# Sequences and Nth Term

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```					                    Homework

Q - How often will it be set?
A - Once a week

Q - How will it be assessed/graded?
A – You will receive an effort grade A, B, C, D, E or
U and an attainment grade 1, 2, 3, 4 or 5. Along
with this you will be told what level you are working
at, depending on which exam questions you
performed well in.

Q – What if it’s not done on time?
A – In the first instance a detention then the
maths department strike system will be followed.

Q – What do I do if I get stuck?
A – Either see your maths teacher for help or go to
homework club.
Basic 4 Rules of Number (5 – 8)
1. Solve the following (show all working):

a)      492 + 39                                      b) 16.91 + 708.7
c)      6230 – 781                                    d) 93.1 – 3.65
e)      187 x 816                                     f) 8.14 x 682.9
g)      45975 ÷ 5                                     h)   7 5715

2. (a) I pay £16.20 to travel to work each week. I work for 45 weeks each year.
How much do I pay to travel to work each year? Show your working.
(b) I could buy one season ticket that would let me travel for all 45 weeks.
It would cost £630. How much is that per week?

Level 5 question
(a) A football club is planning a trip. The club hires 234 coaches. Each coach holds 52
passengers. How many passengers is that altogether? Show your working.
2 marks
(b) The club wants to put one first aid kit into each of the 234 coaches. These first aid
kits are sold in boxes of 18. How many boxes does the club need?               1 mark

Level 6 question - Here are six number cards.                              1   2        3   4    5   6
Arrange these six cards to make the calculations below.
The first one is done for you.
939   =   4    2   3     +   5   1    6

a)                                                          b)   b)

c) Now arrange the six cards to make a difference of 115
3 marks

Level 7 question
a) The ship ‘Queen Mary' used to sail across the Atlantic Ocean. The ship's usual speed
was 33 miles per hour. On average, the ship used fuel at the rate of 1 gallon for every 13
feet sailed. Calculate how many gallons of fuel the ship used in one hour of travelling at
the usual speed. (There are 5280 feet in one mile.) Show your working and write down the
full calculator display.                                                       2 marks

b) Now write your answer correct to 2 significant figures.                                      1 mark

Level 8 question – The table below shows information about some countries.
Country   Populatio               2 a) Which country has the largest population?
Area (km )
n        7           7
1 mark
Canada    3.1 × 10    1.0 × 10      b) Which country has the smallest area?
7            5
France    6.0 × 10    5.5 × 10                                            1 mark
Gambia    1.4 × 10
6
1.1 × 10
4    c) On average, how many more people per km2 are
9            6      there in the UK than in the USA?
India     1.0 × 10    3.3 × 10
7              5
UK         6.0 × 10       2.4 × 10
8              6
USA        2.8 × 10       9.3 × 10
Decimals (5 – 8)
1. Answer the following (Draw a decimal number line to help)
a) How many 1 digit decimals are there between
i) 2.3 and 2.9      ii) 8.5 and 9.8   iii) 0.5 and 6.2                 iv) 3 and 4
b) How many 2 digit decimals are there between
i) 3.42 and 3.49    ii) 8.02 and 8.17 iii) 3.9 and 4.0                 iv) 3.2 and 3.5

2. Round the   following to one decimal place.
a) 4.79    b) 7.43     c) 12.42   d) 13.64    e) 19.99              f) 16.47
3. Round the   following to two decimal places.
a) 0.473   b) 3.721 c) 8.648 d) 15.192        e) 81.196             f) 0.0852
4. Round the   following to the nearest whole number.
a) 2.53    b) 7.43     c) 13.64   d) 23.493   e) 13.642             f) 29.962

Level 5 question
The number 6 is halfway between 4.5 and 7.5                                  6

Fill in the missing numbers below
4.5                        7.5
(a) The number 6 is halfway between 2.8 and .....................                        1 mark

(b) The number 6 is halfway between –1.5 and .....................                       1 mark

Level 6 question – Fill in the missing decimal number.

(a) 15 ‚ ……….. = 15 x 0.1                                                                1 mark

(b) 15 ‚ 1000 = 15 x ………..                                                               1 mark

(b) 15 x 0.01 = 15 ‚ …………..                                                              1 mark

Level 7 question
Each of these calculations has the same answer, 60. Fill in each gap with a number.

2.4 x 25                              60 ÷ 1

= 60
0.24 x ……                                         6 ‚ ……
2 marks

Level 8 question – A housing report gave this information.

    In the year 2001, the population of England was 49.87 million people.
    Most of these people lived in households.
    The total number of households was 20.97 million.
    The average (mean) household size was 2.34 people.

In the year 2001, what percentage of people in England did not live in households?
3 marks
Negative Numbers (5 – 8)
1. Put these numbers into ascending order (smallest to biggest).
a) 5, -9, 2, -3, 1            b) -11, 4, 8, -3, -7           c) 49, -34, 6, -8, 11
2. What number is half way between
a) 5 and -1      b) -6 and -14    c) 7 and -19     d) -7 and -21     e) 2 and -34

3. Work out the following
a) 4 – 7      b) 7 – 13     c) 8 – (-4)       d) 0 – 5     e) -11 – (-9)     f) -34 + 19
g) -20 – 13      h) 11 + (-11)      i) -8 – (-17)     j) 15 – 7     k) -16 – 22      l) 62 – (-41)
4. A deep sea diver needs to return to the surface. He is at a depth of -75 metres
below and rises 40 metres. At what depth is he now?
5. Billy is overdrawn at the bank by £180. His brother Mark is better off than him by
£70. How much money does Mark have?
6. What temperature is 12°C warmer than -4°C?
7. Work out the following
a) 4 x 5      b) -3 x 7      c) -14 x -2       d) 7 x -8      e) -9 x -1     f) -19 x 0
g) 82 ÷ 2      h) -36 ÷ -6       i) -54 ÷ 9      j) 350 ÷ -25       k) -81 ÷ 3     l) -56 ÷ 8

Level 5 question – Write a number in each box to make the calculations correct.

a)                                                      b)

2 marks

Level 6 question – Write the missing numbers in the table. The first row is done for you.

Sum of first               Product of first
First      Second
and second                  and second
number      number
numbers                     numbers
3               6               9                          18
5              –3                                                           1 mark
–8                              –5                                          1 mark

Level 7 question - Write the missing numbers in these multiplication grids.

×         8                                      ×          0.2
a)    9          72                           b)        3                    1.2
–6                    30                                                6

3 marks

Level 8 question – Write a number in each box to make the inequalities true.

a)                              ÷                       <    -1
1 mark

b)               –1        <               ÷                        <         0            1 mark
Primes, Factors and Multiples (5 – 8)
1. Find the Highest Common Factor (HCF) of the following sets of numbers
a) 56 an 42    b) 36 and 54  c) 105 and 75    d) 64 and 80      e) 27, 18 and 99

2. Find the Lowest Common Multiple (LCM) of the following sets of numbers
a) 6 and 9 b) 20 and 8    c) 16 and 12    d) 20 and 24     e) 15, 20 and 24

3. Write down all the prime numbers between 0 and 100.

4. Find all the prime factors of these numbers.
Write your answers in index notation (e.g. 2 x 2 x 2 x 5 = 23 x 5)
a) 8       b) 20    c) 36     d) 42     e) 80    f) 99     g) 108     h) 168                   i) 216

5. Find all the prime factors of the following numbers.
Use the prime factors to find the HCF and the LCM of each set.
a) 70 and 98                 b) 90 and 165             c) 126 and 72

d) 6 and 15                              e) 48 and 88                  f) 124 and 160

Level 5 question –
(a) I am thinking of a number. My number is a multiple of 4.
Tick (       ) the true statement below.
My number       My number                         My number
must be even     must be odd                  could be odd or even

Explain how you know.                                                                         1 mark

b) I am thinking of a different number. My number is a factor of 20
Tick (      ) the true statement below.
My number        My number                           My number
must be even     must be odd                     could be odd or even

Explain how you know                                                                          1 mark

Level 5+ Question -The diagram shows part of a number grid.
The grid has 6 columns. All the prime numbers in the grid are circled.
43   44     45       46   47     48

37   38     39       40   41     42
(a) 35 is not circled. Explain why 35 is not a prime number.
1 mark
31   32     33       34   35     36

(b) There are no prime numbers circled in column Y.
Explain how you know there will never be a prime number
25   26     27       28   29     30

19   20     21       22   23     24           in column Y.
1 mark
13   14     15       16   17     18

7    8       9       10   11     12
(c) There is one prime number circled in column X.
Explain how you know there will never be another prime
1    2       3       4    5       6
number in column X.                         1 mark

column X             column Y
Fractions, Decimals & Percentages (5 – 8)

