Stage 7 Unit 3 Introduction to Algebra and Equations Lesson

3.1.1

3.1.2

3.1.3

Specific Instructional Objectives ➢ Construct simple algebraic expressions by using letters to represent numbers ➢ Know the meanings of the words term, expression and equation ➢ Simplify linear expressions by collecting like terms ➢ Know that algebraic operations follow the same order as arithmetic operations ➢ Expand linear expressions by multiplying a constant over a bracket

Cambridge Learning Objectives 7Ae3 Construct simple algebraic expressions by using letters to represent numbers. 7Ae4 Simplify linear expressions, e.g. collect like terms; multiply a constant over a bracket. 7Ae2 Know that algebraic operations follow the same order as arithmetic operations.

Process Skills

21st Century Skills

Number of Periods

Reasoning and Proving Connecting

Initiative

1

Synthesising

Critical Thinking

2

Initiative

1

Investigative Reasoning

3.1 Algebraic Expressions Background Pupils know and apply the arithmetic laws and able to make general statements about sums, differences and multiples of odd and even numbers. Learning Outcomes ➢ Construct simple algebraic expressions by using letters to represent numbers ➢ Know the meanings of the words term, expression and equation ➢ Simplify linear expressions by collecting like terms ➢ Know that algebraic operations follow the same order as arithmetic operations ➢ Expand linear expressions by multiplying a constant over a bracket

Lesson Development 3.1.1 Class Tasks Trigger

Teaching Approaches/ Instructional Strategies

Teaching Tip

Go through Investigate! on p.46 with pupils. Ask: What are the possible pairs of numbers that could make 10? Invite responses from pupils. Teacher to list number bonds to 10 in a table on the board. x y 0 10 1 9 2 8 3 7 4 6 5 5

As this is the first time students are encountering Algebra, strong focus is placed on developing the concept and guiding the students to understand and be comfortable with the use of Algebra.

Cambridge Lower Secondary Maths

1

Stage 7 Unit 3 Introduction to Algebra and Equations 6 7 8 9 10

4 3 2 1 0

Say: There are several different possible pairs. We can use letter like x and y to represent the different possible numbers added to equal 10. Therefore, the relationship can be written as ‘x + y = 10’. Get pupils to refer to p.47 of textbook. Ask: How many children are there altogether in line? [ 7/8/ 9 / 10] Explain: The number of children is not certain as part of the line is hidden by the bush. Algebraic letters are use as representation of unknown quantities. Hence, let n be the number of children. Think and Share → Slightly introduce the ‘+’ and ‘x’ of algebraic letter? (e.g. Teacher paste an ‘n’ counter on the board, and another, Ask: How many n’s are there? One plus one gives two so we have two n’s. We write n + n = 2n. We can also read this as 2 groups of ‘n’. Teacher can circle the ‘n’ counter on the board to show ‘2 groups’. So it mean when we write 2n, it stands for 2 groups of n. Ask: Can you trying writing the expression for 2 groups of ‘n, plus 3 more?) Explain: Algebra is like learning another language. Instead of using words and numbers, algebra uses letters to represent numbers. These letters are known as variables and can be used to represent any numbers.

Indicate the x and y column on the table drawn on the board. Write on the board: ‘x + y = 10’.

Draw the table on the board and ask pupils to make a comparison between A and B. Inform pupils that 3b means 3 times of b or 3 groups of b. 3 x b is simply written as 3b.

Ask: What is the difference between the two groups? A x+y 3b− 6 𝑥+1 3𝑥 − 5

B x + y = 10 8𝑔 − 7 = 0 𝑥+3 = −2 5

[In group B, there is an equal sign while in group A, there is no equal sign.] Say: In group A, they are called expressions while in group B, they are called equations. Explain: An expression is a combination of numbers and/or variables connected by mathematical operations while an equation is a mathematical statement that says two expressions are equal to each other. In group B, the equal sign shows that the left-hand side and righthand side are equal to each other.

Warm Up

Ask: Let’s express the following phrases/words into algebraic expressions. Recall in algebraic expressions, there are no words but we replace unknown quantities with letters, connected by operations (+, –,

x,  ).

Cambridge Lower Secondary Maths

Get pupils to do individually or in pairs as you make your way around to

2

Stage 7 Unit 3 Introduction to Algebra and Equations Words Add b to y

Algebraic expression b+y

Subtract c from d Multiple e by f

d-c e x f or ef It is usually written as ef e  h or

Divide e by h

e h

It is usually written as Add 5 to the product of a and b

e h

ab + 5

take note of struggling students. Draw the table on the board. Invite lower ability pupils, where possible, to the board to write down their answers. This is to develop their confidence in algebra in successfully doing easier tasks.

