Rule: Start at The 9th term is .
= 12 – 7 = 5 = 25 = –10 = 4 = –14 = 3 + 24 = 27 = 16 ÷ 4 = 4 = 20 ÷ 2 = 10 55.Subtract 6 from the previous term to get the next term. 7 25
4. Look at the coordinate plane.
a) Write the coordinates of each point.
b) Locate the points of the given pairs of coordinates on the coordinate plane. Then, write the letters representing the points.
(7, 1) (0, 0) (3, 4)
Anne spends d minutes on social media every day. Marcus spends 30 minutes less than Anne on social media every week.
a) How much time does Marcus spend on social media in a week?
b) If d = 144, how much time does Marcus spend on social media in a week?
(7d – 30) minutes 978 minutes or 16 hours 18 minutes
How can we solve this problem? Discuss in your group and fill in columns 1 and 2.
1. What I already know that will help me solve the problem 2. What I need to find out and learn 3. What I have learnt
Multiply and subtract numbers.
Write and evaluate algebraic expressions involving multiplication and subtraction. Answer varies.
You will learn to...
• use letters to represent unknown numbers
• write algebraic expressions
• find the value of an algebraic expression by substitution
• find the value of an algebraic expression given in exponent notation
• simplify algebraic expressions with two or more like terms
Let's Learn Let's Learn
Harper is 3 years older than Chloe.
When Chloe is x years old, Harper will be (x + 3) years old. When Harper is y years old, Chloe will be (y – 3) years old. x and y can stand for any whole number. Chloe's age Harper's age
When Chloe is 5 years old, Harper will be (5 + 3) years old. Harper will be 8 years old.
x + 3 and y – 3 are algebraic expressions in terms of x and y
The letters x and y in both the algebraic expressions are known as variables. In algebra, variables are letters which can be used to represent unknown quantities or amounts in algebraic expressions.
Do Let's Do
1. Write an algebraic expression for each of the following.
a) Add p and 2.
2. Emily is 5 years old.
b) Subtract 4 from w.
a) How old will she be in x years' time? Express your answer in terms of x.
b) How old was she y years ago? Express your answer in terms of y.
1. Write an algebraic expression for each of the following.
a) Add 14 and r.
c) Subtract q from 9.
b) Add m and 6.
d) Subtract 32 from x.
2. Taika has 3 more pencils than Tim. If Taika has p pencils, how many pencils does Tim have? Express your answer in terms of p.
3. Darren has y stamps. Ria has 8 more stamps than Darren. Express the number of stamps Ria has in terms of y
Mr Clarke had k sheets of paper at first. He gave 3 sheets to his son.
a) Express the number of sheets of paper he had left in terms of k
Number of sheets of paper left = k – 3
b) If Mr Clarke had 11 sheets of paper at first, how many sheets of paper did he have left? k – 3 = 11 – 3 = He had sheets of paper left. We use substitution to find the value of an algebraic expression.
Substitute k with 11 in the expression ‘k – 3’.
1. Kimberly had $m at first. She saved another $90.
a) Express the amount of money she had altogether in terms of m
She had $( ) altogether.
b) If Kimberly had $200 at first, how much money did she have altogether?
$ + $ 90 = $
She had $ altogether.
2. Amelia had 10 kūmara at first. She bought another p kūmara.
a) Express the number of kūmara she has now in terms of p.
10 + p or p + 10
She has kūmara now.
b) If she bought 12 kūmara, how many kūmara does she have now?
She has kūmara now.
1. There were 20 biscuits in a tin. Mrs Brown ate h biscuits.
a) How many biscuits were left in the tin? Express your answer in terms of h
b) If Mrs Brown ate 4 biscuits, how many biscuits were left in the tin?
2. Find the value of each algebraic expression when n = 3.
a) n + 4 b) n – 2
c) 13 – n d) 18 + n
20 – h = 3 + 4 = 7 = 18 + 3 = 21 = 3 – 2 = 1 = 13 – 3 = 10 m + 90 or 90 + m
a) There are 5 flags on each string.
Number of strings Number of flags 1 1 × 5 = 5 2 2 × 5 = 10 3 3 × 5 = 15 n n × 5 = ?
There are 5n flags on n strings.
If n = 12, how many flags will there be altogether?
5n = 5 × n = 5 × 12 =
n × 5 is the same as 5 × n. They are algebraic expressions. We write n × 5 and 5 × n as 5n 5n is a specialised algebraic notation.
If there are 12 strings, there will be flags altogether.
b) Atarangi uses 1 3 metre of ribbon to tie a gift box.
Number of gift boxes
We multiply the whole number by the numerator.