1. For each question, find the equivalent percentage:
4                            21                                    17          73
a) 0.45   b)       c) 7      d) 0.07    e)      f) 0.483                    g)          h)
5       20                   30                                    52          89
2. For each question, find the equivalent decimal:
12       7                            15                                        2
a) 57%    b)        c)        d) 39% e) 5.4% f)                              g) 0.61% h)
20       25                           26                                        7

3. For each question, find the equivalent fraction:
a) 42%         b) 0.36    c) 0.78 d) 97%         e) 1.38      f) 137%       g) 7%        h) 0.08

Level 5 question
(a) Look at this diagram:                                   What fraction is shaded?
1 mark

2
(b) Shade         of the diagram:                                                                1 mark
5
What percentage of the diagram have you shaded?
1 mark

Level 6 question – Some of the statements below are correct. Circle the correct ones.
9             3                       1                                                       3
                                    = 0.5                             0.75   
30            10                       2                                                       4

1 is equivalent to 10%                         1 is equivalent to 5%
2                                              5
2 marks

Level 7 question - A teacher said to a pupil:
1
To the nearest per cent           is 17%
6

The pupil said:       So, to the nearest per cent, 2 must be 34%
6

Show that the pupil is wrong.                                                                    1 mark

Level 8 question
a) One calculation below gives the answer to the question What is 70 increased by 9%?
Tick () the correct one.
1 mark
70 × 0.9         70 × 1.9     70 × 0.09              70 × 1.09

b) Choose one of the other calculations. Write a question about percentages that this
calculation represents.
calculation chosen: ...........................
question it represents: ..............................                                       1 mark
c) Now do the same for one of the remaining two calculations.
calculation chosen: ...........................question it represents: ............................... 1 mark
Fractions (5 – 8)
1. Evaluate the following (Write each fraction in its simplest form)
3 1           2 2            8 3           1 3             5 1                                         9 2
a)          b)          c)           d)           e)                                      f)       
4 8           5 4            21 7          2 8             9 4                                         11 3

g) 5  3     h) 1  10     i) 4  2     j) 12  3     k) 2  8                                 l) 21  4 
6 7           8 11           8 5           15 4            3 13                                     25 6

2. Solve each question.
1             1                        1                 1                      1                       1
a)    of 28 b)     of 56             c)     of 452   d)       of 48           e)     of 108         f)        of 1475
4             7                        2                12                      9                       25

4                  5                2                 7                      10          2
g)      of 36       h)     of 84     i)     of 111   j)       of 48           k)      of 143 l) of 72
6                  7                3                12                      11          9

Level 5 question – Fill in the missing numbers.
a) 1 of 20 = 1 of .......................                   b)       3 of 100 = 1 of .......................
2           4                                                  4          2

c) 1 of 60 = 2 of .......................                                                                   3 marks
3             3
Level 6 Question –
(a) Add 6 and 6 Now use an arrow ( ) to show the result on the number line
10    5    0                           1                           2

2 marks
(b) How many sixths are there in 3 1 ?                                                                      1 mark
3
1 5
(c) Work out 3                     Show your working                                                       2 marks
3 6
Level 7 Question –
All unit fractions must have
1 , 1 , 1 are all examples of unit fractions.
3 8 5                                                                           a numerator that is 1
The ancient Egyptians used only unit fractions.                         1
For 3 , they wrote the sum 1 + 1                                        3
4                       2    4                                             a denominator that is
an integer greater than 1

(a) For what fraction did they write the sum 1 + 1 ? Show your working.                                     1 mark
2        5
(b) They wrote 9 as the sum of two unit fractions. One of them was 1
20                                                    4
What was the other? Show your working                                   1 mark
(c) What is the biggest fraction you can make by adding two different unit fractions?
Level 8 Question –
I fill a glass with orange juice and lemonade in the ratio 1 : 4. I drink 1 of the contents of the
4
glass, then I fill the glass using orange juice. Now what is the fraction of orange juice in the
glass? Show your working, and write the fraction in its simplest form.
3 marks
Percentages (5 – 8)
1. Evaluate the following (Do Not use a calculator)
a) 10% of 30       b) 25% of 460        c) 1% of 40         d) 15% of 70      e) 75% of 28
f) 30% of 61       g) 17.5% of 520      h) 99% of 57        i) 4% of 91       j) 42% of 310
2. Find the missing number (Do Not use a calculator)
a) ___% of 100 = 45      b) 25% of ___ = 17     c) ___% of 75 = 15          d) 90% of ___ = 180

3. Evaluate the following (you may use a calculator)
a) 4% of 28    b) 17% of 56           c) 71% of 452         d) 65% of 48      e) 9% of 108
f) 61% of 1475 g) 76% of 36           h) 97% of 84          i) 47% of 111     j) 88% of 48
4. Find the missing number (you may use a calculator)
a) 42% of ___ = 189 b) ___% of 208 = 52 c) ___% of 34 = 19 d) 76% of ___ = 79

Level 5 question – Work out the following answers
a) 10% of 84 = ___     5% of 84 = ___       2.5% of 84 = ___                         2 marks

b) The cost of a CD player is £84 plus 17½% tax. What is the total cost of the CD

Level 6 Question – Kate asked people if they read a daily newspaper. Then she wrote this
table to show her results.   No      80 people = 40%
Yes      126 people = 60%
The values in the table cannot all be correct.
The error could be in the number of people.
Complete each table to show what the correct numbers could be.
No          80 people = 40%         No      .......... people = 40%
Yes     ......... people = 60%      Yes         126 people = 60%                  2 marks

Level 7 Question – In a quiz game two people each answer 100 questions. They score one
point for each correct answer. The quiz game has not yet ﬁnished. Each person has
answered 90 questions. The table shows the results so far.
Person A                Person B
60% of the ﬁrst 90      50% of the ﬁrst 90
questions correct       questions correct
B can win.
B cannot win but can draw.
B cannot win or draw.                                                              2 marks

Level 8 Question – Here is part of a newspaper report about wildlife in a country in Africa.

The number of gorillas has fallen by 70% in the
last ten years. Only about 5000 gorillas are left.

About how many gorillas were there in this country ten years earlier?                2 marks
Ratio (5 – 8)
1. Cancel down fully the following ratios.
a) 6 : 4 b) 7 : 28 c) 8 : 20 d) 14 : 8 e) 15 : 35 f) 640 : 720 g) 96 : 24
h) 2 : 4 : 10  i) 7 : 28 : 77    j) 18 : 30 : 54 k) 36 : 63 : 81 l) 28 : 35 : 56
2. Change the following into the same units, then cancel down the ratio.
a) £1 : 50p b) 3Kg : 500g c) 25mm : 6cm d) 45cm : 3m e) 3.5cm : 2.8cm : 7mm

3. In a sandwich bar 120 ham, 30 egg and 10 chicken sandwiches are sold. Find the
ratio of ham to egg to chicken sandwiches sold. Write it in its simplest form.
4. Divide the following amounts by in the given ratios.
a) £40 in the ratio 3 : 2    b) £49 in the ratio 5 : 2    c) £1540 in the ratio 9 : 5
d) £65 in the ratio 7 : 3    e) £200 in the ratio 3 : 7   f) £240 in the ratio 5 : 3 : 2
g) £52.50 is split between Alec, Beth and Chloe in the ratio 3 : 5 : 7. How much does

Level 5 question – Work out the number of boys and girls in each class below.
a) In class 8K, there are 28 pupils.   Number of boys         Number of girls
The ratio of boys to girls is 3 : 1  .....................  ..................... 1 mark
b) In class 8T, there are 9 boys.
Number of boys         Number of girls
The ratio of boys to girls is 1 : 2                                               1 mark
.....................  .....................

Level 6 Question – The screens of widescreen and standard televisions look different.
They have different proportions.

Widescreen      Standard
Ratio of height to                                     Ratio of height to
width is 9 : 16       Television     Television
width is 3 : 4
Keri starts to draw scale drawings of the televisions. For each, the height is 4.5 cm.
What should the width of each scale drawing be?

Widescreen:
Widescreen
Width = ____cm                               Standard
4.5 cm                                                     4.5 cm
television                                                         television

Standard:
Width = ___cm                                                 2 marks

Level 7 Question – The diagram shows a shaded rectangle.
It is divided into four smaller rectangles,
labelled A, B, C and D.                     5 cm    A                                         B

The ratio of area C to area B is 1 : 2
Calculate area A.                         3 cm       C                        D
Not drawn
accurately

5 cm           10 cm        2 marks
Level 8 Question – Films at the cinema and ﬁlms on television are shown at different speeds.