Invite pupils to the board to write down their answers. Try out Worksheet 3.1.1 in Appendix with the students to further strengthen students’ familiarity of algebraic expression through a fun activity.

Activity

Teacher to introduce how in primary school they use bars or to represent an unknown. Teacher to develop that by replacing with a letter through the use of examples and questionings to form an expression. Pupils to refer to Worked Example 3 on p.49. Ask: How will you represent the question using model method? Peter John

x x

2

Teacher to write x in the bars.

Ask: What is unknown here? [John’s age and Peter’s age] Say: We will let John’s age be x years old. Ask: How old is Peter in terms of x? [(x + 2) years old] Ask: Alternatively, the variable selected can also be to represent Peter’s age. We can let y be Peter’s age. How old is John in terms of y? [ (y -2) years old ] y Peter John

2

Go through Worked Example 1, 2 and 4 for pupils to get use to writing algebraic expressions.

Extension

Teacher to erase off x and indicate where is y. Teacher can challenge higher ability learners to try the Think and Share to write the product of length and breadth, and sum of all sides.

Get pupils to read p.49 to identify what is called a term, coefficient, variable and constant.

Cambridge Lower Secondary Maths

3

Stage 7 Unit 3 Introduction to Algebra and Equations Write the linear expression 5𝑥 + 𝑦 − 4𝑧 − 3 on the board and invite pupils to answer the following questions. Ask: How many terms are there in this expression?[4]

5𝑥 + 𝑦 − 4𝑧 − 3 Ask: How many variables are there and identify them? [3; 𝑥, 𝑦 𝑎𝑛𝑑 𝑧] Ask: What is the constant? [-3] Explain: A constant term is a term in the expression which is only a number with no variable. Ask: What is a coefficient? [The numbers placed in front of the variable/letter.]

Highlight to pupils that terms are written in alphabetical order. For lower ability pupils, recall what is a variable first. Remind pupils the sign stays with the coefficient/ number behind it.

Ask: What are the coefficients of 𝑥, 𝑦 𝑎𝑛𝑑 𝑧 ? [𝑥, 𝑦 𝑎𝑛𝑑 𝑧 are 5, 1 and -4 respectively.] Ask: What is the coefficient in the term ‘-t’? [-1] Explain: By convention, the coefficient 1 is not written down. For eg. we do not write 1𝑦, instead it is written as 𝑦. It is understood that there is one ‘y’.

Evaluation

Reflection Invite pupils for answers. Ask: What is the difference between expression and an equation? [ An expression has no equal sign while an equation has an equal sign] Ask: Give one example of an expression and an example of an equation. Based on the expression pupils provided, get pupils to determine how many terms are there and identify the variables, constants and to state the coefficient of the variable. Homework Get pupils to try out Check My Understanding on p.50 Q 1 to 4 to further practice interpretation of a question and construction of algebraic expression. For higher ability pupils, to try out Q 5 to 7 which manipulation of algebraic expressions.

Cambridge Lower Secondary Maths

4

Stage 7 Unit 3 Introduction to Algebra and Equations

Lesson Development 3.1.2 Class Tasks Warm Up

Teaching Approaches/ Instructional Strategies Ask: Can you add 7 oranges to 3 oranges? [Yes, you will get 10 oranges] Ask: Then, can you add 7 oranges to 3 apples? [No] Explain: We can only simply write 7 oranges + 3 apples. And if 𝑥 stands for oranges and y stands for apples, it will be written as 7𝑥 + 3y. As the two terms 7𝑥 and 3y do not have the same variables, they are unlike terms. Like terms are terms that have the same variable. Only like terms can be added or subtracted. Ask: Can 3𝑟 2 + 3𝑟 be simplified further? [No. As 𝑟 2 and 𝑟 are different variables, hence, they are unlike terms and cannot be added together] Pupils to discuss Think and Share on p.51. Invite students for responses. Explain: The two are like terms. The variable 𝑥y is the same as y𝑥. 𝑥y is the same as 𝑥 multiply y. Consider ‘2 x 3’ is the same as writing ‘3 x 2’.

Activity

Say: Simplify the expression 2𝑥 − 3𝑦 + 5𝑥 + 7𝑦 Ask: What can we do first? Can you rearrange the positions of the terms to put the like terms next to each other? 2𝑥 − 3𝑦 + 5𝑥 + 7𝑦 = 2𝑥 + 5 𝑥 + 7𝑦 − 3𝑦 = 7𝑥 + 4𝑦

Step 1: Group the like terms together. The sign stays with the coefficient behind it. Step 2: Deal with each set of like terms at a time. Step 3: Perform algebraic operations which is the same as arithmetic operations.

