2. The Further Mathematics Support Programme
Our aim is to increase the uptake of AS and A level Further Mathematics to ensure that more students reach their potential in mathematics.
The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.
To find out more please visit

3. National Geographic Earthquakes video
Images taken from

4. What is a logarithm?
Invented by John Napier (1550 – 1617) , a Scottish mathematician
They are used to answer the question
“ How many of a number must we
multiply together to get another
number?”
e.g. We know 4 3 =64 so three 4s
must be multiplied together to
get 64. The logarithm is 3.
Image taken from

5. Logarithm notation
The statement 4 3 =64 can also be written in the form
log =3
4 is the base is the logarithm
Use a calculator to check the value of log 4 64
Generally, log 𝑎 𝑐 =𝑏 is equivalent to 𝑎 𝑏 =𝑐

6. Sheet 2

7. Calculator investigation
log log
log 2 8
log 3 27
2log 3 9
log
log
log 2 32− log 2 4
log log 3 27
2log 3 2
log 3 9
log − log 3 9 4

8. Equivalent expressions
log log
log log 3 27
log
log − log 3 9
2log
2log 3 9
log 3 27 2
Correct groupings shown 4
log 2 8 3
log 3 9
log
log 2 32− log 2 4

9. Laws of Logarithms log 𝑎 𝑥+ log 𝑎 𝑦= log 𝑎 𝑥𝑦

10. Logarithms and Indices
Start with the number 243.
Type log into a calculator
Type 3 𝐴𝑁𝑆 (i.e. use your previous answer as the power of 3)
What do you notice?
Repeat the process for the starting number 625 and base 5.
Repeat for a starting number and base of your choice.
Does the same thing happen each time?
What can we say about the functions log 𝑎 𝑥 and 𝑎 𝑥 ?

11. Solving equations involving indices
To solve the equation 3 𝑥 =100:
Identify the base 𝑎 (𝑎=3 in this case)
Take the logarithm (base 𝑎) to both sides:
log 𝑥 = log
Simplify the left hand side: 𝑥= log
Use a calculator to find 𝑥=4.19 (3 s.f.)
Solve the equations:
10 𝑥 = 𝑥 = 𝑥 = 𝑥+5 =400
Answers: 1.70, 3.55, 1.73 and (all to 3 s.f.)

12. National Geographic Earthquakes video
Images taken from

13. Measuring earthquakes
Graph taken from
Other two images taken from

14. Earthquake calculations
How many times stronger is an earthquake:
of magnitude 5 than of magnitude 4?
of magnitude 5 than of magnitude 3?
of magnitude 7 than of magnitude 4?
What if we wanted to know how many
times stronger is an earthquake of
magnitude 6.2 than an earthquake of
magnitude 4.7?
Graph taken from
Answers are 10, 100, 1000 respectively. The next question will be answered later…..

15. Earthquake calculations
The formula for calculating the strength 𝑅 of an earthquake on the Richter scale is: 𝑅= log 𝐼 𝐼 0 where 𝐼 0 is the intensity of a ‘threshold earthquake’, which is the smallest perceptible with a seismometer and 𝐼 is the intensity of the earthquake being measured. For example: If an earthquake is 147 times the intensity of a threshold quake, 𝑅= log 147 𝐼 0 𝐼 0 = log 147=2.17

16. How would this earthquake feel?
Image taken from
2.17 would probably not be perceptible

17. Earthquake calculations
𝑅= log 𝐼 𝐼 0
How many times stronger than a threshold earthquake is an earthquake measuring 5.3 on the Richter scale? Let 𝑘 be the number of times stronger. 5.3= log 𝑘 𝐼 0 𝐼 0 5.3= log 𝑘 Raise both side to power 10: = 10 log 𝑘 =𝑘 So 𝑘= The earthquake is around times stronger.

