PR1ME Mathematics – Year 7 Teacher's Guide (Sample)

100% coverage of New Zealand Mathematics and Statistics Curriculum for Phases 1-3

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Strand(s): Algebra

Vocabulary

Objectives Materials Resources

• CB: pp. 67-68

• Math Pro: Recall

• Do mixed operations involving the four operations without or with parentheses

• Find the sum or difference of two integers

• Develop a rule for a growing pattern in words and make conjectures about further elements in the pattern

• Read and plot coordinates in the 1st quadrant of the coordinate plane

Scheme of Work Unit

Let’s Remember

Unit 1: Algebraic Expressions

Year 7: Algebra — Equations and relationships

New Zealand Curriculum

• algebraic expression

• variable

• CB: pp. 69–70

• PB: p. 41

• Math Pro: Practice

1.1 Knowledge ( ★ )

★ A variable can be used to represent:

–an unknown number, often in formulae.

–a quantity that can vary or change.

–a specific unknown value to be solved.

• Use a letter to represent an unknown number

• Write an algebraic expression in one variable involving addition or subtraction

1.1 Writing algebraic expressions involving addition and subtraction

1.2 Knowledge ( ★ ) and Practices ( ◆ )

★ A variable can be used to represent:

–an unknown number, often in formulae.

–a quantity that can vary or change.

–a specific unknown value to be solved.

substitution

value

◆ Using substitution to find the value of an expression or formula

• Math Pro: Practice • substitute

• algebraic notation

• CB: pp. 70–71

• PB: p. 42

• Use substitution to find the value of an algebraic expression involving addition or subtraction

1.2 Finding the value of an algebraic expression involving addition or subtraction

1.3 Knowledge ( ★ ) and Practices ( ◆ )

★ A variable can be used to represent:

–an unknown number, often in formulae.

–a quantity that can vary or change.

–a specific unknown value to be solved.

• CB: pp. 72–73

• PB: pp. 43–44

• Math Pro: Practice

★ Algebra has its own specialised notation to express relationships and operations concisely.

◆ Using substitution to find the value of an expression or formula

• Write an algebraic expression in one variable involving multiplication

• Use substitution to find the value of an algebraic expression in one variable involving multiplication

1.3 Writing and evaluating algebraic expressions involving multiplication

1.4 Knowledge ( ★ ) and Practices ( ◆ )

★ A variable can be used to represent:

–an unknown number, often in formulae.

–a quantity that can vary or change.

–a specific unknown value to be solved.

★ Algebra has its own specialised notation to express relationships and operations concisely.

◆ Using substitution to find the value of an expression or formula

• CB: pp. 74–75

• PB: p. 45

• Math Pro: Practice

• CB: pp. 75–77

• PB: p. 46

• Math Pro: Practice

• CB: pp. 78–79

• PB: p. 47

• Math Pro: Practice

• like terms

• term

• CB: pp. 80–81

• PB: p. 48

• Math Pro: Practice

• Write an algebraic expression in one variable involving division

• Use substitution to find the value of an algebraic expression in one variable involving division

Writing and evaluating algebraic expressions involving division

Knowledge ( ★ ) and Practices ( ◆ )

1.4

1.5

★ A variable can be used to represent:

–an unknown number, often in formulae.

–a quantity that can vary or change.

–a specific unknown value to be solved.

★ Algebra has its own specialised notation to express relationships and operations concisely.

◆ Using substitution to find the value of an expression or formula

• Write an algebraic expression in one variable involving the four operations

• Use substitution to find the value of an algebraic expression in one variable involving the four operations

1.5 Writing and evaluating algebraic expressions involving more than one operation

1.6 Practices ( ◆ )

◆ Using substitution to find the value of an expression or formula

• Use substitution to find the value of an algebraic expression given in exponent notation

1.6 Finding the value of an algebraic expression given in exponent notation

1.7 Practices ( ◆ )

◆ Simplifying expressions involving any of the four operations by collecting like terms

• Algebra tiles

• Simplify an algebraic expression in one variable with two or more like terms

1.7 Simplifying algebraic expressions with like terms

Objectives

Unit

Unit 2: Algebraic Equations New Zealand Curriculum Year 7: Algebra — Equations and relationships

• algebraic equation

• equation

linear equation

• variable

• CB: pp. 82–84

• PB: p. 49

• Math Pro: Practice

2.1 Practices ( ◆ )

◆ Forming and solving oneand two-step linear equations with integer solutions

• 1 pan balance

• 12 connecting cubes of the same mass

• Understand what an equation is

• Identify an algebraic equation

2.1 Understanding equations

• Identify a linear equation

2.2 Knowledge ( ★ )

★ The solution to an equation satisfies that equation.

★ Solutions to equations can be checked using substitution.

• false

solution

true

• CB: pp. 84–85

• PB: p. 50

• Math Pro: Practice

• Check whether a given number is a solution to an equation using substitution

2.2 True and false equations

2.3 Knowledge ( ★ ) and Practices ( ◆ )

★ A variable can be used to represent:

–an unknown number, often in formulae.

–a quantity that can vary or change.

–a specific unknown value to be solved.

★ The solution to an equation satisfies that equation.

★ Solutions to equations can be checked using substitution.

• solve

• CB: pp. 86–88

• PB: p. 51

• Math Pro: Practice

★ Equations can be solved through trial and error, but this can be an inefficient method.

◆ Forming and solving oneand two-step linear equations with integer solutions

• Use the guess and check method to solve an algebraic equation

2.3 Solving algebraic equations using the guess and check method

Knowledge ( ★ ) and Practices ( ◆ )

2.4

★ A variable can be used to represent:

–an unknown number, often in formulae.

–a quantity that can vary or change.

–a specific unknown value to be solved.

★ The solution to an equation satisfies that equation.

★ Solutions to equations can be checked using substitution.

◆ Forming and solving oneand two-step linear equations with integer solutions

• CB: pp. 89–93

• PB: p. 52

• Math Pro: Practice

• 1 pan balance

• 3 lightweight opaque plastic bags

• Use the balance method to solve an algebraic equation

• 20 connecting cubes of the same mass

2.4 Solving algebraic equations using the balance method

2.5 Knowledge ( ★ ) and Practices ( ◆ )

★ A coordinate plane extends to 4 quadrants that meet at the origin (0, 0).

◆ Identifying and plotting points in the four quadrants of the coordinate plane, using ordered pairs and values from a table

• ordered pair • origin

• CB: pp. 94–99

• PB: pp. 53–55

• Math Pro: Practice

• Coordinate Plane (BM11.1): 1 copy per student, 1 enlarged copy for demonstration

• Read and plot coordinates in all four quadrants of the coordinate plane

• Recognise that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes

2.5 Reading and plotting points in all four quadrants

2.6 Knowledge ( ★ ) and Practices ( ◆ )

★ Linear patterns have a constant increase or decrease, can be described by the rule t = a × n + d , and can be graphed as a straight line on a coordinate plane.

◆ Using tables, graphs in the coordinate plane, and diagrams to recognise the relationship between the ordinal position and its corresponding element in a linear pattern, develop a rule for the pattern in words, and make conjectures about further elements in the pattern

◆ Identifying the constant increase or decrease in a linear pattern, using variables and algebraic notation to represent the rule in an equation, and using the equation to make conjectures

• constant • linear pattern

• CB: pp. 100–103

• PB: pp. 56–57

• sequence

• Math Pro: Practice

• CB: pp. 104–105

• PB: pp. 58–60

• Math Pro: Practice

• CB: pp. 106–109

• PB: pp. 61–62

• Math Pro: Practice

• CB: pp. 110–112

• Math Pro: Assessment

• 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

Linear patterns and equations

Curriculum Year 7: Algebra — Equations and relationships

2.6

3.1 Practices ( ◆ )

◆ Using substitution to find the value of an expression or formula

• Solve word problems using algebraic expressions

3.1 Word problems involving algebraic expressions

3.2 Knowledge ( ★ ) and Practices ( ◆ )

★ A variable can be used to represent: –an unknown number, often in formulae. –a quantity that can vary or change. –a specific unknown value to be solved.

★ The solution to an equation satisfies that equation.

★ Solutions to equations can be checked using substitution.

◆ Forming and solving oneand two-step linear equations with integer solutions

• Solve word problems by forming an algebraic equation

• 40 square tiles per group

• 1 copy of Maths Journal (BM11.2) per student

• Solve non-routine problems involving algebra using the strategy of looking for a pattern

Word problems involving algebraic equations

3.2

3.3 Mind stretcher

The suggested duration for each lesson is 1 hour.

Chapter 11 Algebra

Chapter Overview

Let’s Remember Unit 1: Algebraic Expressions Unit 2: Algebraic Equations Unit 3: Problem Solving

Let's Remember Let's Remember Recall:

1. Doing mixed operations involving the four operations without or with parentheses (CB6 Chapter 3)

2. Find the sum or difference of two integers (CB7 Chapter 7)

3. Develop a rule for a growing pattern in words and make conjectures about further elements in the pattern (CB6 Chapter 15)

4. Read and plot coordinates in the 1st quadrant of the coordinate plane (CB6 Chapter 15)

EXPLORE

Have students read the word problem on CB p. 68. Discuss with students the following questions:

• Do you think you have gained a lot of knowledge from social media?

