Can you make money in this game?
Pay £5
To find out if this game is worth playing, you can
use a decision tree. This looks like a tree diagram
but has some extra symbols that are used.
Roll the dice. If you get 5 or 6,
I’ll give you £20.
is a decision made by the player
If you lose you can pay
another £5 to try for a 6. If
you roll a 6 on this, I’ll give
you £35! How can you lose?
is something that depends on chance
is the amount of money the player wins
or loses by following a particular path
Maximum 2 rolls per turn.
You always begin your calculations on a decision tree from the end and work back towards the start!
At this stage you would expect to win an
average of
At this stage you would expect to win an
average of
If you play and win the
first roll you will have won
£20 less your £5 stake.
which comes to a loss of £4.17 to 2 d.p.
which comes to a profit of £2.22 to 2 d.p.
win
yes
If you play, lose the first
roll and then try for 6 and
win, you will have won
£35 less your £10 stake.
£15
win
£2.22
yes
lose
£25
-£4.17
Try for a 6
Do I play the game?
lose
no
-£10
no
£0
You would choose to play
as the average winnings
for this game are £2.22
and cross off this least
favourable option.
If you play, lose the first
roll and then try for 6
and lose, you will have
paid a total of £10.
-£5
A loss of £4.17 is slightly
better than a loss of £5 so at
this stage you would choose
to have a second roll if you
lost the first one. The double
crossing out shows that this is
the least favourable option.
If you play, lose the
first roll and then
give up, you will have
paid a total of £5.
Your strategy for this game, based on the decision tree would be to always choose to play and, if you lose the first roll, roll again. This would give
you expected winnings of £2.22 per game (your mean average profit, after getting your stake back, over a large number of games). This is called
the expected monetary value of the game.
FMSP Enrichment
What has this got to do with
A level Mathematics?
The calculations on this poster use techniques
that you meet in A level Mathematics. These
techniques relate to two areas of Mathematics;
Decision (or Discrete) Mathematics and Statistics.
Decision Mathematics uses Mathematics to
understand and make predictions about the
world in which we live and interact. Statistics used
mathematics to study of the collection, analysis,
interpretation, presentation, and organisation
of data.
In particular, this poster uses the idea of
expectation from statistics to calculate the average
profit that would be gained by following a
particular strategy. Decision analysis from Decision
Mathematics is used to set up the tree diagram.
In the world of work, being able to use these
topics allows financial managers to develop
strategies for investing money. It provides a way
for managers to decide between two or more
competing ventures.
Studying A level Further Mathematics broadens
the mathematical topics you learn and deepens
your understanding of these topics. It gives you
the chance to tackle many more problems in
Mechanics, Statistics, Decision Mathematics and
Pure Mathematics.
A level Further Mathematics is an excellent
preparation for mathematically rich courses at
University. Students aiming to study for Science,
Engineering, Technology or Mathematics degrees
find that an AS or A level in Further Mathematics
is a real advantage in their first year at university.
A level Mathematics and Further Mathematics
open the door to a wide variety of interesting,
challenging and rewarding careers. Find out
where maths could take you at:
www.furthermaths.org.uk
Let Maths take you further…
Can you make things seem real?
Modern computers are powerful enough to create scenes in films and computer games that seem to be real.
In order to do this they use some mathematics that is based on the geometrical rules you learn in school.
The building blocks
Lighting is key
FMSP Enrichment
e
Finding the lighting angl
What has this got to do with
A level Mathematics?
describe the two directions.
to
rs
cto
ve
e
us
u
yo
is
th
do
To
s a direction that is three units
ibe
scr
de
r
cto
ve
e
th
ns
In 2 dimensio
The calculations on this poster use techniques that
you meet in A level Mathematics. These techniques
are found in the Core (Pure) Mathematics past of
the course and are extended in the A level Further
Mathematics course.
across and one down:
A grid is used to create scenes and objects.
e same. The vector starts
In 3 dimensions the idea is th
end point is two units
point as the one above but its
u.
out of this poster towards yo
A scene without lighting
at the same
In particular, this poster uses vector methods to find
the angle between a light source and an object.
