Further Mathematics Support Programme
Welcome to the FMSP Enrichment Pack
The FMSP is committed to providing opportunities for Key Stage 4 students to
extend and enrich their understanding of mathematics. We run a number of
enrichment days at universities and other educational establishments for Key Stage
4 students across the country as well as working in schools with smaller groups. Our
aim is to encourage students to consider Mathematics and Further Mathematics at A
level.
This pack contains materials to help you to provide an enriching experience for your
students in your classroom. The FMSP has, over the years, produced a wide variety
of materials for Key Stage 4 students. The accompanying pen drive contains over
150 files of enrichment, extension and problem solving materials. We hope that you
will enjoy using these materials and investigate all of the resources available on our
website www.furthermaths.org.uk.
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Further Mathematics Support Programme
Contents
Enrichment Materials
Growing a recursive tree
The Tower of Hanoi Map
The Recursive Photocopier
The Chaos Game
Making Decisions Using Mathematics
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Extension Materials
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Maths Feast
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Comparison Round
Comprehension Round
Four in a Row Round
Matching Round
Practical Round
Problem Solving Round
The Year 10 Maths Competition
Problem Solving Materials
The GCSE Problem Bank
Twenty Problems with Hints and Prompts
Groupwork
Ratio and symmetry patterns
Building Bridges
Epidemic
Willis Tower
Why Study Maths?
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Further Mathematics Support Programme
Enrichment Materials
These consist of several shorter activities to widen your students’ perception of
mathematics and one longer activity that focuses on a particular use of mathematics
in making financial decisions.
Growing a Recursive Tree
This is a set of instructions for students familiar with MSW LOGO (or equivalent).
The programs refer to themselves as part of the code. Students should be
encouraged to notice this.
Once the programs have been saved, typing tree 100 produces the result.
It starts as a twig.
As the programs are edited and added to, the design becomes more tree-like until
students finally produce a fractal tree.
They should be encouraged to ‘tamper’ with the code – scale factors, angles,
lengths – to see if they can create something more natural looking.
Skills required: A working knowledge of the LOGO language
The Tower of Hanoi Map
This consists of a powerpoint presentation and a student sheet.
Students need to be aware of how the Tower of Hanoi puzzle works. They may even
know the formula for the least number of moves for any number of disks.
The idea is that they should draw a map showing all of the possible legal moves
when solving the puzzle.
The Powerpoint has a demonstration of the start of the map on slides 3 and 4.
These slides are animated so it is worth running through them before using them
with a class.
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Further Mathematics Support Programme
Creating the map
Students can use stick drawings, a coordinate system or an actual Tower of Hanoi
puzzle to think about the moves. The coordinate system is the hardest to use for
most students but is very useful for those that like any sort of computer
programming. The coordinates stand for
(position of small disk, position of middle sized disk, position of large disk)
So (1,1,1) means all 3 disks are on peg 1 and (1,3,2) means that the small disk is on
peg 1, the middle disk is on peg 3 and the large disk is on peg 2.
The map is constructed by thinking what moves are possible from each
arrangement (represented by nodes on the graph). If you can move from one
arrangement to another by one legal move, then a line is drawn between them.
Lines should be one unit long.
Students will need to be very organised and draft/redraft their map.
The final map should look like a Sierpinsky triangle if it is done correctly.
Slide 5 shows the repeating patterns that are produced.
Slide 6 shows how a student could change a map for a 3 disk puzzle into a map for
a 4 disk puzzle.
Skills required: An understanding of the rules of the Tower of Hanoi puzzle. A great
deal of patience in setting up the map!
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Further Mathematics Support Programme
The Recursive Photocopier
This consists of a powerpoint presentation, a student instruction sheet and a student
worksheet.
Give the students the sheets then show them slide 3. They complete the worksheet
for setting 1.
Slide 4 shows the answer. Each click runs through an animation showing the result
of each copy (there are sound effects). Before getting too far, ask the students what
the pattern will look like after several thousand copies. You can mention that the
diagonal line is the attractor for the rule or that it is the limit of the rule. These
phrases would be being used quite loosely but that’s fine for this activity.
Show the students slide 5 and 2 run-throughs on slide 6 to get the idea. They then
complete the worksheet for setting 2. Ask if they can think what the attractor looks
like for this rule.
Continue with slide 6 until a Sierpinsky triangle is formed.
Interestingly enough, it is the same pattern as that produced for The Tower of Hanoi
Map!
Skills required: Not many! The ability to follow instructions and an example.
