KS3 Maths · Key Stage 3 · Unit 3: Equations, Sequences and Angles

One-step equations, nth terms and angle facts

Years 7–8About 80 minutesNo calculator85 marks

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NameClassDate

Before you start

Recall

You will need these in every question below.

A1
Work out $- 5 + 12$ and $- 3 \times 4$ .
[2 marks]
Answer $\dots\dots$ and $\dots\dots$

A2
Simplify $4 n + 3 - n + 5$ .
[2 marks]
Answer
A3
Work out the value of $5 n - 2$ when $n = 6$ .
[1 mark]
Answer
A4
How many degrees are there in a full turn? How many degrees are there on a straight line?
[2 marks]
Answerfull turn $\dots\dots$ straight line $\dots\dots$
A5
Write down the next two terms of this sequence.
$19, 24, 29, 34, \dots$
[1 mark]
Answer

Examples, then your turn

Core

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

Example 1

Solve $x - 7 = - 3$ .

Hint Undo the subtraction: add 7 to both sides. Then check your answer in the equation.

B1
Solve $y + 9 = 2$ .
[2 marks]
Answer $y = \dots\dots$

Example 2

Solve $\frac{x}{3} = - 8$ .

Hint Undo the division: multiply both sides by 3.

B2
Solve $\frac{n}{5} = - 9$ .
[2 marks]
Answer $n = \dots\dots$

Example 3

Find the $n$ th term of the sequence $15, 19, 23, 27, \dots$

Hint The terms go up by 4, so the rule starts with

4 n

. Compare

4 n

with the terms to find what to add.

B3
Find the $n$ th term of the sequence $15, 23, 31, 39, \dots$
[2 marks]
Answer

Example 4

Three angles meet at a point. Work out the size of angle $x$ .

Not drawn accurately

Hint Angles at a point add up to

36 0^{\circ}

B4
Four angles meet at a point. Work out the size of angle $y$ .
[2 marks]
Not drawn accurately
Answer $y = \dots\dots$

Practice

Core

Each question asks for something different. Show your working.

C1
Solve $12 = p - 15$ .
[2 marks]
Answer $p = \dots\dots$

C2
A rectangle has an area of 63 cm² and a width of 7 cm. Its length is $l$ cm.
Write down an equation in $l$ . Solve it to find the length.
[2 marks]
Answer $l = \dots\dots$
C3
The $n$ th term of a sequence is $6 n - 4$ .
(a)
Work out the 15th term.
[1 mark]
Answer
(b)
Is 100 a term in the sequence? You must show your working.
[2 marks]
Answer
Total for C3: 3 marks
C4
The $n$ th term of a sequence is $12 - 3 n$ . Write down the first four terms.
[2 marks]
Answer
C5
Square tables are pushed together in a row. The diagram shows how many people can sit at 1, 2 and 3 tables.
(a)
Find an expression for the number of people who can sit at $n$ tables.
[2 marks]
Answer
(b)
How many tables are needed to seat 50 people?
[2 marks]
Answer
Total for C5: 4 marks
C6
Two straight lines cross. Work out the sizes of angles $a$ , $b$ and $c$ .
[3 marks]
Not drawn accurately
Answer $a = \dots\dots$ $b = \dots\dots$ $c = \dots\dots$
C7
Four angles meet at a point.
Not drawn accurately
(a)
Work out the value of $x$ .
[2 marks]
Answer $x = \dots\dots$
(b)
Work out the size of the largest angle.
[1 mark]
Answer
Total for C7: 3 marks
C8
Here are four equations.
$A : x + 10 = 4 B : 2 x = 12 C : \frac{x}{3} = 2 D : 9 - x = 15$
Which two equations have the solution $x = - 6$ ?
[2 marks]
Answer $\dots$ and $\dots$
C9
The 3rd term of a linear sequence is 17. The 8th term is 47. Find the $n$ th term of the sequence.
[3 marks]
Answer

Test-style questions

Stretch

Answer every question in the space given. Show your working: most of the marks are for method.