Cinema                            Television
24 pictures per second              25 pictures per second

At the cinema a ﬁlm lasts 175 minutes. How many minutes does the same ﬁlm last on
television?                                                      2 marks
Number Review 1 (5 – 8)
Level 5 questions
1. a)Write the missing numbers.
50% of 80 = .........  5% of 80 = ........       1% of 80 = ..........          2 marks
b) Work out 56% of 80
2. Look at this diagram it may help you work out some of these fraction calculations.
1 1
1
12
a)                                     1 mark
1
6          1
12
12 4
1 1
 
1

b)                                       1 mark
12

1
3 4
3
1
1 1
 
4
c)                                       1 mark
3 6
Level 6 Questions
1. a) Give an example to show the statement below is not correct.
When you multiply a number by 2, the answer is always greater than 2               1 mark
b) Now give an example to show the statement below is not correct.
When you subtract a number from 2, the answer is always less than 2                1 mark
c) Is the statement below correct for all numbers?
The square of a number is greater than the number itself.
Explain how you know.                                                            1 mark
2. I think of a number. I multiply this number by 8, then subtract 66. The result is twice
the number that I was thinking of. What is the number I was thinking of?
2 marks
Level 7 Questions
1. A three-digit number is multiplied by a two-digit number. How many digits could the
answer have? Write the minimum number and the maximum number of digits that the
2. Look at these number cards.         0.2         2       10         0.1      0.05   1

a) Choose the two cards which give the lowest possible answer when they are multiplied.
____ x ____ =                                                     2 marks
b) Choose the two cards that give the answer 100 when divided.
____ ÷ ____ =                                                     1 mark

Level 8 Questions
1. a) Is 3100 even or odd? Explain your answer.                                       1 mark
b) Tick ( ) the number below that is the same as 3100 × 3100
3200       6100      9200     310 000     910 000               1 mark
2. People were asked if they were considering changing what they eat.
29% of the people asked said yes.
Of these, 23% said they were considering becoming vegetarian. What percentage of the
people asked said they were considering becoming vegetarian?            1 mark
Number Review 2 (5 – 8)
Level 5 questions
1 a) Show that 9 × 28 is 252                                                    1 mark
b) What is 27 × 28?
2. a) Nigel pours 1 carton of apple juice and 3 cartons of orange juice into a big jug.
What is the ratio of apple juice to orange juice in Nigel's jug?
apple juice : orange juice = ..................... : ..................... 1 mark
b) Lesley pours 1 carton of apple juice and 1 ½ cartons of orange juice into another
big jug. What is the ratio of apple juice to orange juice in Lesley's jug?
apple juice: orange juice = ..................... : .....................  1 mark
c) Tandi pours 1 carton of apple juice and 1 carton of orange juice into another big jug.
She wants only half as much apple juice as orange juice in her jug.
What should Tandi pour into her jug now?                                          1 mark
Level 6 Questions
1. Write the missing numbers in the table. The first row is done for you.
First     Second         Sum of first and       Product of first and
number     number          second numbers          second numbers

3           6                  9                      18
5          –3                                                          1 mark
–8                            –5                                       1 mark
2. Find the values of t and r.
2 t                             2 5
a)                             b)                                   2 marks
3 6                             3 r
Level 7 Questions
1. a) Circle the best estimate of the answer to 72.34 ÷ 8.91
6         7        8         9        10        11                   1 mark
b) Circle the best estimate of the answer to 32.7 x 0.48
1.2       1.6      12        16       120       160                   1 mark

c) Estimate the answer to 8.62  22.1 . Give your answer to 1 sig. figure     1 mark
5.23

d) Estimate the answer to 28.6  24.4                                         1 mark
5.67  4.02
2. A newspaper printed this information about the world’s population.
If the world was a village of 100 people,
6 people would have 59% of the total wealth.
The other 94 people would have the rest.
On average, how many times as wealthy as one of the other 94 people would one of
these 6 people be?                                                        2 marks

Level 8 Question
1. a) Each side of a square is increased by 10%
By what percentage is the area increased?                                 2 marks
Number Review 3 (5 – 8)
Level 5 questions
1. Fill in the missing numbers
1            1             3         1                                   1         2
of 20 =     of ___        of 100 = of ___                               of 60 =   of ___           3 marks
2            4             4         2                                   3         3
2. Complete the sentences.
a) ............................... out of 10 is the same as 70%                                        1 mark
b) 10 out of 20 is the same as ..............................%                                         1 mark
c) ..................... out of ..................... is the same as 5%                                1 mark

Level 6 Questions
1. a) How many eighths are there in one quarter?                                                        1 mark
3 1
b) Now work out                                                                                      2 marks
4 8                         6      5     4      3        2      1
2. a) Look at these numbers     1  2    3    4   5    6
Which is the largest? _____     Which is equal to 64? ____                                         2 marks
b) Which two of the numbers below are not square numbers?
24 25 26 27 28             ____ and ____                                                      1 mark

Level 7 Questions                                  Birth rate per 1000 population
1. Look at the table                                   1961             1994
England               17.6
Wales                17.0              12.2
a) In England, from 1961 to 1994, the birth rate fell by 26.1%. What was the birth rate in
England in 1994? Show your working.                                         2 marks
b) In Wales, the birth rate also fell. Calculate the percentage fall from 1961 to 1994.
2. A square of area 64cm2 is cut to make two rectangles, A and B.
The ratio of area A to area B is 3 : 1
Work out the dimensions of rectangles A and B.
Rectangle A: ......... cm by ............ cm
Rectangle B: ......... cm by ............ cm
Area = 64cm2                                                                    2 marks

Level 8 Questions                                                               Earth         Mercury
1. Look at the table:                       Mass (Kg)                                    24             23
5.98x10        3.59x10
2                                            -8
Atmospheric pressure (N/m )                                     2x10
12
a) The atmospheric pressure on Earth is 5.05 × 10 times as great as the atmospheric
pressure on Mercury. Calculate the atmospheric pressure on Earth.      1 mark
b) What is the ratio of the mass of Earth to the mass of Mercury?
c) The approximate volume, V, of a planet with radius r is given by V  4 πr 3
3
Assume the radius of Mercury is 2400 km. Calculate the volume of Mercury.
Give your answer, to 1 significant figure, in standard form.          2 marks
Number Review 4 (5 – 8)
Level 5 questions
1. I am thinking of a number.     My number multiplied by 15 is 315
What is my number?             My number multiplied by 17 is 357          2 marks
2.
a) 240 people paid the entrance fee on Monday. How much
money is that altogether? Show your working.     2 marks
b) The museum took £600 in entrance fees on Friday. How many
people paid to visit the museum on Friday? Show your working.
2 marks
3. Here is a list of numbers:
–7              –5        –3       –1       0        2        4      6
a) What is the total of all eight of the numbers on the list?                       1 mark
b) Choose the three numbers from the list which have the lowest possible total.
Write the three numbers and their total.
...... + ...... + ...... = ...... You must not use the same number more than once. 2 marks

Level 6 Questions
1. Calculate 57.3 × 2.1      Show your working.                            2 marks
1
2. a) Some of the fractions below are smaller than     Tick ( ) them.
1
9
1          4                       1           1
10          9          2           100          8        1 mark
b) To the nearest per cent, what is 1 as a percentage? Tick ( ) the correct percentage.
9
0.9%          9%            10%           11%        19%    1 mark
1
c) Complete the sentence by writing a fraction.       is half of .............    1 mark
9
Level 7 Questions
1. People who live to be 100 years old are called centenarians. In 1998 there were 135 000
centenarians. The ratio of male to female was 1 : 4. How many female centenarians were
there in 1998? Show your working.                                           2 marks
No       80 people = 40%        Then she wrote this table to show her results.
Yes      126 people = 60%       The values in the table cannot all be correct.
a) The error could be in the number of people.
No     80 people = 40%        Complete each table to show what the correct
Yes     ....... people = 60%   numbers could be.                       1 mark

b)     No     80 people = ......... %    The error could be in the percentages.
Yes    126 people = ......... %   Complete the table with the correct percentages.
2 marks
Level 8 Questions
1. I start with any two consecutive integers. I square each of them, then I add the two
squares together. Prove that the total must be an odd number.              3 marks
2. A shop had a sale. All prices were reduced by 15%. A pair of shoes cost £38.25 in the
sale. What price were the shoes before the sale? Show your working.        2 marks
Sequences and Nth Term (5 – 8)
Find the next 5 terms of the following sequences and describe the rule:
c) 3, 6, 9, 12, 15, ………….
d) 4, 7, 10, 13, 16, …………
e) 23, 21, 19, 17, 15, …………
f) 1, 2, 4, 8, 16, …………
g) 8000, 4000, 2000, 1000, …………

Find the nth term and the 20th term for each of these sequences:
a) 5, 7, 9, 11, 13, ……               b) 7, 11, 15, 19, 23, ……
c) -1, 1, 3, 5, 7, ……                d) 9, 12, 15, 18, 21, ……
e) 5, 8, 13, 20, 29, ……              f) -2, 1, 6, 13, 22, ……

Level 5 question
(a) Here is a number chain:
2         4         8         16         32         64
What could the rule be?                                                            1 mark
(b) Some number chains start like this:
1          5
Show three different ways to continue this number chain. For each chain write down the
next three numbers. Then write down the rule you are using.