Vectors underpin a large amount of the mathematics
needed to create computer games and special
effects. Vectors are used to “project” images from
a constructed three dimensional world on to a two
dimensional background. They are also used to
analyse the trajectories of objects moving in those
worlds and check for collisions or other interactions.
rs.
e the dot product of the vecto
us
u
yo
rs
cto
ve
o
tw
n
ee
tw
To find the angle be
ing done in this example:
See if you can see what is be
In the world of work, being able to use these
topics allows games designers and other computer
imaging specialists to construct worlds that are
more and more convincing. Vectors are widely used
in air traffic control systems, weather prediction,
construction, cosmology and electronics. In fact
vectors have a massive number of applications in
the real world.
ctors is 4.
The dot product of the two ve
ch vector.
You also need the length of ea
For
To make it look convincing textures, shading and
lighting are added.
Standard textures are distorted using matrices to
fit the polygons.
the length is
Studying A level Further Mathematics broadens
the mathematical topics you learn and deepens
your understanding of these topics. It gives you
the chance to tackle many more problems in
Mechanics, Statistics, Decision Mathematics and
Pure Mathematics.
The same scene with lighting.
the length is
To give a scene atmosphere and make it look as
real as possible, the lighting has to be correct.
For
For a flat shape in a 3D environment, the angle
between a line at 90° to its surface and the light
source will determine how bright that shape
should appear. The smaller the angle, the more lit
the shape will be.
The angle is calculated using
A level Further Mathematics is an excellent
preparation for mathematically rich courses at
University. Students aiming to study for Science,
Engineering, Technology or Mathematics degrees
find that an AS or A level in Further Mathematics is
a real advantage in their first year at university.
rs is 76.5°
The angle between the vecto
A level Mathematics and Further Mathematics
open the door to a wide variety of interesting,
challenging and rewarding careers. Find out where
maths could take you at
Which shade would you choose for these two vectors?
Angle
A small angle gives a
light shading
A larger angle gives a
darker shading
Shade
0° - 15°
15° - 30°
30° - 45°
45° - 60°
60° - 75°
75° - 90°
www.furthermaths.org.uk
Let Maths take you further…
Are you connected to a Mongolian yak herder?
The Oracle of Bacon
Hollywood actor Kevin Bacon has
appeared in a large number of films. The
Oracle of Bacon links all actors, male and
female, to Kevin Bacon through the films
they have each appeared in. Jennifer
Lawrence has a Bacon number of 1 as
they both appeared in the film X-Men:
First Class. Damian Lewis has a Bacon
number of 2 as he was in the film The
Situation with John Slattery who was in
the film Sleepers with Kevin Bacon. Try it
out at oracleofbacon.org. Can you find
someone with a Bacon number
greater than 6?
You’d be surprised how close you are !
FMSP Enrichment
What has this got to do with
A level Mathematics?
The calculations on this poster use techniques that
you meet in A level Mathematics. These techniques
relate to the area of Mathematics called Decision or
Discrete Mathematics, which uses Mathematics to
understand and make predictions about the world in
which we live and interact.
Think about everyone you’ve ever met or
been introduced to. That includes friends,
family, classmates, teachers… a lot of
people already. The connections of who’s
met who might look something like this,
but with lots more points and lines.
In particular, this poster uses Graph Theory which
allows us to study networks. This allows engineers
to analyse paths through networks to identify
bottlenecks, useful when studying flows of cars along
road networks and essential when studying movement
of people through buildings in case of fire or other
emergencies. These topics are developed further at A
level in critical path analysis and, if you study Further
Mathematics you may find yourself studying Game
Theory, Linear Programming and Matchings.
Now imagine similar groups of
acquaintances in other parts of the world
In the world of work, being able to use these topics
allows managers to make decisions about how to
assign roles within an organisation and which parts
of a process are critical to completing a project on
time; they are used by PR agencies and marketing
departments to predict how effective different
advertising campaigns will be; civil engineers and
town planners use network flows to model traffic
flows which indicate where to put roads and
junctions and even how long to phase traffic lights.
Six degrees of separation
This is the theory that everyone is six or fewer steps away, by way of acquaintance, from any
other person in the world, so the chain of “knows... who knows...” statements can be made
to connect any two people in a maximum of six steps.
So you may know someone who knows someone who knows someone... who knows a
Mongolian yak herder.
It only takes a few connections across
countries to link the groups together.
Who uses this?