To aid visual learners, demonstrate solving the question using algebra discs to visualise the adding and subtracting of terms in x and y. Get volunteers to help with the demonstration. Ask: In front of you are two type of discs. What are on the sides of each of the disc? [One has 𝑥 and −𝑥. The other has 𝑦 and −𝑦.] Ask: Using the discs, how would you represent the question. x

x

x

x

x

x

x

-y

-y

-y

y

y

y

y

y

y

y

Teaching Tip Remind pupils they can add or subtract LIKE TERMS but they cannot add or subtract UNLIKE TERMS. Note to pupils that all constants are like terms. So, first, they need to remind themselves to identify the like terms.

Write on the board and share with the pupils the thinking process in simplifying the expression. Using the steps written here, help pupils break down the question into smaller problems to solve.

Remind pupils that the sign in front of each term stays with the term when rearranging their positions!

Ask: Recall, what is a zero pair? [ +1 – 1 = 0] Extend: In the same way, −𝑥 + 𝑥 = 0 and −𝑦 + y = 0 Ask: How many zero pairs are there? [3] Extend: Remove all the zero pairs and write down what is left as the answer.

Cambridge Lower Secondary Maths

5

Stage 7 Unit 3 Introduction to Algebra and Equations Get pupils to go through and discuss Worked Example 5 to familiarise with simplifying of algebraic expressions. Post Activity Discussion

Teacher to discuss Worked Example 6 on patterns involving algebra. Ask: What is the pattern observed? [Adding two blocks below will give the expression of the block on top.] Discuss Check My Understanding on p.53 Q4a and Q4d Ask: Analyse the diagram. Where should you start to solve the pattern? [ We can start at 12a and subtract 5a away, giving 7a in the unknown box in the second row.] [ We can also start at 5a and subtract 2a, giving 3a in the bottom right row.] Explain: If we go from top to down, instead of adding, the reverse which is subtraction will be done. Ask: In Q 4d), where should you start to solve the pattern? [From the top at 15 𝑥 + 25.] Ask: Which is the following step to solve the pattern? 15 𝑥+ 25 – 5 𝑥 + 2 OR 15 𝑥+ 25 – 5 𝑥 – 2 [15𝑥+ 25 – 5𝑥 – 2 is correct. In 15 𝑥+ 25 – 5 𝑥 + 2, only ‘5𝑥’ is subtracted off. In 15𝑥+ 25 – 5𝑥 – 2, both 5x and 2 is taken away from 15𝑥 + 25.]

Evaluation

Get all pupils to try out Check My Understanding on p.52 Q2 which may involves simplifying algebraic expressions up to 8 terms, Q3 which involves constructing and simplifying algebraic expressions. Higher ability students to try out Q5 and 6 which involves analysing of algebraic pattern and working out backwards to get to the answer.

Lesson Development 3.1.3 Class Tasks Trigger

Teaching Approaches/ Instructional Strategies

Teaching Tip

Instruct students to take out a piece of rough A4 size paper and to do the following steps. 1. Label the width of the rectangle ‘a’ 2. Draw a line parallel to the width such that it splits the original rectangle into 2 smaller rectangles of different sizes. (exact dimension is not important) 3. Label the longer length ‘b’ and the shorter length ‘c’ 4. Express the area of the original rectangle in the form of a product. Ask: How do we find area of the original rectangle? [ Width x Length = a x (b + c) = a(b + c)] 5. Cut or tear properly along the line they have drawn. Hold the torn pieces close to each other, one piece in each hand. Ask: Did we throw away any paper? [No] Ask: Is the area of the original rectangle still the same? [Yes] Ask: Looking at these 2 pieces, are there other ways to express the area of the original rectangle? [Find the area of the two rectangles using the terms a, b and c.] 6. Express the area of the two rectangles as ab + ac

Cambridge Lower Secondary Maths

a c

b

6

Stage 7 Unit 3 Introduction to Algebra and Equations Say: Since the area remains the same, the two expressions are equal. 7. Equate the two areas. a(b + c) = ab + ac. Say: This expansion a(b + c) = ab + ac shows the distributive law of multiplication. Emphasize to students that a x (b + c) can also be written as a(b + c). a(b – c) can also be written as ab – ac. Activity

Say: Distributive Law of Multiplication helps to expand algebraic expressions. Multiply each term in the bracket by the term right outside the bracket. After multiplying, the brackets will be removed. Then, the like terms can be group together. Key Rule: You must multiply EVERY TERM inside the bracket by the term outside the bracket. ’a’ is the coefficient of the terms ‘x’ and ‘y’.

a( x + y) = ax + ay Ask: How many terms are there in the bracket? [2] Explain: So ‘a’ the term outside the bracket needs to multiply the first term in the bracket, x, and then add to the product of a and the second term in the bracket, y.

a(x – y) = ax – ay

Write on the board

Drawing of arrows will help pupils to remember and not leave out any terms.