18. Other applications of logarithms
The decibel (dB) is a measure of sound. A decibel is one- tenth of a bel, named after Alexander Graham Bell who patented the first telephone.
Decibels are used as a measure in acoustics, music environmental quality and electronics.
The formula for calculating decibels is given by 𝐷𝑏=10 log 𝐼 𝐼 where 𝐼 is the ‘threshold sound’ that can barely be perceived by the human ear and 𝐼 is sound intensity, measured in Watts per square metre (W/m2). .
Image taken from

19. Other applications of logarithms
The measure of the acidity of a liquid is called the pH of the liquid. Solutions with a pH of less than 7 are acidic; solutions with a pH of greater than 7 are alkaline. A pH of 7, such as in pure water, indicates a neutral solution.
The formula for pH is 𝑝𝐻=− log 𝐻 + where 𝐻 + is the hydrogen ion concentration (moles per litre).
Image taken from

20. Other applications of logarithms
In accounting and finance, the value 𝐴 of an investment under compound interest at annual interest rate 𝑟 calculated over 𝑚 periods per year is 𝐴=𝑃 1+ 𝑟 𝑚 𝑚𝑡 where 𝑃 is the principal investment.
To find the time taken to treble an initial amount, with interest calculated monthly at a rate of 4% the formula would become 3𝑃=𝑃 𝑡 which simplifies to 3= 𝑡 .
This can be solved using logarithms to obtain 𝑡=27.5 years.
Image taken from

21. Review
Was the A level material introduced in this session more or less demanding than you expected?
How does the topic of logarithms relate to other subjects you are considering studying post-16?
Has studying this topic made you more likely to study A level Mathematics / Further Mathematics?

22. Acknowledgements http://earthquake.usgs.gov/learn/photos.php
2011/003/Teams/Geneve/Homework_13/Logarithmic_scale
hQuarter/Chapter10/Lesson2/09Decibles.gif
act.gif
rates.gif

24. The Further Mathematics Support Programme
Our aim is to increase the uptake of AS and A level Further Mathematics to ensure that more students reach their potential in mathematics.
The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.
To find out more please visit

25. Pascal’s triangle What is the next row in this triangle?
What patterns can you spot in the rows and diagonals? 1 2 3 4 6 5 10 15 20

26. Expanding binomials
A binomial expression has two terms, for example 𝑝+𝑞
By expanding the brackets, find the binomial expansions:
(𝑝+𝑞) 1 =
(𝑝+𝑞) 2 =
(𝑝+𝑞) 3 =
(𝑝+𝑞) 4 =
Write down any patterns that you notice.

27. Expanding binomials
A binomial expression has two terms, for example 𝑝+𝑞
By expanding the brackets, find the binomial expansions:
(𝑝+𝑞) 1 = 𝑝+𝑞
(𝑝+𝑞) 2 = 𝑝 2 +2𝑝𝑞+ 𝑞 2
(𝑝+𝑞) 3 = 𝑝 3 +3 𝑝 2 𝑞+3𝑝 𝑞 2 + 𝑞 3
(𝑝+𝑞) 4 = 𝑝 4 +4 𝑝 3 𝑞+6 𝑝 2 𝑞 2 +4𝑝 𝑞 3 + 𝑞 4
Write down any patterns that you notice.

28. Expanding binomials
How quickly can you predict and write down the expansion of (𝑝+𝑞) 5 ?

29. Expanding binomials
How quickly can you predict and write down the expansion of (𝑝+𝑞) 5 ?
𝑝+𝑞 5 = 𝑝 5 +5 𝑝 4 𝑞+10 𝑝 3 𝑞 𝑝 2 𝑞 3 +5𝑝 𝑞 4 + 𝑞 5

30. Number patterns The nCr button Calculate the values:
3C0, 3C1, 3C2 and 3C3
How do they relate to the binomial expansions and the number pattern shown to the left?

31. Calculating (𝑝+𝑞) 9
𝑝 9 +9 𝑝 8 𝑞+36 𝑝 7 𝑞 𝑝 6 𝑞 𝑝 5 𝑞 𝑝 4 𝑞 𝑝 3 𝑞 𝑝 2 𝑞 7 +9𝑝 𝑞 8 + 𝑞 9

32. Catching the flu
Medical records show that the probability of an adult in the UK catching the flu in any year is 8%.
Three adults are monitored for a one year period.
Produce a tree diagram to show the outcomes of whether or not each of the three adults get the flu.
Work out the probability that all of the adults catch the flu;
two of the adults catch the flu;
one of the adults catch the flu;
none of the adults catch the flu.