Facebook, Twitter, LinkedIn and other social networking sites are particularly interested in degrees
of separation. Medical scientists use the ideas to study the spread of disease.
Twitter
4.67 degrees
of separation
1 friends away
2 friends away
3 friends away
4 friends away
5 friends away
6 friends away
7 friends away
Studying A level Further Mathematics broadens the
mathematical topics you learn and deepens your
understanding of these topics. It gives you the chance
to tackle many more problems in Mechanics, Statistics,
Decision Mathematics and Pure Mathematics.
A level Further Mathematics is an excellent
preparation for mathematically rich courses at
University. Students aiming to study for Science,
Engineering, Technology or Mathematics degrees
find that an AS or A level in Further Mathematics is a
real advantage in their first year at university.
A level Mathematics and Further Mathematics open
the door to a wide variety of interesting, challenging
and rewarding careers. Find out where maths could
take you at:
www.furthermaths.org.uk
Let Maths take you further…
Could you make the jump?
The Stunt
In the film Fast and Furious 7, a Lykan Hypersport supercar is
driven out of the window high up in the Etihad towers complex
in Abu Dhabi. The car flies through the air before crashing
through the window of a neighbouring building 50m away.
How long does the car spend in the air?
The car starts off travelling at 27.8 metres per seco
nd (the same as 100 km/h) as it
leaves one building. There is no air resistance to
slow it down so it will continue to
move horizontally at that speed.
The target building is 50 metres away.
Using the formula
time =
distance
speed
time =
50
27.8
time = 1.8s
The car spends 1.8 seconds in the air.
How far does the car drop?
It looks good with CGI but is it
possible?
With a bit of mathematical modelling, we can find out.
We can make a good start with one key fact:
•The speed of the car as it leaves the building
The car leaves the building horizontally so it has
no speed in the vertical direction at
all to start with. It will accelerate downwards with
an acceleration of 9.8 metres per
second squared.
Using the formula
s = ut+ 1 at2
2
The Lykan Hypersport can accelerate from 0 to 100
kilometres per hour in 2.8 seconds.
where s is the drop distance, u is the starting vert
ical speed (0), t is the time spent in
the air (1.8 s) and a is the acceleration (9.8 m/s²).
Building a mathematical model
s = 0+ 1 x 9.8 x 1.82
2
To make it possible to do the calculations, we’ll ignore
some things that won’t make a big difference to the
results. We’ll make some assumptions:
s = 15.9m
The car drops 15.9 m vertically (about 4½ floors).
•The car is travelling at 100 km/h as it leaves the
first building
•There is no air resistance on the car (so it doesn’t
slow down when flying through the air)
•The windows won’t slow the car down as it
smashes through them
•The car is a particle so we don’t have to worry
about it flipping over
FMSP Enrichment
What has this got to do with
A level Mathematics?
The calculations on this poster use techniques
that you meet in A level Mathematics. These
techniques relate to the area of Mathematics
called Mechanics, which uses Mathematics to
understand and make predictions about the
physical world.
In particular, these calculations use the equations
of motion (often referred to as the suvat equations)
which allow us to use three of five quantities,
displacement (distance), initial velocity (speed), final
velocity, acceleration and time, to find a fourth.
These concepts are developed further at A level to
consider the effects of air resistance, friction and
when acceleration changes during the motion.
In the world of work, being able to use these
calculations allows stunt directors to ensure that
the stunts they want to perform are possible;
racing teams to produce the best set-up for their
cars; and scientists to predict the paths of planets
and space ships.
Studying A level Further Mathematics broadens
the mathematical topics you learn and deepens
your understanding of these topics. It gives you
the chance to tackle many more problems in
Mechanics, Statistics, Decision Mathematics and
Pure Mathematics.
A level Further Mathematics is an excellent
preparation for mathematically rich courses at
University. Students aiming to study for Science,
Engineering, Technology or Mathematics degrees
find that an AS or A level in Further Mathematics is
a real advantage in their first year at university.
A level Mathematics and Further Mathematics
open the door to a wide variety of interesting,
challenging and rewarding careers. Find out where
maths could take you at:
www.furthermaths.org.uk
Let Maths take you further...
So, it looks possible but would you try it? Do you trust the model or do you think it should take account of some more of the features of the situation?