Level 6 question - Uri makes a sequence of shapes using square tiles as in the diagram.
The number of square tiles in shape
number n is 2n + 1.
Uri makes a different sequence of
shapes. This time the number of tiles in
shape number n is 3n + 1. Draw what the          shape        shape            shape
number       number           number
first 3 shapes might look like.                    1            2                3

Level 7 question
Look at this part of a number line. Copy and fill in the 2 missing numbers
-7  ........  1      5         9    ........ 17

Finish the sentence:
The numbers on this number line go up in steps of ..............................   2 marks

Level 8 question - Each pattern below shows a square grid that is 2 squares high. Only
one square at each end of the top row is
shaded. All squares in the bottom row are

Imagine one of these patterns that has n squares in the bottom row. Write an expression
for the fraction of the pattern that is shaded.                            2 marks
Simplifying (5 – 8)

1) 3r + 2r – 2r                                                 2) 6p + p – 3p
3) 8u + 5u – 2u + u                                             4) 6h – 2h – h – h
5) 8j – 2j + j + 3j – 4j                                        6) 3x – 2y – 4x + 8y

a)            a3  a 4                      b)           a7  a6                c)        2a 3  3a 2
x9                               p4
d)            5a 2  a 7                    e)                                  f)
x2                               p3

g)
12 a 7
4a 2
h)           a 
5 3
n)        2a 4
Level 5 question

Simplify these expressions.
5k + 7 + 3k  ...........................                              1 mark
k + 1 + k + 4  .........................                              1 mark
Level 6 question

Write the correct operations (+ or – or × or ÷) in these statements.
a .............. a  0
a .............. a  1
a .............. a  2a
2
a .............. a  a                               2 marks

Level 7 question

Write these expressions as simply as possible.
9 – 3k + 5k  ............................                                      1 mark
2
k + 2k + 4k  ............................                                      1 mark
3k × 2k  ............................                                          1 mark
9k 2
 ............................                                        1 mark
3k

Level 8 question
a2 – b2
(a)          Show that                           simplifies to a + b            1 mark
a–b
a 3b 2
(b)          Simplify the expression                                            1 mark
a 2b 2
a 3b 2 – a 2b 3
(c)          Simplify the expression
a 2b 2
Manipulation (5 – 8)

Expand these brackets
a) 3(4+a)       b) 6(2-3a)                                                     c) a(a+3)      d) a(2a+3b)
e) 3a(5a-2b) f) (x+2)(x+5)                                                     g) (x-3)(x+4) h) (2x+1)(3x+5)
i) (5x-2)(5x+2) j) (3a+2)2                                                     k) (p+3q)(2p-5q)

Factorise
a) 4x + 24                     b) 3a2 + 9a                     c) 8b – 12             d) 4p2 + 3p

Level 5 question
1.       Brackets
Jenny wants to multiply out the brackets in the expression 3(2a + 1)
She writes:
3(2a + 1) = 6a + 1
Show why Jenny is wrong.                                                        1 mark

Level 6 question
2.       Bracket multiplication
Multiply out the brackets in these expressions.
y (y – 6)  ..........................................................                1 mark

(k + 2)(k + 3)  ..................................................             1 mark

Level 7 question
3.       Factors again
2
(a)            Ring the expression below that is the same as y + 8y + 12
(y + 3) (y + 4)        (y + 7) (y + 1)
(y + 2) (y + 6)
(y + 1) (y + 12)       (y + 3) (y + 5)                               1 mark

(b)            Multiply out the expression (y + 9) (y + 2)

Level 8 question
4.       Factorising

(a)            Complete these factorisations.
2
x + 7x + 12 (x + 3)(........... + ...........)                      1 mark
2
x – 7x – 30 (x + 3)(........... – ...........)                      1 mark

(b)            Factorise these expressions.
2
x + 7x – 18                                                           2 marks
2
x – 49                                                                 1 mark
Solving Equations (5 – 8)
Solve these equations
1) x – 5 = 15      2) e – 7 = 10            3) u – 5 = 5             4) 8 + t = 23
5) 16 – y = 8      6) 34 – r = 18           7) 4x = 16               8) 5r = 25
9) 2w = 18         10) 6y = 36              11) 5m = 30              12) 2a = 16
13) 2x + 3 = 15    14) 3x + 1 = 13          15) 2x + 6 =12           16) 6x – 8 = 10
17) 25 = 5x + 15   18) 10 = 8x – 14         19) 6x – 3 = 10          20) 7 + 4r = 3

Solve these equations
1) 4x + 3 = 2x + 11            2) 5x + 7 = 3x + 11                   3) 2x – 4 = 5x – 19
4) 6x – 2 = 3x + 10            5) 7x + 4 = 10x – 20                  6) 9x + 7 = 15x + 1
7) 8x + 3 = 2x + 21            8) 3x – 6 = 10 – x

Level 5 question
1.      Solving
Solve these equations.
32x + 53 = 501                          1 mark
375 = 37 + 26y                          1 mark

Level 6 question
2.      Finding y
Solve this equation.
3y + 14  5y + 1                     2 marks

Level 7 question
3.      x and y
Solve these simultaneous equations using an algebraic method.
3x + 7y 18
x + 2y 5

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

Level 8 question
4.      Equation solving
Solve this equation.
5(2 y – 3)
3                       2 marks
3y
Substitution (5 – 8)
Let a = 4, b = 1, c = 5 and d = 10
1) a + b + c          2) abc                              3) 3(d + c)                      4) a(c – b)
5) (a + b)(d – c)     6) 2a + 3c                          7) 5d – 3c                       8) abc – bd

9) 5a                      10)
d                      11) 50                           12) 5a
2                            c                            2c                               2b

Let a = 2, b = -1 and c = -3
1) 5c                2) 4b                                3) 3c – a                        4) 10a + c
5) a + c             6) a – c                             7) b + c                         8) bc

Level 5 question
1.       Values
Look at the three expressions below.
8+k                                          3k                                          2
k
When k  10, what is the value of each expression?
2
8 + k ...........................           3k  ........................... k ...........................   2 marks

Level 6 question
2. Heron of Alexandria
About 2000 years ago, a Greek mathematician worked out this formula to find the area of any triangle.
For a triangle with sides a, b and c

Area          ss  as  bs  c
where        s abc
2
A triangle has sides, in cm, of 3, 5 and 6
Use a  3, b  5 and c  6 to work out the area of this triangle.                                              2 marks

Level 7 question
3.       Algebra
(a)      Find the values of a and b when p  10
3 p3
a                                                              1 mark
2
2 p 2  p – 3                                            1 mark
b
7p

Level 8 question
4.       Equation
Look at this equation.
y      60
 x – 10
(a)        Find y when x  19
There are two answers. Write them both.

y = ......................... or y = .........................        1 mark
Coordinates (5 – 8)
Question 1 – Draw a grid with axis from -5
to 5 and plot these points.

Question 2 – Write the coordinates of the
points labelled A to K.

Level 5 question                                                               y
1.       Midpoint                                                                      A
P is the midpoint of line AB.                                    120

What are the coordinates of point P?
P is (................ , ................)                                                   P

1 mark

B
0                                              120         x

Level 6 question                                                               y
2. (b)   Q is the midpoint of line MN.
The coordinates of Q are ( 30, 50 )
M

What are the coordinates of points M and N?
M is (................ , ................) 1 mark
Q (30, 50)
N is (................ , ................)       1 mark

N
0                                                          x
Level 7 question
y

3.Points and rules                                                                                4
(a)The grid shows six points labelled A, B, C, D, E and
F.Complete the table to show which points have                                                    3
B           A
coordinates that match the rules below.
2
The first one is done for you.
Rule       A         B         C        D            E     F                C
1

x3                                                                                            0                                               x
–4   –3           –2   –1        0           1       2   3       4
y  –3                                                                                         –1
F
xy
–2

–3
D                              E

–4
Forming Equations (5 – 8)

Level 5 question                                                                                               Expression
Statement

1. A ruler costs k pence. A
pen costs m pence.
Match each statement with                                                                                         5k

the correct expression for the
The total cost
amount in pence.                                         of 5 rulers
The ﬁrst one is done for you.                                                                                     5m

3 marks
5 – 5m
The total cost
of 5 rulers and 5 pens

500 – 5m

5k + m
How much more 5 pens cost
than 5 rulers

5(k + m)

The change from £5,                                         5m – 5 k
in pence, when you buy 5 pens

5k – 5 m

Level 6 question
2.       Look at the information.
x=4             y = 13
Complete the rules below to show different ways to get y using x.
The first one is done for you.
To get y, multiply x by .........2......... and add ..........5.........
This can be written as y =......... 2x + 5.........