• Do you think everything that you read on social media is true?

• When you read something interesting on social media, how do you tell the difference between fact and opinion?

• Have you or anyone you know ever been bullied on social media? How did you or the person deal with it?

Have students form groups to complete the tasks in columns 1 and 2 of the table. Let students know that they do not have to solve the word problem. Ask the groups to present their work.

Tell students that they will come back to this word problem later in the chapter.

Unit 1: Algebraic Expressions

1.1 Writing algebraic expressions involving addition and subtraction

Let's Learn Let's Learn

Objectives:

• Use a letter to represent an unknown number

• Write an algebraic expression in one variable involving addition or subtraction

Resources:

• CB: pp. 69–70

• PB: p. 41

Vocabulary:

• algebraic expression

• variable

Stage: Abstract Representation

Have students look at the table on CB p. 69.

Ask: What does the table show? (Chloe’s and Harper’s ages) Are the two girls of the same age? (No) When Chloe is 2 years old, how old is Harper? (5) When Chloe is 4 years old, how old is Harper? (7) Who is older? (Harper) How much older is Harper than Chloe? (3 years older)

Say: Harper is 3 years older than Chloe.

Ask: When Chloe is 5 years old, how can we find Harper’s age? (Add 3 to 5.)

Write: 5 + 3 = ____

Ask: How old will Harper be? (8 years old)

Say: Since Harper is 3 years older than Chloe, whatever Chloe’s age is, we can add 3 to her age to find Harper’s age. Let us use the letter x to represent Chloe’s age. x can stand for any whole number. We say x is a variable. If Chloe is x years old, we add 3 to x to find Harper’s age.

Write: Chloe’s age = x Harper’s age = x + 3

Say: ‘x + 3’ is an algebraic expression in terms of x.

Write: algebraic expression

Say: In an algebraic expression, any letter can be used to represent an unknown number. In this example, x represents Chloe’s age and x + 3 represents Harper’s age.

Write: Harper’s age = y

Say: Let us use the letter y to represent Harper’s age. y is also called a variable as it can represent any number.

Ask: If Harper is y years old, should we add or subtract 3 to find Chloe’s age? (Subtract)

Unit 1 Algebraic Expressions

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 1.1

Writing algebraic expressions involving addition and subtraction

Harper is 3 years older than Chloe.

Chloe's ageHarper's age

When Chloe is 5 years old, Harper will be (5 + 3) years old. Harper will be 8 years old.

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.

+ 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.

Write: Chloe’s age = y – 3

Say: ‘y – 3’ is an algebraic expression in terms of y. Have a student write down his/her age and his/her parents’ ages on the board. Guide students to write algebraic expressions to represent the age of each of the student’s parents in terms of the student’s age.

Let's Do Let's

Tasks 1 and 2 require students to write algebraic expressions in one variable involving addition or subtraction.

Let's Practise Practise

Tasks 1 to 3 require students to write algebraic expressions in one variable involving addition or subtraction.

1.2 Finding the value of an algebraic expression involving addition or subtraction

Let's Learn Let's Learn

Objective:

• Use substitution to find the value of an algebraic expression involving addition or subtraction

Resources:

• CB: pp. 70–71

• PB: p. 42

Vocabulary:

• substitute

• substitution

• value

(a) Stage: Abstract Representation

Write: Mr Clarke had k sheets of paper at first. He gave 3 sheets to his son.

Ask: How many sheets of paper did Mr Clarke have at first? (k sheets) What does k represent?

(The number of sheets of paper that Mr Clarke had at first) How many sheets of paper did he give to his son? (3 sheets)

Say: Let us find the number of sheets of paper Mr Clarke had left. To do that, we subtract 3 from the number of sheets of paper Mr Clarke had at first.

Write: Number of sheets of paper left = k – 3

Say: He had (k – 3) sheets of paper left. ‘k – 3’ is an algebraic expression in terms of k. For struggling students, get them to replace k with a number, e.g. 10, in the scenario. Guide them to realise that they have to subtract 3 from 10 to get the answer. Similarly, when the number is replaced by the unknown number k, they will have to subtract 3 from k

(b) Stage: Abstract Representation

Say: If Mr Clarke had 11 sheets of paper at first, we can find the number of sheets of paper he had left by using substitution to find the value of the algebraic expression ‘k – 3’. We substitute k with 11 in the expression ‘k – 3’.

1. Write an algebraic expression for each of the following. a) Add p and 2. b) Subtract 4 from w

2. Emily is 5 years old.

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.

Let's Practise

1. Write an algebraic expression for each of the following.

a) Add 14 and r b) Add m and 6.

c) Subtract q from 9. 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

1.2 Finding the value of an algebraic expression involving addition or subtraction

Let's Learn

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 =

Substitute k with 11 in the expression ‘k – 3’.

He had sheets of paper left. We use substitution to find the value of an algebraic expression.

Write: Number of sheets of paper left = k – 3 = 11 – 3 = 8

Say: He had 8 sheets of paper left.

Ask: If Mr Clarke had 9 sheets of paper at first, how many sheets of paper did he have left? (6)

Let's Do Let's

Tasks 1 and 2 require students to write an algebraic expression in one variable involving addition, and use substitution to find its value.

Let's Practise Let's Practise

Task 1 requires students to write an algebraic expression in one variable involving subtraction, and use substitution to find its value.

Task 2 requires students to use substitution to find the values of algebraic expressions involving addition or subtraction.

Let's Do Do

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 She has kūmara now.

Let's Practise

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 she bought 12 kūmara, how many kūmara does she have now? She has kūmara now. 16

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.

1.3 Writing and evaluating algebraic expressions involving multiplication

Let's Learn Let's Learn

Objectives:

• Write an algebraic expression in one variable involving multiplication

• Use substitution to find the value of an algebraic expression in one variable involving multiplication

Resources:

• CB: pp. 72–73

• PB: pp. 43–44

Vocabulary:

• algebraic notation

(a) Stage: Pictorial Representation

Have students look at the strings of flags on CB p. 72.

Ask: How many flags are there on each string? (5) How many strings of flags are there? (3)

Stage: Abstract Representation

Draw the table as shown on CB p. 72 on the board but omit the values in the right column. Guide students to complete the table.

Say: We know that there are 5 flags on each string. So, we multiply the number of strings by the number of flags on each string to find the total number of flags. The total number of flags on 1 string is 1 × 5 = 5.

Write ‘1 × 5 = 5’ in the row corresponding to 1 string in the table.

Ask: How can we find the total number of flags on 2 strings? (Multiply 2 by 5.) How many flags are there on 2 strings? (10)

Write ‘2 × 5 = 10’ in the row corresponding to 2 strings in the table.

Ask: How can we find the total number of flags on 3 strings? (Multiply 3 by 5.) How many flags are there on 3 strings? (15)

Write ‘3 × 5 = 15’ in the row corresponding to 3 strings in the table.

Ask: How can we find the total number of flags on n strings? (Multiply n by 5.)

Say: Since there are 5 flags on each string, we multiply n strings by 5 to find the total number of flags on n strings.

Write ‘n × 5’ in the row corresponding to n strings in the table.

Explain to students that n × 5 is the same as 5 × n They are called algebraic expressions. They are written as 5n and 5n is a specialised algebraic notation.

Say: There are 5n flags on n strings.

1.3 Writing and evaluating algebraic expressions involving multiplication

Let's Learn Let's Learn

a) There are 5 flags on each string.

There are 5n flags on n strings. If n = 12, how many flags will there be altogether?

5n = 5 × n = 5 × 12

b) Atarangi uses 1 3 metre of ribbon to tie a gift box.

= If there are 12 strings, there will be flags altogether. 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.

Ask: What does n represent? (The number of strings)

Say: n = 12 means that there are 12 strings of flags. Let us find the total number of flags on 12 strings by substituting n with 12 in the expression ‘5n’.

Write: Total number of flags on 12 strings = 5n = 5 × n = 5 × 12

Elicit the answer from students. (60)

Say: So, there are 60 flags altogether on 12 strings.

Explain to students that if we have two variables a and b, a × b is the same as b × a and we write both expressions as ab, in alphabetical order.

(b) Stage: Pictorial Representation

Draw the bar model and the table as shown in (b) on CB p. 72 on the board.

Say: We know that each gift box uses 1 3 metre of ribbon. So, we multiply the number of gift boxes by the length of ribbon used for each gift box to find the total length of ribbon. The length of ribbon used for 1 gift box is 1 × 1 3 = 1 3 metre.

(Continued on the next page)

Write ‘1 × 1 3 = 1 3 ’ in the row corresponding to 1 gift box in the table.

