KS3 Maths · Year 8 · Chapter 2: Equations and formulae · Lesson 3

Expressions with several variables

Year 8About 70 minutesNo calculator49 marks

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Quick questions to start.

A1
Needed today
Simplify $a \times a \times b$ .
[1 mark]
Answer
A2
Needed today
Simplify $3 x \times 4 y$ .
[1 mark]
Answer
A3
Last lesson
Factorise $12 m^{2} - 8 m$ as much as possible.
[1 mark]
Answer
A4
Year 7
Work out the value of $2 a + 3 b$ when $a = 4$ and $b = - 2$ .
[1 mark]
Answer

Example, then your turn

Year 8

Two letters are dealt with the same way as one. Multiply the numbers, then each letter: $ab \times a = a^{2} b$ . To factorise, take out the biggest number and every letter that is in all the terms. $x y$ and $y x$ are like terms. Your teacher works through each example with you. Then try the one beside it.

Example 1

Expand $3 ab (2 a - 5 b)$ .

Hint

3 ab \times 2 a = 6 a^{2} b

. Then

3 ab \times (- 5 b)

B1
Expand (i) $4 x y (3 x + y)$ (ii) $2 pq (5 - 3 p)$ .
[2 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$

Example 2

Expand $5 x (x^{2} - 3 x y + 2 y^{2})$ .

Hint Three terms in the bracket, so three terms in the answer.

B2
Expand $2 m (3 m^{2} + mn - 4 n^{2})$ .
[2 marks]
Answer

Example 3

Factorise $3 pq + p q^{2}$ .

Hint

p

is in both terms, and so is

q

B3
Factorise (i) $a^{2} b + 5 ab$ (ii) $7 x y - x^{2} y$ .
[2 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$

Example 4

Factorise $10 a^{2} b - 15 a b^{2} + 5 ab$ as much as possible.

Hint The HCF of

10

15

and

5

5

. Every term has an

a

and a

b

. The last term becomes

1

, not

0

B4
Factorise $12 x^{2} y + 8 x y^{2} - 4 x y$ as much as possible.
[2 marks]
Answer

Example 5

Expand and simplify $2 x (x + 3 y) - y (4 x - y)$ .

Hint

- y \times (- y) = + y^{2}

. Then

6 x y

and

- 4 x y

are like terms.

B5
Expand and simplify $3 a (a - 2 b) + 2 b (4 a + b)$ .
[3 marks]
Answer

Practice

Year 8

C1
A formula used in science is $E = \frac{1}{2} m (v^{2} - u^{2})$ . Expand the brackets in this formula.
[2 marks]
Answer $E = \dots\dots\dots$
C2
A rectangle is $3 a$ cm by $(2 a + 5 b)$ cm. Write an expression for its area, without brackets.
[2 marks]
Not drawn accurately
Answer

C3
Find the missing term: $4 ab (□ - 3 b) = 8 a^{2} b - 12 a b^{2}$ .
[2 marks]
Answer
C4
Max says $3 pq (2 p - q) = 6 p^{2} q - 3 p q^{2}$ . Check this by working out both sides when $p = 2$ and $q = 5$ .
[2 marks]
C5
Factorise $9 x^{3} y - 6 x^{2} y^{2}$ as much as possible.
[2 marks]
Answer
C6
Factorise $8 π r^{2} + 6 π r h$ as much as possible.
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Mia factorises $20 x^{2} y - 30 x y^{2}$ and gets $10 x (2 x y - 3 y^{2})$ . She says, "That is factorised as much as possible."
Mia is wrong. Explain why, and factorise the expression as much as possible.
[3 marks]
AnswerCorrect answer $\dots\dots\dots$

D2
A cuboid has a square base, $x$ cm by $x$ cm. Its height is $h$ cm.
Not drawn accurately
(a)
Show that the total surface area of the cuboid is $2 x^{2} + 4 x h$ cm².
[2 marks]
(b)
Factorise $2 x^{2} + 4 x h$ as much as possible.
[2 marks]
Answer
(c)
Use your answer to part (b) to work out the surface area when $x = 5$ and $h = 8$ .
[1 mark]
Answer
Total for D2: 5 marks
D3
The shape is made from two rectangles. Rectangle $A$ is $2 a$ cm by $(a + b)$ cm. Rectangle $B$ is $3 b$ cm by $a$ cm.
Not drawn accurately
(a)
Write an expression for the area of the shape. Simplify it and factorise it as much as possible.
[3 marks]
Answer
(b)
Work out the area when $a = 4$ and $b = 3$ .
[1 mark]
Answer
Total for D3: 4 marks
D4
Expand and simplify $4 p (p + 3 q) + 2 q (p - 3 q)$ . Factorise your answer as much as possible.
[3 marks]
Answer
D5
Show that $3 a (2 a + b) - a (4 a - 3 b) = 2 a (a + 3 b)$ .
[3 marks]

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Work out the value of $x^{2} y - x y^{2}$ when $x = 8.3$ and $y = 3.3$ . Do not use a calculator.
[4 marks]
Hint 1 · What to try
Factorise $x^{2} y - x y^{2}$ first.
Hint 2 · The first line
$x^{2} y - x y^{2} = x y (x - y)$ , and $x - y = 8.3 - 3.3 = 5$ .
Hint 3 · The full method
Work out $8.3 \times 3.3$ , then multiply by $5$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can expand a bracket with two letters outside or inside.
I can factorise when two letters are common to every term.
I can expand and factorise formulae, including ones with a half or $π$ .
I can use factorising to show two expressions are equal.

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