Remind pupils to take note of the sign. Include the sign of the number when applying distributive law.

Say: Let’s draw those arrows again. Ask: How does the expansion go? [the a is multiplying the x and -y.] Say: Notice how important it was to get the sign correct for the negative term.

Ask: For the 2 equations, what similarity do you see? Invite students for answers. Explain: They are ‘expanding’ the expression by multiplying the factor (or coefficient) in front of the bracket to the terms. After that, the brackets are removed as they are no longer needed. Post Activity Discussion

Get pupils to expand the following expressions and invite pupils to write down their answers on the board (or on their own mini whiteboards). - (a + b) = -a – b - (a - b) = -a + b - (-a - b) = a + b Ask: What can you observe when the terms in the bracket is multiplied by a – 1? [All the terms signs of the terms in the bracket are reversed] Demonstrate to pupils on how to simplify -5(2a – b). Illustrate on the board.

-5(2a – b)

Cambridge Lower Secondary Maths

Remind pupils that the terms in bracket are multiplied by -1 and not just by a negative sign.

Highlight to pupils that -5(2a-b) = -5 x (2a-b). 3 multiply 4 can also be written as

7

Stage 7 Unit 3 Introduction to Algebra and Equations

= (-5 x 2a) – (-5 x b) = -10a – (-5b) = -10a + 5b

3(4) with the use of brackets.

Say: Recall from previous topic, –(-5b) is the same as +5b. Alternatively, get pupils to simplify 5(2a – b), first, using algebra discs. Explain: 5(2a – b) means 5 groups of 2a – b. 5(2𝑎 − 𝑏) = (2𝑎 − 𝑏) + (2𝑎 − 𝑏) + (2𝑎 − 𝑏) + (2𝑎 − 𝑏) + (2𝑎 − 𝑏)

Get pupils to use algebra discs to lay out the discs and simplify the expression. 5(2𝑎 − 𝑏) = (2𝑎 − 𝑏) + (2𝑎 − 𝑏) + (2𝑎 − 𝑏) + (2𝑎 − 𝑏) + (2𝑎 − 𝑏) = 10𝑎 − 5𝑏

Ask: What happens when instead (2a-b) is multiplied by 5 it is multiplied by -5? [Due to the negative sign, all the discs are flip.] −5(2𝑎 − 𝑏) = −10𝑎 + 5𝑏

Challenge pupils to illustrate the movement and operation of terms in the examples using algebra discs or by drawing.

Get pupils to flip all the discs.

Get pupils to go through Worked Example 7 to expose to the different type of questions. Evaluation

Reflection Simplify 4𝑞 − 2𝑞 − 5(𝑝 − 𝑞 + 3) together with the help of the pupils. 4𝑞 − 2𝑞 − 5(𝑝 − 𝑞 + 3) = 4𝑞 − 2𝑞 − 5 × 𝑝 − 5 × (−𝑞) − 5 × 3 = 4𝑞 − 2𝑞 − 5𝑝 + 5𝑞 − 15 = 4𝑞 − 2𝑞 + 5𝑞 − 5𝑝 − 15 =7𝑞 − 5𝑝 − 15 =−5𝑝 + 7𝑞 − 15

Step 1: Expand by multiplying each term in the bracket by -5. Step 2: Group the like terms together. Step 3: Perform algebraic operations which is the same as arithmetic operations.

Higher ability learners can try the investigation activity and try to work backwards to form the algebraic expression. The test if the expression works.

Students can arrange the terms by alphabetical order of the variables for some systematic structure. Students can also work out each product on the side of their paper (solving smaller problems before solving the big one), before filling in the product of each term in the bracket into their main working.

Homework Get all pupils to try out Check My Understanding on p.55 Q 1 and 2 involving two terms and Q 3 and 4 which involves more than two terms.

Cambridge Lower Secondary Maths

8

Stage 7 Unit 3 Introduction to Algebra and Equations Lesson

3.2.1

3.2.2

Specific Instructional Objectives ➢ Substitute positive integers into simple linear expressions/ formulae. ➢ Derive and use simple formulae

Cambridge Learning Objectives 7Ae5 Derive and use simple formulae, e.g. to change hours to minutes. 7Ae6 Substitute positive integers into simple linear expressions/ formulae.

