1. All the basics you need to know for the Level 3/4 course.......
Level 3/4 Mathematics
All the basics you need to know for the Level 3/4 course

2. I can .... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97
Created by Mr. Lafferty

3. multiply small decimal numbers together ? 0.04 x 0.7
Question 1
multiply small decimal numbers together ? 0.04 x 0.7
answer

4. 0.028 multiply numbers together then count number of decimal places.
Answer to Question 1
multiply numbers together
then count number of decimal places.
0.028

5. divide small decimal numbers
Question 2
divide small decimal numbers
15.3 ÷ 0.03
answer

6. Answer to Question 2
Do not divide by a decimal scale it up to a whole number
Scale the number to be divided
by the same amount
1530 ÷ 3 = 510

7. tell you the meaning of the words
Question 3
tell you the meaning of the words
Profit
Loss
Salary
Overtime
answer

8. Answer to Question 3
Profit: When you sell something for MORE than you bought it. Loss : When you sell something for LESS than you bought it. Salary : How much a person is paid to do a job. Can be measured weekly, monthly or annually. Overtime : Extra work outside normal work time. Usually at night or weekends.

9. tell you the meaning of the words
Question 4
tell you the meaning of the words
Hire Purchase
Foreign Exchange
answer

10. Answer to Question 4
Hire Purchase : When you pay for goods by putting down a deposit and then paying the rest up in small fixed payments over a period of time. Foreign Exchange : Converting one currency to another.

11. explain the meaning of Significant Figures
Question 5
explain the meaning of Significant Figures
answer

12. Answer to Question 5
A measure of both:
Quantity
Accuracy

13. explain the term double negative.
Question 6
explain the term double negative.
(-4) – (-6)
answer

14. Double negative changes to a positive
Answer to Question 6
Double negative changes to a positive
(-4) – (-6) = (-4) + 6 = 2

15. tell you the instrument that helps us to add and subtract integers.
Question 7
tell you the instrument that helps us to add and subtract integers.
answer

16. Answer to Question 7
thermometer

17. explain the rules of multiplication and division of integers.
Question 8
explain the rules of multiplication and division of integers.
answer

18. Different signs = negative answer Same signs = positive answer
Answer to Question 8
Different signs = negative answer Same signs = positive answer

19. explain BODMAS 3 – ( 2 x (-4))2
Question 9
explain BODMAS 3 – ( 2 x (-4))2
answer

20. B - Brackets O – Other M - Multiplication A - Addition S - Subtraction
Answer to Question 9
3 – ( 2 x (-4))2
B - Brackets O – Other M - Multiplication A - Addition S - Subtraction
3 – (-8)2
3 – 64
– 61

21. explain what an integer is.
Question 10
explain what an integer is.
answer

22. A positive or negative number including zero.
Answer to Question 10
A positive or negative number including zero.

23. scientific notation / standard form
Question 11
explain the term
scientific notation / standard form
and convert to it.
answer

24. (a) 5.68 x 106 (b) 6.23 x 10-3 Answer to Question 11 a x 10n
a between 1 and 10
n is an integer
(a) x 106
(b) x 10-3

25. write numbers in normal form
Question 12
write numbers in normal form
(a) x 104
(b) x 10-5
answer

26. Answer to Question 12
Normal Form
(a)
(b)

27. Question 13 write these numbers to 3 significant figures 92418400
answer

28. Answer to Question 13
3 significant figures
0.0406

29. round decimals to any significant figures
Question 14
round decimals to any significant figures
(a) ( to 2 s.f.)
(b) ( to 4 s.f.)
answer

30. Answer to Question 14
(a) 93 ( to 2 s.f.)
(b) ( to 4 s.f.)

31. Question 15
write numbers to any decimal place (a) ( to 2 dp) (b) ( to 4 dp)
answer

32. Higher Mathematics: ©HappySmile Productions
Answer to Question 15
(a) ( to 2 dp) (b) ( to 4 dp)
Higher Mathematics: ©HappySmile Productions