To get y, multiply x by .............................. and add ..............................
This can be written as y = .............................                                             1 mark

To get y, multiply x by .............................. and subtract ..............................
This can be written as y = ..............................                                            1 mark

To get y, divide x by ....................... and add .......................
This can be written as y = ...........................                                               1 mark
Level 8 question
3.          a and b
In this question, a and b are numbers where a  b + 2
The sum of a and b is equal to the product of a and b
Show that a and b are not integers.                                                      3 marks
Linear Graphs (5 – 8)

Draw the following graphs on the same axis where x goes from
-5 to 5. Don’t forget you will need to draw a table first!
a) y = 4x + 5                 b) y = 3x + 3
c) y = 3x – 3                 d) y = 2x + 1
e) y = 2x                     f) x = 3
g) y = -2

y
Level 6 question

4
1. The graph shows the straight line with equation y  3x
–4

(a) A point on the line y  3x – 4 has an x-coordinate of
50
What is the y-coordinate of this point?       1 mark                   –4           0                 4     x

(b)A point on the line y  3x – 4 has a y-coordinate of 50
What is the x-coordinate of this point?       1 mark

–4

Level 7 question                   A         y = 3x – 4        B            y=4          C       x=–5
2. Here are six different
equations, labelled A to F

D         x + y = 10        E       y = 2x + 1        F          y =x2
these equations.

(a) Which graph goes through the point (0, 0)?               1 mark

(b) Which graph is parallel to the y-axis?                   1 mark

(c) Which graph is not a straight line?                      1 mark

(d) Which two graphs pass through the point ( 3 , 7 )?       2 marks

Level 8 question

3.         The graph of the straight line with equation y  x + 1 passes through the point (0, 1)

(a)        Write the equations of two different straight lines that also pass through the point (0, 1)
2 marks
Algebra Review 1 (5 – 8)
Level 5 question
1.       Simplify these expressions.
5k + 7 + 3k  ...............................................................................................          1 mark

k + 1 + k + 4  .............................................................................................    1 mark

2.             Some people use yards to measure length.
The diagram shows one way to change yards to metres.
number of                                                                                           number of
yards               × 36                    × 2.54                       100                    metres

(a)      Change 100 yards to metres.                                                                                            1 mark
Level 6 question
3.       Write the missing numbers.
6x + 2 = 10
so       6x + 1 = .............                                                  1 mark

1      –  2y = 10
2
so             (1      – 2y) = .............                                     1 mark

4.       Multiply out this expression.
5(x + 2) + 3(7 + x)                                                     2 marks
Level 7 question
5.       Look at the triangle.
Not drawn
accurately
a°
2b°

a°                                    b°

Work out the value of a                                                             3 marks

6.             Multiply out these expressions.
5(x + 2) + 3(7 + x)                                                                 2 marks

(x + 2)(x + 5)                                        2 marks
Level 8 question
7.       I am thinking of a number.
When I subtract 25 from my number, then square the answer,
I get the same result as
when I square my number, then subtract 25 from the answer.

What is my number?
You must show an algebraic method.                          2 marks
8.             For each equation below, when x increases by 3, what happens to y?
Complete the sentences.
y=x
When x increases by 3, y increases by................
y = 2x
When x increases by 3, y increases by................ 2 marks
Algebra Review 2 (5 – 8)
Level 5 question
1.       Look at this sequence of patterns made with hexagons.

pattern number 1                      pattern number 2

pattern number 3
To ﬁnd the number of hexagons in pattern number n you can use these rules:
Number of grey hexagons                   n+1
Number of white hexagons                  2n

Altogether, what is the total number of hexagons in pattern number 20?                   2 marks
Level 6 question
2.       Look at this equation.
14y – 51  187 + 4y
Is y  17 the solution to the equation?
Yes                     No
Show how you know.                                                            1 mark
Level 7 question
3. (a)   Draw lines to match each nth term rule to its number sequence.
nth term                                   Number sequence

4n                                  4, 7, 12, 19, …

(n + 1)2                               4, 8, 12, 16, …

n2 + 3                                4, 9, 16, 25, …

n (n + 3)                             4, 10, 18, 28, …
2 marks

(b)       Write the first four terms of the number sequence using the nth term rule below.

n3 + 3
,            ,           ,
2 marks
Shapes and Their Properties (5 – 8)
1. Fill in the blanks with the correct Triangle names:
a) A triangle with 3 equal sides and 3 equal angles is a(n) ________ triangle.
b) A triangle with no equal sides or angles is a(n) _______ triangle.
c) A triangle with a 90º angle is called a(n) __________ triangle.
d) A triangle with 2 equal angles and 2 equal sides is a(n) ________ triangle.

______       ______      ______ _______ _______               _______
3. What is a Re-Entrant Quadrilateral? Draw an example.
4. What does the word ‘regular’ mean when used to describe a polygon?
(e.g. Regular Pentagon, Regular Octagon, etc)

Level 5 question
(a) A triangle has three equal sides. Write the sizes of the angles in this triangle.
................°, ................°, ................°              (1 mark)
(b) A right-angled triangle has two equal sides. Write the sizes of the angles in this
triangle. ................°, ................°, ................°               (1 mark)

Level 6 question - Any quadrilateral can be split into 2 triangles.
a) Explain how you know that the angles inside a quadrilateral add up to 360°
(1 mark)
(b) What do the angles inside a pentagon add up to?                             (1 mark)

(c) What do the angles inside a heptagon (7-sided shape) add up to?

Level 7 question -The diagram shows a rectangle that just touches an equilateral triangle.

(   a) Find the size of the angle marked x
(1 mark)

(b) Now the rectangle just touches the equilateral triangle so that ABC is a straight line.

Show that triangle BDE is isosceles.
(2 marks)

Level 8 question – A pupil has three tiles.
One is a regular octagon, one is a regular
hexagon, and one is a square. The side length
of each tile is the same. The pupil says the
hexagon will fit exactly like this. Show
calculations to prove that the pupil is wrong                                   (3 marks)
Angles and Angle Facts (5 – 8)
1. Use a protractor to accurately measure the following angles.
(trace them into
extend the lines)

2. Find each missing angle.

Level 5 question - The diagram shows triangle PQR.
Work out the sizes of angles a, b and c

a = ………………°               b = ………………°                      c = ………………°

(3 marks)

Level 6 question - Look at the diagram, made from four straight lines.
Work out the sizes of the angles marked with letters.

a = .........................°           b = .........................°
c = .........................°           d = .........................°

(3 marks)

Level 7 question - The diagram shows a rhombus.
The midpoints of two of its sides are joined with a
straight line.
What is the size of angle p?

p = .............................°
(2 marks)

Level 8 question – AC is the diameter of a circle
and B is a point on the circumference of the
circle. What is the size of angle x?

x = .............................°
(2 marks)
Isometric Drawing (5 – 8)
1) The following solids have been partially completed on isometric paper. On Isometric
paper, copy and complete the solid.

Level 5 question
1. a) This cuboid is made from 4 small cubes.
On isometric paper draw a cuboid that is twice
as high, twice as long and twice as wide.

(2 marks)

b) Graham made this cuboid from 3 small cubes.
Mohinder wants to make a cuboid which is twice
as high, twice as long and twice as wide as Graham's
cuboid. How many small cubes will Mohinder need
altogether?
(1 mark)

Level 6 and above questions
1) I join six cubes face to face to make each 3-D shape below. I can then join the 3-D
shapes to make a cuboid. Draw this cuboid on isometric paper.

2) Four cubes join to make an L-shape.
The diagram shows the L-shape after quarter turns in one direction.
On isometric paper, draw the L-shape after the next quarter turn in the same direction
Nets and Plan Views (5 – 8)
1. Below are the plan (top), side and front elevations (views) of a solid. For each one
identify (name) the solid and sketch a net of the solid.

Level 5 question
(a) Alex is making a box to display a shell. The base of the box is shaded.

He draws the net of the box like this:

Alex wants to put a lid on the box. He must add one more square to his
net. On each diagram below, show a different place to add the new
square. Remember, the base of the box is shaded.

(3 marks)

(b) Alex makes a different box with a lid hinged at the top.
The base of his box is shaded. Complete the net below.

(3 marks)
Level 6 and above question - The diagram shows a model made
with nine cubes. Five of the cubes are grey.
The other four cubes are white.
(a) The drawings below show the four side-views of the model.                        side–view B
side–view C
Which side-view does each drawing show?

side–view A
side–view D

(1 mark)

(b) Complete the top-view                  (c) Imagine you turn the model upside
of the model by shading                    down. What will the new top-view
the squares which are                      of the model look like? Complete
grey.                                      the new top-view of the model by
grey.
top - view
(1 mark)                                 (1 mark)                               new top - view
Area and Perimeter (5 – 8)
1. Find the Area and the Perimeter for each shape. Don’t forget your units!