Process Skills

21st Century Skills

Number of Periods

Initiative

Critical Thinking

0.75

Synthesising

Critical Thinking

0.75

3.2 Deriving and Using Formulae Background Pupils know and apply the arithmetic laws and able to make general statements about sums, differences and multiples of odd and even numbers. Learning Outcomes ➢ Substitute positive integers into simple linear expressions/ formulae. ➢ Derive and use simple formulae

Lesson Development 3.2.1 Class Tasks Warm Up

Teaching Approaches/ Instructional Strategies

Teaching Tip

Say: The process of substituting each variable with a given value to find the value of the algebraic expression is called evaluation. For this, you need to remember BODMAS and Negative Numbers! Work out the following example with the pupils’ input. Example What is the value of a – 3b when a = 3, b = 2? a – 3b Ask: Recall, what does a stands for? [3] = 3 – 3x2 =3–6 Ask: What is the relationship between -3 = -3 and b? [-3 x b]

Illustrate on the board and ask questions to help pupils to recall the concepts.

Say: Substitute in the values into the expression. So [-3xb] will now be [-3x2]

Post Activity Discussion

Get pupils to work out Example 1 to familiarise with substituting values. Ask: In the question, what is the value of 2 − x(3 − y) where x = 2 and 𝑦 = 6, will it matter whether expansion or substitution of values is done first? Get pupils to evaluate the expressions on their own first using both ways.

Cambridge Lower Secondary Maths

Invite two pupils onto the front to solve on the board, one to solve using expansion first and the other pupil to solve using substitution first.

9

Stage 7 Unit 3 Introduction to Algebra and Equations Ask: Do you get the same answer using either method? [Yes. If no, get pupils to check for careless mistakes.] Expansion done first: 2 − x(3 − y) = 2 − 3x − x(−y) = 2 − 3x + xy = 2 − 3(2) + 2(6) = 2 − 6 + 12 =8

Pupils to recall that when simplifying expressions that involve addition and subtraction operations only, pupils need to start from left to right.

Substitution done first: 2 − x(3 − y) = 2 − 2(3 − 6) = 2 − 2(−3) =2+6 =8 [No, it does not matter as the answer will be the same but it is easier to solve when substitution is done first] Evaluation

Get pupils to try out Check My Understanding on p.57, Q 1 to 4 for practice in substituting into an algebraic expression. Higher ability pupils can try out Q5 to 6 which requires them to think critically and spot logical patterns and Q7 is a good checkpoint to check for tudents’ misconception on algebraic operations.

Lesson Development 3.2.2 Class Tasks Trigger

Teaching Approaches/ Instructional Strategies

Teaching Tip

Ask: How do you find the area of a rectangle? [ By taking its length x breadth] Explain: By choosing letters to represent the length and breadth we will be able to construct the formula of finding the area of a rectangle. For example, Let A be area of rectangle, b the length and l the length. Area of rectangle = breadth x length A = b x l A = bl Ask: What other formulas do you know of? Invite pupils for answers. Say: These formulas are in fact in the form of an algebraic equation which consists of algebraic terms. Conclude: To form a formula, you need to choose letters to be the variable to represent a quantity. Get pupils to work out Example 2 to familiarise with translating words into algebraic expressions.

Cambridge Lower Secondary Maths

10

Stage 7 Unit 3 Introduction to Algebra and Equations Activity

Invite pupil to give their input and work out Investigate! activity on p.59 as a class. Ask: What is 𝑥 representing in this question? [No. of hours] Ask: What is the value of 𝑥 in part a) and how do you find which company is more expensive)? [As 𝑥 = 1, substitute into both formulae. When 𝑥 =1, Company M→ 𝐶 = 2𝑥 + 5 = $7 Company N → 𝐶 = 3𝑥 + 1 = $4 Hence, Company M is more expensive for hiring of bicycle for 1 hour.]

Invite pupils to come to the board and present their workings, verbalising their steps aloud.

Ask: Will Company M still be more expensive to hire in part b)? [Substitute 𝑥 = 6 into both formulae Company M→ 𝐶 = 2(6) + 5 = $17 Company N → 𝐶 = 3(6) + 1 = $19 Hence, Company N is more expensive for hiring of bicycle for 6 hours.] Ask: For part c), Instruct students to fill up the table below by substituting 𝑥 values into the expressions. No. of hours 𝑥hours 1 2 3 4

Company M 𝐶 = 2𝑥 + 5 7 9 11 13

Company N 𝐶 = 3𝑥 + 1 4 7 10 13

[ When x=4, it will cost the same to rent from both companies.] Invite pupils to attempt part d) and share their answers on the board. [Company M→ 𝐶 = 2(8) + 5 = $21 Company N → 𝐶 = 3(8) + 1 = $25 Company N is mor expensive] Explain: After 4 hours onwards, it will be more expensive to rent from Company N as it increases by $3/hr and Company M increases by $2/hr. Evaluation

Reflection Ask: What are formulae? How can we use them? [It is an algebraic equation which consists of algebraic terms.] Ask: When should we write formulae? [When we need to repeatedly go through the same operations but trying it out with many different values] Ask: Cite an example scenario. Make up a problem and get your partner to solve it using algebra. Homework Get all pupils to try out Check My Understanding on p.55 Q 1 and 2 involving two terms and Q 3 and 4 which involves more than two terms.