33. Question 16
explain how to get triangular numbers and write down the first 6 triangular numbers.
answer

34. Answer to Question 16
1, 3, 6, 10, 15, 21

35. explain how to get square numbers and write down the first 10
Question 17
explain how to get square numbers and write down the first 10
answer

36. Answer to Question 17
1, 4, 9, 16, 25, 36 49, 64, 81, 100

37. decompose a number into product of primes e.g. 18
Question 18
decompose a number into product of primes e.g. 18
answer

38. Answer to Question 18 18 3 6 2
= 3 x 6
= 3 x 3 x 2
All Prime !

39. Question 19
explain what the LCM is between a set of numbers e.g. LCM for 3, 4 and 5
answer

40. Answer to Question 19
the lowest number that 3, 4, 5 all divide into without leaving a remainder is 60

41. explain what the HCF is between a set of numbers
Question 20
explain what the HCF is between a set of numbers
e.g. HCF for 25, 50 and 100
answer

42. Answer to Question 20
the highest number that divides into 25, 50, 100 without leaving a remainder is 25

43. know the basic angle properties at Level E
Question 21
know the basic angle properties at Level E
answer

44. Answer to Question 21 115o 95o 120o Two angles making a
Angles round a point
Add up to 360o
115o
Two angles making a
straight line add to 180o
angles opposite each other
at a cross are equal.
34o
3 angles in a triangle ALWAYS
add up to 180o.
50o
40o
65o
90o
146o
145o

45. explain the term alternate angles
Question 22
explain the term alternate angles
answer

46. two angles that form a Z - shape
Answer to Question 22
two angles that form a Z - shape
= equal angles

47. explain the term corresponding angles
Question 23
explain the term corresponding angles
answer

48. two angles that form a F - shape
Answer to Question 23
two angles that form a F - shape
= equal angles

49. explain the term interior angle of a shape
Question 24
explain the term interior angle of a shape
answer

50. Answer to Question 24 Pentagon 72o Hexagon 60o Octagon 45o
Interior Angles
Interior Angle

51. explain the term exterior angle of a shape
Question 25
explain the term exterior angle of a shape
answer
Higher Mathematics: ©HappySmile Productions

52. Answer to Question 25 This is called the “Exterior angle” Pentagon B A

53. find the exterior angle of a shape
Question 26
find the exterior angle of a shape
Pentagon
Find Exterior angle B A E O Q
answer

54. Exterior angle = 180o – interior angle
Answer to Question 26
Exterior angle = 180o – interior angle
Pentagon
This is called the
“Exterior angle” B A E C D O Q
Exterior angle = 180o – 108o = 72o

55. find the interior angle of a shape
Question 27
find the interior angle of a shape
Octagon
45o
Find the interior angles
answer

56. Interior angle = (180o – 45o) = 135o
Answer to Question 27
Octagon
45o
Interior angle = (180o – 45o) = 135o

57. use angle properties to calculate missing angles
Question 28
use angle properties to calculate missing angles
Find all missing angles ao , bo and co
d = 115o co ao bo
answer

58. Answer to Question 28
ao = co = 115o bo = 65o

59. remove a single bracket (a) -2 ( 8y – 3) (b) 4b ( – 5b + a)
Question 29
remove a single bracket (a) -2 ( 8y – 3) (b) 4b ( – 5b + a)
answer

60. Answer to Question 29
(a) -16y + 6 (b) -20b2 + 4ab

61. Question 30
remove a single bracket and simplify (a) 2 (x + 5) - 7 (b) 3(a + 4) + 2(a – 1)
answer

62. (a) 2x + 10 -7 = 2x + 3 (b) 3a + 12 + 2a – 2 = 5a + 10
Answer to Question 30
(a) 2x =
2x + 3
(b) 3a a – 2 =
5a + 10

63. evaluate algebraic expressions
Question 31
evaluate algebraic expressions
a = 3 ; b = 4 and c = -1
answer

64. Answer to Question 31

65. factorise algebraic expressions
Question 32
factorise algebraic expressions
(a) 4y + 80
(b) 9x – 6x2
answer