Ask: How can we find the total length of ribbon used for 2 gift boxes? (2 × 1 3 = 2 3 )

Write ‘2 × 1 3 = 2 3 ’ in the row corresponding to 2 gift boxes in the table.

Repeat the above procedure for 3 gift boxes and p gift boxes.

Write ‘p × 1 3 = p 3 ’ in the row corresponding to p gift boxes in the table.

Explain to students that p × 1 3 can be written as p 3 , and that 1 3 × p is the same as p × 1 3 and is also written as p 3

Say: p 3 metres of ribbon is used for p gift boxes.

Remind students that p × 1 3 or 1 3 × p are algebraic expressions and p 3 is a specialised way to write these expressions. We say it is a specialised algebraic notation.

Stage: Abstract Representation

Ask: What does p represent? (The number of gift boxes)

Say: Let us find the total length of ribbon used for 8 gift boxes by substituting p with 8 in the expression ‘ p 3 ’.

Write: Total length of ribbon used for 8 gift boxes = p 3 = 8 3 = 2 2 3

Say: So, if there are 8 gift boxes, Atarangi will use a total of 2 2 3 metres of ribbon.

Let's Do Let's

Task 1 requires students to write algebraic expressions and algebraic notations in one variable involving multiplication.

Task 2 requires students to write an algebraic expression in one variable involving multiplication, and use substitution to find its value.

Let's Practise Let's Practise

Tasks 1 and 2 require students to write an algebraic expression in one variable involving multiplication.

Tasks 3 and 4 require students to use substitution to find the values of algebraic expressions in one variable involving multiplication.

1. Write an algebraic expression and notation for each of the following. MultiplyAlgebraic expressionAlgebraic notation a) s and 4 s × 4 b) 8 and c c)

Let's Practise Let's Practise q and 1 6

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. a) 4m b)

4. Find the value of each algebraic expression when n = 24.

EXPLORE

Have students go back to the word problem on CB p. 68.

Ask: Can you solve the problem now? (Answer varies.) What else do you need to know? (Answer varies.)

Students are not expected to be able to solve the problem now. They will learn more skills in subsequent lessons and revisit this problem at the end of the chapter.

Let's Learn Let's Learn

Objectives:

• Write an algebraic expression in one variable involving division

• Use substitution to find the value of an algebraic expression in one variable involving division

Resources:

• CB: pp. 74–75

• PB: p. 45

(a) Stage: Pictorial Representation

Have students look at the bags of oranges on CB p. 74.

Ask: How many bags are there? (4) Ryan puts an equal number of oranges into each bag. Do we know how many oranges he has? (No)

Stage: Abstract Representation

Say: Since Ryan puts an equal number of oranges in each of the 4 bags, we can divide the total number of oranges by the number of bags to find the number of oranges in each bag.

Ask: If there are y oranges altogether, how can we find the number of oranges in each bag? (Divide y by 4.)

Say: Since there are y oranges, we divide y by 4 to find the number of oranges in each bag.

Write: Number of oranges in each bag

= y ÷ 4

= y 4

Say: We write y ÷ 4 as y 4 . If there are y oranges altogether, there will be y 4 oranges in each bag. Remind students that y ÷ 4 is an algebraic expression and y 4 is a special way of writing y ÷ 4. We say it is a specialised algebraic notation.

(b) Stage: Abstract Representation

Say: If y = 48, we can find the number of oranges in each bag by substituting y with 48 in the expression ‘ y 4 ’.

Write: y 4 = 48 4

Elicit the answer from students. (12) Say: If there are 48 oranges altogether, there will be 12 oranges in each bag.

1.4 Writing and evaluating algebraic expressions involving division Let's Learn Let's Learn

There are 4 paper bags. Ryan puts an equal number of oranges

Let's Do Let's Do

expressionAlgebraic notation

b) If h = 36, what is the length of each side of the square?

Explain to students that if we have two variables a and b, a ÷ b is written as a b . Have students see that if both variables are the same, a ÷ a = 1.

So, a ÷ a = a a = 1.

Let's Do Let's Do

Task 1 requires students to write algebraic expressions and their corresponding algebraic notations in one variable involving division.

Task 2 requires students to write an algebraic expression in one variable involving division, and use substitution to find its value.

Let's Practise Let's Practise

Task 1 requires students to write an algebraic expression in one variable involving division.

Task 2 requires students to use substitution to find the values of algebraic expressions in one variable involving division.

1.5 Writing and evaluating algebraic expressions involving more than one operation

Let's Learn Let's Learn

Objectives:

• Write an algebraic expression in one variable involving the four operations

• Use substitution to find the value of an algebraic expression in one variable involving the four operations

Resources:

• CB: pp. 75–77

• PB: p. 46

(a) Stage: Pictorial Representation

Have students look at the pictures and read the scenario in (a) on CB p. 75.

Ask: How many boxes are there? (3) Joel puts w toy cars into each box. Do we know exactly how many toy cars there are in each box? (No) How many toy cars are left? (2)

Stage: Abstract Representation

Say: Let us express the total number of toy cars Joel has in terms of w. First, let us find the number of toy cars in 3 boxes. To find the number of toy cars in 3 boxes, we multiply the number of boxes by the number of toy cars in each box.

Write: Number of toy cars in 3 boxes

= w + w + w = 3 × w = 3w

Say: There are 3w toy cars in the 3 boxes. There are 2 toy cars left. We have to add 2 to the number of toy cars in the 3 boxes to get the total number of toy cars.

Write: Total number of toy cars = 3w + 2

Say: Joel has (3w + 2) toy cars.

Say: Let us find the number of toy cars Joel has if w = 4. We can find the answer by substituting w with 4 in the expression ‘3w + 2’.

Write: 3w + 2 = 3 × 4 + 2

Have a student work out the answer on the board.

Remind students of the rules of order of operations. In this case, we have to multiply before adding.

Say: Joel has 14 toy cars.

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. a)

1.5 Writing and evaluating algebraic expressions involving more than one operation

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 =

= 4, find the number of toy cars Joel has.

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

(b) Stage: Abstract Representation

Have students read the word problem in (b) on CB p. 75.

Ask: How much money did Daniel have? ($350) How much money did he save? ($x) What did he do with the remaining amount of money? (He spent it on 6 games.) What do we have to do? (Express the amount of money he spent on each game in terms of x.) What do we do first? (Find the amount Daniel spent on 6 games.)

Say: To find the amount of money Daniel spent on 6 games, we subtract $x from $350.

Write: Total amount spent on 6 games = $(350 – x)

Ask: How can we find the amount of money Daniel spent on each game? (Divide the amount of money spent on 6 games by 6.)

Say: We can write $(350 – x) divided by 6 as a fraction.

Write: Amount of money spent on each game = $ 350 – x 6

Say: The amount of money Daniel spent on each game is $ 350 – x 6

Say: If x = 50, we can find the amount of money Daniel spent on each game by substituting x with 50 in the expression ‘$ 350 – x 6 ’.

Write: $ 350 – x 6 = $ 350 – 50 6 Have a student work out the answer on the board. ($50)

Say: Daniel spent $50 on each game.

Let's Do Let's Do

Task 1 requires students to write an algebraic expression in one variable involving addition and division.

Task 2 requires students to use substitution to find the value of an algebraic expression in one variable involving multiplication and subtraction.

Let's Practise Let's Practise

Task 1 requires students to write an algebraic expression in one variable involving subtraction and division.

Task 2 requires students to use substitution to find the values of algebraic expressions in one variable involving the four operations.

Task 3 requires students to write algebraic expressions in one variable involving the four operations.

THINK ABOUT IT

Have students work in groups to discuss the tasks. Ask the groups to present their answers.

Explain that Joseph spent all his money on 2 books, which is to say that he spent $(y + 10) on 2 books.

So, the amount of money he spent on each book is $ y + 10 2 . Conclude that Sarah is correct.

For struggling students, get them to replace y with a number, e.g. 4, in the scenario. Guide them to realise that adding 10 after dividing y by 2 will give them a different answer.

Reiterate to students that the algebraic expression should correctly represent the information given in the problem.

Make use of the examples presented by the groups to let students understand the importance and usefulness of algebraic expressions.

1.6 Finding the value of an algebraic expression given in exponent notation

Let's Learn Let's Learn

Objective:

• Use substitution to find the value of an algebraic expression given in exponent notation

Resources:

• CB: pp. 78–79

• PB: p. 47

(a) Stage: Pictorial Representation

Draw a square of sides s centimetres on the board as shown in (a) on CB p. 78. Point to the square. Say: This is a square.

Ask: What is the length of each side of the square? (s cm)

Stage: Abstract Representation

Say: Let us find the area of the square.

Ask: What is the formula for the area of a square?

(Area of square = Length × Length)

Write: Area of the square = s × s = s2 square centimetres

Say: We write s × s as s2. The area of the square is s2 square centimetres.