Cambridge Lower Secondary Maths

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Stage 7 Unit 3 Introduction to Algebra and Equations Lesson 3.3.1

3.3.2

Specific Instructional Objectives ➢ Represent simple functions using words, symbols and mappings ➢ Use letters to represent unknown numbers or variables.

Cambridge Learning Objectives 7As3 Represent simple functions using words, symbols and mappings 7Ae1 Use letters to represent unknown numbers or variables.

Process Skills

21st Century Skills

Analysis Synthesising

Critical Thinking

Number of Periods

2

3.3 Functions and Mapping Background Pupils know and apply the arithmetic laws and able to make general statements about sums, differences and multiples of odd and even numbers. Learning Outcomes ➢ Represent simple functions using words, symbols and mappings ➢ Use letters to represent unknown numbers or variables.

Lesson Development 3.3.1 and 3.3.2 Class Tasks Warm Up

Teaching Approaches/ Instructional Strategies

Teaching Tip

Ask: You will like to play a bowling game. There is a one-time charge of $5 for rental of shoes and socks and it costs $3 for every set of game. How much will you need to pay to play 4 sets of bowling? [ 4 x 3 +5 = $17] Get pupils to refer to p.60 and teacher to highlight the functional relationship between cost of petrol and number of litres. Explain: A function machine takes the input, a known variable, and produces an output. Ask: In our example, what is the function machine doing? [ The input is 4 and it is multiplied by 3 and added to 5 giving an output of $17.] Say: When buying snacks from a vending machine, you will need to enter the code tagged to the snack into the vending machine. For example, You can enter the code D3 for a packet of potato chips or C4 for a chocolate bar. A vending machine operates the same way as a mathematical function.

Use an analogy to explain function machine.

Explain: A function can only assign one output value to an input value. As seen in the vending machine, for every input there is exactly one output for example, D3→ potato chips, C4→ chocolate bar. A function takes an input value and apply one or more operations to the input and assign an output value.

Cambridge Lower Secondary Maths

12

Stage 7 Unit 3 Introduction to Algebra and Equations

Activity

Say: Functions can be represented by drawing mapping diagrams. Ask: How will we express our earlier example on charges of a bowling game using a mapping diagram? X3

Explain: 4

17

+5

Teacher to draw out on the board and explain.

𝑥 is the input. We write it as ‘𝑥 is mapped onto 3 𝑥 + 5’ or ‘𝑥 → 3 𝑥 + 5’.

Ask: Is it the same to do ‘+5’ and then ‘x 3’? Explain. Allow students to discuss and invite them to explain. [It is not the same. It will give an input of 27.] Explain: This is because the function is to multiply 3 to the number of sets of games played and not to the rental of the shoes and socks. Ask: If you have $26, how many sets of bowling games can you play? Allow students to discuss and invite them to explain. [ I will take $26 – 5 = $21, and divide it by $3. Hence, 21 ÷ 3 = 7 games.] Explain: In other words, we inverse the operations and work backward and can you represent it on the mapping diagram?

?

X3

+5

÷3

-5

Pupils may use other heuristics like guess and check or trial and error to find values that satisfies the given constraints if they’re not sure how to derive a formula.

26

Get pupils to go through Example 1 on p.61 to practise the use of function machine.

This can relate back to the Investigate they did on p.55.

Get pupils to try out Q1 Investigate! activity on p.63. Higher ability pupils can try Q2 of the activity. They can use guess and check to identify possible patterns that gives the pairs of values.

Post Activity Discussion

Ask: Going back to our vending machine analogy, is it possible for a function to have different inputs with the same output? [Yes] Explain: It is still a function. Because each input has one output. input output C4

chocolate

D3

chips

D2

Cambridge Lower Secondary Maths

13

Stage 7 Unit 3 Introduction to Algebra and Equations Ask: Is it possible, in a function, for the same input to have different output? [No] Explain: Vending machine is spoilt in this case as it does not work in such a way, the same for a function. This does not represent a function. Why? input output C4

chocolate

D3

chips

Encourage students to think of the features of a function machine which they have learnt.

cookie [You should get the same output for the same input. A function applies the same operations/ rules to the input, hence each input has only one output value.]