66. (a) 4y + 80 = 4(y + 20) (b) 9x – 6x2 = 3x(3 – 2x)
Answer to Question 32
(a) 4y + 80 = 4(y + 20)
(b) 9x – 6x2 = 3x(3 – 2x)

67. Question 33
simplify algebraic expressions (a) -6x + 7y – 16x + 11y (b) 2b – 3b2 – 5b + 5b2
answer

68. Answer to Question 33
(a) -22x + 18y (b) -3b + 2b2

69. find a percentage of quantity (with a calculator)
Question 34
find a percentage of quantity (with a calculator)
What is 17.5 % of £450
answer

70. Answer to Question 34
= £76.50
17.5 % of £450

71. explain the term appreciation with reference to percentages
Question 35
explain the term appreciation with reference to percentages
The average house prices in Glasgow have appreciated by 50% over the past 10 years. If you bought the house for £ ten years ago.
How much would the house be worth now?
answer

72. when an item has increased in value by a certain percentage.
Answer to Question 35
when an item has increased in value by a certain percentage.
Appreciation = 50% x £
= 0.50 x £80 000
= £
New value = Old Value + Appreciation
= £ £40 000
= £

73. explain the term depreciation with reference to percentages
Question 36
explain the term depreciation with reference to percentages
A Mini Cooper cost £ in 2005.
At the end 2006,
it will have depreciated by 20%
What will the mini cooper worth at end 2006?
answer

74. when an item has decreased in value by a certain percentage.
Answer to Question 36
when an item has decreased in value by a certain percentage.
End 2006
Depreciation = 20% x £15 000
= 0.2 x £15 000
= £3 000
New value = Old value - Depreciation
= £ £3000
= £12 000

75. express a value as a percentage of another value.
Question 37
express a value as a percentage of another value.
Frances scored 13 out of 20 in a Maths test.
What was her percentage score?
answer

76. Answer to Question 37
scored 13 out of 20

77. work backwards to find initial value.
Tricky
Question 38
work backwards to find initial value.
After a 10% increase the price of a house
is £
What was the price before the increase.
answer

78. 110 % = £88 000 We have : 1 % : Price before is 100% :
Answer to Question 38
100 % % = £88 000
Deduce from question :
110 % = £88 000
We have :
1 % :
Price before is 100% :
£800 x 100 = £80 000

79. work backwards to find initial value.
Tricky
Question 39
work backwards to find initial value.
The value of a car depreciated by 15%.
It is now valued at £2550.
What was it’s original price.
answer

80. Answer to Question 39 85 % = £2 550 We have : 1 % :
100 % % = £2 550
Deduce from question :
85 % = £2 550
We have :
1 % :
Price before is 100% :
£30 x 100 = £3 000

81. find a percentage of quantity (with out a calculator)
Question 40
find a percentage of quantity (with out a calculator)
What is 17.5 % of £600
answer

82. 10%  600 ÷ 10 = £ 60 5%  half of 10% = £ 30 2.5%  half of 5% = £ 15
Answer to Question 40
10%  600 ÷ 10 = £ 60
5%  half of 10% = £ 30
2.5%  half of 5% = £ 15
17.5%
£105

83. identify the main parts of a circle.
Question 41
identify the main parts of a circle.
Name the main parts of the circle O
answer

84. Answer to Question 41 O
radius
Circumference
Diameter

85. Question 42
explain the connections between the radius, diameter and the circumference of a circle. O
answer

86. Answer to Question 42
radius O
Circumference
Diameter

87. Question 43
explain the difference between the circumference and the area of a circle.
answer

88. Circumference is the name given to the perimeter of the circle
Answer to Question 43
Circumference is the name given to the perimeter of the circle
(the length round the outside of the circle)
The area is the inside of the circle

89. calculate the perimeter and area for a circle.
Question 44
calculate the perimeter and area for a circle.
Find the area and perimeter of the circle ?
4cm
answer

90. Answer to Question 44
4cm

91. explain how to calculate the area of a composite shape.
Question 45
explain how to calculate the area of a composite shape.
20cm
5 cm
answer