Level 5 question – Here is a rectangle.
a) A square has the same area as this rectangle.
What is the side length of this square?
(1 mark)
b) A different square has the same perimeter as this rectangle. What is the side length
of this square?                                                          (1 mark)

Level 6 question - The diagram shows a shaded parallelogram drawn inside a rectangle.
3 cm
(not drawn accurately)

What is the area of the shaded parallelogram?
5 cm

3 cm                                                   (2 marks)
10 cm

Level 7 question - The diagram shows two circles and a square, ABCD. A and B are the
centres of the circles. The radius of each circle is 5 cm.
(not drawn accurately)
A
5 cm

Calculate the area of the shaded part of the square.

B   5 cm                     D

(3 marks)

C

Level 8 question – Two right-angled triangles are joined together to make a larger
triangle ACD.
a) Show that the perimeter of triangle ACD is 78 cm.
(2 marks)

b) Show that triangle ACD is also a right-angled
triangle.
(2 marks)
Volume and Surface Area (5 – 8)
1. Find the volume and surface area of the following objects.                     (Don’t forget the correct units)

2. Dog food comes in two types of tins. A square based tin of side 8.5cm and
height 15cm and a circular based tin of radius 5cm and height 13cm.
a) Calculate the Volume of each tin. Which one holds more and by how much?
b) A label will be made for each tin. It will go around the tin covering everything
except the top and bottom lids. How much paper will each tin need for its label?

Level 5 question – This shape is made from 4 cubes put together.
Volume              3 The table shows information about the shape.
4 cm
Surface Area            2
18 cm
The same four cubes are then used to make this new shape.
Copy and Complete the table for the new shape.
Volume                          3
......... cm
Surface Area                     2
......... cm                                                       (2 marks)

Level 6 question - The drawing shows 2 cuboids that have the same volume.
a) What is the volume of cuboid A?
(2 marks)

b) Work out the value of the length marked x
(1 mark)

Level 7 question
a) Look at the triangular prism.                                Work out the volume of the prism.
(Not drawn accurately) 6cm                                                           (1 mark)
10cm

4cm

b) One face of another prism is made from 5 squares.
Each square has side length 3cm. (not drawn accurately) 3cm
Work out the volume of the prism.                                                          10cm
(1 mark)

Level 8 questions
2) A cylinder has a radius of 2.5 cm.
1) Six cubes each have a surface area
The volume of the cylinder, in cm3, is
of 24 cm2. They are joined together
4.5π. What is the height of the
to make a cuboid. What could the
cylinder?                          2.5 cm
surface area of this cuboid be?
(3 marks)
Write them both.                                                        height

(2 marks)
Transformations & Loci (5 – 8)
1. Trace the diagram into your book. By measurement, reflect the Flag A in the mirror
line M1. Label this A1. Then, by measurement, reflect the Flag A1 in mirror line M2.
Label this A2.

2. Draw each shape, Reflect each shape along the marked side (●) without the use of a
mirror. Then Draw the shape again and Rotate each shape about the marked point (●)
180º clockwise and 90º anticlockwise.

Level 5 question
(a) I have a square piece of card. I cut along the dashed line to
make two pieces of card. Do the two pieces of card have the

(b) The card is shaded grey on the front, and black on the back.
I turn piece A over to see its black side. Which of the shapes
front of
piece A

below shows the black side of piece A?

(1 mark)

Level 6 question

The grid shows an arrow. Copy the grid carefully into
your book, then draw an enlargement of scale factor 2
of the arrow. Use point C as the centre of enlargement.
(2 marks)

Level 7 question
In a wildlife park in Africa, wardens want to
know the position of an elephant in a certain area.
They place one microphone at each corner of a
4 km by 4 km square. Each microphone has a range of 3½ km.
The elephant is out of range of microphones A and B.
Where in the square could the elephant be?
(Draw an 8cm by 8cm square and use the scale 2cm = 1km)
Show the region accurately on the diagram, and label the region.            (2 marks)
Level 8 question- A picture has a board behind it.
The drawings show the dimensions of the
rectangular picture and the rectangular board.
(a) Show that the two rectangles are not mathematically similar.
Shape and Space Review 1 (5 – 8)
Level 5 questions

1. These two congruent triangles make a parallelogram.
a) Draw another congruent b) Draw another congruent              c) Draw another congruent
triangle to make a triangle     to make a bigger                   triangle to make a
rectangle.    (1 mark)          triangle. (1 mark)                 different bigger triangle.
(1 mark)

2. Three shapes ﬁt together at point B. Will ABC make a straight line?
93º
A          24º                                Not drawn
B
61º
accurately      (1 mark)
C

Level 6 questions
3. The diagram shows three straight lines.
Work out the sizes of angles a, b and c
a = ................° because …………………………….        (1 mark)
b = ................° because …………………………….        (1 mark)
c = ................° because …………………………….        (1 mark)

4. Kevin is working out the area of a circle with radius 4
He writes: Area = π × 8.
4

Explain why Kevin’s working is wrong.

Level 7 questions
5. Look at the triangle.
Work out the value of a

(3 marks)

6. The diagram shows a shaded rectangle.
It is divided into four smaller rectangles,
labelled A, B, C and D.
The ratio of area C to area B is 1 : 2
Calculate area A

(2 marks)

Level 8 question
7. The diagram shows a square inside a triangle.
DEF is a straight line. The side length of square
ABCE is 12 cm. The length of DE is 15 cm.
Show that the length of EF is 20 cm. (3 marks)
Shape and Space Review 2 (5 – 8)
Level 5 question
1. Some statements in the table are true. Some are false. Beside each statement, write
true or false. For true statements you must draw an example. The ﬁrst one is done for
you.                                                                         (3 marks)
Statement                            Write true or false. If true, draw
an example.
Some triangles have one right
true
angle and two acute angles.

Some triangles have three right
angles.
Some triangles have three acute
angles.
Some triangles have one obtuse
angle and two acute angles.
Some triangles have two obtuse
angles and one acute angle.

Level 6 questions
2. The diagrams show nets for dice.
Each dice has six faces, numbered 1 to 6
Write the missing numbers so that the
numbers on opposite faces add to 7
(2 marks)

3. This shape has been made from two                           35°

congruent isosceles triangles. What is
the size of angle p?                                   p

(Not drawn accurately)
35°
(2 marks)

Level 7 question
4. The diagram shows two circles and a square, ABCD.               A
5 cm

(Not drawn accurately)
A and B are the centres of the circles.
The radius of each circle is 5 cm.
B  5 cm            D

Calculate the area of the shaded part of
the square.                 (3 marks)                            C

Level 8 question
- The height of the
5. Engineers have worked
tower is 56 m.
on the leaning tower of
- The angle of tilt is
Pisa to make it safe. A
5.5°
website gave this
- The tower leans
5.2 m from the
tower before the work.
perpendicular.
Give calculations to show
that the information cannot all be true.                (2 marks)
Shape and Space Review 3 (5 – 8)
Level 5 question
1. Look at these angles

a) One of the angles measures 120°
Write its letter.        (1 mark)
b) Using a protractor accurately draw an angle measuring 157°. Label the angle 157°.
(2 marks)
c) 15 pupils measured two angles. Here are their results. Use the results to decide what
each angle is most likely to measure.
Angle A                            Angle B              Angle A is ..............°
Angle         Number of            Angle         Number of How did you did
measured as         pupils          measured as         pupils     decide? (1 mark)
6°               1                45°               5
37°               2                134°              3        Angle B is ……………º
38°              10                135°              4        How did you
39°               2                136°              3        Decide? (1 mark)

Level 6 questions
2. Each diagram shows an enlargement of
scale factor 2. Where is the centre of
enlargement in these diagrams?
Mark each one with a cross.
(2 marks)

3. The diagram shows a rectangle.
(Not drawn accurately)
Work out the size of angle a
(3 marks)

Level 7 questions
4. Calculate the area of this triangle.
The picture is not drawn to scale.                                            D    (3 marks)
6 cm

5. ABC and ACD are both right-angled triangles.
C
(Not drawn accurately)
a) Explain why the length of AC is 10 cm.                              6 cm
(1 mark)
b) Calculate the length of AD. Show your working. (2 marks)               B                        A
8 cm

Level 8 question                                                                                800 km
6. A satellite passes over both the north and south poles,                                  N
and it travels 800 km above the surface of the Earth.
The satellite takes 100 minutes to complete one orbit.
Assume the Earth is a sphere and that the diameter
of the Earth is 12,800 km. Calculate the speed of the
satellite, in kilometres per hour. Show your working.    (3 marks)
Shape and Space Review 4 (5 – 8)
Level 5 question                                 b) Where could you put point E so
1. a)Where should you put point D so                that shape ABCDE is a trapezium?
that shape ABCD is a square?                   Mark point E on the grid below.
Mark point D on the grid. (1 mark)                                           (1 mark)

Now write the coordinates of point E
(……….. , ………..)    (1 mark)
Level 6 question
2. The drawing shows an isosceles triangle. (not drawn accurately)
(a) When angle b is 70º, what is the size of angle a?

