KS3 Maths · Key Stage 3 · Unit 2: Powers, Shape and Probability

Powers and roots, transformations, constructions, Pythagoras and probability

Year 7About 85 minutesCalculator allowed86 marks

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Before you start

Recall

You will need these in every question below.

A1
Point $P$ is at $(2, - 1)$ . Point $Q$ is 4 units to the right of $P$ and 3 units up from $P$ . Write down the coordinates of $Q$ .
[1 mark]
Answer $(\dots\dots, \dots\dots)$

A2
Write down the equation of the vertical line that passes through the point $(3, 5)$ .
[1 mark]
Answer
A3
Work out $2.5 \times 2.5$ .
[1 mark]
Answer
A4
Write the fraction $\frac{18}{30}$ in its simplest form.
[1 mark]
Answer
A5
Work out $1 - \frac{3}{8}$ and $1 - 0.35$ .
[2 marks]
Answer $\dots\dots$ and $\dots\dots$

Examples, then your turn

Core

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

Example 1

A right-angled triangle has shorter sides of 7 cm and 9 cm. Work out the length of the hypotenuse. Give your answer to 1 decimal place.

Hint Square the two shorter sides and add them. That gives the square of the hypotenuse, so take the square root at the end.

B1
A right-angled triangle has shorter sides of 6 cm and 11 cm. Work out the length of the hypotenuse. Give your answer to 1 decimal place.
[3 marks]
Answer

Example 2

Rotate triangle $A$ through $9 0^{\circ}$ clockwise about the origin $O$ . Write down the coordinates of the vertices of the image.

Hint Use tracing paper. Put your pencil on

O

and turn the paper a quarter turn clockwise. A point

(x, y)

moves to

(y, - x)

B2
Rotate triangle $T$ through $9 0^{\circ}$ anticlockwise about the origin $O$ . Write down the coordinates of the vertices of the image.
[2 marks]
Answer

Example 3

A bag holds 5 red, 3 blue and 4 green counters. One counter is taken at random. Find the probability that it is not blue. Give your answer as a fraction in its simplest form.

Hint Count the counters that are not blue. Divide by the total number of counters.

B3
A bag holds 7 red, 2 white and 6 black counters. One counter is taken at random. Find the probability that it is not black. Give your answer as a fraction in its simplest form.
[2 marks]
Answer

Example 4

$60$ lies between two whole numbers that are next to each other. Which two?

Hint List the square numbers until you pass 60. The square root lies between the roots of the two squares either side of 60.

B4
$3150$ lies between two whole numbers that are next to each other. Which two?
[2 marks]
Answer $\dots\dots$ and $\dots\dots$

Practice

Core

Each question asks for something different. Show your working.

C1
Answer both parts.
(a)
Work out $3^{4} - 4^{3}$ .
[2 marks]
Answer
(b)
Write down both solutions of $x^{2} = 324$ .
[1 mark]
Answer
Total for C1: 3 marks

C2
$A$ is the point $(0, 0)$ and $B$ is the point $(6, 0)$ . A point $P$ is exactly 5 units from $A$ and exactly 5 units from $B$ .
(a)
$P$ lies on the perpendicular bisector of $A B$ . Write down the equation of this line.
[1 mark]
Answer
(b)
There are two possible points $P$ . Find the coordinates of both.
[2 marks]
Answer $(\dots, \dots)$ and $(\dots, \dots)$
Total for C2: 3 marks
C3
Point $A (- 2, 3)$ is translated by the vector $(5 - 4)$ . Its image is $A^{'}$ .
(a)
Write down the coordinates of $A^{'}$ .
[1 mark]
Answer $(\dots\dots, \dots\dots)$
(b)
Write down the vector of the translation that takes $A^{'}$ back to $A$ .
[1 mark]
Answer
Total for C3: 2 marks
C4
Reflect shape $S$ in the line $x = 1$ . Write down the coordinates of the vertices of the image.
[2 marks]
Answer
C5
The shaded triangle is enlarged by scale factor 2, centre $C (- 1, 0)$ . Write down the coordinates of the vertices of the image.
[3 marks]
Answer
C6
Ali uses compasses to construct the perpendicular bisector of a line $A B$ . The line $A B$ is 9 cm long.
(a)
The arcs he draws from $A$ and from $B$ must cross. The radius must be more than a certain length. What is that length?
[1 mark]
Answer
(b)
Point $C$ is on the perpendicular bisector. $A C = 7.5$ cm. Write down the length of $B C$ .
[1 mark]
Answer
Total for C6: 2 marks
C7
A ladder 6.5 m long leans against a vertical wall. The foot of the ladder is 2.5 m from the bottom of the wall, on level ground. How far up the wall does the ladder reach?
[3 marks]
Not drawn accurately
Answer
C8
A triangle has sides of 48 cm, 55 cm and 73 cm. Is the triangle right-angled? You must show your working.
[2 marks]
Answer
C9
A spinner is spun 80 times. It lands on red 28 times.
(a)
Work out the relative frequency of red. Give your answer as a decimal.
[1 mark]
Answer
(b)
The spinner is spun 500 more times. Use your answer to part (a) to predict how many of these land on red.
[1 mark]
Answer
Total for C9: 2 marks

Test-style questions

Stretch

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

D1
Sam says, " $2^{5}$ and $5^{2}$ are equal, because they both use a 2 and a 5."
Sam is wrong. Show that he is wrong.
[2 marks]