33. Catching the flu Has flu 0.08 3 0.08 Has flu No flu 0.08 0.08 2 ×0.92
0.08×
0.92
Has flu
0.08
×0.92
Has flu
0.92
0.08
No flu
No flu
0.08×
0.92
Has flu
No flu
0.08
0.92
0.08×
No flu
0.92
0.92 3

34. Catching the flu p(all catch the flu) = 0.08 3
p(two catch the flu) = 3× ×0.92
p(one catches the flu) = 3×0.08×
p(none catch the flu) =
What would the tree diagram look like if there were four adults?

35. Catching the flu p(all catch the flu) = 0.08 3
p(two catch the flu) = 3× ×0.92
p(one catches the flu) = 3×0.08×
p(none catch the flu) =
Compare this to 𝑝 3 +3 𝑝 2 𝑞+3𝑝 𝑞 2 + 𝑞 3
With 𝑝=0.08 and 𝑞=0.92 the terms are the same
Generally,
p(𝑟 adults catch the flu)=3Cr× 0.08 𝑟 × −𝑟

36. Catching the flu Use the expansion
(𝑝+𝑞) 4 = 𝑝 4 +4 𝑝 3 𝑞+6 𝑝 2 𝑞 2 +4𝑝 𝑞 3 + 𝑞 4
to find the probability that in a group of four adults:
(i) all catch the flu (ii) one catches the flu
𝑝+𝑞 5 = 𝑝 5 +5 𝑝 4 𝑞+10 𝑝 3 𝑞 𝑝 2 𝑞 3 +5𝑝 𝑞 4 + 𝑞 5
to find the probability that in a group of five adults:
(i) none catch the flu (ii) three catch the flu
Can you write down a formula for the probability of 𝑟 adults out of a group of 𝑛 adults catching the flu?

37. Catching the flu Four adults p(all catch the flu) = 0.08 4
p(one catches the flu) = 4×0.08×
Five adults
p(none catch the flu) =
p(three catch the flu) = 10× ×
p(𝑟 adults out of 𝑛 catching the flu) = nCr× 𝑝 𝑟 × 𝑞 𝑛−𝑟

38. Catching the flu
The assumptions in carrying out this activity include:
The adults catch the flu independently of each other;
The adults either catch the flu or don’t catch the flu (no cases of ‘just having a few symptoms’);
The probability of catching the flu is 8% for each of the individual adults (so none are more or less prone to catching it)
There are a fixed number of adults, in this case 3, in the group (if there were an unlimited number of adults you couldn’t produce a tree diagram)

39. Sheet 1

40. Binomial Distribution assumptions
There are a fixed number of trials
The trials are independent of each other
There are two possible outcomes – you can call these ‘success’ and ‘failure’
The probability of ‘success’ 𝑝 is the same for each trial.

41. Sheet 2

42. Review
Was the A level material introduced in this session more or less demanding than you expected?
How does the topic of binomial expansions and the binomial distribution relate to other subjects you are considering studying post-16?
Has studying this topic made you more likely to study A level Mathematics / Further Mathematics?

44. The Further Mathematics Support Programme
Our aim is to increase the uptake of AS and A level Further Mathematics to ensure that more students reach their potential in mathematics.
The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.
To find out more please visit

45. Audiology
Audiology is a science that involves identifying and assessing hearing loss and balance disorders.
The results of hearing tests are shown on a graph called an audiogram.
An audiogram uses two scales:
- vertical: the volume of the sound (measured in
decibels, dB)
- horizontal: the frequency of the sound (measured
in hertz, Hz) which determines the pitch (how
high/low the sound is when it is heard)

46. An audiogram grid
Image taken from

47. An introduction to logarithms
Using your calculator work out:
log log log log log
Comment on your answers
Find the value of 𝑎 if (i) log 𝑎 = (ii) log 𝑎 = 7
(iii) log 𝑎 = 0 (iv) log 𝑎 = -1
Complete the following sentence:
To increase the value of log 𝑎 by 1, the value of 𝑎 must be
……………………………….
(Note: Base 10)
Answers are 1, 2, 3, 4, 5. Each time the number increases by a multiple of 10 the answer increases by 1.
, , 1, 0.1
Multiplied by 10

