GCSE Maths · Higher · Chapter 24: Algebraic fractions and functions · Lesson 1

Algebraic fractions

Grades 7–9About 50 minutesNo calculator42 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Factorise $x^{2} - 9 x$ .
[1 mark]
Answer
A2
From chapter 18
A cumulative frequency graph shows 100 values. At what cumulative frequency do you read the upper quartile?
[1 mark]
Answer
A3
From chapter 7
Describe the region of points less than 2 cm from a point $P$ .
[1 mark]
Answer
A4
From chapter 14
Write $9200$ in standard form.
[1 mark]
Answer

Grades 7–8

Your teacher works through each example with you. Then try the one beside it.

Example 1

Simplify fully $\frac{x ^{2} - 4 x - 21}{x ^{2} - 49}$ .

Hint Factorise the top and the bottom. Cancel a whole bracket, never a single term.

Example 2

Write $\frac{5}{x + 2} + \frac{3}{x - 4}$ as a single fraction.

Hint Use

(x + 2) (x - 4)

as the denominator. Put a bracket round every new top.

B2
Write $\frac{4}{x - 1} - \frac{3}{x + 5}$ as a single fraction in its simplest form.
[3 marks]
Answer

Grades 7–8

C3
Simplify fully $\frac{3 x ^{2} - 10 x - 8}{x ^{2} - 16}$ .
[3 marks]
Answer
C4
Simplify fully $\frac{x ^{2} - 4 y ^{2}}{x ^{2} + 2 x y}$ .
[2 marks]
Answer
C5
Write $\frac{x + 1}{3} + \frac{2 x - 1}{4}$ as a single fraction.
[2 marks]
Answer
C6
Simplify fully $\frac{x ^{2} - 4}{3 x} \times \frac{6 x ^{2}}{x + 2}$ .
[2 marks]
Answer
C7
Simplify fully $\frac{x + 1}{x ^{2} - 9} \div \frac{2 x + 2}{x - 3}$ .
[2 marks]
Answer
C8
Solve $\frac{x + 3}{2} - \frac{x - 1}{5} = 2$ .
[2 marks]
Answer

Grades 7–9

D1
Zara says, " $\frac{2}{x} + \frac{3}{x + 1} = \frac{5}{2 x + 1}$ , because you add the tops and add the bottoms."
Show that Zara is wrong.
[2 marks]

D2
Kim cycles 30 km at an average speed of $x$ km/h. She then cycles 12 km uphill at an average speed of $(x - 3)$ km/h. The whole ride takes 3 hours.
(a)
Show that $x^{2} - 17 x + 30 = 0$
[3 marks]
(b)
Solve $x^{2} - 17 x + 30 = 0$ to find Kim's speed on the first part of the ride. Give a reason why you reject the other solution.
[3 marks]
Answer $\dots\dots$ km/h
Total for D2: 6 marks
D3
$\frac{3}{x + 2} + \frac{a}{x - 1}$ can be written as $\frac{5 x + 1}{( x + 2 ) ( x - 1 )}$ , where $a$ is a whole number. Work out the value of $a$ .
[3 marks]
Answer $a = \dots\dots$

Grade 9 and beyond

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can simplify an algebraic fraction by factorising and cancelling.
I can multiply, divide, add and subtract algebraic fractions.
I can solve an equation with algebraic fractions in it.

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