(b) Stage: Abstract Representation

Say: Suppose the length of each side of the square is 18 centimetres.

Write: s = 18

Say: Let us find the area of the square.

Write: s2 = 182

Ask: What is the value of 182? (324)

Write: s2 = 182 = 324

Say: The area of the square is 324 square centimetres.

Let's Do Let's Do

Task 1 requires students to write an algebraic expression in exponent notation and find the value of the algebraic expression.

Let's Practise Let's Practise

Tasks 1 and 2 require students to write an algebraic expression in exponent notation and find the value of the algebraic expression.

Task 3 requires students to find the values of algebraic expressions given in exponent notation.

1.7 Simplifying algebraic expressions with like terms

Let's Learn Let's Learn

Objective:

• Simplify an algebraic expression in one variable with two or more like terms

Materials:

• Algebra tiles

Resources:

• CB: pp. 80–81

• PB: p. 48

Vocabulary:

• like terms

• term

(a) Stage: Concrete Experience

Have students work in groups. Distribute algebra tiles to each group.

Have students read the problem and look at the pictures of bags in (a) on CB p. 80.

Say: 3 of the bags have red apples. There are m red apples in each bag.

Have students show three long rectangular tiles, with each long rectangular tile representing the number of red apples in each bag.

Guide students to count the total number of red apples: m, 2m, 3m.

Ask: How many red apples are there in these bags? (3m)

Say: 2 of the bags have green apples. There are m green apples in each bag.

Have students show another two long rectangular tiles, with each long rectangular tile representing the number of green apples in each bag. Point out to students that the tiles used to represent the red apples and the tiles used to represent the green apples should be of the same size.

Guide students to count the total number of green apples: m, 2m.

Ask: How many green apples are there in these bags? (2m)

Say: Let us find the total number of apples. Guide students to count the total number of apples by counting the total number of rectangular tiles: m, 2m, 3m, 4m, 5m.

Stage: Pictorial Representation

Draw the part-whole bar model with 3 units as shown in (a) on the page on the board and relate it to the algebra tiles. Label each unit ‘m’ as shown on the page.

Ask: How many red apples are there? (3m) Label the 3 units ‘3m’ as shown on the page.

Say: There are 2 bags of green apples. There are m green apples in each bag. Add 2 units to the bar model and label each unit ‘m’ as shown on the page.

Ask: How many green apples are there? (2m) Label the 2 units ‘2m’ as shown on the page.

Say: From the bar model, we can see that we have to add 3m and 2m to get the total number of apples.

Stage: Abstract Representation

Write: 3m + 2m

Say: 3m and 2m are the terms of the algebraic expression ‘3m + 2m’.

Write: term

Explain that 3m is m + m + m and 2m is m + m

Write: 3m + 2m = m + m + m + m + m = 5m

Say: The total number of apples is 5m. When we simplify the expression ‘3m + 2m’, we get ‘5m’.

Have students see that the terms 3m and 2m have the same variable and exponent. Explain to them that such terms are called like terms.

Write: like terms

Say: Let us now find how many more red apples than green apples there are. We can draw a comparison bar model to show the difference between the number of red and green apples.

(Continued on the next page)

Draw the comparison bar model as shown on CB p. 80.

Say: From the bar model, we can see that we have to subtract 2m from 3m to find the difference in the number of apples.

Write: 3m – 2m = m

Say: There are m more red apples than green apples.

(b) Stage: Concrete Experience

Have students continue to work in groups and use the algebra tiles from (a). Write the algebraic expression in (b) on CB p. 81 on the board.

Ask: How many terms are there in this algebraic expression? (Three) What are the three terms?

(6x, 4x and 3x)

Say: We want to simplify the algebraic expression.

Ask: What is the first term of the expression? (6x)

Have students use six long rectangular tiles to represent 6x. Explain to students that each long rectangular tile represents x, an unknown number.

Say: We want to subtract 4x from 6x.

Have students remove 4 long rectangular tiles.

Ask: How many rectangular tiles are left? (2) Remind students that the 2 rectangular tiles represent 2x.

Say: Next, we add 3x.

Have students add 3 long rectangular tiles to the 2 long rectangular tiles.

Ask: How many rectangular tiles are there altogether now? (5) What do we get when we simplify the expression ‘6x – 4x + 3x’? (5x)

Stage: Pictorial Representation

Draw 6 bags and label each bag ‘x’ as shown in (b) on the page and relate it to the algebra tiles.

Say: Each bag has x apples. We have 6 bags, so we have 6x apples.

Write: 6x

Cross out 4 of the bags.

Ask: If we remove 4 bags of x apples from 6 bags of x apples, what is the algebraic expression? (6x – 4x)

Write: 6x – 4x

Draw another 3 bags of x apples on the board. Ask: If we add 3 more bags of x apples, what is the algebraic expression? (6x – 4x + 3x)

Write: 6x – 4x + 3x

Stage: Abstract Representation

Say: Let us simplify the expression ‘6x – 4x + 3x’. We should add and subtract from left to right. So, we subtract 4x from 6x first.

Ask: What is 6x – 4x? (2x)

Write: 6x – 4x + 3x = 2x + 3x

Ask: What is 2x + 3x? (5x)

Write: 6x – 4x + 3x = 2x + 3x = 5x

Say: When we simplify the expression ‘6x – 4x + 3x’, we get ‘5x’.

Reiterate to students that when an algebraic expression with like terms has only addition and subtraction operations, we work from left to right. Point out to them that it is similar to how we carry out the order of operations involving whole numbers.

Let's Do Let's Do and Let's Practise Let's Practise

Task 1 requires students to simplify algebraic expressions in one variable with two or more like terms.

Unit 2: Algebraic Equations

2.1 Understanding equations

Let's Learn Let's Learn

Objectives:

• Understand what an equation is

• Identify an algebraic equation

• Identify a linear equation

Materials:

• 1 pan balance

• 12 connecting cubes of the same mass

Resources:

• CB: pp. 82–84

• PB: p. 49

Vocabulary:

• algebraic equation

• equation

• linear equation

• variable

(a) Stage: Concrete Experience

Set up a pan balance. Place a block of 2 connecting cubes on the left pan facing students as shown in (a) on CB p. 82.

Ask: How many cubes are there on the left pan? (2)

Write: 2

Add a block of 4 connecting cubes to the left pan.

Say: There are 2 + 4 cubes on the left pan.

Write: 2 + 4

Ask: Is the scale balanced? (No)

Say: In order to balance the scale, we have to put connecting cubes on the right pan. Place one connecting cube at a time onto the right pan until the scale is balanced.

Ask: Is the scale balanced? (Yes) How many cubes are on the right pan? (6)

Say: 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.

Stages: Pictorial and Abstract Representations

Have students look at the scale on the page and relate the scale to the earlier activity.

Write: 2 + 4 = 6

Say: 2 + 4 = 6 is an equation. An equation is a mathematical sentence where the values on both sides of the equal sign are the same.

Write: equation

Guide students to relate the left side of the equation, 2 + 4, to the pictorial representation — 2 represents the block of 2 red cubes, 4 represents the block of 4 yellow cubes. There are 6 cubes altogether on the left pan.

Unit 2 Algebraic Equations

You will learn 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

2.1 Understanding equations

Let's Learn 2 4 6

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

Next, guide students to relate the right side of the equation, 6, to the pictorial representation — 6 represents the block of 6 orange cubes. Say: The left side, 2 + 4, has the same value as the right side, 6.

Write: 4 + x = 10 3z – 2 = 4

Say: 4 + x = 10 and 3z – 2 = 4 are also equations. These equations have an unknown number that is represented by a letter. Such equations are called algebraic equations.

Write: algebraic equation

(b) Stage: Abstract Representation

Write: 4 – 3p 2q3 + 5

2p + 7 = 4 6 – q2 = 2

Say: 4 – 3p and 2q3 + 5 are algebraic expressions. Algebraic expressions do not have equal signs. 2p + 7 = 4 and 6 – q2 = 2 are algebraic equations. Algebraic equations have equal signs.

Ask a student to write two algebraic expressions on the board. Then, ask another student to write two algebraic equations on the board.

Say: The letters representing unknown numbers in the algebraic expressions or equations are called variables.

Write: variable

Guide students to identify the variables in the algebraic expressions and equations.

(c) Stage: Abstract Representation

Point to the algebraic equations 2p + 7 = 5 and 4 – 3p = 10. Have students identify the exponent of the variable. Remind students that p = p1

Say: When the exponent of the variable in an algebraic equation is 1, we also call it a linear equation.

Write: linear equation

Say: 2p + 7 = 5 and 4 – 3p = 10 are linear equations.

Write: 2q3 + 5 = 7 and 6 – q2 = 2

Have students identify the exponent of q in each equation above and conclude that these 2 equations are not linear equations.

Let's Do Let's

Task 1 requires students to identify algebraic equations.

Task 2 requires students to identify the variables in algebraic expressions and equations.

