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Lesson 3: �Basic algebra

Objectives

Use correct algebraic notation
Expand brackets
Factorise algebraic expressions
Understand multiplicative algebraic structure using an area representation

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Buying carpet

2

Sam is buying new carpet for the living room.

n

4

n = length customer wants to buy

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Area of carpet

3

Area of carpet =

4n

Length is measured in metres.

Area is measured in square metres.

4

n

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Variables

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KEY IDEA

A variable is an unknown value�represented by a letter �(e.g. n).

The value of a variable can change.

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Buying extra

5

n

4

2

The owner recommends buying an extra 2 metres.

4(n + 2)

4n + 8

4n

8

Tell students that the owner of the carpet shop recommends that Sam buys an extra 2 metres of carpet (to allow for error when fitting the carpet). Check that students are happy that the diagram shows that the area calculation needed is 4(n + 2).

Tell students that Sam says the calculation should be 4n + 8. Ask students if they can explain Sam’s thinking.

Establish that the owner and Sam are both describing the same thing but in different ways. The owner is describing a calculation that involves multiplying two lengths, whereas Sam is describing a calculation that involves adding two areas. Emphasise where the two expressions are featured in the diagram.

Check that students understand the vocabulary of factorising. Establish that:

4 and (n + 2) are factors

4(n + 2) is the expression in factorised form

4n + 8 is the expression in expanded form.

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Factorised and expanded expressions

6

KEY IDEA

Expanded expression

brackets multiplied out

Factorised expression

two or more terms �multiplied together

4(n + 2)

4n + 8

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Completing the diagrams

7

YOUR

TURN

Handout�available

Take turns to fill in the missing information on each diagram.

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Completing the diagrams

8

YOUR

TURN

Handout�available

9

3n

3n²

2

3n

n

2n

3

2n

2

n²

3n

3

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Matching expressions to diagrams

9

YOUR

TURN

Handout�available

Use the diagrams to identify the factorised and expanded forms of the expression.
In the empty cells, either place a card or write an expression.

Give each pair a copy of the ‘Expressions’ cards. (Students will need scissors to cut the expressions cards up if they haven’t already been cut into cards prior to the lesson. Alternatively, they can copy the cards onto sticky notes). Point out that the column on the left of the diagrams has the heading ‘Factorised’ and the one to the right has the heading ‘Expanded’. If needed, remind them that, in the example of Sam buying carpet, 4(n + 2) is the factorised form of the expression and 4n + 8 is the expanded form.

Ask students to use the completed area models to help identify the factorised and expanded forms of each expression, and to use this information to place the cards in the appropriate cells. Tell them that once the cards are placed, they should write the missing expressions in any empty cells.

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Completing rows A to D

10

REVIEW

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Completing rows E to J

11

REVIEW

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Tony expanding brackets

12

Tony is expanding the bracket in this expression:

(n + 4)²

He writes:

Explain why Tony’s solution is incorrect.

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Expanding brackets

13

(n + 4)²

n² + 8n + 16

n

4

n²

4n

16

= (n + 4)(n + 4)

Factorised

Expanded

Diagram

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Rows D and G

14

REVIEW

What do you notice?

Can you factorise 2n² + 2n another way?

2(n² + n)

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Common factors and highest common factors

15

KEY IDEA

An expression is factorised by taking out a common factor of the terms.

To factorise an expression fully,�take out the

highest common factor of the terms.

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Cameron’s factorising

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Cameron is factorising this expression: 4n² + 6n

He writes:

Explain why Cameron hasn’t factorised fully.

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Factorising fully

17

YOUR

TURN

Factorise fully:

12n² + 18n

12n²

18n

2n

3

6n

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Practice question

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REVIEW

(a) Expand 4e(e + 2)

……………………………

(2)

(a) 4e² + 8e

(b) Factorise fully 9x² + 6x

……………………………

(2)

(b) 3x(3x + 2) �

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Lesson review: �Basic algebra

Objectives

Use correct algebraic notation
Expand brackets
Factorise algebraic expressions
Factorise fully (both number/letters) in the same algebraic expression

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Suggested further steps/areas to work on

Factorise quadratics of the form x² + bx + c

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Lesson 3:�Credits

Photo acknowledgements

Shutterstock.com: Vincent Giordano Photo

Text acknowledgements

Pearson Education Ltd: Pearson Edexcel �GCSE (9-1) In Mathematics (1MA1) Foundation (Non-Calculator) Paper 1F, June 2018 and Paper 3F, November 2019

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