We write p × 1 3 as p 3 p 3 is a specialised algebraic notation.
If p = 8, what is the total length of ribbon Atarangi will use?
p 3 = 8 3 = If there are 8 gift boxes, Atarangi will use a total of metres of ribbon.
1. Write an algebraic expression and notation for each of the following.
Multiply Algebraic expression Algebraic notation
a) s and 4 s × 4
b) 8 and c c)
Let's Practise Let's Practise q and 1 6
2. There are 2 bags of marbles. Each bag has r marbles.
a) Express the total number of marbles in terms of r.
Total number of marbles = 2 × =
b) If each bag contains 9 marbles, how many marbles are there altogether?
2 = 2 × = There are marbles altogether.
1. Matua Thompson had w students. He gave each student 9 lollies. How many lollies did Matua Thompson give out? Express your answer in terms of w
2. Mr Patel has v students. He gave 1 8 of a pie to each of his students. How much pie was given to them? Express your answer in terms of v.
3. Find the value of each algebraic expression when m = 5.
4. Find the value of each algebraic expression when n = 24.
>> Look at EXPLORE on page 68 again. Can you solve the problem now? What else do you need to know?
There are 4 paper bags. Ryan puts an equal number of oranges into each bag.
a) If there are y oranges altogether, there will be y 4 oranges in each bag.
We write y ÷ 4 as y 4 .
b) If y = 48, how many oranges will there be in each bag?
y 4 = 48 4 = If there are 48 oranges altogether, there will be oranges in each bag.
Let's Do Let's Do
y ÷ 4 is an algebraic expression, y 4 is a specialised algebraic notation.
1. Write an algebraic expression and notation for each of the following.
2. The perimeter of a square is h centimetres.
a) What is the length of each side of the square?
The length of each side of the square is 4 centimetres.
b) If h = 36, what is the length of each side of the square? 4 = 4
The length of each side of the square is centimetres.
1. Kelly has 14 litres of juice. She pours all the juice equally into s jugs. What is the volume of juice in each jug?
Express your answer in terms of s. litres
2. Find the value of each algebraic expression when m = 8.
Let's Learn Let's Learn
a) Joel has some toy cars. He puts w toy cars each into 3 boxes and has 2 toy cars left.
Express the number of toy cars Joel has in terms of w
Number of toy cars in the 3 boxes = w + w + w = 3 × w = 3w
Joel has (3w + 2) toy cars.
There are 3w toy cars in the 3 boxes. There are 2 toy cars left.
If w = 4, find the number of toy cars Joel has.
3w + 2 = 3 × 4 + 2 = 12 + 2 = 14
Joel has 14 toy cars.
b) Daniel had $350. He saved $x and spent the remaining amount of money equally on 6 games. Express the amount of money he spent on each game in terms of x.
Total amount spent on 6 games = $(350 – x)
Amount of money spent on each game = $ 350 – x 6 (350 – x) ÷ 6 = 350 – x 6
If x = 50, how much money did he spend on each game?
$ 350 – x 6 = $ 350 – 50 6 = $ 300 6 = $ He spent $ on each game.
1. Sam had 20 balloons. He bought another j balloons. Then, he tied all the balloons equally into 9 bunches. How many balloons were there in each bunch? Express your answer in terms of j.
2. Find the value of 4p – 5 when p = 6. 4p – 5 = 4 × – 5 = – 5 =
First, I add. Then, I divide.
1. The sum of the page numbers on two facing pages is m. What is the lesser page number? Express your answer in terms of m.
2. Find the value of each algebraic expression when n = 7.
3. Kahu had a apples. After her mother gave her some apples, Kahu has four times as many apples as she had at first.
a) How many apples does Kahu have after her mother gave her some apples?
b) Kahu’s father gave her another 35 apples. How many apples does Kahu have now?
c) Kahu packs all of the apples equally into 9 bags. Express the number of apples in each bag in terms of a
Sarah and David write an algebraic expression for this problem.
Joseph had $y. His mother gave him another $10. Then, he spent all the money equally on 2 books. Express the amount of money he spent on each book in terms of y.
Who is correct?
Why do you say so?
Sarah is correct. She adds 10 to y first, then divides the sum by 2.
Who is wrong?
Why do you say so?
David is wrong. Joseph spent the $10 from his mother equally on 2 books so he should spend $ y 2 + 5 on each book.
What did you learn about writing algebraic expressions?
The algebraic expression should correctly represent the information given in the problem.
Think of a time in your daily life when algebraic expressions can be useful.
I save $x every week. I can use an algebraic expression to calculate my savings after a month, 2 months and so on.