Let's Practise Let's Practise

Task 1 requires students to identify algebraic equations.

Variables are letters which represent unknown quantities or amounts in algebraic expressions or equations. c) When the exponent of the variable in an

Task 2 requires students to identify algebraic expressions and equations, and their variables.

Task 3 requires students to identify linear equations.

2.2 True and false equations

Let's Learn Let's Learn

Objective:

• Check whether a given number is a solution to an equation using substitution

Resources:

• CB: pp. 84–85

• PB: p. 50

Vocabulary:

• false

• solution

• true

(a) Stage: Abstract Representation

Write: 6 + x = 9

Say: Substitute x with 3 in the equation.

Write: 6 + x = 9

6 + 3 = 9 9 = 9

Ask: What is the value of the expression on the left side of the equal sign? (9) What is the value of the expression on the right side of the equal sign? (9)

Say: When x = 3, the values of the expressions on both sides of the equal sign are equal. We say the equation 6 + x = 9 is true for x = 3. This means that x = 3 is a solution of the equation 6 + x = 9.

(b) Stage: Abstract Representation

Write: 6x = 9

Say: Substitute x with 3 in the equation.

Write: 6x = 9

6 × 3 = 18 18 ≠ 9

Ask: What is the value of the expression on the left side of the equal sign? (18) What is the value of the expression on the right side of the equal sign? (9)

Point to ‘18 ≠ 9’ on the board.

Say: This is read as ‘18 is not equal to 9’. When x = 3, the values of the expressions on both sides of the equal sign are not equal. We say the equation 6x = 9 is false for x = 3. This means that x = 3 is not a solution of the equation 6x = 9.

Variable(s): Variable(s):

2.2 True and false equations Let's Learn Learn

Let's Do Let's and Let's Practise Let's Practise

Tasks 1 and 2 require students to check whether a given number is a solution to an equation using substitution.

2.3 Solving algebraic equations using the guess and check method

Let's Learn Let's Learn

Objective:

• Use the guess and check method to solve an algebraic equation

Resources:

• CB: pp. 86–88

• PB: p. 51

Vocabulary:

• solve

(a) Stage: Abstract Representation

Write: 9 + m = 16

Say: Let us try to find a value for m that makes the equation true.

Explain to students that to find the value of m, we need to make guesses and this method of finding the value of the variable by trial and error is the guess and check method.

Say: In the guess and check method, we usually start with small numbers. Let us start with m = 2.

Write: m = 2

9 + m = 9 + 2 = 11

Ask: What is the value on the right side of the equation? (16) Is 11 equal to 16? (No)

Write: 11 < 16, so m ≠ 2.

Ask: Should our next guess be greater than or less than 2? (Since 11 is less than 16, our next guess has to be greater than 2.)

Repeat the above procedure using m = 8 and conclude that since the answer, 17, is not equal to 16, the value of m cannot be 8.

Ask: Should our next guess be greater than or less than 8? (Since 17 is greater than 16, our next guess should be less than 8.)

Say: Since 17 is close to 16, the value of m must be close to 8. Let us try m = 7.

Ask a student to substitute m with 7 in the expression ‘9 + m’ and work out the answer on the board.

Have students see that when m = 7, we get the answer 16. This is the same as the value on the right side of the equal sign.

Say: So, 7 is the correct value of the variable, m. m = 7 is a solution of the equation 9 + m = 16.

To solve an algebraic equation, we find the value of the variable in the equation. We have found the value of m in the equation, so we have solved the equation.

(b) Stage: Abstract Representation

Follow the procedure in (a).

(c) Stage: Abstract Representation

Follow the procedure in (a) on TG p. 87.

Let's Do Let's Do

Tasks 1 and 2 require students to use the guess and check method to solve algebraic equations.

Task 3 requires students to determine if a given value of a variable is a solution of an algebraic equation.

Let's Practise Let's

Tasks 1 and 2 require students to determine if a given value of a variable is a solution of an algebraic equation.

Task 3 requires students to use the guess and check method to solve algebraic equations.

Let's Learn Let's Learn

Objective:

• Use the balance method to solve an algebraic equation

Materials:

• 1 pan balance

• 3 lightweight opaque plastic bags

• 20 connecting cubes of the same mass

Resources:

• CB: pp. 89–93

• PB: p. 52

(a) Stage: Concrete Experience

Set up the pan balance as shown in (a) on CB p. 89. Use a bag of 4 connecting cubes to represent x.

Say: The scale is balanced. We want to find out the number of cubes in the bag. We can do so by first removing 6 cubes from the left pan.

Remove 6 connecting cubes from the left pan.

Ask: Is the scale still balanced? (No)

Say: For the scale to stay balanced, we need to remove some cubes from the right pan as well. Remove one connecting cube at a time from the right pan until the scale is balanced.

Ask: How many cubes have been removed from the right pan? (6) How many cubes remain on the right pan? (4)

Next, have students recall that in an equation, the values on the left side and the right side of the equal sign are the same.

Say: Since the bag of cubes on the left pan balances the remaining 4 cubes on the right pan, this means that the number of cubes in the bag must be the same as the number of cubes on the right pan.

Take out the 4 cubes from the bag and reveal them to students to show that this is true.

Say: There are 4 cubes in the bag. Return the 4 cubes to the bag and revert to the original set up at the beginning of the activity.

Stages: Pictorial and Abstract Representations

Say: We can draw a bar model to represent the number of cubes on the scale. There are x cubes in the bag. There are x cubes and 6 cubes on the left pan.

Draw a part-whole bar model with two parts as shown in the thought bubble in (a) on the page but do not label the total.

Say: There are 10 cubes on the right pan. Since the scale is balanced, the number of cubes on

the left pan is equal to the number of cubes on the right pan.

Label the total, 10, in the bar model.

Say: We can write an equation to represent the bar model.

Write: x + 6 = 10

Say: To solve the equation, we have to find the value of x that will make the values on both sides of the equal sign the same. Let us subtract 6 from both sides of the equation.

Write: x + 6 = 10

x + 6 – 6 = 10 – 6

Relate the subtraction of 6 from both sides of the equation to the removal of 6 cubes from both pans in the earlier activity.

Write: x + 6 = 10

x + 6 – 6 = 10 – 6 x = 4

Say: x = 4 is a solution of the equation x + 6 = 10. We can check if x = 4 is a solution of the equation by substituting x with 4 in x + 6. Ask a student to substitute x with 4 in the expression x + 6 on the board. He/she should get the answer 10.

(b) Stage: Concrete Experience

Set up the pan balance as shown in (b) on CB p. 90. Use 3 bags containing 2 connecting cubes each to represent 3x

Say: All the bags contain the same number of cubes. We want to find the number of cubes in one bag. First, we remove 4 cubes each from the left pan and the right pan to keep the scale balanced.

Remove 4 connecting cubes each from the two pans.

Say: There are 3 bags of cubes on the left pan, so we put the cubes on the right pan into 3 equal groups.

Put the 6 cubes on the right pan into 3 groups of 2.

Say: Each bag of cubes on the left pan is equal to each group of cubes on the right pan. We want to find the number of cubes in one bag, so we remove 2 bags from the left pan and 2 groups of cubes from the right pan. Remove 2 bags of cubes from the left pan and 2 groups of 2 cubes from the right pan.

Ask: Is the scale balanced? (Yes) What is remaining on the left pan? (1 bag of cubes) What is remaining on the right pan? (1 group of cubes or 2 cubes)

Explain to students that the bag of cubes on the left pan balances the 2 cubes remaining on the right pan, so the number of cubes in the bag must be the same as the number of cubes on the right pan.

Take out the 2 cubes from the bag and reveal them to students to show that this is true.

Say: There are 2 cubes in the bag.

Revert the scale to the original set up at the beginning of the activity.

Stages: Pictorial and Abstract Representations

Say: We can draw a bar model to represent the number of cubes on the scale. There are x cubes in each bag. There are 3x cubes and 4 cubes on the left pan.

Draw a part-whole bar model as shown in the thought bubble in (b) on CB p. 90 but do not label the total.

Say: There are 10 cubes on the right pan. Since the scale is balanced, the number of cubes on the left pan is equal to the number of cubes on the right pan.

Label the total, 10, in the bar model.

Say: We can write an equation to represent the bar model.

Write: 3x + 4 = 10

Say: To solve the equation, we first subtract 4 from both sides of the equation.

Write: 3x + 4 = 10

3x + 4 – 4 = 10 – 4

Relate the subtraction of 4 from both sides of the equation to the removal of 4 cubes from both pans in the earlier activity.

Write: 3x + 4 = 10

3x + 4 – 4 = 10 – 4

3x = 6

Say: We can get x on one side of the equation by dividing both sides of the equation by 3.

Write: 3x ÷ 3 = 6 ÷ 3

Relate the division by 3 on both sides of the equation to the putting of cubes into 3 equal groups on both pans in the earlier activity. Write ‘x = 2’ in the next line of the working.