(b) When angle a is 70º, what is the size of angle b?

Level 7 question
3. Some pupils want to plant a tree in the school’s garden. The tree must be at least 12m
from the school buildings. It must also be at least 10m from the centre of the round
pond.
a) Trace the diagram into your book. Show accurately on the plan the region in which the
tree can be planted. Shade in this region.                                 (3 marks)

(b) The pupils want to make a gravel path of width 1m around the pond. Calculate the area
of the path. Show your working.                                              (2 marks)

Level 8 question
4. The diagram shows two circles that intersect at P and Q.                    P

B is the centre of the larger circle.
C is the centre of the smaller circle.
ABCD is a straight line.
A                                   D
B             C

Prove that the line through A and P
is a tangent of the smaller circle.    (2 marks)                           Q
Shape and Space Review 5 (5 – 8)
Level 5 questions
1. Look at this shape made from six cubes. Four cubes are white
Two cubes are grey.
Part of the shape is rotated
through 90° to make the shape below.

After another rotation of 90°, the shape is a cuboid.
Draw this cuboid on the isometric paper.

(2 marks)
10cm                     8cm

2. The diagram shows a rectangle 18cm long and 14cm wide.
It has been split into four smaller rectangles.            10cm

a) Write the area of each small rectangle on the diagram.            ............... cm 2    ............... cm 2

One has been done for you.                       (1 mark)
b) What is the area of the whole rectangle?        (1 mark)   4cm         40cm 2             ............... cm 2

c) What is 18 × 14?                                (1 mark)

Level 6 questions
3. The squared paper shows the nets of
cuboid A and cuboid B.
a) Do the cuboids have the same surface area?
Show calculations to explain how you know.
(1 mark)
b) Do the cuboids have the same volume?
Show calculations to explain how you know. (2 marks)

Level 7 question
4. a) The grid shows an arrow. Copy the grid, and draw an enlargement of
scale factor 2 of the arrow. Use point C as the centre of enlargement.
(2 marks)
b) The sketch below shows two arrows. The bigger arrow is an enlargement of scale
1.5 of the smaller arrow.

Write down the three missing values.
(3 marks)

Level 8 question
5. The diagram shows ﬁve points joined with four straight lines.
BCE and ADE are isosceles triangles.
The total length of the four straight lines is 40 cm.
What is the length of EA? (3 marks)
Shape and Space Review 6 (5 – 8)
Level 5 question                                                I fold square A in half to make
1. a) I have a square piece of paper.                           rectangle B
A        8 cm                                        B
The diagram shows information
8 cm
make square C.           C
Complete the table below to show the area and perimeter of each shape.
Area               Perimeter
Square A                   2                         cm
cm
Rectangle B                2                         cm                                       (3 marks)
cm
Square C                   2                         cm
cm
b) I start again with square A. Then I fold it in half to make triangle D.                           D

What is the area of triangle D?                        (1 mark)
c) One of the statements below is true for the perimeter of triangle D.
Put a ring around the correct one.
 The perimeter is less than 24cm
 The perimeter is 24cm
 The perimeter is greater than 24cm

Level 6 question
2. The drawing shows how shapes A and B fit together
to make a right-angled triangle.                                                    A
B

Work out the size of each of the angles in shape B.

(3 marks)

Level 7 question
C                        B
60°

3. ABCD is a parallelogram
(Not drawn accurately)                                                    h

Work out the sizes of angles h and j
80°

(2 marks)
j
D                  A

Level 8 question
4. a) The triangles below are similar.
(Not drawn accurately)
What is the value of p? Show your working.
(2 marks)
B
b) Triangles ABC and BDC are similar.                                       x

6 cm
What is the length of CD?              (1 mark)
x
A           C                        D
3 cm
Frequency Tables (5 – 8)
1. In a taste test, 30 people were asked to select their favourite sausage from four brands. The brands
were Porkers (P), Sizzlers (S), Yumbos (Y), and Bangers (B). These are the results:
P, B, B, B, B, Y, P, Y, B, Y, P, P, S, B, B, B, S, Y, P, B, P, Y, P, S, P, Y, B, Y, B, P
Use a frequency table to organise this data.

2. A survey was done to see how many letters were delivered to each house on North Street on one
day. The results are shown in the table below.
Number of letters               Frequency
0                           4
1                           6
2                          10
3                           4
4                           3
5                           1
6                           2
a) How many houses had 3 letters delivered?
b) What was the highest amount of letters that were delivered to a single house?
c) How many houses were there on the street altogether?
d) How many houses has 4 or more letters delivered? How many letters was this in total?
e) Overall how many letters were delivered on North Street on this day?

Level 6 question                                                            Two-way table

1.On World Book Day, each pupil in                                Boys            Girls      Total
Year 7 chose one book to read.
Some pupils chose a fiction book. Some
chose a non-fiction book.                         Fiction                                     77

Non-fiction book     36

Total           68                        142

(a) Complete the two-way table.                     2 marks
(b) What percentage of boys chose a non-fiction book?
Level 7 question
2.        Altogether, I have 10 bags of sweets.
The mean number of sweets in the bags is 41
The table shows how many sweets there are in 9 of the bags.
Number of sweets                     Frequency
in a bag
39                                   3
40                                   2
41                                   1
42                                   1
43                                   0
44                                   2

Calculate how many sweets there are in the 10th bag.
2 marks
Averages (5 – 8)
1) Lisa and Lucy have both completed 10 French homeworks.
These are their marks out of 10.
Lisa      7, 8, 7, 8, 7, 8, 8, 7, 7, 8
Lucy      3, 9, 10, 4, 5, 9, 10, 3, 10, 10
a) Calculate the mean, median, mode and range for each person.
b) Say who you think is better at French and why?

2) The midday temperatures in two different seaside resorts for a week during July
were:
Skegness - 22C, 21C, 23C, 24C, 26C, 24C, 24C
Eastbourne - 28C, 30C, 24C, 20C, 19C, 26C, 30C
a) Find the median midday temperature and the range for Skegness and
Eastbourne.
b) Describe the differences between the two resorts.

3) The number of goals scored in eleven Premier League matches one Saturday was:
1, 0, 0, 2, 3, 5, 2, 4, 5, 2, 3
Find the modal number of goals scored and the mean, median and range.

Level 5 question
1.(a)    Look at these three numbers.

9                  11               10

Show that the mean of the three numbers is 10                     1 mark
Explain why the median of the three numbers is 10                 1 mark

(b)      Four numbers have a mean of 10 and a median of 10, but none of the numbers is 10
What could the four numbers be?
Give an example.                                                 1 mark
Level 6 question
2.       Statistics
Here are three number cards.
The numbers are hidden.
?           ?          ?
The mode of the three numbers is 5
The mean of the three numbers is 8
What are the three numbers?
Level 8 question
3.       Here is information about a data set.
There are 100 values in the set.
The median is 90
The mean is 95
I increase the highest value in the data set by 200

Now what are the median and the mean of the data set?             2 marks
Averages from a frequency table (5 – 8)
Find the mean from the following frequency tables:

Level 7 questions
1.       A pupil investigated how the teachers at his school travel to work.
The table shows the results.
Number of teachers                   Number of teachers
who travel by car                 who do not travel by car
18                                   7
(a)      What percentage of these teachers travel by car?                    1 mark

(b)        18 teachers travel by car. Some of these teachers travel together
Write the missing frequency in the table below.
Number of teachers                         Number of cars
in one car
1
2                                          4
3                                          2
1 mark
(c)        What is the mean number of teachers in each car?                             2 marks

2. Owls eat small mammals. They regurgitate the bones and fur in balls called pellets. The table shows
the contents of 62 pellets from long-eared owls.
Number of mammals
1     2         3         4        5        6
found in the pellet
Frequency
9    17        24         6        5        1
(a) Show that the total number of mammals found is 170                        1 mark

(b) Calculate the mean number of mammals found in each pellet.