Evaluation

Reflection Invite pupils for answers. Ask: What is a function machine? [ A function machine performs mathematical operations on an input and assign an output.] Ask: What are the features of the function machine have you learnt? [ - For every input, it has only one same output. - It can perform more than one operation. - The order of operation must be correct following the relationship of the input to the output. Swapping the order of operation may lead to a wrong output assigned. -A mapping diagram shows a function.] Homework Get pupils to try out Check My Understanding on p.62 Q 1 to 5 to further practise on the function machine. For higher ability pupils, to try out Q 6 which involves determining the order of operation in the function machine.

Cambridge Lower Secondary Maths

14

Stage 7 Unit 3 Introduction to Algebra and Equations Lesson

3.4.1

3.4.2 3.4.3

Specific Instructional Objectives ➢ Construct simple linear equations with integer coefficients ➢ solve linear equations where unknowns are on one side only

Cambridge Learning Objectives

7Ae7 Construct and solve simple linear equations with integer coefficients (unknown on one side only), e.g. 2x = 8, 3x + 5 = 14, 9 – 2x = 7.

Process Skills

21st Century Skills

Investigative Synthesising

Critical Thinking

1

Synthesising

Critical Thinking

1

Synthesising

Critical Thinking

2

Number of Periods

3.4 Constructing and Solving Equations Background Pupils know and apply the arithmetic laws and able to make general statements about sums, differences and multiples of odd and even numbers. Learning Outcomes ➢ construct simple linear equations with integer coefficients ➢ solve linear equations where unknowns are on one side only

Lesson Development 3.4.1 Class Tasks Recall

Warm Up

Teaching Approaches/ Instructional Strategies

Teaching Tip

Ask: What is an equation? [It is a mathematical statement that says two expressions are equal to each other.] Explain: The left-hand side (LHS) is equal to the right-hand side (RHS). Ask: What does solving the equation means? Say: This means finding the value of the variable(s) (or unknowns) which makes LHS equals to the RHS. Let’s try using the guess and check method first, by substituting different values for the variable. Example: Find the value of 𝑥 in 3𝑥 + 1 = 10. Ask: What will we do first? [ Substitute values into 𝑥 until the RHS is equals to the LHS which is 10.] Say: Where 𝑥 = 1, 3×1+1= 3+1 Ask: Which operations will you do first? =4 [multiplicaton] Where 𝑥 = 2, 3×2+1= 6+1 =7 Where 𝑥 = 3, 3×3+1= 9+1 = 10 So 𝑥 = 2, is the solution to the equation 3𝑥 + 1 = 10.

Encourage pupils to keep increasing the value of 𝑥 till they get 10.

Get pupils to try out Worked Example 1 to familiarise with trial and error method in solving equation.

Cambridge Lower Secondary Maths

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Stage 7 Unit 3 Introduction to Algebra and Equations

Evaluation

Get pupils to try out Check My Understanding on p.65 Q 1 to 2 find the solution to the equation using guess and check method which involves substituting of values for the variables in the equation. Higher ability pupils can challenge themselves by trying out Q3.

Lesson Development 3.4.2 Class Tasks Warm Up

Teaching Approaches/ Instructional Strategies

Teaching Tip

Ask: What can you say by looking at the scale? [The scale is balanced as the left side weighs as heavy as the right side.]

Say: For the scale to remain balance, what is taken or added on LHS, the same must be done on the RHS. Ask: So how many black balls o you think the green bag is as heavy as? [9 balls] Ask: What happens if I remove 1 black ball from the left side? What will happen to the scale? If I want to re-balance the scale, what should I remove on the right side? Extend: Think of the balance as an equation. Whatever you do to one side of the equation, you must do exactly the same to the other side to keep the equation in balance.

Activity

Offer a strategy to struggling learners: cover the black ball on the left with 1 finger and the same on the right to find out how much the green bag to worth.

Teacher to demonstrate solving the question on the board. Example: Find the value of 𝑥 in 𝑥 + 1 = 10. Say: In order to find the value of the variable 𝑥, our aim is to have the variable 𝑥 alone on one side of the equal sign. Ask: How do we make 𝑥 be the only one left on one side? [Subtract 1 from both sides.] 𝑥 + 1 = 10 𝑥 + 1 − 1 = 10 − 1 𝑥=9

By convention, x is made the subject on the LHS of = sign.

It helps the visual learners to understand better as the teachers explain with the use of scale balancing throughout the question.

Cambridge Lower Secondary Maths

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Stage 7 Unit 3 Introduction to Algebra and Equations

Say: Check your answer using substitution. Check: 9 + 1 = 10 Get pupils to refer to p.66 of textbook to go through Example 2 to familiarise with steps to find the value of the variables.

Evaluation

Get pupils to complete Check My Understanding on p.66 Q 1-3 for practice on solving equation involving one operation.

