KS3 Maths · Key Stage 3 · Unit 6: Coordinates, Graphs and Data

Coordinates, straight-line graphs, volume and surface area, and pie charts

Year 8About 85 minutesCalculator allowed93 marks

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Before you start

Recall

You will need these in every question below.

A1
Work out the value of $3 x - 4$ when $x = - 2$ .
[2 marks]
Answer

A2
A rectangle is 8.5 cm long and 6 cm wide. Work out its area.
[1 mark]
Answer
A3
What fraction of a full turn is $12 0^{\circ}$ ? Give your answer in its simplest form.
[1 mark]
Answer
A4
Work out $7 - (- 5) + (- 3)$ .
[1 mark]
Answer
A5
Work out $\frac{2}{5}$ of 360.
[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

Find the midpoint of $A (- 3, 4)$ and $B (5, - 2)$ .

Hint The midpoint is halfway across and halfway up. Find the mean of the two

x

-coordinates and the mean of the two

y

-coordinates.

B1
Find the midpoint of $P (- 6, - 1)$ and $Q (2, 9)$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$

Example 2

Find the equation of the straight line through $(1, 2)$ and $(4, 14)$ .

Hint The gradient is the rise divided by the run. Then put one point into

y = m x + c

to find

c

B2
Find the equation of the straight line through $(2, 1)$ and $(5, - 8)$ .
[3 marks]
Answer $y = \dots\dots$

Example 3

A cylinder has a radius of 5 cm and a height of 12 cm. Work out its volume. Give your answer to 1 decimal place.

Hint A cylinder is a prism. Its volume is the area of the circle on the end times the height.

B3
A cylinder has a diameter of 16 cm and a height of 7 cm. Work out its volume. Give your answer to 1 decimal place.
[2 marks]
Answer

Example 4

72 pupils choose a school lunch: 30 choose pasta, 18 choose curry and 24 choose salad. Work out the angle for each sector of a pie chart.

Hint The whole circle is

36 0^{\circ}

, shared between 72 pupils. Find the angle for one pupil first.

B4
45 pupils choose a sport: 18 choose football, 12 choose netball and 15 choose hockey. Work out the angle for each sector of a pie chart.
[3 marks]
Answerfootball $\dots\dots$ netball $\dots\dots$ hockey $\dots\dots$

Practice

Core

Each question asks for something different. Show your working.

C1
$A$ , $B$ and $C$ are three corners of the parallelogram $A B C D$ . Write down the coordinates of $D$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$

C2
$M (2, - 1)$ is the midpoint of the line $A B$ . $A$ is the point $(- 4, 3)$ . Find the coordinates of $B$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
C3
Write down the equation of the straight line drawn on the grid.
[2 marks]
Answer $y = \dots\dots$
C4
Find the equation of the line that is parallel to $y = 4 x - 3$ and passes through the point $(2, 11)$ .
[2 marks]
Answer $y = \dots\dots$
C5
The points in this table all lie on one straight line.
(a)
Find the equation of the line.
[2 marks]
Answer $y = \dots\dots$
(b)
Write down the coordinates of the point where the line crosses the $x$ -axis.
[1 mark]
Answer $(\dots\dots, \dots\dots)$
Total for C5: 3 marks
C6
A cuboid measures 8 cm by 6 cm by 2.5 cm. Work out its total surface area.
[2 marks]
Answer
C7
A cylinder stands upright on a table. Its diameter is 10 cm and its height is 4 cm.
(a)
What shape is the plan view of the cylinder?
[1 mark]
Answer
(b)
Work out the area of the front elevation.
[1 mark]
Answer
(c)
Work out the area of the plan view. Give your answer to 1 decimal place.
[2 marks]
Answer
Total for C7: 4 marks
C8
The pie chart shows how 120 pupils travel to school.
(a)
How many pupils walk to school?
[2 marks]
Answer
(b)
How many more pupils walk than cycle?
[1 mark]
Answer
Total for C8: 3 marks
C9
A cube has a volume of 343 cm³. Work out its total surface area.
[2 marks]
Answer

Test-style questions

Stretch

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

D1
Leo says, "The Swimming sectors are the same size, so the same number of pupils chose swimming in each year."
60 pupils in Year 7 and 150 pupils in Year 8 were asked their favourite sport. Each year group's results are shown in its own pie chart. In both pie charts the Swimming sector is $4 8^{\circ}$ . Explain why Leo is wrong, and work out how many more Year 8 pupils than Year 7 pupils chose swimming.
[3 marks]
Answer $\dots\dots$ more

