CONTENTS

Pythagoras and Trigonometry…………………………………2

Standard Form……………………………………………………………9

Handling Data………………………………………………………………14

Algebra……………………………………….…………………………………21

Level 8 Check List ……………….…………………………………28

SATs Aim for 8 1

Pythagoras and TrigonometryLevel 7

1. Boat

A boat sails from the harbour to the buoy. The buoy is 6km to the east 4km to the north of the harbour.

HAR BOUR

LAND

BUOYLIGHTHOUSE

SEA

N

NO T TO S CALE

Calculate the shortest distance between the buoy and the harbour. Give your answer to 1 decimal place. Show your working.

............................... km2 marks

2. Prism

(a) Look at this triangle:

NOT TO SCALE

Show working to explain why angle x must be a right angle.

1 mark

SATs Aim for 8 2

6cm

10cm 8cm

x º

(b) What is the volume of this prism?

7cm 6cm

8cm

10cm

NOT TO SCALE

You must show each step in your working.

…………………........... cm3

2 marks

(c) Prisms A and B have the same crosssectional area.

NOT TO SCALE

Complete the table:

Prism A Prism B

Height 5cm 3cm

volume 200cm3 .....……..... cm3

1 mark

Total 4 marks

SATs Aim for 8 3

5cm

A

3cm

B

3. Cupboard

A cupboard needs to be strengthened by putting a strut on the back of it like this.

stru t

150 cm

190 cm

(a) Calculate the length of the diagonal strut. Show your working.

..................................... cm2 marks

(b) In a small room the cupboard is in this position.

SATs Aim for 8 4

210 cm

150 cm

80 cm

165 cm

VIE W LO O K IN GD O W N O NTH E R O O M

Calculate if the room is wide enough to turn the cupboard like this and put it in its new position.

N E WP O SIT IO N

Show your working.

3 marks

Level 8

4. Triangles

ABC and ACD are both rightangled triangles.

SATs Aim for 8 5

6 cm

8 cmAB

C

D

6 cmNot drawn accurately

(a) Explain why the length of AC is 10 cm.

1 mark

(b) Calculate the length of AD

Show your working.

…………… cm2 marks

6 cm

8 cm AB

C

D

6 cm

x

y

Not drawn accurately

SATs Aim for 8 6

(c) By how many degrees is angle x bigger than angle y?

Show your working.

…………… °3 marks

Total 6 marks

5. Sailing

In this question you will get no marks if you work out the answer through scale drawing.

(a) Cape Point is 7.5km east and 4.8km north of Arton.

SATs Aim for 8 7

Cape Po int

A rton

4.8 km

7.5 km

N

NO T TOSC ALE

land

Calculate the direct distance from Arton to Cape Point.

Show your working.

………………… km2 marks

Bargate is 6km east and 4km north of Cape Point.

Bargate

C ape Po int

4 km

6 km

N

NO T TOSC ALE

land

(b) Steve wants to sail directly from Cape Point to Bargate.On what bearing should he sail?

Show your working.

SATs Aim for 8 8

……………… °2 marks

(c) Anna sails from Cape Point on a bearing of 048°.She stops when she is due north of Bargate.

How far north of Bargate is Anna? Show your working.

……………… km3 marks

6. Solving x

Look at the diagram:A

B C

D

3x º

x º

NOT TO SCALE

Side AB is the same length as side AC. Side BD is the same length as side BC. Calculate the value of x

Show your working.

2 marks

Standard FormLevel 8

7. Newton

Sir Isaac Newton (1642–1727) was a mathematician, physicist and astronomer.

SATs Aim for 8 9

In his work on the gravitational force between two bodies he found that he needed to multiply their masses together.

(a) Work out the value of the mass of the Earth multiplied by the mass of the Moon.

Give your answer in standard form.

M ass of Earth = 5 .98 x 10 kg

M ass of M oon = 7.35 x 10 kg

24

22

.............................. kg2 marks

Newton also found that he needed to work out the square of the distance between the two bodies.

(b) Work out the square of the distance between the Earth and the Moon.

Give your answer in standard form.

D istance be tw een Earth and M oon = 3 .89 x 10 km5

.............................. km²2 marks

Newton’s formula to calculate the gravitational force

(F) between two bodies is FGm m

R 1 2

2 where

G is the gravitational constant, m1 and m2 are the masses of the two bodies, and R is the distance between them.

SATs Aim for 8 10

(c) Work out the gravitational force (F) between the Sun and the Earth using the formula

F Gm m

R 1 2

2 with the

information in the box below.

Give your answer in standard form.

m m kg

R km

G

1 255 2

2 16 2

20

119 10

2 25 10

6 67 10

.

.

.

gravitational force ..............................2 marks

8. Earth and Moon

Wendy is making a scale model of the Earth and the Moon for a museum. She has found out the diameters of the Earth and the Moon, and the distance between them in centimetres.

Diameter of the Earth 1.28 109 cmDiameter of the Moon 3.48 108 cmDistance between Earth and Moon 3.89 1010 cm

(a) How many times bigger is the diameter of the Earth than the diameter of the Moon?