P B Chapter 11: Exercise 1.5, page 46
A square has sides of s centimetres.
a) Express the area of the square in terms of s.
Area of the square = s × s = s2 square centimetres
The area of the square is s2 square centimetres.
b) If the length of each side of the square is 18 centimetres, what is the area of the square?
s2 = 182
The area of the square is square centimetres.
1. A cube has edges of p centimetres.
a) Express the volume of the cube in terms of p.
Volume of the cube = Edge × Edge × Edge = × × = cubic centimetres
The volume of the cube is cubic centimetres.
s2 is a specialised algebraic notation of s × s
b) If the cube has edges of 6 centimetres, what is the volume of the cube?
The volume of the cube is
1. Ms Halley had y students. She gave each student 2y stickers.
a) How many stickers did Ms Halley give out? Express your answer in terms of y
y × 2y = 2y2
Ms Halley gave out 2y2 stickers.
b) If Ms Halley had 21 students, how many stickers did she give out?
2y2 = 2 × 212 = 882
Ms Halley gave out 882 stickers.
2. There were n runners for a charity run. Each runner raised $10n for the charity.
a) How much money did the runners raise for the charity? Express your answer in terms of n.
n × 10n = 10n2
The runners raised $10n2 for the charity.
b) If there were 30 runners, what was the total amount of money raised?
10n2 = 10 × 302 = 9000
The runners raised $9000 for the charity.
3. Find the value of each algebraic expression when n = 9.
a) n2 b) 2n3
n2 = 92 = 81
2n3 = 2 × 93 = 1458
c) 4n2 d) 9n2
4n2 = 4 × 92 = 324
9n2 = 9 × 92 = 729
P B Chapter 11: Exercise 1.6, page 47
a) Whaea Lisa has 3 bags of red apples and 2 bags of green apples. There are m apples in each bag. Find the total number of apples in terms of m = 5m
The total number of apples is 5m.
3m and 2m are the terms of the algebraic expression ‘3m + 2m’.
The terms 3m and 2m have the same variable, m, and exponent. We call such terms like terms
How many more red apples than green apples are there? Express your answer in terms of m. m is the same as m1 The exponent of m is 1.
3m – 2m = 1m = m
1m is the same as m
There are m more red apples than green apples.
b) There are x apples in each bag.
Simplify 6x – 4x + 3x.
1. Simplify each expression.
When there are only addition and subtraction operations, we work from left to right.
1. Simplify each expression.
I have learnt to... use letters to represent unknown numbers write algebraic expressions find the value of an algebraic expression by substitution find the value of an algebraic expression given in exponent notation simplify algebraic expressions with two or more like terms
• understand and identify algebraic equations and linear equations
• use substitution to determine whether a given number makes an equation true or false
• solve algebraic equations
• read and plot points in all four quadrants of a coordinate plane
• identify the constant increase or decrease in a linear pattern
• use variables and algebraic notation to represent the rule of a linear pattern in an equation, and use the equation to make conjectures
a) We place some cubes on both sides of a scale.
The total number of cubes on the left of the scale is equal to the total number of cubes on the right. The scale is balanced.
We can use the equation, 2 + 4 = 6, to show this relationship between the number of cubes on both sides of the scale. ‘2 + 4’ has the same value as ‘6’.
An equation is a mathematical sentence where the values on both sides of the equal sign (=) are the same.
4 + x = 10 and 3z – 2 = 4 are also equations. There are unknown quantities or amounts in these equations and they are represented by letters. These equations are called algebraic equations.
b) 4 – 3p and 2q3 + 5 are algebraic expressions.
2p + 7 = 4 and 6 – q2 = 2 are algebraic equations.
The letters and in both the algebraic expressions and algebraic equations are known as variables
Variables are letters which represent unknown quantities or amounts in algebraic expressions or equations.
c) When the exponent of the variable in an algebraic equation is 1, we also call it a linear equation.
2p + 7 = 5 and 4 – 3p =10 are linear equations.
2q3 + 5 = 7 and 6 – q2 = 2 are not linear equations. p = p1 exponent variable
1. Are the following algebraic equations? Write Yes or No.
2. Write the variable(s) for each of the following.
1. Identify and circle the algebraic equation in each of the following.
2. Identify each of the following as algebraic expression or algebraic equation. Then, write the variable(s). a) 5 + 9 + r b) 5 + z = 10
algebraic expression algebraic equation
Variable(s):
3. Identify and circle the linear equations.
33 – 2y + 8
Variable(s):
a) Given the equation 6 + x = 9, evaluate 6 + x when x is substituted with 3.
6 + 3 = 9
9 = 9
Each expression on either side of the equal sign evaluates to 9.