Say: x = 2 is a solution of the equation 3x + 4 = 10. We can check if x = 2 is a solution of the equation by substituting x with 2 in 3x + 4.

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.

Check: When

3f = 3f = f = Let's Do 15 10 3 5

1. Use the balance method to solve these equations. Write +

a) g + 11 = 16

g + 11 = 16 g = b) w – 7 = 9

w – 7 = 9 w =

3f – 19 = 17

3f – 19 = 17

Ask a student to substitute x with 2 in the expression 3x + 4 on the board. He/she should get the answer 10.

(c) Stage: Abstract Representation

Follow the procedure in (b). Guide students to solve the equation by first adding 5 to both sides of the equation and then dividing by 5 on both sides. Write the steps on the board.

Have students check if q = 3 is a solution of the equation by substituting q = 3 into 5q – 5 and checking if they get 10 as the answer.

Let's Do Let's Do

Task 1 requires students to use the balance method to solve algebraic equations.

David is correct. When David uses the balance method to solve 2x – 10, he adds 10 to both sides of the equation.

When solving an algebraic equation using the balance method, we perform the same operation on 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.

I

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.

THINK ABOUT IT

Have students work in groups to discuss the tasks. Ask the groups to present their answers.

Have students observe that Sarah adds 10 to one side of the equation whereas David adds 10 to both sides of the equation. Get them to substitute 5 and 10 for x in 2x – 10 to see if they get the answer 10. Conclude that David is correct.

Reiterate to students that when solving an algebraic equation using the balance method, we perform the same operation on both sides of the equation.

Make use of the examples presented by the groups to let students understand the importance and usefulness of knowing how to solve algebraic equations.

Task 1 requires students to use the balance method to solve algebraic equations.

2.5 Reading and plotting points in all four quadrants

Let's Learn Let's Learn

Objectives:

• Read and plot coordinates in all four quadrants of the coordinate plane

• Recognise that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes

Materials:

• Coordinate Plane (BM11.1): 1 copy per student, 1 enlarged copy for demonstration

Resources:

• CB: pp. 94–99

• PB: pp. 53–55

Vocabulary:

• ordered pair

• origin

(a) Stages: Pictorial and Abstract Representations

Have students observe the axes on the coordinate plane in (a) on CB p. 94.

Point to the origin on the coordinate plane.

Say: The x-axis and y-axis cross each other at a right angle. The point at which the axes cross is called the origin.

Write: origin

Say: We can describe the location of any point on the coordinate plane by describing the distance of the point from the origin along the x-axis and the y-axis. The distance of the point from the y-axis is its x-coordinate.

Write: x-coordinate

Say: The distance of the point from the x-axis is its y-coordinate.

Write: y-coordinate

Say: To describe the location of a point, we use an ordered pair. An ordered pair is made up of two numbers, the x-coordinate and the y-coordinate of that point.

Write: ordered pair

Point to the origin.

Say: The coordinates of the origin are (0, 0).

Write: (0, 0)

Say: The x-axis and y-axis divide the coordinate plane into four sections called quadrants. The quadrants are numbered anticlockwise and are called the 1st quadrant, 2nd quadrant, 3rd quadrant and 4th quadrant. Have students look at the points shown on the coordinate plane.

Say: We use coordinates to describe the location of a point on a coordinate plane. The signs of the

Let's Learn Let's 2.5 Reading and plotting points in all four quadrants

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

x-coordinate and y-coordinate determine the quadrant in which a point lies.

Have students look at point A on the coordinate plane on CB p. 94.

Say: Point A is in the 1st quadrant. Let us find the coordinates of point A.

Have students place their finger at the origin and move it along the x-axis until they reach the vertical gridline on which point A lies.

Ask: What is the x-coordinate of the point where point A is located? (3)

Have students move their finger upwards from 3 on the x-axis along the gridline to the point where point A is located.

Ask: What is the y-coordinate of the point? (2)

Say: The coordinates of point A are (3, 2).

Write: 1st quadrant, point A (3, 2)

Say: When writing the coordinates of a point, we write the x-coordinate first, followed by the y-coordinate within brackets.

Write: (x, y)

Have students look at point B on the coordinate plane on CB p. 94.

Say: Point B is in the 2nd quadrant. Let us find the coordinates of point B.

(Continued on the next page)

Using the same procedure as for point A above, guide students to find the coordinates of point B.

Have students find the coordinates of points C and D on their own.

Add ‘+’ and ‘–’ signs next to the coordinates of points A to D written on the board. Guide students to observe the relationship between the signs of the x- and y-coordinates and the quadrants. Lead students to conclude that the points in the 1st quadrant have positive x- and y-coordinates, the points in the 2nd quadrant have a negative x-coordinate and a positive y-coordinate, the points in the 3rd quadrant have negative x- and y-coordinates, and the points in the 4th quadrant have a positive x-coordinate and a negative y-coordinate.

Write: Point E (–1, –2)

Ask: In which quadrant will point E lie? (3rd quadrant)

Say: We can tell that point E will lie in the 3rd quadrant because both its x- and y-coordinates are negative.

Name the coordinates of some points in the other quadrants and have students identify the quadrant in which each point will lie.

(b) Stages: Pictorial and Abstract Representations Have students look at points P and Q on the coordinate plane in (b) on CB p. 95.

Ask: How are points P and Q related? (They are reflections of each other.)

Say: Point Q is the reflection of point P across the x-axis.

Ask: What are the coordinates of point P? (3, 2)

Write: Point P (3, 2)

Ask: What are the coordinates of point Q? (3, –2)

Write: Point Q (3, –2)

Ask: Compare the ordered pairs. What do you notice? (The x-coordinates are the same. The y-coordinates have the same numerical value but different signs.)

Remind students that the distance of a point from the y-axis is its x-coordinate.

Say: The x-coordinates of points P and Q are the same. So, points P and Q are the same distance from the y-axis.

Remind students that the distance of a point from the x-axis is its y-coordinate.

Say: The y-coordinates of points P and Q 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 of the x-axis. Repeat the above procedure to guide students to realise that point S is the reflection

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.

of point P across the y-axis and their coordinates differ only by signs.

Ask students to provide examples of other points that are related by reflections across the x-axis and/or y-axis.

c) Plot and label these

Point A B

x-coordinate –3–4

y-coordinate 4 –5

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)?

(c) Stages: Pictorial and Abstract Representations Distribute a copy of Coordinate Plane (BM11.1) to each student.

Stick an enlarged copy of Coordinate Plane (BM11.1) on the board.

Say: We can plot points on a coordinate plane if we know their coordinates. Let us plot point A at (–3, 4) on the coordinate plane. Write: A (–3, 4)

Say: Since the x-coordinate of point A is negative and the y-coordinate is positive, point A lies in the 2nd quadrant of the coordinate plane.

Ask: What is the x-coordinate of point A? (–3) Start from the origin and move 3 units to the left along the x-axis on the coordinate plane. Have students do the same on their coordinate plane.

Ask: What is the y-coordinate of point A? (4) Move 4 units up and then mark the point. Have students do the same on their coordinate plane.

Say: This is point A at (–3, 4).

Repeat the above procedure and guide students to plot point B (–4, –5) on the coordinate plane. Have students plot point C (4, –6) on their own.

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.

–4–3–2–1 1234

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) H (–1, 3)

2. Write x-axis or y-axis

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).

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).

Let's Do Let's

Task 1 requires students to read and plot coordinates in all four quadrants of the coordinate plane.

Task 2 requires students to recognise that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

1. Look at the coordinate plane below.

a) Write the coordinates of each point.

3.

Plot and label these points on the coordinate plane below. Point ABCDEFGH x-coordinate 1511–4–246 y-coordinate 226–2–2–4–4–2

(–4, 3) (2, 5) (–3, 0)

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, –2) (4, –4) (3, 2)

2. Write the coordinates of each point.

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

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

Let's Practise Let's Practise

(–8, 0) (5, 1) (0, 0) (–2, –3)

Task 1 requires students to read and plot coordinates in all four quadrants of the coordinate plane.

Task 2 requires students to recognise that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

Task 3 requires students to plot coordinates in all four quadrants of the coordinate plane. Students are also expected to identify the quadrant in which each point lies.

2.6 Linear patterns and equations

Let's Learn Let's Learn

Objectives:

• 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

Resources:

• CB: pp. 100–103

• PB: pp. 56–57

Vocabulary:

• constant

• linear pattern

• sequence

(a) Stage: Abstract Representation

Have students read the scenario in (a) on CB p. 100.

Copy the table on the page on the board but leave out the amount of savings in the bottom row.

Ask: How much savings does William have in week 0? ($5) How much does William save each week? ($10) How much savings does William have in week 1? ($15)

Fill the ‘Amount of savings’ row in the table as students answer each question.

Say: $10 was added to the amount of savings in week 0 to get $15.

Write ‘Week 0 = $5’ and ’Week 1 = $10 + $5 = $15’ below the table. This will allow students to start to observe the repeated addition within the pattern as it grows.