The Chaos Game
This consists of a powerpoint presentation, a student sheet, a Geogebra file and an
Excel file.
Slide 3 of the powerpoint shows how the game is played. The students will need to
have some way of randomly selecting the colours red, blue and yellow with equal
probabilities.
Students put a dot somewhere in the triangle. They then use a dice or a spinner to
select one of the three vertices. A new dot is put half way from the current dot to the
selected vertex. This becomes the current dot and the process is repeated as many
times as possible. Each time the student should halve the distance from the last
point plotted to the randomly selected vertex. This could be done as a whole class
with the Geogebra file "The Chaos Game", using the mid-point tool to speed up the
process. There is also an Excel file that has results collected from a very large
number of FMSP enrichment events. For this file, the selected vertices are referred
to as 1, 2 and 3 and the new points are coordinates calculated from these
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Further Mathematics Support Programme
selections. It is possible to add new random numbers and drag the formula down to
create the new points. Drawing a scatter graph of the coordinates produces a real
"wow" moment.
The pattern is the Sierpinsky triangle that was also produced by The Tower of Hanoi
Map and The Recursive Photocopier setting 2.
Skills required: The ability to halve distances. Patience – a large number of points
are needed to get the pattern!
Making Decisions Using Mathematics
This consists of a powerpoint and a teacher guide.
This is an extended activity that requires some careful preparation. It is described in
detail in the teachers’ guide.
Skills required: An understanding of probability including tree diagrams. The ability
to understand the idea of expectation.
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Extension Materials
These are aimed at students who are working towards GCSE Mathematics and
would benefit from exposure to mathematics beyond the GCSE specifications.
All of the materials have the following format:
 Starts from a topic or idea that is in GCSE Mathematics and features a ‘What
you should know’ section;
 A ‘New idea’ that is an extension to GCSE Mathematics;
 A task for students to attempt so they can investigate the idea;
 Ideas for further investigation suggested in a ‘Take it further section’;
 A brief explanation of how this topic is developed at A level;
 All the materials are based on a single sheet of A4 (except for NA 11, SSM 4
and SSM 6) which can be copied and given to students;
Teachers’ notes, including solutions, are available for free from the Integral Online
Resources website. To access these resources, schools/colleges must register with
the FMSP. Registration is free and will also provide access to other resources and
information about local events to support teachers and students. To register go to
www.furthermaths.org.uk.
These materials could be used:
 As a whole-class activity when students have finished studying the topic;
 As extra materials to stretch and challenge some of the more able students
within a class;
 As a regular, possibly optional, homework task;
 As a basis for study in lunchtime or at an after-school mathematics club.
The tasks are not intended to be linear. Students will benefit from the investigations
even if they do not ‘complete’ them.
In each of the three sections the resources are ordered by accessibility; however, it
is inevitable that some students will progress further with some tasks than with
others.
The materials are based on interesting mathematical ideas and are not designed to
provide extension to every aspect of the GSCE specifications.
The activities are:
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Further Mathematics Support Programme
NA 1
NA 2
NA 3
NA 4
NA 5
NA 6
NA 7
NA 8
NA 9
NA 10
NA 11
GTM 1
GTM 2
GTM 3
GTM 4
GTM 5
GTM 6
Number and Algebra
GCSE topic
Graphs of quadratic equations
Surds
Interpreting graphs
Trial and improvement
Functions
Indices
Simultaneous equations
Solving quadratic equations
Adding fractions
Inequalities
Plotting curves *
* this activity features a separate extension sheet
Geometry, Trigonometry and Measures
GCSE topic
Trigonometry 1
Trigonometry 2
Circles
Pythagoras’ theorem *
Loci
Vectors *
* these activities require two sheets to be copied for
the students
Probability and Data Handling
GCSE topic
PDH 1 The mean from a frequency table
PDH 2 Interquartile range
PDH 3 Tree diagrams
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Further Mathematics Support Programme
Maths Feast
The Year 10 Maths Feast is an enjoyable and challenging team competition testing
mathematical, team-working and communication skills. Each year the format of the
competition changes slightly so that the rounds remain interesting and exciting.
The materials in this pack are either from the competition itself or are materials
produced to support teams preparing for the competition. They can be used in
mathematics classes to enrich and extend students’ learning.
There are several rounds requiring different skills and strategies for success.
All solutions are provided for these materials.
Comparison Round
In this round students are given two statements A and B e.g. A 7th Fibonacci
number, B 7th Prime number and have to select the correct option from A < B, A = B
and A > B.