The equation is true for x = 3. This means that x = 3 is a solution of the equation 6 + x = 9.
When we substitute the variable with a value in an algebraic equation and number sentence is true, the equation is true.
b) Given the equation 6x = 9, evaluate 6x when x is substituted with 3.
6 × 3 = 9
18 ≠ 9
Variable(s): Variable(s): r 9 18 18 9 d y algebraic equation algebraic expression z
The expression on the left side of the equal sign evaluates to but the right side of the expression is . is not equal to .
The equation is false for x = 3. x = 3 is not a solution of the equation 6x = 9.
1. Given the equation y + 6 = 20,
a) is the equation true or false when y = 12?
b) is y = 12 the solution of the equation?
a) When y = 12, y + 6 = + 6 = is not equal to 20, so the equation is when y = 12.
b) y = 12 is / is not a solution of the equation.
2. Given the equation 4p + 3 = 35,
a) is the equation true or false when p = 8?
b) is p = 8 the solution of the equation?
a) When p = 8, 4p + 3 = 4 × + 3 = + 3 = is equal to 35, so the equation is when p = 8.
b) p = 8 is / is not a solution of the equation.
Practise Let's Practise
1. Is the equation k + 24 = 33 true or false when k = 9?
When k = 9, k + 24 = 9 + 24 = 33 33 is equal to 33.
The equation k + 24 = 33 is when k = 9.
2. Is the equation 5h – 13 = 20 true or false when h = 6?
When h = 6, 5h – 13 = 5 × 6 – 13 = 30 – 13 = 17
17 is not equal to 20.
The equation 5h – 13 = 20 is when h = 6.
a) Find the value of m in the equation 9 + m = 16.
Guess
m = 2
m = 8
m = 7
Check
9 + m = 9 + 2 = 11 ✗
11 < 16, so m ≠ 2.
The next guess should be greater than 2.
9 + m = 9 + 8 = 17 ✗
17 > 16, so m ≠ 8.
The next guess should be greater than 2 but less than 8. Since 17 is close to 16, the value of m must be close to 8.
9 + m = 9 + 7 = 16 ✓
m = 7 is a solution of the equation 9 + m = 16.
9 + m = 16 is a linear equation.
To solve an equation, we find the value of the variable in it.
b) Solve 2 x + 5 = 15.
Guess
x = 4
x = 5
Check
2 x + 5 = 2 × 4 + 5 = 8 + 5 = 13 ✗
13 < 15, so x ≠ 4.
13 is close to 15, so the next guess should be a greater number that is close to 4.
2 x + 5 = 2 × 5 + 5 = 10 + 5 = 15 ✓
x = is a solution of the equation 2 x + 5 = 15.
c) Solve 9 – 3n = 0.
n = 1 9 – 3n = 9 – 3 × 1 = 9 – 3 = 6 ✗
6 > 0, so n ≠ 1. The next guess should be a number greater than 1. n = 3 9 – 3n = 9 – 3 × 3 = 9 – 9 = 0 ✓
n = is a solution of 9 – 3n = 0.
1. Use the guess and check method to solve q + 9 = 20.
When q = 15, q + 9 = + 9 = is greater than 20, so q = 15 is not a solution of q + 9 = 20.
When q = , q + 9 = + 9 = q = is a solution of q + 9 = 20.
2. Solve 5y – 22 = 13.
When y = 5, 5y – 22 = 5 × – 22 = – 22
When y = , 5y – 22 = 5 × – 22 = – 22 = y = is a solution of 5y – 22 = 13.
3. Is s = 2 a solution of 4 s – 16 = 0?
When s = 2, 4 s – 16 = 4 × – 16 = – 16 =
Write a solution or not a solution.
s = 2 is of 4 s – 16 = 0.
Let's Practise
For tasks 1 and 2 below, show your work clearly. Then, write a solution or not a solution.
1. Is w = 9 a solution of w + 12 = 21? w = 9 is of w + 12 = 21.
2. Is d = 8 a solution of 3d – 34 = 10? d = 8 is of 3d – 34 = 10.
When w = 9, w + 12 = 9 + 12 = 21
When d = 8, 3d – 34 = 3 × 8 – 34 = 24 – 34 = –10
3. Solve these equations using the guess and check method.
Answers vary. Sample:
a) h + 21 = 38 b) 4q – 17 = 19
When h = 17, h + 21 = 17 + 21 = 38
So, h = 17 is a solution.
When q = 9, 4q – 17 = 4 × 9 – 17 = 36 – 17 = 19
So q = 9 is a solution.
a) Solve x + 6 = 10.
The scale is balanced.
To find the value of x, remove the same number of unit cubes from both sides until only x is left on one side.