Ask: How much savings does William have in week 2? ($25)

Say: $10 was added to the amount of savings in week 1 to get $25.

Write ’Week 2 = $10 + $10 + $5 = $25’ to continue to allow students to observe the repeated addition within the pattern. Repeat the above procedure for the remaining weeks.

Ask: From the table, how much does William’s savings increase each week? ($10)

Say: We can see from the pattern that the numbers or the amount of savings are arranged in a particular order. This is called a sequence. 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.

Ask: So, what is the constant increase in this linear pattern? ($10)

Draw arrows from one column to the next in the table and write ‘+ $10’ below each arrow.

2.6

Linear patterns and equations

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?

Number of weeks 01234

Amount of savings $5$15$25$35$45

$10

We notice that the amount of money saved

The savings increase by $10 each

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.

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.

Refer to the page to see how the table should look like. Next, direct students’ attention to the equations written below the table.

Say: We know William’s savings increase by $10 each week. Let’s look at the equations to study the relationship between the number of $10 and the number of weeks.

Ask: How can we get the total amount being added to $5 each week? (By multiplying $10 by the number of weeks)

Write the relationship between the amount of savings and the number of weeks as shown in the thought bubble at the bottom of the page on the board.

Say: Let’s define the variables. Let x represent the number of weeks and y represent the amount of savings. Here, y is the dependent variable and x is the independent variable because the amount of savings (y) depends on the number of weeks (x). We can write an equation to show the relationship between the two variables.

Write: y = 10x + 5

Ask: How can we use the equation y = 10x + 5 to find how much money will William have after saving for 10 weeks? (Week 10 = 10 × 10 + 5)

How do we find the answer? (Multiply 10 by 10 to get 100, then add 5 to get 105.) How much money does William need to buy the gift for his mother? ($100) Will William be able to buy the gift after saving for 10 weeks? (Yes)

(b) Stage: Abstract Representation

Have students look at the pattern in (b) on CB p. 101.

Copy the table on the page on the board but do not fill the table yet.

Ask: What is the first term in the pattern? (0) Fill in the position '1st' and the number ‘0’ in the table on the board.

Continue to ask students similar questions and fill in the table on the board as students answer each question.

Have students observe the numbers in the table and see that the numbers increase by 1 each time. Draw arrows from one column to the next in the table and write ‘+ 1’ below each arrow. Refer to the page to see how the table should look like.

Ask: What is the constant increase in the pattern? (1)

Have students look at the table again. Explain to students that we can find a relationship between each number and its position. Have them observe that each number is 1 less than its position number.

Write: 1st ter m = 1 – 1 = 0

2nd ter m = 2 – 1 = 1

3rd term = 3 – 1 = 2

4th ter m = 4 – 1 = 3

5th ter m = 5 – 1 = 4

Say: Let x represent the position and y represent the number. We can write an equation to show the relationship between the two variables.

Write: y = x – 1

Ask: How can we use the equation y = x – 1 to find the 25th term? (25th term = 25 – 1) So, what is the 25th term? (24)

Let's Do Let's

Task 1 requires students to identify whether a linear pattern shows a constant increase or decrease, and then find the constant increase or decrease.

2.

a) Complete the table.

b) Write

height of the 8th shelf from the ground? centimetres

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.

= x + 5 y = 5x

Task 2 requires students to complete a linear pattern in a table, identify whether the pattern shows a constant increase or decrease, find the constant increase or decrease use variables and algebraic notation to represent the rule of the linear pattern in an equation, and use the equation to make a conjecture.

Let's Practise Let's Practise

Task 1 requires students to identify whether a linear pattern is increasing or decreasing, and then find the constant increase or decrease.

Task 2 requires students to identify the correct equation for the given linear pattern.

3. Look at the pattern below.

a) Draw the figure that comes next in the pattern.

Figure 1 Figure 2 Figure 3Figure 4

b) Complete the table.

Figure 1234

The constant in the pattern is

Number of triangles 1 increase

c) Write increase or decrease in the first blank. Then, find the constant increase or decrease.

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.

y = x + 1 20 101 2345

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

11: Algebra

Task 3 requires students to draw the figure that comes next in the linear pattern, complete the pattern in a table, identify whether the pattern shows a constant increase or decrease, find the constant increase or decrease, use variables and algebraic notation to represent the rule of the linear pattern in an equation, and use the equation to make conjectures.

Unit 3: Problem Solving

3.1 Word problems involving algebraic expressions

Let's Learn Let's Learn

Objective:

• Solve word problems using algebraic expressions

Resources:

• CB: pp. 104–105

• PB: pp. 58–60

Have students read the word problem on CB p. 104.

1. Understand the problem. Pose the questions in the thought bubble in step 1.

2. Plan what to do.

Say: We can write an algebraic expression and then substitute the given value of m into the algebraic expression to find the length of lace Amelia had left.

3. Work out the Answer

Say: We multiply m by 3 to find the total length of lace Amelia has.

Write: 3 × m

Elicit the answer from students. (3m)

Write: 3 × m = 3m

Say: We subtract the length of lace used from 3m centimetres to find the length of lace Amelia had left.

Write: Length of lace left = (3m – 45) cm

Say: The length of lace Amelia had left is (3m – 45) centimetres.

Say: We want to find the length of lace Amelia had left when m = 24.

Write: When m = 24, 3m – 45 = (3 × 24) – 45

Have a student work out the answer on the board.

Ask: What is the length of lace left? (27 cm) Say: Amelia had 27 centimetres of lace left.

4. Check if your answer is correct.

Say: We can add the length of the lace left to the length of the lace used to check our answer.

Write: 27 + 45 = ______

Elicit the answer from students. (72)

Say: So, the total length of lace is 72 centimetres. Now, we find the length of each piece of lace.

Unit 3 Problem Solving

3.1 Word problems involving algebraic expressions

Let's Learn

Have a student work out the length of each piece of lace.

Write: 72 ÷ 3 = ______

Elicit the answer from students. (24) Say: Each piece of lace was 24 centimetres long. Since m = 24, we can conclude that our answer is correct.

5. + Plus Solve the problem in another way. Have students try to solve the problem in a different way.

Have 1 or 2 students share their methods. If students are unable to solve the problem in a different way, explain the method shown on CB p. 105.

Ask: Which method do you prefer? Why? (Answers vary.)

Let's Do Let's

Task 1 requires students to solve a word problem using an algebraic expression.

Let's Practise Let's Practise

Tasks 1 to 3 require students to solve word problems using algebraic expressions.

3.2 Word problems involving algebraic equations

Let's Learn Let's Learn

Objective:

• Solve word problems by forming an algebraic equation

Resources:

• CB: pp. 106–109

• PB: pp. 61–62

1. Have students read the word problem on CB p. 106

1. Understand the problem. Pose the questions in the thought bubble in step 1.

2. Plan what to do. Point out to students that they can form an equation in terms of z to solve the problem.

3. Work out the Answer

Say: The number of guppies and angelfish is 38. So, z and 21 make 38. Write: z + 21 = 38

Guide students to solve the equation by subtracting 21 from both sides of the equation.

Ask: What is the value of z? (17) Say: So, Evan has 17 guppies.

4. Check if your answer is correct. Guide students to check if they get 38 when they substitute z with 17 in the expression z + 21.

5. + Plus Solve the problem in another way. Have students try to solve the problem in a different way. Have 1 or 2 students share their methods. If students are unable to solve the problem in a different way, explain the method shown on CB p. 106.

Ask: Which method do you prefer? Why? (Answers vary.)

2. Have students read the word problem on CB p. 107

1. Understand the problem. Pose the questions in the thought bubble in step 1.

2. Plan what to do.

Point out to students that they can form an equation in terms of y to solve the problem.

3. Work out the Answer

Ask: How many eggs did the baker buy? (5y)

Say: The baker had 5y eggs. He used 12 eggs and had 18 eggs left. So, 5y take away 12 is 18.

Write: 5y – 12 = 18

Guide students to solve the equation by first adding 12 to both sides of the equation and then dividing by 5 on both sides.

Ask: What is the value of y? (6)

Say: So, there were 6 eggs in each carton at first.

4. Check if your answer is correct. Guide students to check if they get 18 when they substitute y with 6 in the expression 5y – 12.

5. + Plus Solve the problem in another way. Have students try to solve the problem in a different way. Have 1 or 2 students share their methods. If students are unable to solve the problem in a different way, explain the method shown on CB p. 108. Ask: Which method do you prefer? Why? (Answers vary.)

Let's Do Let's Do

Tasks 1 to 3 require students to solve word problems by forming algebraic equations.

Let's Practise Let's Practise

Tasks 1 to 4 require students to solve word problems by forming algebraic equations.

CREATE YOUR OWN

Have students work in pairs. Get students to create a word problem and exchange the word problem with their partner. Ask students to solve the word problem from their partner. Have a few pairs of students present their work. They should first explain how they decide what numbers to use and their partner has to explain the solution.