D2
An isosceles triangle has two sides of 17 cm and a base of 16 cm. The height $h$ is drawn from the top to the middle of the base.
Not drawn accurately
(a)
Work out the height $h$ .
[3 marks]
Answer
(b)
Work out the area of the triangle.
[1 mark]
Answer
(c)
A square has the same area as the triangle. Work out the perimeter of the square. Give your answer to 1 decimal place.
[2 marks]
Answer
Total for D2: 6 marks
D3
There are 60 people at a cinema. 35 are adults and the rest are children. 22 of the adults buy popcorn. 16 of the children do not buy popcorn.
(a)
One person is chosen at random. Find the probability that they are a child who buys popcorn. Give your answer as a fraction in its simplest form.
[2 marks]
Answer
(b)
One person is chosen at random. Find the probability that they buy popcorn.
[1 mark]
Answer
(c)
This time an adult is chosen at random. Find the probability that this adult does not buy popcorn.
[2 marks]
Answer
Total for D3: 5 marks
D4
Describe fully the single transformation that maps triangle $A$ onto triangle $B$ .
[3 marks]
Answer
D5
Point $P (7, 5)$ is enlarged with centre $(1, 2)$ . Its image is $P^{'} (3, 3)$ .
(a)
Work out the scale factor of the enlargement.
[2 marks]
Answer
(b)
Point $Q (4, 11)$ is enlarged in the same way. Find the coordinates of its image.
[2 marks]
Answer $(\dots\dots, \dots\dots)$
Total for D5: 4 marks
D6
A garden is drawn on a grid. Each square is 1 m long. A lamp is at $L (1, 0)$ and a tree is at $T (7, 0)$ .
A pond must be less than 5 m from the lamp, and closer to the lamp than to the tree.
Which of the points $P$ , $Q$ , $R$ and $S$ could be in the pond? You must show how you decide.
[4 marks]
Answer $\dots\dots\dots\dots$
D7
Show that a triangle with sides of 3.6 cm, 7.7 cm and 8.5 cm is right-angled.
[2 marks]
D8
Jess rolls a six-sided dice 300 times. She gets a six 81 times.
(a)
Work out the relative frequency of a six. Give your answer as a decimal.
[1 mark]
Answer
(b)
If the dice were fair, how many sixes would you expect in 300 rolls?
[1 mark]
Answer
(c)
Do you think the dice is fair? Give a reason.
[1 mark]
Answer
Total for D8: 3 marks
D9
A box is a cube. Its volume is 512 cm³. Work out the total area of its six faces.
[3 marks]
Answer
D10
A bag holds only red, blue and green counters. The probability of taking a red counter is 0.25. The probability of taking a blue counter is 0.4. There are 21 green counters in the bag.
How many counters are in the bag altogether?
[3 marks]
Answer

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
Two square numbers are next to each other in the list of square numbers. Their difference is 37. What is the larger of the two square numbers?
[3 marks]
Hint 1 · What to try
Try some pairs of square numbers that are next to each other, such as 16 and 25, and look at their differences.
Hint 2 · The first line
The differences go 3, 5, 7, 9, … The difference between $n^{2}$ and the square before it is $2 n - 1$ .
Hint 3 · The full method
$2 n - 1 = 37$ , so $n = 19$ . The larger square is $1 9^{2} = 361$ .
Answer

E2
A rectangle has a perimeter of 34 cm. Its diagonal is 13 cm long. What is the area of the rectangle?
[4 marks]
Hint 1 · What to try
Call the sides $a$ and $b$ . Write down what the perimeter tells you, and what Pythagoras tells you about the diagonal.
Hint 2 · The first line
$a + b = 17$ and $a^{2} + b^{2} = 169$ . Try whole numbers that add to 17.
Hint 3 · The full method
$5 + 12 = 17$ and $5^{2} + 1 2^{2} = 25 + 144 = 169$ . So the sides are 5 cm and 12 cm, and the area is $60$ cm².
Answer
E3
A bag holds only red and blue counters. The probability of taking a red counter is $\frac{3}{7}$ . Six more red counters are put in the bag. Now the probability of taking a red counter is $\frac{3}{5}$ .
How many blue counters are in the bag?
[4 marks]
Hint 1 · What to try
The number of blue counters does not change. Write the first bag as $3 k$ red and $4 k$ blue.
Hint 2 · The first line
After adding 6 red, the bag has $3 k + 6$ red out of $7 k + 6$ counters, and this fraction equals $\frac{3}{5}$ .
Hint 3 · The full method
$5 (3 k + 6) = 3 (7 k + 6)$ gives $15 k + 30 = 21 k + 18$ , so $k = 2$ . There are $4 k = 8$ blue counters. Check: $\frac{12}{20} = \frac{3}{5}$ .
Answer
E4
A point $P$ is reflected in the line $x = 3$ . The image is then reflected in the line $y = 1$ . The final point is $(- 4, 6)$ . Find the coordinates of $P$ .
[3 marks]
Hint 1 · What to try
Work backwards from the final point, undoing the last reflection first.
Hint 2 · The first line
Reflecting $(- 4, 6)$ in $y = 1$ gives $(- 4, - 4)$ , since $6$ is 5 above the line.
Hint 3 · The full method
Reflecting $(- 4, - 4)$ in $x = 3$ gives $(10, - 4)$ , since $- 4$ is 7 to the left of the line. So $P$ is $(10, - 4)$ .
Answer $(\dots\dots, \dots\dots)$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out squares, cubes, square roots and cube roots, and say which two whole numbers a root lies between.
I can translate, reflect, rotate and enlarge a shape on a grid, and describe a transformation fully.
I can use a perpendicular bisector and the distance from a point to decide where a locus or a region is.
I can use Pythagoras' theorem to find a missing side and to test whether a triangle is right-angled.
I can find a probability from counts or a two-way table, and use relative frequency to make a prediction.

Answers and mark scheme

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