Pupils can check out these links for revision: https://www. bbc.co.uk/ bitesize/guides/ zbybkqt/revision/2 https://www. mathsisfun.com/ algebra/addsubtractbalance.html

Lesson Development 3.4.3 Activity

Say: Let’s take a look at the example from the previous lesson. Example: Find the value of 𝑥 in 3𝑥 + 1 = 10. Teacher to demonstrate on solving of the question on the board. Ask: On the LHS, what are the two operations? [3 multiplying 𝑥 and addition of 1 to 3 𝑥.] Say: We will have the terms with the variables be on LHS, in this question 3𝑥, and make the constants (terms without variables) on the RHS, in this case 1. So, what is our first step? [Subtract 1 from both sides.] 3𝑥 + 1 = 10 3𝑥 + 1 − 1 = 10 − 1 3𝑥 = 9

This is a common way. It is also fine to place the variables on the RHS instead

Ask: How do we ‘remove’ the coefficient 3? [Divide both sides by 3] 3𝑥 ÷ 3 = 9 ÷ 3 𝑥=3

Cambridge Lower Secondary Maths

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Stage 7 Unit 3 Introduction to Algebra and Equations

Say: Check your answer by substituting your answer into the given expression and see if you get back the original statement. Check: 3 x 3 + 1 = 10 ✔︎ Get pupils to refer to p.67 of textbook to go through Example 3 to familiarise with steps involving two operations to find the value of the variables.

Post Activity Discussion

Ask: In finding the value of 𝑥 in 3𝑥 + 1 = 10. Can the first step be to divide by 3 on both sides instead? Discuss and invite responses from the pupil – Challenge the higher ability pupils. Teacher to explain on the board. Yes, it is possible but it is more tedious and pupils tend to make careless mistake as 3 is only multiplied to 1 on not to all the terms on the LHS. Ask: What is the operation between 3 and 𝑥 ? [multiplication; 3 x 𝑥 ] Ask: So what do we do to ‘remove’ the coefficient 3? [Divide both side by 3] 3𝑥 + 1 = 10 3𝑥 + 1 10 = 3 3 3𝑥 1 10 + = 3 3 3 3𝑥 10 1 = − 3 3 3 9 𝑥= =3 3

Evaluation

Get pupils to try out Check My Understanding on p.68 Q 1 to 3. In Q 3, pupils will need to construct their own equations.

Evaluation

Reflection Teacher to work out Q 4e of Check My Understanding with inputs from pupils. Say: To solve such an equation using guess and check will be tedious and time consuming. From now onwards, we are to use the balancing equation method. Teacher can ask ‘what is the next step?’ 67 = 3(4𝑥 − 3) − 8𝑥 67 = 12𝑥 − 9 − 8𝑥 67 + 9 = 12𝑥 − 8𝑥 − 9 + 9 76 = 4𝑥 76 4𝑥 = 4 4 𝑥 = 19 Check:

Cambridge Lower Secondary Maths

Step 1: First, remove brackets by expansion. Ask: What is next? Step2: Make all terms with the variable on one side and all the constant terms on the other side. Say: The rule is what is done on the LHS, is done on the RHS. Ask: What is next? Step3: Remove coefficient of x. Ask: What is next?

Higher ability pupils to attempt Investigate! activity on p.68

Draw out the arrows. Remind pupils that the negative sign belongs to the term behind it.

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Stage 7 Unit 3 Introduction to Algebra and Equations RHS=3(4𝑥 − 3) − 8𝑥 =3(4 × 19 − 3) − 8 × 19 =3(76 − 3) − 152 = 3(73) − 152 = 219 − 152 =67 =LHS

Step 4: To check by substituting x value into the initial equation.

Homework Get pupils to try out Revision on p.70 Q 5 – 8. For further practice on construction of word equation from word problems, get pupils to try on Check My Understanding on. p.69 Q 5.

Cambridge Lower Secondary Maths

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Appendix Unit 3 Lesson 3.1.1

Addition and Subtraction of Integers Worksheet 3.1.1 Objective To practise symbol literacy Method Give each individual or pair of students a set of cards. Teacher to read out a cue card. Ask students to select and hold up the matching algebraic expression. Teacher to visually check for correct responses. Student Cards (enlarge before making sets)

𝑛+8

8−𝑛

8÷𝑛

𝑛÷8

𝑛−8

8𝑛

8𝑛 + 1

8(𝑛 + 1)

Cues (read by the teachers) • A number n less than 8 • 8 more than a number n • A number n decreased by 8 • The sum of a number 8 and n • 8 less than a number n • The difference between a number n and 9 • 8 divided by a number n • The quotient of a number n and 8 • 8 subtracted from a number n • 8 times a number n increased by 1 • The product of a number n and 8 • A number n increased by 8 • 8 times a number n • The product of number n increased by 1 and 8

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