The scale stays balanced as the same number of unit cubes have been removed from both sides.
x = 4 is a solution of x + 6 = 10.
Check:
b) Solve 3x + 4 = 10.
The scale is balanced.
First, remove the same number of unit cubes from both sides. The scale stays balanced.
Then, divide each side by 3 to find the value of x
The scale remains balanced.
x = 2 is a solution of 3x + 4 = 10.
Check:
c) Solve 5q – 5= 10.
We can carry out the same operations on both sides until only q is left on one side.
5q – 5 + 5 = 10 + 5
5q = 15
5q ÷ 5 = 15 ÷ 5 q = 3
Add 5 to both sides.
Divide by 5 on both sides.
q = 3 is a solution of 5q – 5 = 10.
Check:
When q = 3, 5q – 5 = 5 × 3 – 5
Let's Do Do
1. Use the balance method to solve these equations. Write + , –, × or ÷ in each . Then, write the missing numbers.
a) g + 11 = 16 g + 11 = 16 g = b) w – 7 = 9 w – 7 = 9 w =
Solve 2x – 10 = 10.
Add 10 to the left side of the equation.
2x – 10 + 10 = 10
2x = 10
Divide by 2 on both sides.
2x ÷ 2 = 10 ÷ 2 x = 5
Add 10 to both sides of the equation.
2x – 10 + 10 = 10 + 10 2x = 20
Divide by 2 on both sides.
2x ÷ 2 = 20 ÷ 2 x = 10
David Sarah
Who is wrong? Why do you say so? Who is correct? Why do you say so?
David is correct. When David uses the balance method to solve 2x – 10, he adds 10 to both sides of the equation.
Sarah is wrong. When Sarah uses the balance method to solve 2x – 10, she adds 10 to the left side of the equation only.
What did you learn about solving an algebraic equation using the balance method?
When solving an algebraic equation using the balance method, we perform the same operation on both sides of the equation.
Think of a time in your daily life when you need to solve an equation.
I want to make some bracelets to sell at an arts and crafts fair. I want to sell each bracelet at $3. The total cost of the materials is $20. Using x to represent the number of bracelets, I can solve the equation 3x – 20 = 80 to find out the number of bracelets I have to sell to make $80.
Let's Practise Let's Practise
1. Use the balance method to solve these equations.
a) q + 32 = 51
q + 32 – 32 = 51 – 32 q = 19
c) 3s + 12 = 45
3s + 12 – 12 = 45 – 12
3s = 33
3s ÷ 3 = 33 ÷ 3 s = 11
e) 4u – 17 = 31
4u – 17 + 17 = 31 + 17
4u = 48
4u ÷ 4 = 48 ÷ 4 u = 12
b) r – 26 = 26
r – 26 + 26 = 26 + 26 r = 52
d) 6t + 7 = 61
6t + 7 – 7 = 61 – 7
6t = 54
6t ÷ 6 = 54 ÷ 6 t = 9
f) 7v – 38 = 18
7v – 38 + 38 = 18 + 38
7v = 56
7v ÷ 7 = 56 ÷ 7 v = 8
P B Chapter 11: Exercise 2.4, page 52
a) The x-axis and the y-axis divide the coordinate plane into four quadrants.
The origin is the point where the x-axis and the y-axis cross at a right angle.
The coordinates of the origin are (0, 0).
The signs of the x-coordinate and y-coordinate determine the quadrant in which a point lies.
Point A is in the 1st quadrant.
The coordinates of point A are (3, 2).
Point B is in the 2nd quadrant.
The coordinates of point B are (–2, 1).
Point C is in the 3rd quadrant.
The coordinates of point C are (–3, –2).
Point D is in the 4th quadrant.
The coordinates of point D are (2, –3).
The coordinates of point E are (–1, –2).
In which quadrant will point E lie?
3rd quadrant
The coordinates of a point are always written in the form (x, y). (x, y) is called an ordered pair of coordinates.
Both the x-coordinate and y-coordinate of point E are negative.
b) Look at the coordinate plane below.
Point Q is the reflection of point P across the x-axis.
The coordinates of point P are (3, 2).
The coordinates of point Q are (3, –2).
Compare the ordered pairs (3, 2) and (3, –2).
The x-coordinates are the same. So, points P and Q are the same distance from the y-axis and on the same side.
The y-coordinates have the same numerical value but different signs. So, points P and Q are the same distance from the x-axis but on opposite sides.
Point S is the reflection of point P across the y-axis.
The coordinates of point P are (3, 2).
The coordinates of point S are (–3, 2).
Compare the ordered pairs (3, 2) and (–3, 2).