92. Answer to Question 45
Area = rectangle + semicircle
20cm
5 cm

93. calculate the area of any triangle. Find the area below.
Question 46
calculate the area of any triangle. Find the area below.
10cm
4cm
answer

94. Altitude h outside triangle this time.
Answer to Question 46
Altitude h outside triangle this time.
10cm
4cm

95. calculate the area of a rhombus. Find the area below.
Question 47
calculate the area of a rhombus. Find the area below.
5cm
2cm
answer

96. Answer to Question 47
5cm
2cm

97. calculate the area of a kite and inverted kite. Find the area below.
Question 48
calculate the area of a kite and inverted kite. Find the area below.
9cm
4cm
7cm
answer

98. Answer to Question 48
9cm
4cm
7cm

99. calculate the area of a parallelogram. Find the area below.
Question 49
calculate the area of a parallelogram. Find the area below.
9cm
3cm
answer

100. Answer to Question 49
9cm
3cm

101. calculate the area of a trapezium. Find the area below.
Question 50
calculate the area of a trapezium. Find the area below.
5cm
6cm
4cm
answer

102. Answer to Question 50
5cm
6cm
4cm

103. Question 51
calculate the area of composite shapes. Find the area shaded grey below.
10cm
11cm
8cm
4cm
answer

104. Answer to Question 51
10cm
11cm
8cm
4cm

105. write down the area formula for
Question 52
write down the area formula for
Any triangle
Rhombus
Parallelogram
Kite
Trapezium
answer

106. The area formula for Any triangle Rhombus Parallelogram Kite Trapezium
Answer to Question 52
The area formula for
Any triangle
Rhombus
Parallelogram
Kite
Trapezium

107. write down in words Pythagoras Theorem for right-angled triangles.
Question 53
write down in words Pythagoras Theorem for right-angled triangles.
answer

108. c b a Answer to Question 53 Two shorter sides squared
and added together
equal to the longest side c b a

109. calculate the longest side given the two shorter sides.
Question 54
calculate the longest side given the two shorter sides. 8 12 c
answer

110. Answer to Question 54 8 12 c

111. Question 55
calculate a shorter side given the longest side and the other shorter side.
20cm
12cm
a cm
answer

112. Check answer ! Always smaller than hypotenuse
Answer to Question 55
20cm
12cm
a cm
Check answer ! Always smaller than hypotenuse

113. write down the three different versions of Pythagoras Theorem
Question 56
write down the three different versions of Pythagoras Theorem c b
answer a

114. c b a Answer to Question 56 Finding hypotenuse c Finding
Finding shorter side a
Finding
shorter side b a c b

115. write down the rule for a linear pattern.
Question 57
write down the rule for a linear pattern.
answer

116. Answer to Question 57
b = 3s + 8

117. write down the steps for finding the rule for a linear pattern.
Question 58
write down the steps for finding the rule for a linear pattern.
answer

118. 2. Write down part of formula
Answer to Question 58
1. Find the difference
2. Write down part of formula
3. Find the correction factor.
4. Write down the full formula

119. convert hour and minutes to decimal hours.
Question 59
convert hour and minutes to decimal hours.
48 minutes to decimal is
2hr 15 minutes to decimal is
answer

120. = 0.8 hr = 2.25 hr Answer to Question 59
To change minutes to a decimal divide by 60
= 0.8 hr
= 2.25 hr

121. convert decimal hours to hours and minutes.
Question 60
convert decimal hours to hours and minutes.
0.7 hrs to minutes is
3.4 hrs to hours and minutes is
answer

122. 3.4 hrs to hours and minutes is
Answer to Question 60
To change decimal time to minutes ‘multiply by 60’
0.7 hrs to minutes is
= 42 mins
3.4 hrs to hours and minutes is
= 3 hrs 24 mins

123. calculate the speed of an object.
Question 61
calculate the speed of an object.
Daniel drove from his house to the Blackpool,
a distance of 135 miles.
It took him 2hrs 15mins.
What was his average speed?
answer