(c) There are about 10 000 long-eared owls in Britain.
On average, a long-eared owl regurgitates 1.4 pellets per day.
Altogether, how many mammals do the 10 000 long-eared owls eat in one day?
Level 8 question
4. A teacher asked fifty pupils in Year 9 the question ‘How much time did you spend on homework last
night?’ The results are shown in the frequency table
Time spent on homework (mins)         Frequency
0  time  30                    6
Show that an estimate of the mean time
30  time  60                   14
spent on homework is 64.8 minutes.
60  time  90                   21
2 marks.
90  time  120                    9
Total                       50
Displaying Data (5 – 8)
1. Twenty students are given a mark out of 5 for a short test. The results were:
Mark                Frequency
a) Draw a bar chart to show this
0                       2              information.
1                       1
2                       9              b) Draw a pie chart to show this
3                      13              information.
4                       8
5                       3

2. a) Draw a stem and leaf diagram to show this information.
142, 157, 136, 149, 163, 139, 140, 158, 139, 151, 132, 148, 143
b) Write down all of the numbers that are represented in
the stem and leaf diagram shown.
c) Find the median number from the stem and leaf
diagram shown.

Level 5 question
1.       Look at this information.
In 1976, a man earned £16 each week.

The pie chart shows how he spent his money.

Other
Clothes

Entertainment                        Rent

Food

(a)         How much did the man spend on food each week?                            1 mark

(b)         Now look at this information.
In 2002, a man earned £400 each week.

The table shows how he spent his money.
Rent                                     £200
Food                                     £100
Entertainment                            £50
Other                                    £50
Draw a pie chart to show how the man spent his money.
Remember to label each sector of the pie chart.                                   2 marks
Interpreting Data (5 – 8)

Level 5 question
1.       Teachers
Question 1             Question 2               Question 3

Should teachers        Should teachers       Should teachers tell
give homework?           make jokes            pupils off if they
in lessons?         forget their books?

Key

Yes     Don't know          No

(a)      What was the least common answer to Question 2?                            1 mark
(b)     What was the modal answer to Question 3?                                   1 mark
(c)      About what proportion of pupils answered ‘yes’ to Question 1?              1 mark
Level 6 question
2.I went for a walk.
The distance-time graph shows information about my
walk.
Distance
Tick () the statement below that describes my walk.           travelled

1

Time taken
I was walking faster and faster.
I was walking slower and slower.
I was walking north-east.
I was walking at a steady speed.
I was walking uphill.

1 mark
Level 7 question
3.       Building
A teacher asked 21 pupils to estimate the height of a building in metres.
The stem-and-leaf diagram shows all 21 results.
6     5   represents 6.5 m            6      5       9
7      0       2      6      8     8
8      3       3      5      7     7     9
9      0       5      5      5
10     4       8
11     2       7

(a)       Show that the range of estimated heights was 5.2 m.                 1 mark

(b)       What was the median estimated height?                               1 mark

(c)      The height of the building was 9.2 m.
What percentage of the pupils over-estimated the height?          1 mark
Level 8 question
4.       Acorns
Two groups of pupils collected a sample of acorns from the same oak tree.
The box plots summarise the two sets of results.

Group A

Group B

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Length (mm)

(a)       Explain how the box plots show the median of group B is 3 mm more than the median
of group A.
1 mark
(b)       Which group has the bigger inter-quartile range?
A                     B

1 mark

(c)       The results from the two groups of pupils are very different.
Give a reason why the results might have been different.
1 mark
Scatter Graphs (5 – 8)
Maths                            9          8        1         4         2        10        6     8
Science                          8          7        2         5         1         9        7     10
a) Plot the points on the scatter graph.
b) Comment on the correlation of the graph and what it means.
c) Draw a line of best fit on the graph.
d) Use the line of best fit to estimate what score someone should get in science if they
get 7 in maths.

Level 6 question

1.         The scatter graph shows information about trees called poplars.

6

5

4
Height of
tree (m)

3

2

1

0
0          1           2           3           4           5
Diameter of tree trunk (cm)

(a)        What does the scatter graph show about the relationship between the diameter of the tree trunk
and the height of the tree?                                                             1 mark

(b)        The height of a different tree is 3 m. The diameter of its trunk is 5 cm.
Use the graph to explain why this tree is not likely to be a poplar.                     1 mark

(c)        Another tree is a poplar. The diameter of its trunk is 3.2 cm.
Estimate the height of this tree.                                                        1 mark
Calculating Probabilities (5 – 8)
1) I have a pack of playing cards. I pick a card at random. What is the probability that
the card I select is:
a) A king.           b) a diamond       c) a black card    d) the jack of clubs
2) There are four different coloured writing pads. In a pack there are 3 blue, 8 red, 1
white and 4 green. What is the probability when I open a new pack I randomly pick a:

3) A dice is thrown and a coin is flipped, fill in the sample space diagram to find all
possible outcomes. Then answer the questions that follow:
1            2             3           4           5         6
H                      H, 2
T                                               T, 4
Find the probability of getting:
a) a head                      b) a four             c) an even number
d) a two or a four               e) a head and a 5?

Level 5 question
1. (a)   Jo has these 4 coins.

Jo is going to take one of these coins at random.
Each coin is equally likely to be the one she takes.
Show that the probability that it will be a 10p coin is 1                   1 mark
2
(b)      Colin has 4 coins that total 33p.
He is going to take one of his coins at random.
What is the probability that it will be a 10p coin?
You must show your working.                                        1 mark
Level 6 question
2.       A computer is going to choose a letter at random from an English book.
The table shows the probabilities of the computer choosing each vowel.
Vowel           A            E            I         O            U
Probability     0.08         0.13         0.07      0.08         0.03
What is the probability that it will not choose a vowel?           2 marks
Level 7 question
3.       A bag contains counters that are red, black, or green.
1 of the counters are red
3
1 of the counters are black
6
There are 15 green counters in the bag.
How many black counters are in the bag?                            2 marks
Level 8 question
4.       I have three fair dice, each numbered 1 to 6
I am going to throw all three dice.
What is the probability that all three dice will show the same number?    2 marks
Handling Data Review 1 (5 – 8)

Level 5 question
1.       I buy 12 packets of cat food in a box.
The table shows the different varieties in the box.       Variety        Number of packets
(a)     I am going to take out a packet at random from the         Cod                  3
box.                                                              Salmon                 3
What is the probability that it will be cod?                Trout                3
1 mark                                                              Tuna                 3
(b)      My cat eats all the packets of cod.
I am going to take out a packet at random from the ones left in the box.
What is the probability that it will be salmon?                      1 mark

(c)         A different type of cat food has 10 packets in a box.
The probability that the variety is chicken is 0.7
What is the probability that the variety is not chicken?              1 mark
Level 6 question
2. In a bag there are only red, blue and green counters.              Colour
Number of
(a) I am going to take a counter out of the bag at random.              of                    Probability
counters
Complete the table.                                                  counters
Red           6

2 marks
1
Blue
5
Green          6
Level 7 question
3.       The table shows the number of boys and girls in two different classes.
Class 9A           Class 9B
Boys                           13                 12
Girls                          15                 14
A teacher is going to choose a pupil at random from each of these classes.
In which class is she more likely to choose a boy?
You must show your working.                            2 marks
Level 8 question
4.       100 students were asked whether they studied French or German.

French                                German
39          27           30

4
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
(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?
27 26                      27 26                   27 27
                                                
100 100                    100 99                  100 100
27 26                       27 27
                            
100 100                     100 100             1 mark
Handling Data Review 2 (5 – 8)

Level 5 question
1. (a)   There are four people in Sita’s family.
Their shoe sizes are 4, 5, 7 and 10
What is the median shoe size in Sita’s family?                        1 mark

(b)            There are three people in John’s family.
The range of their shoe sizes is 4
Two people in the family wear shoe size 6
John’s shoe size is not 6 and it is not 10
What is John’s shoe size?                                      1 mark
Level 6 question
Here are his results.
football cricket       football hockey         swimming
hockey     swimming football netball           football

(a)        Is it possible to work out the mean of these results?
Yes                     No
Explain how you know.                                   1 mark

(b)          Is it possible to work out the mode of these results?
Yes                     No
Explain how you know.                                   1 mark
Level 7 question
3. Chris read the first 55 numbers from a book of random numbers. As he read each number he recorded
it in the diagram.
0     5    9    9    8    3    4    1
Key
1     6    3    1    0    3
2     8    2                                        1      3   represents 13
3     1    1    6    9    3
4     6    9    9    4    7    0
5     5    7    7    6
6     0    2    8    4    8    0    3       5
7     6    8    0    1    5    4
8     6    6    9    2    8    5    7
9     6    7    8    0    0
(a) What was the largest number he recorded?                      1 mark
(b) Explain how Chris could change the diagram to make it easier for him to ﬁnd the median of his data
set.                                                               1 mark
Level 8 question
4.         The mean of a set of numbers is zero. For each statement below, tick ( ) the correct box.
Must          Could        Cannot
be true        be true      be true
All the numbers in the set are zero.

The sum of the numbers in the set is zero.

There are as many positive numbers as negative
numbers in the set.
2 marks

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