The x-coordinates have the same numerical value but different signs. So, points P and S are the same distance from the y-axis but on opposite sides.
The y-coordinates are the same. So, points P and Q are the same distance from the x-axis and on the same side.
Look for other points that are related by reflections across the x-axis and/or y-axis.
c) Plot and label these points on the coordinate plane.
Since the x-coordinate of point A is negative and the y-coordinate is positive, point A lies in the 2nd quadrant.
Since both the x-coordinate and the y-coordinate of point B are negative, point B lies in the 3rd quadrant.
To plot point A, we start from the origin and move 3 units to the left as the x-coordinate is negative. Next, we move 4 units up since the y-coordinate is positive.
To plot point B, we start from the origin and move 4 units to the left as the x-coordinate is negative. Next, we move 5 units down since the y-coordinate is negative.
How would you plot point C (4, –6)?
Let's Do Let's Do
1. Look at the coordinate plane below.
a) Write the coordinates of each point.
Locate the x-coordinate followed by the y-coordinate of each point.
(3, 1)
(–3, 3)
(–3, –2)
b) Locate the points of the given pairs of coordinates on the coordinate plane. Then, write the letters representing the given points.
(–2, 2)
(–4, –3)
c) Plot and label these points on the coordinate plane.
G (2, –4)
2. Write x-axis or y-axis.
H (–1, 3)
a) Point J is located at (–7, –6) on the coordinate plane. After a reflection across the , point J' is now located at (–7, 6).
b) Point K is located at (–4, 5) on the coordinate plane. After a reflection across the , point K' is now located at (4, 5).
x-axis x-axis / y-axis y-axis y-axis / x-axis
c) Point L is located at (3, –8) on the coordinate plane. After a reflection across the and then across the , point L' is now located at (–3, 8).
1. Look at the coordinate plane below.
a) Write the coordinates of each point.
(–4, 3)
(2, 5)
(–3, 0)
(–2, –3)
b) Locate the points of the given pairs of coordinates on the coordinate plane. Then, write the letters representing the given points.
(–3, 5)
(4, –4)
2. Write the coordinates of each point.
(4, –2)
(3, 2)
(–8, 0)
a) Point Q is located at (8, 0) on the coordinate plane. After a reflection across the y-axis, point Q' is now located at .
b) Point R is located at (5, –1) on the coordinate plane. After a reflection across the x-axis, point R' is now located at .
(5, 1)
c) Point S is located at the origin on the coordinate plane. After a reflection across the y-axis and then across the x-axis, point S' is now located at
(0, 0)
3. a) Plot and label these points on the coordinate plane below.
b) State the quadrant in which each point lies.
1st quadrant
4th quadrant
3rd quardrant
B Chapter 11: Exercise 2.5, pages 53–55
a) William records the amount of money that he saves to buy a gift for his mother. He starts with $5 in his savings and continues to save $10 each week. The gift costs $100. Will William be able to buy the gift after saving for 10 weeks?
We notice that the amount of money saved each week follows a pattern. In the pattern, the arrangement of numbers is in a particular order. This is called a sequence.
The savings increase by $10 each week.
Week 0 = $5
Week 1 = $10 + $5 = $15
Week 2 = $10 + $10 + $5 = $25
Week 3 = $10 + $10 + $10 + $5 = $35
Week 4 = $10 + $10 + $10 + $10 + $5 = $45
When a sequence increases or decreases by the same amount each time, the difference between the consecutive terms is constant. This type of pattern is called a linear pattern.
The constant increase in this linear pattern is $10.
Week 0: (10 × 0) + 5
Week 1: (10 × 1) + 5
Week 2: (10 × 2) + 5
Week 3: (10 × 3) + 5
Week 4: (10 × 4) + 5
Let x be the number of weeks and y be the amount of savings.
In the above pattern, we get the amount of savings (y) when we multiply 10 by the number of weeks and add 5.
To represent the relationship between the amount of savings (y) and the number of weeks (x), William writes the equation y = 10x + 5.
Using the equation, y = 10x + 5, we can find the amount of savings in a later week.
Week 10 = 10 × 10 + 5 = 100 + 5 = 105
William will have saved $105 after 10 weeks. So, he will be able to buy the gift.
b) Look at the pattern below. What is the 25th term?
1st term = 1 – 1 = 0
2nd term = 2 – 1 = 1
3rd term = 3 – 1 = 2
4th term = 4 – 1 = 3
5th term = 5 – 1 = 4
Let x be the position and y be the number.
The constant increase in the pattern is 1.
Look at the pattern. The nth term is always 1 less than its position.
We can use the equation y = x – 1 to represent the relationship between the number (y) and its position (x).