124. Answer to Question 61

125. calculate the time taken for an object.
Question 62
calculate the time taken for an object.
How long did the bus journey take
if it travelled a total distance of 60 km
at an average speed of 40 km/hr.
answer

126. Answer to Question 62

127. calculate the distance travelled for an object.
Question 63
calculate the distance travelled for an object.
A racing car travelled at 50 km/hr.
What is the distance covered in 6 hours ?
answer

128. Answer to Question 63

129. write down the three formulae linking distance, speed and time.
Question 64
write down the three formulae linking distance, speed and time.
answer

130. D S T D = S T Answer to Question 64
Simple way to remember the 3 formulae !

131. Question 65
simplify ratios
answer

132. Answer to Question 65

133. express ratios in unity form
Question 66
express ratios in unity form
answer

134. Answer to Question 66

135. solve questions involving ratio
The ratio of boys to girls is 4:5.
If there are 16 boys,
how many girls are there.
answer

136. boys girls 4 5 x 4 x 4 16 20 Answer to Question 67
The ratio of boys to girls is 4:5.
If there are 16 boys, how many girls are there.
boys
girls 4 5
x 4
x 4 16 20

137. work out portions using ratios.
Question 68
work out portions using ratios.
Ryan and Kerry share a raffle win of £400
in the ratio 3:5.
How much does each get ?
answer

138. Answer to Question 68 Step 1 : Since the ratio is 3:5, there are :
Ryan and Kerry share a raffle win of £400 in the ratio 3:5.
How much does each get ?
Step 1 : Since the ratio is 3:5, there are :
3+5 = 8 shares
Step 2 : Each share is worth :
Step 3 : Ryan gets 3 x 50 = £150
Check !
= 400
Kerry gets 5 x 50 = £250

139. Question 69 draw a graph of direct proportion and
state the key points of the graph.
answer

140. Answer to Question 69 Two quantities which are in DIRECT PROPORTION
always lie on a straight line
passing through the origin.

141. solve problems involving
Question 70
solve problems involving
direct proportion
The cost of 6 cakes is £4.20.
Find the cost of 5 cakes.
answer

142. Are we expecting more or less
Answer to Question 70
Are we expecting more or less
Easier method
Cakes Pence
6  420 5
(less)
19-Apr-17

143. Question 71 solve problems involving indirect (inverse) proportion
It takes 10 worker 12 months to build a house.
How long should it take 8 men.
answer

144. Are we expecting more or less
Answer to Question 71
Are we expecting more or less
Easier method
Workers months
10  12 8
10  12
1  12 x 10 = 120
(more)

145. add two simple fractions
Question 72
add two simple fractions
answer

146. Answer to Question 72 20 + 18 24 ÷2 ÷2

147. subtract two simple fractions
Question 73
subtract two simple fractions
answer

148. Answer to Question 73 25 - 6 30

149. multiply two simple fractions
Question 74
multiply two simple fractions
answer

150. Answer to Question 74

151. divide two simple fractions
Question 75
divide two simple fractions
answer

152. Answer to Question 75

153. add two mixed fractions
Question 76
add two mixed fractions
answer

154. Answer to Question 76 3 + 4 6

155. subtract two mixed fractions
Question 77
subtract two mixed fractions
answer

156. Answer to Question 77 21 - 16 24

157. multiply two mixed fractions
Question 78
multiply two mixed fractions
answer

158. Answer to Question 78

159. divide two mixed fractions
Question 79
divide two mixed fractions
answer

160. Answer to Question 79

161. solve simple equations
Question 80
solve simple equations
answer

162. Answer to Question 80

163. solve simple equations
Question 81
solve simple equations
answer

164. Answer to Question 81

165. solve more complex equations
Question 82
solve more complex equations
answer

166. Answer to Question 82

167. solve equations with brackets
Question 83
solve equations with brackets
answer

168. Answer to Question 83

169. solve equations with double brackets
Question 84
solve equations with double brackets
answer

170. Answer to Question 84

171. solve equations with fractions
Question 85
solve equations with fractions
answer

172. Answer to Question 85

173. solve equations with brackets & fractions
Question 86
solve equations with brackets & fractions
answer