D1
Tom says, "The $n$ th term of the sequence $13, 16, 19, 22, \dots$ is $n + 3$ , because the terms go up by 3."
Tom is wrong. Explain why, and find the correct $n$ th term.
[2 marks]

D2
$P QR$ is a straight line. Work out the size of angle $P QS$ . You must show your working.
[3 marks]
Not drawn accurately
Answer
D3
Five adult cinema tickets cost £42.50 altogether.
(a)
One adult ticket costs £ $t$ . Write down an equation in $t$ and solve it.
[2 marks]
Answer $t = \dots\dots$
(b)
A child ticket costs £3.25 less than an adult ticket. Kim buys 1 adult ticket and 2 child tickets. How much does she pay?
[2 marks]
Answer
Total for D3: 4 marks
D4
Jo makes grids of squares from sticks. Every grid is 3 squares tall. Here are the first three patterns.
(a)
Find an expression for the number of sticks in Pattern $n$ .
[2 marks]
Answer
(b)
How many sticks are in Pattern 25?
[1 mark]
Answer
(c)
Jo has 160 sticks. She makes the largest pattern she can. Which pattern does she make, and how many sticks are left over?
[3 marks]
AnswerPattern $\dots\dots$ $\dots\dots$ sticks left over
Total for D4: 6 marks
D5
$A B$ and $C D$ are straight lines that cross at $O$ .
Not drawn accurately
(a)
Work out the value of $y$ .
[2 marks]
Answer $y = \dots\dots$
(b)
Work out the size of angle $A O D$ . Give a reason for your answer.
[2 marks]
Answer
Total for D5: 4 marks
D6
Ana's bank balance is £ $b$ . She pays in £35. Her balance is now $-$ £18.
Write down an equation in $b$ . Solve it to find Ana's balance before she paid in.
[3 marks]
Answer $b = \dots\dots$
D7
The $n$ th term of a sequence is $40 - 7 n$ . Work out the first term of the sequence that is negative. Give the term, not its position.
[3 marks]
Answer
D8
Four angles meet at the point $O$ . One of them is a right angle. Work out the size of the smaller of the two angles marked with $x$ . You must show your working.
[4 marks]
Not drawn accurately
Answer
D9
An equilateral triangle has a perimeter of 57 cm.
(a)
Each side is $s$ cm. Write down an equation in $s$ and solve it.
[2 marks]
Answer $s = \dots\dots$
(b)
A square has the same perimeter as the triangle. Work out the length of one side of the square.
[1 mark]
Answer
Total for D9: 3 marks

Challenge

UKMT level

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
Here are the first terms of two sequences.
Sequence $A$ : $3, 10, 17, 24, \dots$
Sequence $B$ : $103, 98, 93, 88, \dots$
Both sequences carry on in the same way. What is the largest number that is in both sequences?
[4 marks]
Hint 1 · What to try
Find the $n$ th term of each sequence.
Hint 2 · The first line
Sequence $A$ is $7 n - 4$ and sequence $B$ is $108 - 5 n$ . A number in both is 3 more than a multiple of 7 and 3 more than a multiple of 5.
Hint 3 · The full method
The numbers in both are $3$ , $38$ , $73$ , $108$ , … Sequence $B$ never reaches $108$ , because it starts at $103$ and goes down. So the largest is $73$ .
Answer

E2
What is the size of the smaller angle between the hour hand and the minute hand of a clock at 8:24?
[3 marks]
Hint 1 · What to try
In one hour the minute hand turns $36 0^{\circ}$ and the hour hand turns $3 0^{\circ}$ . Find how far each turns in one minute.
Hint 2 · The first line
The minute hand turns $6^{\circ}$ a minute, so at 8:24 it is $14 4^{\circ}$ round from 12. The hour hand turns $0. 5^{\circ}$ a minute, so it is $240 + 12 = 25 2^{\circ}$ round from 12.
Hint 3 · The full method
The angle between them is $252 - 144 = 10 8^{\circ}$ . This is less than $18 0^{\circ}$ , so it is the smaller angle.
Answer
E3
A purse holds 9 coins. Every coin is either a 20p coin or a 50p coin. The coins are worth £3.00 altogether. How many 50p coins are in the purse?
[3 marks]
Hint 1 · What to try
Call the number of 50p coins $f$ . Then there are $9 - f$ coins of 20p.
Hint 2 · The first line
If all 9 coins were 20p they would be worth 180p. That is 120p short of 300p.
Hint 3 · The full method
Swapping a 20p coin for a 50p coin adds 30p. So $120 \div 30 = 4$ coins must be 50p.
Answer
E4
The $n$ th term of a sequence is $4 n + 3$ . How many terms of the sequence are between 150 and 250?
[3 marks]
Hint 1 · What to try
Find the first term above 150 and the last term below 250.
Hint 2 · The first line
$4 \times 37 + 3 = 151$ is the first term above 150. $4 \times 61 + 3 = 247$ is the last term below 250.
Hint 3 · The full method
The terms between them are the 37th up to the 61st, which is $61 - 37 + 1 = 25$ terms.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can solve one-step equations, including ones with negative numbers and with the unknown on the right.
I can find and use the $n$ th term of a linear sequence.
I can use angles on a straight line, angles at a point and vertically opposite angles to find missing angles.

Answers and mark scheme

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