Show your working.

............................... times1 mark

(b) In Wendy’s scale model the diameter of the Earth is 50cm.

What should be the distance between the Earth and the Moon in Wendy’s model?

Show your working.

SATs Aim for 8 11

............................... cm2 marks

9. Storm

S peed o f ligh t is abou t 1 .1 x 10 km per hour

S peed o f sound is about 1.2 x 10 km pe r hour

9

3

(a) Calculate the speed of light in km per second.Give your answer in standard form.

Show your working.

.............................. km per second2 marks

(b) How many times as fast as the speed of sound is the speed of light? Give your answer to an appropriate degree of accuracy.

Show your working.

2 marks

(c) Gary sees a flash of lightning. 25 second later he hears the sound of thunder.

Calculate how far away he is from the lightning.(You do not need to include the speed of light in your calculation).

Show your working.

SATs Aim for 8 12

.............................. km2 marks

10. Values

(a) Which of these statements is true? Put a tick () by the correct one.

4 × 103 is a larger number than 43 ………………

4 × 103 is the same size as 43 ………………

4 × 103 is a smaller number than 43………………

Explain your answer.

1 mark

(b) One of the numbers below has the same value as 3.6 × 104

Put a tick () under the correct number.

363 364 (3.6 × 10)4 0.36 × 103 0.36 × 105

…… …… ……… ……… ………1 mark

(c) One of the numbers below has the same value as 2.5 × 10–3

Put a tick () under the correct number.

25 × 10–4 2.5 × 103 –2.5 × 103 0.00025 2500

.............. ............... ............... .............. ...........1 mark

(d) (2 × 102) × (2 ×102) can be written simply as 4 × 104

SATs Aim for 8 13

Write these values as simply as possible:

(3 × 102) × (2 × 10–2)

1 mark

4

8

102106

1 mark

SATs Aim for 8 14

Handling DataLevel 7

11. Experiment

Barry is doing an experiment. He drops 20 matchsticks at random onto a grid of parallel lines.

Barry does the experiment 10 times and records his results. He wants to work out an estimate of probability.

(a) Use Barry’s data to work out the probability that a single matchstick when dropped will fall across one of the lines.

Show your working.

..............................2 marks

(b) Barry continues the experiment until he has dropped the 20 matchsticks 60 times.

About how many matchsticks in total would you expect to fall across one of the lines?

Show your working.

............................... matchsticks2 marks

SATs Aim for 8 15

Num ber of the 20 m atchsticks that have fallen across a line

5 7 6 4 6 8 5 3 5 7

12. Tests

200 pupils in Year 9 at Oakdene School took two mathematics tests in January.

Each test had a maximum of 60 marks. The graph shows the pupils’ results.

Cum

ula

tive

% fr

eque

ncy

Marks0 10 20 30 40 50 60

0

1020

304050607080

90100

Test 1Test 2

(a) Put a next to each statement that is true.Put a x next to each statement that is not true.

For Test 1A The majority of pupils obtained fewer than 30

marks. ..........

B The top mark obtained was 60. ...........

C 20 pupils obtained 12 marks or fewer. ...........1 mark

(b) Put a next to each statement that is true.Put a x next to each statement that is not true

For Test 2A No pupil obtained fewer than 5 marks. .........

B The majority of pupils obtained more than 40 marks. .........

C 55 pupils obtained 34 marks or fewer. .........1 mark

SATs Aim for 8 16

(c) What was the median number of marks obtained on Test 1?

.............................. marks1 mark

(d) What was the lowest mark obtained by a pupil in the top 20% of Test 2?

.............................. marks1 mark

(e) Anita came top in both tests.What was her total number of marks?

.............................. marks1 mark

Level 8

13. Languages

100 students were asked whether they studied French or German.

3 9 2 7 3 0

4

F r e n c h G e r m a n

27 students studied both French and German.

(a) What is the probability that a student chosen at random will study only one of the languages?

1 mark

(b) What is the probability that a student who is studying German is also studying French?

1 mark

SATs Aim for 8 17

(c) Two of the 100 students are chosen at random.

Circle the calculation which shows the probability that both the students study French and German?

10027

10027

9926

10027

10026

10027

10027

10027

10026

10027

1 mark

14. Farm

40 students worked on a farm one weekend. The cumulative frequency graph shows the distribution of the amount of money they earned. No one earned less than £15.

(a) Read the graph to estimate the median amount of money earned.

Median £ ..............................1 mark

(b) Estimate the percentage of students who earned less than £40.

.............................. %1 mark

(c) Show on the graph how to work out the interquartile range of the amount of money earned.

Write down the value of the interquartile range.

Interquartile range £ ..............................1 mark

SATs Aim for 8 18

4 0

3 0

2 0

1 0

00 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0

CumulativeFrequency

Am ount of m oney earned (£)

(d) 30 of the students work on the farm another weekend later in the year. The tables below show the distribution of the amount of money earned by the students.