25th term = 25 – 1 = 24
The 25th term is 24.
1. Look at each linear pattern below. Write increase or decrease in the first blank of each sentence. Then, find the constant increase or decrease.
a) 3, 6, 9, 12, 15
The constant in the pattern is
b) 20, 16, 12, 8, 4
The constant in the pattern is .
c) 10, 8, 6, 4, 2
The constant in the pattern is .
d) 4, 9, 14, 19, 24
The constant in the pattern is . Let's Do Let's Do
2. Ava builds a bookshelf. The first shelf is 30 centimetres above the ground. She adds shelves to the bookshelf at a constant distance.
a) Complete the table.
of the shelf from the ground (cm)
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 height of the shelf from the ground (y) and the number of shelves in the bookshelf (x).
d) What is the height of the 8th shelf from the ground? centimetres
Let's Practise Let's Practise
1. Look at each linear pattern below. Write increase or decrease in the first blank of each sentence. Then, find the constant increase or decrease.
a) 17, 15, 13, 11, 9
The constant in the pattern is .
b) 51, 54, 57, 60, 63
The constant in the pattern is
2. Look at the pattern below. 5, 10, 15, 20, 25
Then, circle the correct equation for the pattern.
y = x + 5 y = 5x
3. 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 triangles (y) and the figure number (x).
e) How many triangles are there in figure 19?
There are triangles in figure 19.
f) How many triangles are there in figure 100?
There are triangles in figure 100.
I have learnt to...
understand and identify algebraic equations and linear equations use substitution to determine whether a given number makes an equation true or false solve algebraic equations read and plot points in all four quadrants of a coordinate plane identify the constant increase or decrease in a linear pattern use variables and algebraic notation to represent the rule of a linear pattern in an equation, and use the equation to make conjectures
You will learn to...
• solve word problems using algebraic expressions and algebraic equations
• solve non-routine problems involving algebra
Let's Learn Learn
Amelia had 3 pieces of lace. Each piece of lace was m centimetres long. Amelia used 45 centimetres of lace to decorate a gift. If m = 24, find the length of lace Amelia had left.
Understand the problem. 1 Plan what to do.
Whakaaro 2 Work out the Answer Whakaatu 3 Check if your answer is correct. Tirohia 4
How many pieces of lace were there? What was the length of each piece of lace?
How much lace did Amelia use? What do I have to find?
I can write an algebraic expression to find the length of lace Amelia had left.
3m – 45 = 3 × 24 – 45 = 72 – 45 = 27
Amelia had 27 centimetres of lace left.
27 + 45 = 72
The total length of the lace was 72 centimetres.
72 ÷ 3 = 24
Each piece of lace was 24 centimetres long.
My answer is correct.
+ Plus Solve the problem in another way.
Tāpiri
3 × 24 = 72
The total length of the lace was 72 centimetres. 72 – 45 = 27
Amelia had 27 centimetres of lace left.
Compare the methods in steps 3 and 5. Which method do you prefer? Why?
1. Understand 2. Plan 3. Answer 4. Check 5. Plus
Solve the word problem. Show your work clearly.
Next, try solving the problem in a different way. Which method do you prefer? Why?
1. Understand 2. Plan 3. Answer 4. Check 5. Plus
1. Sally had a string that was y metres long. Sally’s string was twice as long as Olivia’s string. Olivia used 1 metre of string to tie a parcel. Find the length of string that Olivia had left when y = 5. Express your answer as a decimal. Sally Olivia
Solve the word problems. Show your work clearly.
1. There are 16 cranberries and x blueberries in a bowl. 9 children share all the berries equally. If x = 11, find the number of berries that each child will get.
2. A library issued p books each to 4 students. It also issued 5 books to a teacher. If p = 2, find the total number of books issued by the library.
3. Dan spent 20 minutes less than r hours to paint 5 vases. He spent the same amount of time to paint each vase. Find the time that Dan took to paint each vase when r = 3.
1. Evan has z guppies and 21 angelfish. He has 38 fish altogether. How many of his fish are guppies?
Understand the problem. 1 Plan what to do. Whakaaro 2 Work out the Answer. Whakaatu 3 Check if your answer is correct. Tirohia 4
How many guppies are there? How many angelfish are there? How many fish are there altogether? What do I have to find?
I can form an equation in terms of z to solve the problem.
2. A baker bought 5 cartons of eggs. Each carton had y eggs. He used 12 eggs to make some sponge cakes and had 18 eggs left. How many eggs were there in each carton at first? Understand the problem.
1 Plan what to do. Whakaaro
3
2 Work out the Answer. Whakaatu
How many cartons of eggs did the baker buy?