D2
The straight line $L$ passes through $A (- 1, 9)$ and $B (3, - 7)$ .
(a)
Work out the gradient of $L$ .
[1 mark]
Answer
(b)
Find the equation of $L$ .
[2 marks]
Answer $y = \dots\dots$
(c)
Does the point $(10, - 36)$ lie on $L$ ? You must show how you decide.
[1 mark]
Answer
Total for D2: 4 marks
D3
A water tank is a cylinder. It has a diameter of 1.2 m and a height of 1.5 m.
(a)
Work out the volume of the tank. Give your answer in m³ to 2 decimal places.
[2 marks]
Answer
(b)
$1$ m³ is 1000 litres. Water flows into the empty tank at 15 litres a minute. How long does the tank take to fill? Give your answer to the nearest minute.
[2 marks]
Answer
(c)
A second tank is a cuboid with a square base 1.1 m by 1.1 m. It holds the same volume of water as the cylinder. Work out its height. Give your answer to the nearest centimetre.
[2 marks]
Answer
Total for D3: 6 marks
D4
A tent is a triangular prism. Its cross-section is an isosceles triangle with a base of 2.4 m, a height of 1.6 m and sloping sides of 2 m. The tent is 3 m long.
Not drawn accurately
(a)
Work out the volume of the tent.
[2 marks]
Answer
(b)
The tent is made from fabric: the two ends, the two sloping sides and the floor. Work out the total area of fabric.
[3 marks]
Answer
Total for D4: 5 marks
D5
The grid shows the triangle $P QR$ .
(a)
Work out the area of triangle $P QR$ .
[2 marks]
Answer
(b)
Find the coordinates of the midpoint of $P R$ .
[1 mark]
Answer $(\dots\dots, \dots\dots)$
(c)
$P QR S$ is a rectangle. Write down the coordinates of $S$ .
[1 mark]
Answer $(\dots\dots, \dots\dots)$
Total for D5: 4 marks
D6
A taxi charges a fixed fee plus the same amount for each mile. A 4-mile journey costs £9.60. A 10-mile journey costs £18.60.
(a)
Work out the cost for each mile.
[1 mark]
Answer
(b)
Work out the fixed fee.
[1 mark]
Answer
(c)
Write a formula for the cost, $C$ pounds, of a journey of $m$ miles.
[1 mark]
Answer $C = \dots\dots$
(d)
Lin has £24. What is the greatest whole number of miles she can travel?
[2 marks]
Answer
Total for D6: 5 marks
D7
A pie chart shows the favourite fruit of a group of people. The Apple sector has an angle of $8 4^{\circ}$ and stands for 21 people. The Banana sector has an angle of $13 2^{\circ}$ .
(a)
How many people are in the group?
[2 marks]
Answer
(b)
The rest of the people chose grapes or oranges in the ratio $1 : 3$ . How many people chose oranges?
[2 marks]
Answer
Total for D7: 4 marks
D8
A cuboid is 9 cm long and 4 cm wide. Its total surface area is 202 cm². Work out the height of the cuboid, and then its volume.
[4 marks]
Answerheight $\dots\dots$ cm volume $\dots\dots$ cm³
D9
The line $L$ has equation $y = 3 x - 6$ .
(a)
Write down the coordinates of the point where $L$ crosses the $y$ -axis.
[1 mark]
Answer $(\dots\dots, \dots\dots)$
(b)
Work out the coordinates of the point where $L$ crosses the $x$ -axis.
[1 mark]
Answer $(\dots\dots, \dots\dots)$
(c)
$L$ and the two axes make a triangle. Work out its area.
[2 marks]
Answer
Total for D9: 4 marks
D10
$A$ is the point $(- 5, 2)$ . The midpoint of $A B$ is $M (1, - 3)$ . $N$ is the midpoint of $M B$ . Find the coordinates of $N$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$

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
Three corners of a parallelogram are at $(0, 0)$ , $(4, 1)$ and $(1, 3)$ . There are three different places the fourth corner could be. Work out the sum of the $x$ -coordinates of all three places.
[3 marks]
Hint 1 · What to try
Sketch the three points. Any two of them could be the ends of a diagonal, or the ends of a side.
Hint 2 · The first line
Opposite sides of a parallelogram are the same move. So the fourth corner is one point plus a second point, minus the third point.
Hint 3 · The full method
The three places are $(4, 1) + (1, 3) - (0, 0) = (5, 4)$ , then $(0, 0) + (4, 1) - (1, 3) = (3, - 2)$ , then $(0, 0) + (1, 3) - (4, 1) = (- 3, 2)$ . The $x$ -coordinates add to $5 + 3 - 3 = 5$ .
Answer

E2
The line $y = m x + 12$ makes a triangle with the positive $x$ -axis and the positive $y$ -axis. The area of the triangle is 48 square units. Work out $m$ .
[3 marks]
Hint 1 · What to try
The line crosses the $y$ -axis at $(0, 12)$ , so the height of the triangle is 12.
Hint 2 · The first line
The area is $\frac{1}{2} \times base \times 12 = 48$ , so the base is 8. The line crosses the $x$ -axis at $(8, 0)$ .
Hint 3 · The full method
The line goes from $(0, 12)$ to $(8, 0)$ : down 12 as it goes 8 across. So $m = - \frac{12}{8} = - 1.5$ .
Answer $m = \dots\dots$
E3
A cuboid has a square base. Its height is 3 cm more than the side of the base. Its volume is 324 cm³. Work out its total surface area.
[4 marks]
Hint 1 · What to try
Call the side of the base $s$ . The volume is $s \times s \times (s + 3)$ .
Hint 2 · The first line
Try whole numbers for $s$ : $s = 5$ gives $25 \times 8 = 200$ , which is too small.
Hint 3 · The full method
$s = 6$ gives $36 \times 9 = 324$ . The base and top are $36$ cm² each and the four sides are $6 \times 9 = 54$ cm² each, so the surface area is $72 + 216 = 288$ cm².
Answer
E4
A pie chart has four sectors. The Red sector is $9 0^{\circ}$ and the Blue sector is $12 6^{\circ}$ . The Green sector stands for 18 people and the Yellow sector for 22 people. How many people were asked?
[3 marks]
Hint 1 · What to try
Green and Yellow together use the angle that Red and Blue leave over.
Hint 2 · The first line
$360 - 90 - 126 = 14 4^{\circ}$ , and this stands for $18 + 22 = 40$ people.
Hint 3 · The full method
Each person has $144 \div 40 = 3. 6^{\circ}$ , so the whole circle is $360 \div 3.6 = 100$ people.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can plot and read points in all four quadrants and find a midpoint.
I can find the gradient and equation of a straight line, and spot parallel lines.
I can work out the volume and surface area of prisms and cylinders, and draw plans and elevations.
I can draw a pie chart and read numbers back off one.

Answers and mark scheme

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