48. Volume of sound (dB)
The decibel (dB) is a measure of sound. A decibel is one-tenth of a bel, named after Alexander Graham Bell who patented the first telephone.
Decibels are used as a measure in audiology when carrying out hearing tests.
The formula for calculating decibels is is given by dB= 10 log 𝐼 𝐼 where 𝐼 0 is the ‘threshold sound’ that can barely be perceived by the human ear and 𝐼 is sound intensity, measured in Watts per square metre (W/m2).
𝐼 0 = 10 −12 W/m2
log is short for ‘logarithm’ – this is a function that you can calculate using the log button on your calculator.
Image from

49. Threshold of hearing 0.000 000 000 001W/m2
Volume of sound (dB)
Using the formula dB=10 log 𝐼 𝐼 with 𝐼 0 = 10 −12 W/m2, find the sound level in decibels of the following:
Complete the following sentence:
Each time the intensity 𝐼 increases by a multiple of 10,
the sound level increases by ….. decibels
Normal speech W/m2
Motorway traffic W/m2
Whisper W/m2
Pneumatic drill W/m2
Normal speech = 10log( /10^-12) = 60dB
Motorway traffic = 10log(0.0001/10^-12) = 80dB
Whisper = 10log( /10^-12) = 20dB
Pneumatic drill= 10log(0.001/10^-12) = 90dB
Threshold of hearing = 10 log( /10^-12) = 0
Each time the intensity I increases by a multiple of 10, the sound level increases by 10 decibels.
Threshold of hearing W/m2

50. Frequency of sound (Hz)
Frequency refers to whether a sound is very low or very high

51. Frequency of sound (Hz)
On a piano keyboard the lower pitched notes are at the left and the higher pitched notes are at the right.
The image shows the frequencies of some notes on a piano.
Moving one octave (8 white keys) to the right doubles the frequency of the note. This is why, on the horizontal scale of the audiogram, each increment corresponds to a doubling of the measurement in Hz.
Image taken from

52. Frequency on an audiogram
The horizontal scale increases by a multiple of 2 in each equally spaced interval.
The normal hearing range of frequencies for humans is 20Hz to 20,000Hz (or 20kHz). Humans are most sensitive to sounds in the 2kHz – 5kHz range.
Other animals have different ranges.
Image taken from

53. How to interpret an audiogram
Video taken from

54. Review
Was the A level material introduced in this session more or less demanding than you expected?
How does the mathematics involved in the topic of Audiology relate to other subjects you are considering studying post-16?
Has studying this topic made you more likely to study A level Mathematics / Further Mathematics?

55. The spread of disease
Video linked from

56. Spreading a rumour…
I told someone a rumour today. Assuming that on each of the following days each person who knows the rumour tells one more person:
- how long would it be before 50 people know the
rumour? 100 people? everyone in the UK
(which has population of approximately 64 million)?
- what if each person who knows the rumour tells two
more people each day? three more people? …..
What assumptions have been made?

57. Spreading a rumour… Each person tells one new person each day 1 2 3 41 2 3 4 5 6 7 8
No. of people who know at the start of the day 16 32 64
128
256
For all of the UK to know, need 2 𝑛 = and so 𝑛=26 days
Day 1 2 3 4 5 6 7 8
No. of people who know at the start of the day 9 27 81
243
729
2187
6561
When each person tells one new person each day:
50 people would know by day 6; 100 people would know by day 7
When each person tells two new people each day:
50 people would know by day 4; 100 people would know by day 5; All the UK would know by day 17
When each person tells three new people each day:
50 people would know by day 3; 100 people would know by day 4; All the UK would know by 13
Each person tells three new people each day
Each person tells two new people each day
Day 1 2 3 4 5 6 7 8
No. of people who know at the start of the day 16 64
256
1024
4096
16384
65536

58. Spreading a rumour… Each person tells three new people each day
Each person tells two new people each day
Each person tells one new person each day

59. The growth and decline of EVD
Map taken from

60. Review
Was the A level material introduced in this session more or less demanding than you expected?
How does the topic of exponential growth relate to other subjects you are considering studying post-16?
Has studying this topic made you more likely to study A level Mathematics / Further Mathematics?

61. Acknowledgements http://www.hearinglink.org/what-is-an-audiogram
e/Lessons/FourthQuarter/Chapter10/Lesson2/09Decibles.gif

63. The Further Mathematics Support Programme
Our aim is to increase the uptake of AS and A level Further Mathematics to ensure that more students reach their potential in mathematics.
The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.
To find out more please visit

64. What is theme of this session?
Images taken from:

65. Centre of mass in 2D
Imagine a two-dimensional shape cut out of thin card.
If you wanted to balance the shape so that it lies flat on the tip of your finger, where should you place your finger?
Would this be the same for each shape?

