KS3 Maths · Year 8 · Chapter 2: Equations and formulae · Lesson 1

Multiplying out brackets

Year 8About 70 minutesNo calculator50 marks

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Date

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Quick questions to start.

A1
Needed today
Expand $4 (2 y - 5)$ .
[1 mark]
Answer
A2
Needed today
Simplify $x^{2} \times x$ .
[1 mark]
Answer
A3
Last lesson
A price of $£200$ goes up by $10%$ , and then by a further $10%$ . Work out the new price.
[1 mark]
Answer
A4
Year 7
Simplify $7 a + 3 b - 2 a + b$ .
[1 mark]
Answer

Example, then your turn

Year 8

Multiply the term outside by EVERY term inside the bracket. Multiply the numbers, then the letters: $x \times x = x^{2}$ and $x^{2} \times x = x^{3}$ . Your teacher works through each example with you. Then try the one beside it.

Example 1

Simplify (i) $3 x \times 5 x$ (ii) $2 a \times 4 a^{2}$ (iii) $3 m \times 2 m \times m$ .

Hint Multiply the numbers together. Then count how many of each letter are multiplied together.

B1
Simplify (i) $4 y \times 7 y$ (ii) $5 k^{2} \times 2 k$ (iii) $2 p \times 3 p \times 2 p$ .
[3 marks]
Answer(i) $\dots$ (ii) $\dots$ (iii) $\dots$

Example 2

Expand $3 x (2 x - 7)$ .

Hint

3 x \times 2 x

and

3 x \times (- 7)

B2
Expand (i) $5 a (3 a + 4)$ (ii) $- 4 n (2 - 3 n)$ .
[2 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$

Example 3

Expand $2 t^{2} (5 t - 3)$ .

Hint

t^{2} \times t = t^{3}

B3
Expand (i) $y^{2} (6 y + 1)$ (ii) $3 m^{2} (4 - 2 m)$ .
[2 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$

Example 4

Expand and simplify $2 x (x + 5) - 3 (x - 4)$ .

Hint The

- 3

multiplies both terms:

- 3 \times x = - 3 x

and

- 3 \times (- 4) = + 12

B4
Expand and simplify $4 y (y - 2) - 5 (y - 3)$ .
[2 marks]
Answer

Example 5

The shape is made from two rectangles. Write an expression for its area. Simplify it as much as possible.

Not drawn accurately

Hint Find the area of each rectangle (width

\times

height), then add.

B5
The shape is made from two rectangles. Write an expression for its area. Simplify it as much as possible.
[2 marks]
Not drawn accurately
Answer

Practice

Year 8

C1
Expand $x (x^{2} - 3 x + 5)$ .
[2 marks]
Answer
C2
Simplify $6 a^{2} - 2 a (3 a - 4)$ .
[2 marks]
Answer

C3
Expand and simplify $2 p (3 p + 1) + p (p - 6) - 4 (p + 2)$ .
[3 marks]
Answer
C4
Ola expands and simplifies $3 n (n - 4) + 2 n$ and gets $3 n^{2} - 10 n$ . Check her answer by putting $n = 5$ into both expressions.
[2 marks]
C5
Find the missing number: $x (x + □) - 2 (x - 3) = x^{2} + 3 x + 6$ .
[2 marks]
Answer
C6
A cuboid is $x$ cm by $2 x$ cm by $(x + 3)$ cm. Write an expression for its volume, without brackets.
[2 marks]
Not drawn accurately
Answer

Exam-style questions

GCSE Foundation

D1
Ella says, " $3 x (x - 2) - x (x + 4)$ simplifies to $2 x^{2} - 2 x$ ."
Ella is wrong. Explain her mistake, and simplify the expression correctly.
[3 marks]
AnswerCorrect answer $\dots\dots\dots$

D2
The diagram shows a small rectangle inside a large rectangle. The large rectangle is $4 x$ cm by $(x + 6)$ cm. The small rectangle is $x$ cm by $(2 x - 3)$ cm.
Not drawn accurately
(a)
Write an expression for the shaded area. Simplify it as much as possible.
[3 marks]
Answer
(b)
Work out the shaded area when $x = 5$ .
[2 marks]
Answer
Total for D2: 5 marks
D3
$n$ is a whole number. Show that $2 n (n + 3) - 2 n (n - 3)$ simplifies to $12 n$ . Explain why its value is always a multiple of $12$ .
[3 marks]
D4
Simplify $2 y (y^{2} - 4 y + 3) - y^{2} (y - 5)$ .
[3 marks]
Answer
D5
$3 x (2 x + a) + 2 (x - b)$ simplifies to $6 x^{2} + 11 x - 8$ . Work out the values of $a$ and $b$ .
[4 marks]
Answer $a = \dots\dots$ $b = \dots\dots$

Extension

Stretch

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

E1
Rectangle $A$ is $x$ cm by $(x + 10)$ cm. Rectangle $B$ is $x$ cm by $(2 x - 3)$ cm. The area of $B$ is $30$ cm² more than the area of $A$ . $x$ is a whole number. Find $x$ .
[4 marks]
Hint 1 · What to try
Write an expression for (area of $B$ ) $-$ (area of $A$ ) and simplify it.
Hint 2 · The first line
$2 x^{2} - 3 x - (x^{2} + 10 x) = x^{2} - 13 x$ , so you need $x^{2} - 13 x = 30$ , which is $x (x - 13) = 30$ .
Hint 3 · The full method
$x - 13$ must be positive. Try $x = 14$ , then $x = 15$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can expand a bracket with a letter or a power outside.
I can expand several brackets and collect like terms.
I can write an area or a volume as an expression without brackets.
I can find a mistake in an expansion and check by substituting.

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