174. Answer to Question 86
Multiply EVERY term by 12

175. solve simple inequations (inequalities)
Question 87
solve simple inequations (inequalities)
answer

176. Answer to Question 87

177. solve harder inequations (inequalities)
Question 88
solve harder inequations (inequalities)
answer

178. Answer to Question 88

179. explain the meaning of the terms
Question 89
explain the meaning of the terms
Mean
Median
Mode
Range
answer

180. Answer to Question 89 The Mean
Sum of all the data ÷ by the number of data values
The Median
(put the data in order then find the MIDDLE value)
The Mode
(the number that appears the most)
4. Range
Highest value - the lowest value

181. construct a frequency table
Question 90
construct a frequency table
78 17
answer

182. Answer to Question 90 10-19 6 20-29 5 30-39 3 40-49 50-59 60-69 70-79
Choose suitable
Class interval
Class Intervals
Tally
Frequency
10-19 6
20-29 5
30-39 3
40-49
50-59
60-69
70-79 6 5 1 6

183. and find the mean from it.
Question 91
No of Siblings (S)
Freq.
(f)
add a third column
to a frequency table
and find the mean from it. 9 1 13 2 6 3 1 5 1
Totals 30
answer

184. Answer to Question 91 S x f 9 0 x 9 =0 1 13 1 x 13 = 13 2 6 2 x 6 = 12
No of Siblings (S)
Freq.
(f)
S x f 9
0 x 9 =0 1 13
1 x 13 = 13 2 6
2 x 6 = 12 3 1
3 x 1 = 3 5 1
5 x 1 = 5
Totals 30 33

185. cumulative frequency graph.
Question 92
Day
Freq.
(f)
Cum. Freq.
Total so far 1 2 2
find the median
from a
cumulative frequency graph. 2 3 5 3 1 6 4 6 12 5 5 17 6 8 25 7 4 29
answer

186. Answer to Question 92
Day
Freq.
(f)
Cum. Freq.
Total so far 1 2 2 2 3 5 3 1 6 4 6 12 5 5 17 6 8 25 7 4 29

187. construct a Pie-chart Question 93 Rugby Football Cricket Ice Hockey 7590 45 60
Favourite Sport
Squash 30
answer

188. Answer to Question 93 Rugby Football Cricket Ice Hockey 75 90 45 60
Favourite Sport
Squash 30
Total
300
Rugby
Football
Cricket
Ice Hockey
Squash
90o
108o
54o
72o
36o

189. construct a stem-leaf diagram
Question 94
construct a stem-leaf diagram 12 40 57 54 55 13 15 32 41 21 23 29 51
answer

190. Answer to Question 94 12 40 57 54 55 13 15 32 41 21 23 29 51 1 1 3 9 2 4 5 7 2 2 2 3 4 5
stem
leaves
n = 20 Key : means 23

191. construct a scattergraph and describe the main features
Question 95
construct a scattergraph
and describe the main features
answer

192. Answer to Question 95
When two quantities are strongly connected we say
there is a strong correlation between them.
Best fit line x x
Best fit line x x
Strong positive
correlation
Strong negative
correlation
No correlation

193. explain the term probability
Question 96
explain the term probability
in words and numbers
answer

194. The chance of an event happening
Answer to Question 96
The chance of an event happening
Probability can be thought of as a fraction or decimal.
It always lies between 0 and 1.
0 meaning impossible ( could not happen)
1 meaning certain ( will definitely happen)

195. Question 97 evaluate a probability
There are 3 red and 4 green balls in a bag.
What is the probability a green ball is picked.
answer

196. Answer to Question 97 There are 3 red and 4 green balls in a bag.
What is the probability a green ball is picked.

197. Online Revision Exercises
Integers
Circle
Angles
Fractions
Type of numbers
Ratio & Proportion
Scientific Notation 1 2
Equations & Inequalities
Algebra
DST
Decimals
Area of Quadrilaterals
Statistics
Linear Patterns 1 2
Percentages
Pythagoras 1 2
Level F exam type question