66. Finding the centre of mass
Use lines of symmetry on the shape to help to identify the position of the centre of mass
e.g. for a square
The centre of mass is located here

67. Centre of mass of a triangle
Measure the distance of the centre of mass along each median (dotted line) – what do you notice?
The centre of mass of a triangle lies two-thirds of the way from the vertex along the medians (the dotted lines shown).

68. Centre of mass of a semicircle
Measure the distance of the centre of mass along the line of symmetry of the semicircle.
It can be shown that the centre of mass lies exactly 4𝑟 3𝜋 along the line of symmetry from the centre of the circle, where 𝑟 is the radius.

69. A balancing act…
The object below is made of thin card and is hanging from a string at the point marked with a cross.
Would the object balance horizontally?
What factors would you need to take in to account to make this decision?

70. The moment of a force
The moment of a force is the turning effect of the force.
It is defined as force x perpendicular distance from the pivot and is measured in Newton Metres (Nm)
The force here is the weight, given by mass x g, where g = 9.8m/s is the acceleration due to gravity and mass is in kg.
For example, the moment of the force of the man of mass 80kg on the see-saw pictured is x 1.5 = 1176Nm
Image taken from
distance
=1.5m
Force = 80 x 9.8 = 784N

71. Balancing moments
A man of mass 80kg and a child of mass 35kg are positioned on a see-saw as shown in the diagram.
Would the see-saw balance?
If not, how could you make the see-saw balance?
Images taken from
1.4m
90cm

72. Balancing moments
The moment of the man is 0.9 x 784 = 705.6Nm and would turn the see-saw in a clockwise direction.
The moment of the child is 1.4 x 343 = 480.2Nm and would turn the see-saw in an anticlockwise direction.
The moment of the man is larger so the see-saw would turn clockwise.
To balance the see-saw, the man would need to stand at distance 𝑥 from the pivot where 784 × 𝑥=480.2
𝑥= 0.61m
Images taken from
1.4m
90cm
Force = 35 x 9.8 = 343N
Force = 80 x 9.8 = 784N

73. A balancing act…
Will this shape balance horizontally if suspended from the point marked with a cross?
Start by finding the centre of mass of the rectangle and the triangle.
9 cm
14 cm
11 cm

74. A balancing act…
The centre of mass of the rectangle and the triangle are marked with crosses. Their weights are (11 x 14) x 9.8 = N and ( 1 2 x 11 x 9) x 9.8 = 485.1N respectively.
The moment of the weight of the rectangle is x 7 = Nm
The moment of the weight of the triangle is x 3 = Nm
The object would not be balanced. To make it balance, call the length of the rectangle 𝑥, then
(11 x 𝑥) x 9.8 x 7 =
This gives 𝑥 = 1.93cm
9 cm
14 cm
11 cm
7cm
3cm

75. Find the values of 𝑥 and 𝑦
6 cm
10 cm
The orange and blue objects both hang horizontally when suspended from the points marked with a black cross.
Find the values of 𝑥 and 𝑦.
5 cm
𝑥 cm
6 cm
10 cm
5 cm
𝑦 cm

76. Find the values of 𝑥 and 𝑦
6 cm
10 cm
Left rectangle: (6 x 𝑥) x 3 = 18𝑥
Right rectangle: (10 x 5) x 5 = 250
So 18𝑥 = 250 ⇒ 𝑥 = 13.9 cm
5 cm
𝑥 cm
5 cm
3 cm
6 cm
10 cm
Triangle: (0.5 x 𝑦 x 6) x 2 = 6𝑦
Rectangle: (10 x 5) x 5 = 250
So 6𝑦 = 250 ⇒ 𝑦 = 41.7 cm
5 cm
𝑦 cm
5 cm
2 cm

77. Review
Was the A level material introduced in this session more or less demanding than you expected?
How does the topic of centres of mass relate to other subjects you are considering studying post-16?
Has studying this topic made you more likely to study A level Mathematics / Further Mathematics?