Students should fill in the blanks with whole numbers. They can solve the word problem by forming an algebraic equation.

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 Practise hhhhhh

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?

CREATE YOUR OWN

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?

Kim planted 16 pots of Kowhai plants.

11:

Let's Learn Let's Learn

Objective:

• Solve non-routine problems involving algebra using the strategy of looking for a pattern

Materials:

• 40 square tiles per group

• 1 copy of Maths Journal (BM11.2) per student

Resource:

• CB: pp. 110–112

Have students read the problem on CB p. 110. Copy the figures on the page on the board.

1. Understand the problem. Pose the questions in the thought bubble in step 1.

2. Plan what to do.

Say: We can count the number of coloured and non-coloured squares in each figure and list the numbers in a table. This will help us look for a pattern to find the relationship between the columns.

3. Work out the Answer.

Draw the table as shown in step 3 on the page on the board but do not fill in the numbers of the coloured squares.

Ask: How many coloured squares are there in figure 1? (12)

Write the answer in the table. Repeat the above question for figure 2 and figure 3.

Say: The number of coloured squares starts at 12 and increases by 4 each time.

Guide students to relate the number of coloured squares, multiples of 4 and the figure number.

Write:

of

1 12 = 3 × 4

Explain that figure 1 has 12 coloured squares and 12 = 3 × 4 = (1 + 2) × 4. Write ‘12 = 3 × 4 = (1 + 2) × 4’ in the row for figure 1 in the table. Similarly, guide students to write ‘16 = 4 × 4 = (2 + 2) × 4’ for figure 2 and ‘20 = 5 × 4 = (3 + 2) × 4’ for figure 3.

Ask: Which number in each number sentence corresponds to the figure number in each row in the table? (The number that is added to 2, within the parentheses.) How many coloured squares would figure y have? ((y + 2) × 4)

Write: Figure y: (y + 2) × 4

Ask: How do we find the number of coloured squares in figure 8? (Substitute y = 8 into the algebraic expression (y + 2) × 4.)

Write: When y = 8, (y + 2) × 4 = (8 + 2) × 4

Elicit the answer from students. (40)

Say: The number of coloured squares in figure 8 is 40.

4. Check if your answer is correct. Have students work in groups. Distribute 40 square tiles to each group. Ask students to use the square tiles to form figure 8. Tell students to set aside 4 squares for the corners.

Ask: How many squares are left? (36) How do we find the number of squares on each side? (Divide 36 by 4.)

Write: 36 ÷ 4 = _____

Elicit the answer from students. (9)

Ask: How can we find the number of squares along each side of figure 8? (Add 2 to 9.)

Write: 9 + 1 + 1 = Elicit the answer from students. (11)

Say: The number of coloured squares along each side of figure 8 is 11. The answer is correct.

5. + Plus Solve the problem in another way. Have students try to solve the problem in a different way.

Have 1 or 2 students share their methods. If students are unable to solve the problem in a different way, explain the method shown on CB p. 111.

Ask: Which method do you prefer? Why? (Answers vary.)

Strategy: Look for a pattern

Number of twenty-dollar notes

Number of ten-dollar notes Total amount 13 0 $20 × 13 + $10 ×

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

Let's Do Let's

Guide students to make a table listing a few combinations of twenty-dollar notes and ten-dollar notes and calculate the total amount of money. Have students write an algebraic expression to represent the total amount for x twenty-dollar notes. From the table, students should observe that the total amount decreases by $10 for every twenty-dollar note that is replaced by a ten-dollar note and use this pattern to work out the number of ten-dollar notes and twenty-dollar notes that make up $190.

EXPLORE

Have students go back to the word problem on CB p. 68. Get them to write down in column 3 of the table what they have learnt that will help them solve the problem, and then solve the problem.

Have a student present his/her work to the class.

Maths Journal Maths Journal

Have students work on the tasks in Maths Journal (BM11.2) independently to check and reinforce their understanding.

Use the rubric provided on page 324 of the blackline masters to score students’ work.

Practice Book Chapter 11: Answers

Exercise 1.1

1. a) 9 + d or d + 9

b) 12 – h

2. 4 + a or a + 4

3. s – 5

4. 6 + r or r + 6

5. 3 + p or p + 3

6. (w – 2) kilograms

Exercise 1.2

1. a) (15 – b) oranges were left in the basket.

b) 3 oranges were left in the basket.

2. a) 8

b) 1

c) 14

d) 7

Exercise 1.3

1. a) e × 6 or 6 × e; 6e

b) 12 × f; 12f

c) g × 1 12 ; g 12

2. a) Cindy used 10m beads altogether.

b) She used 90 beads altogether.

3. a) w 10 cakes were given to all the children.

b) 3 cakes were given to all the children.

4. a) 21

b) 42

c) 35

d) 56

5. a) 5

b) 2

c) 1

Exercise 1.4

1. a) The mass of sugar in each bag is 27 m kilograms.

b) The mass of sugar in each bag is 9 kilograms.

2. a) 5

b) 3

c) 1

d) 2 1 2

Exercise 1.5

1. a) Claire had $(8d + 100) at first.

b) Claire had $116 at first.

2. a) 18

b) 4

c) 9

d) 13

Exercise 1.6

1. a) 16 b) 27

c) 100 d) 137

e) 53 f) 45

2. a) The area of the rectangle is 2p2 square centimetres.

b) The area of the rectangle is 32 square centimetres.

Exercise 1.7

1. a) 2a b) 5n

c) –y d) 27z

e) c f) 7z

g) 4p h) 8r

i) 4b j) –3h

Exercise 2.1

1. Algebraic equation Algebraic expression Variable

a) 6e + 12 ✓ e

b) 5b = 10 ✓ b

c) 9c + 46 19 ✓ c

d) 6x2 = 24 ✓ x

2. Algebraic equation Linear equation

a) 24 = 3r3

b) 23 + 7n = 9 ✓

c) g2 – 1 = 8

d) 4k + 30 = –14 ✓

Exercise 2.2

1. a) false

b) true

2. e = 7 is a solution to the equation 9e – 54 = 9.

Exercise 2.3

1. a) Yes

b) No

2. a) p = 8 is a solution.

b) m = 5 is a solution.

c) n = 11 is a solution.

d) y = 2 is a solution.

Exercise 2.4

1. a) b = 113 b) a = 45

c) r = 30 d) q = 42

e) p = 1 f) c = 13

g) x = 11 h) z = 28

i) w = 8

position (x).

d) What is the 48th term in the pattern?

4. Look at the pattern below.

Exercise 2.5

a) Draw the figure that comes next in the pattern.

1. a) (3, 5); (–4, 2); (0, –5)

b) D; G; H

c) 2nd quadrant; 3rd quadrant; 1st quadrant; 4th quadrant

1

2. a) (4, –2)

b) (–6, 0)

b) Complete the table.

c) (3, –5)

d) (2, 4)

3. a)

2

3. a) Plot and label these points on the coordinate plane below. Point PQRST x-coordinate –4–2024 y-coordinate 2–30–32 –4 –3 –2 –1 1 2

3

Figure 1 2 3 4

Number of squares

3 5 7

c) Write increase or decrease in the first blank. Then, find the constant increase or decrease. The constant in the pattern is

b) c) increase; 2 d) y = 2x + 1 e) 99 f) 157

d) Write the equation to show the relationship between the number of squares (y) and the figure number (x).

2 y = 2x + 1

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.

Chapter 11: Algebra

Exercise 3.1

1. Connor had 7 bananas left.

2. There were 110 biscuits in each jar.

3. The total length of the 6 wooden planks is 13 metres.

4. 4 litres of water should be present in each container.

5. The mass of rice in each bag was 4 kilograms.

c) W

b) State the quadrant in which each point lies. P Q S T

Exercise 2.6

P S 2nd quadrant 3rd quadrant 4th quadrant 1st quadrant

b) 2nd quadrant; 3rd quadrant; 4th quadrant; 1st quadrant

c) Join the points P to T in alphabetical order. What letter do you get?

1. a) increase; 1 b) decrease; 2 c) decrease; 3 d) increase; 5

2. Circle: y = x + 4

3. a)

Chapter 11: Algebra

Position 1st 2nd 3rd 4th 5th

Number 3 6 9 12 15 3 × 1 3 × 2 3 × 3 3 × 4 3 × 5

b) increase; 3

c) y = 3x

d) 144

6. The length of the pink ribbon is 15 metres.

Exercise 3.2

1. 88 people attended the party.

2. The length of stick A is 45 centimetres.

3. There were 7 apples in each box.

4. She gave 2 stickers to each student.

BM11.1 Coordinate Plane

Maths Journal Maths Journal

1. Explain how to find the value of an algebraic expression involving multiplication and that of another algebraic expression involving division.

2. Write an algebraic expression involving more than one operation.

3. Discuss the usefulness of algebra. How do you think you can make use of what you have learnt about algebra?