There are 10 questions covering a wide range of topics.
Comprehension Round
In this round, students are given a poster containing some mathematical information
about a topic they will not have covered. They then have to answer questions using
the information from that poster.
There are 10 questions of increasing difficulty.
Four in a Row Round
In this round, students try to answer a variety of problems. Getting 4 right in a row
gives bonus marks.
There are 16 questions.
Matching Round
In this round, students create groups of cards that they think go together. There are
eight groups.
Each correct group scores two marks.
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Further Mathematics Support Programme
There are some blank cards; for extra marks students can create their own cards.
Each additional card, marked in the correct position will score 1 mark, up to a
maximum of 6 marks.
Practical Round
In this round students are given a practical activity to complete.
There are two activities included in the pack:
 A mathematical origami activity where the students fold Columbus cubes from
squares of paper.
 An activity where the students have to produce the largest box possible with a
sheet of paper
Problem Solving Round
In this round, students attempt to solve at least 4 problems from 6. The best 4
solutions contribute to the final score.
The Year 10 Maths Competition
This competition is the forerunner of the Maths Feast. All of the resources used for
both the heats and the finals from 2011 to 2014 are included.
All answers are provided in this pack.
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Further Mathematics Support Programme
Problem Solving Materials
The FMSP has a commitment to providing materials and training to help schools
improve their students’ problem solving skills. Included in this pack are two series of
problem solving materials.
The GCSE Problem Bank
This consists of a Powerpoint file of 30 problems that can be accessed by selecting
from the menu on the third slide. These problems have been written by the FMSP
and are for a range of abilities. Problems marked H are suitable for higher tier
students, problems marked F are suitable for foundation tier students. Problems
marked H/F are suitable for both.
A book of all of the problems is also included as is a book of worked solutions.
Twenty Problems with Hints and Solutions
This consists of a Powerpoint file of 30 problems that can be accessed by selecting
from the menu on the third slide. These problems have been written by the FMSP
and should be used with the hints and prompts given in the accompanying book
(also in this pack). The aim of this series was to help teachers develop the
questioning skills required to improve their students’ problem solving ability without
giving away the answers.
All solutions are provided.
Groupwork
These activities are designed to promote groupwork. Each activity requires a set of
cards to be printed and cut up. Some of these activities are suitable for Key Stage 3
students. There is a suggested way to use the cards but you may use them as you
wish.
Ratio and Symmetry Patterns
This is best done in groups of 4.
The students are provided with a 8 × 8 grid to colour in. They are also given a set of
clue cards. Clues to what that design should look like are written on the cards.
In their groups, the students
 Deal out the cards
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Further Mathematics Support Programme
 Take it in turns to select one piece of information that they think is important
and read it to the rest of the group
 They can write something down on the group’s answer sheet but they must
not show anyone any of their cards (even after they have read them out)
 As a group, they try to find the design.
There are two student answer sheets provided. One has detailed instructions on it,
the other only has the grid.
There are 3 different sets of clue cards (and 3 different patterns).
As an extension, students can try to work out the fewest number of cards required to
uniquely define the pattern.
Building Bridges
This activity is run in the same way as the Ratio and Symmetry Patterns activity.
The activity requires the group to draw a picture of a suspension bridge using clues
about its symmetry. They are required to do some simple calculations to find out
things like cable spacing.
The activity is a good introduction to groupwork for students as it does have some
mathematical content but at this stage it is not too challenging.
Epidemic
This activity is run in the same way as the Ratio and Symmetry Patterns and
Building Bridges activities.
Students have to use probability (preferably through tree diagrams) to calculate the
amount of anti-viral medicine they should order to combat a flu epidemic.
This is a far more challenging activity than Building Bridges.
Willis Tower
This activity is run in the same way as the Ratio and Symmetry Patterns, Building
Bridges and Epidemic activities.
Students have to calculate the pinnacle height of the Willis Tower (formerly the
Sears Tower) above street level.
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Further Mathematics Support Programme
This activity requires students to use trigonometry to find the heights of various parts
of the Willis Tower before combining these to get the overall pinnacle height.
This is a challenging activity.
Why Study Maths?
The FMSP works to encourage students to study Mathematics and Further
Mathematics at A level. This pack contains a Powerpoint presentation and an
accompanying leaflet that can be used at parents and student open evenings.
The presentation has detailed notes at the bottom of key slides. Some slides may be
omitted depending on the audience.
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