N o. of s tu den ts

> 25 and < 30> 30 and < 35> 35 and < 40> 40 and < 45> 45 and < 50> 50 and < 55> 55 and < 60

1234

1073

N o . of students

< 25< 30< 35< 40< 45< 50< 55< 60

0136

10202730

M oney earned (£ )M o ne y ea rne d (£ )

Draw a cumulative frequency graph using the axes below.

2 marks

(e) Put a by any statement below which is true.

Put a x by any statement below which is false.

A. Three of the students earned less than £35 each. ..............................

B. The median amount earned is between £40 and £45. ........................

C. Most of the 30 students earned more than £50 each. .........................

1 mark

SATs Aim for 8 19

0 5 10 15 20 25 30 35 40 45 50 55 600

5

10

15

20

25

30

Cum

ulat

ive

freq

uenc

y

Am ount of money earned (£)

15. Tests

(a) Thirty pupils took a maths test. The frequency graph shows the pupils' results.

88

44

00 10

10

20 30 40 50

Frequency

M arks

5

3

Complete the cumulative frequency graph of the pupils' results

2 marks

SATs Aim for 8 20

10 20 30 40 50

C um ulativeFrequency

M arks

28

24

20

16

12

8

4

0

(b) The 30 pupils also took a science test.

The cumulative frequency graph below shows their results.

10 20 30 40 50

Cum ulativeFrequency

M arks

28

24

20

16

12

8

4

0

On the axes below, draw a frequency graph to show their science results.

8

4

00 10 20 30 40 50

Frequency

M arks2 marks

Total 4 marks

SATs Aim for 8 21

Algebra SkillsLevel 7

16. Sixty

(a) Each of these calculations has the same answer, 60

Fill in each gap with a number.

= 60

2.4 × 25

60 ÷ 1

0 .24 × ..........

6 ÷ ..........

2 marks

(b) Solve these equations to find the values of a, b and c.

= 60

5 – 40a

4 + 80b

c + 242

a = .................... b = .................... c = ....................3 marks

SATs Aim for 8 22

(c) Solve these simultaneous equations to find the values of x, and y.

= 60

x y + 8

4 x + 4y

Show your working.

x = .................... y = ....................3 marks

17. Rearrange

(a) The subject of the equation below is p

p = 2 ( e + f )

Rearrange the equation to make e the subject.

e =2 marks

(b) Rearrange the equation r =21

(c – d) to make d the

subject. Show your working.

d =2 marks

SATs Aim for 8 23

18. Graphs

Look at this graph:

10

8

6

4

2

0

–2

1086420–2

A

B

y

x

(a) Show that the equation of line A is 2x + y = 8

1 mark

(b) Write the equation of line B.

1 mark

(c) On the graph, draw the line whose equation is y = 2 x + 1

Label your line C.1 mark

SATs Aim for 8 24

(d) Solve these simultaneous equations.

y = 2x + 1

3y = 4x + 6

Show your working.

x = ....………….….. y = .....………….....

3 marks

Total 6 marks

Level 8

19. Rectangles

The two rectangles below have the same area.

y – 3y + 1

y + 5 y + 10

N ot draw n accura tely

Use an algebraic method to find the value of y

You must show your working.

y = ............4 marks

SATs Aim for 8 25

20. Triangular Numbers

Denise and Luke are using the expression n n( )1

2 to generate

triangular numbers.

For example, the triangular number for n = 4 is 4 4 1

2( )

, which

works out to be 10.

(a) Denise wants to solve the inequality 300 < n n( )12

< 360

to find the two triangular numbers between 300 and 360.

What are these two triangular numbers?

You may use trial and improvement.

.............................. and ..............................2 marks

(b) Luke wants to find the two smallest triangular numbers

which fit the inequality n n( )1

2> 2700

What are these two triangular number?

You may use trial and improvement.

.............................. and ..............................2 marks

SATs Aim for 8 26

21. Number Games

Class 9H were playing a number game.

(a) Lena called Elin’s number x and formed an equation:

4x – 5 = 2x + 1

Solve this equation and write down the value of x.Show your working.

x = ...............................2 marks

(b) Call Aled’s number y and form an equation.

............................... = ...............................1 mark

Work out the value of Aled’s number.

Aled’s number is ...............................1 mark

SATs Aim for 8 27

Elin sa id :

"M ultip lying m y num ber by 4 and then subtracting 5 gives the sam e answ er as m ultiply ing m y num ber by 2 and then adding 1."

Aled said:

"M ultip lying m y num ber by 2 and then adding 5 g ives the sam e answer as subtracting m y num ber from 23."

(c) Lena thought of two numbers which she called a and b.

She wrote down this information about them in the form of equations:

a + 3b = 25

2a + b = 15

Work out the values of a and b.

Show your working.

a = ...............................

b = ...............................3 marks

22. Simplify

(a) Show that baba

–– 22

simplifies to a + b

1 mark

(b) Simplify the expression 22

23

baba

1 mark

(c) Simplify the expression 22

3223 –ba

baba

Show your working.

2 marks

SATs Aim for 8 28