How many eggs were there in each carton?
How many eggs did he use?
How many eggs were left?
What do I have to find?
4
I can form an equation in terms of y to solve the problem. Check if your answer is correct.
There were 6 eggs in each carton at first.
My answer is correct.
+ Plus Solve the problem in another way.
Tāpiri
I can draw a bar model.
12 used 18 left
12 + 18 = 30
There were 30 eggs altogether. y = 30 ÷ 5 = 6
There were 6 eggs in each carton at first.
Compare the methods in steps 3 and 5. Which method do you prefer? Why?
1. Understand 2. Plan 3. Answer 4. Check 5. Plus
Let's Do Let's Do
Form an algebraic equation to solve each word problem. Show your work clearly.
1. A florist sold g tulips on Monday. She sold 8 fewer roses than tulips on that day. If she sold 21 roses on Monday, how many tulips did she sell?
2. Kyle baked 29 muffins. He placed p muffins each in 4 boxes and had 5 muffins left over. How many muffins were there in each box?
3. Daniel has 6 bags of cement. Each bag of cement is h kilograms. After using 10 kilograms of it to make an outdoor sink, he has 14 kilograms of cement left. What was the mass of each bag of cement at first?
Let's Practise Let's Practise
Form an algebraic equation to solve each word problem. Show your work clearly.
1. Jane had m chickens in a coop. She bought another 17 chickens and now has 40 chickens altogether. How many chickens did she have at first?
2. Martin had 3 boxes of light bulbs. Each box contained j light bulbs. There were 30 light bulbs. How many light bulbs were there in each box at first?
3.Jake saved $y in August. Don saved 3 times as much as Jake in August. If Don saved another $180, he would have saved $600. How much money did Jake save in August?
4. Sam is d years old. Rupert is twice as old as Sam. Ian is 15 years older than Rupert. If Ian is 27 years old,
a) write an equation in terms of d. b) what is Sam’s age?
2d + 15 = 27 6 years
Kim planted e pots of Kowhai plants. After she gave away pots of Kowhai plants, there were pots of Kowhai plants left. How many pots of Kowhai plants did Kim plant?
Read the word problem. Write the missing numbers. How did you decide what numbers to use?
Next, solve the word problem. Show your work clearly. What did you learn?
Answers vary. Sample: e – 4 = 12
e – 4 + 4 = 12 + 4
e = 16
Kim planted 16 pots of Kowhai plants.
Let's Learn
Look at the pattern below. Find the number of coloured squares in figure 8.
2
1 Plan what to do. Whakaaro
Understand the problem.
How many coloured squares are there in each figure? What do I have to find?
I can look for a pattern to find the relationship between the figure number and the number of coloured squares.
Work out the Answer. Whakaatu 3
Figure Number of coloured squares
When
The number of coloured squares in figure 8 is 40.
Check if your answer is correct.
Tirohia 4 + Plus Solve the problem in another way.
Tāpiri 5
Use 40 coloured squares to form figure 8. Set aside 4 squares for the corners before dividing the remaining squares equally among the 4 sides.
40 – 4 = 36
36 ÷ 4 = 9
9 + 1 + 1 = 11
There will be 11 squares along each side of figure 8.
My answer is correct.
The coloured squares in each figure are made up of 4 equal strips of squares.
The number of coloured squares in figure 8 is 40. Compare the methods in steps 3 and 5. Which method do you prefer? Why?
Cassie has some ten-dollar notes and twenty-dollar notes. She uses 13 notes to make up $190. Find the number of ten-dollar notes and the number of twenty-dollar notes she uses.
Strategy: Look for a pattern
Number of twenty-dollar notes
of ten-dollar notes
× 13 + $10 × 0 = $260
× 12 + $10 × 1 = $250
× 11 + $10 × 2 = $240 x 13 – x
x + $10(13 – x)
From the table, every time we replace a twenty-dollar note with a ten-dollar note, it will result in a decrease of $10 in the total amount.
Starting from 13 twenty-dollar notes and 0 ten-dollar notes, the total amount is $260. When we replace a twenty-dollar note with a ten-dollar note, the total amount decreases by $10.
$260 – $190 = $70
$70 ÷ $10 = 7
There are seven $10 in $70.
Therefore, x = 13 – 7 = 6
When x = 6,
$20 × 6 + $10 × (13 – 6) = $120 + $10 × 7 = $120 + $70 = $190
She uses 7 ten-dollar notes and 6 twenty-dollar notes.
I have learnt to... solve word problems using algebraic expressions and algebraic equations solve non-routine problems involving algebra
>> Look at EXPLORE on page 68 again. Fill in column 3. Can you solve the problem now?