78. Acknowledgements
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ak0.pinimg.com/736x/5c/88/38/5c e31f920dbe438bf821ab35d.jpg
short-man-in-suit-svg-it6Lo3-clipart.png
ak0.pinimg.com/236x/24/1c/9c/241c9ce6b9a262d5f56a705043b9dfad.jpg

80. The Further Mathematics Support Programme
Our aim is to increase the uptake of AS and A level Further Mathematics to ensure that more students reach their potential in mathematics.
The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.
To find out more please visit

81. Finding area Find the area of these shapes: 112m 10m 22m 45m 22m 5m
Pink area: 0.5x(10+15)x12 – 0.5 x 5 x 6.538….=133.7m2
Blue area: 112 x 218 – 98 x x 23 = m2
22m
98m
15m

82. Finding area
Elizabeth Tower is the part of the Houses of Parliament that houses Big Ben. The tower is 96.3m tall in total, with the section below the clock being 55m high. The base of the tower is a 12m square and the section containing the clock face is 6m in height.
Estimate the area of the pictured front side of Elizabeth Tower.
What assumptions have you made in your calculations?
Image taken from
Assuming the front of Big Ben is a flat face consisting of a rectangle and a triangle: (55+6) x x 12 x 35.3 = 943.8m2

83. Le Pont d’Avignon
The Pont Saint Benezet is a bridge in Avignon, France and is often called the Pont d’Avignon.
It originally had 22 arches along its 900m length but after a major flood in 1668 many of the arches collapsed and only 4 are left.
Each arch of the bridge has a span of between 30.8m to 33.5m and the height of each arch is approximately 13m. The bridge is 4m wide and 15m high.
How could you estimate the volume of the stone used to make one arch of the Pont d’Avignon?
Image taken from

84. Le Pont d’Avignon
15m
13m
Image taken from
32m 8m

85. Finding the area under a curve
The area under a curve can be approximated by splitting the region in to trapezia with the parallel sides vertical.

86. Finding the area under a curve
Using a larger number of trapezia increases the accuracy of the approximation.

87. Finding the area under a curve
If each strip has a width ℎ, the areas of the trapezia are:
1 2 ×ℎ× 𝐴+𝐵
1 2 ×ℎ× 𝐵+𝐶
1 2 ×ℎ× 𝐶+𝐷
1 2 ×ℎ× 𝐷+𝐸
1 2 ×ℎ× 𝐸+𝐹
1 2 ×ℎ× 𝐹+𝐺 A B C D E F G ℎ
This is called the Trapezium Rule
This simplifies to 1 2 ×ℎ× 𝐴+2 𝐵+𝐶+𝐷+𝐸+𝐹 +𝐺
In words this is the same as saying 𝒉 𝟐 ′𝒆𝒏𝒅 𝒔 ′ +𝟐 × ′ 𝒎𝒊𝒅𝒅𝒍𝒆𝒔′

88. The Trapezium Rule Area is approximately ℎ 2 ′𝑒𝑛𝑑 𝑠 ′ +2 × ′ 𝑚𝑖𝑑𝑑𝑙𝑒𝑠′
The area under this curve is approximately
( =2.75
𝒚= 𝒙 𝟑 + 𝒙 𝟐 +𝒙+𝟏
Using strips of width ℎ=0.5 𝒙 𝒚 -1
-0.5
0.625 1
0.5
1.875 4

89. Applications of the Trapezium Rule in Architecture
Saint Chapelle, also called the Holy Chapel, is located in Paris.
It is famous for having over 600m2 of stained glass windows including 15 windows in the Upper Chapel that date from the 13th century.
The 15 windows each measure 15.4m in height and have a width of 4.25m at the base.
Image taken from

90. Applications of the Trapezium Rule in Architecture
The Golden Gates to the Royal Palace in Fes, Morocco are a beautiful example of symmetry in architecture.
An architect wants to design a set of three interior doors that will look similar to the ones on the Royal Palace. The central door will have the measurements shown. The smaller doors will have length and width that are three quarters of the central door
3.2m
1.8m
Image taken from
1.45m 2m

91. Review
Was the A level material introduced in this session more or less demanding than you expected?
How does the topic of approximating the area under a curve relate to other subjects you are considering studying post-16?
Has studying this topic made you more likely to study A level Mathematics / Further Mathematics?

92. Acknowledgements http://www.aviewoncities.com/london/bigben.htm
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