Further Mathematics Support Programme
www.furthermaths.org.uk
Enrichment Training
MEI Conference
July 2011
Abigail Bown FMSP
Further Mathematics Support Programme
Let Maths take you Further…
What is Enrichment?
Why?
•Develop mathematical thinking and problemsolving skills
•Offer challenging and engaging activities
•Enrich the experience of the mathematics
curriculum for all learners
•Show rich mathematics in meaningful contexts
•(stolen from the Nrich website)
Questions
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Ways of displaying differentiation in a topic
Resources to extend students
Ideas for lunchtime activities
Activities that make students think for themselves
Rich starting points for lessons
Contacts for trips and events
Ideas to introduce less teacher led lessons
Opening up interesting areas of maths to Gifted
Students
Ideas to Enrich the students I teach
Aims of the session……
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To go back with at least one activity that
you can embed into your order of teaching
1
Enrichment Talks/Activities
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Practical Mechanics (with an extension to
Decision and Statistics) (15 mins)
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Dragon Maths Quiz (15 mins)
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Carousel of Activities (20 mins)
Resources and Ideas
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More Maths Grads Box
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NCETM
Sharing Ideas and Practise
Nrich Curriculum Planning
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Mapping Documents
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stemNRICH
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Arbelos posters, books and activities for
secondary mathematics - extension and
enrichment resources for the classroom or a
maths club.
Rich Starting Points (e.g. 3 & 8)
Mathematics on the Simpsons!
Tarsia Jigsaw Software
Jigsaw examples
A Level Pictures and Puzzles
Useful Links
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Maths Careers Website (do ipods really
shuffle?) (Making Monsters)
MA Applications of Mathematics (Benford’s
law)
Royal Statistical Society Teaching Resources
Understanding Uncertainty (crisps)
FMSP Website
TSM Website
Ron Knott’s website
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2
Funding and Events
in case you’re inspired!
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RSS Free School Lectures
IMA Education Grant
LMS Holgate Lectures
LMS Education Grants
3
28/07/2011
The Dragon Maths Competition
is about to start ....
Dragon Maths
1. Answer as many questions as you can in the time allowed.
2. One person from each team takes your answer to a marker.
3. The marker will say yes if the answer is correct and give you
the next question.
4. The marker will say no if the answer is wrong and give you
the choice of passing or trying again. If you pass you cannot
go back to that question.
5. There is also a bonus question on your desk, which can be
attempted at any time during the competition.
6. No calculators allowed - write exact answers only. E.g. 3√2
7. Remember to write your team number on all sheets.
Each correct answer = 5 points
Fully correct bonus = 20 points (+ pro rata)
Question 1
Question 2
x2 – 6x = (x - a)2 – b
Find the value of a and the value of b.
Answer
Team
The sum of numbers A, B, C and D is 123.
Two of the numbers are consecutive.
The other two are respectively double the
first two.
Find the value of A, B, C and D.
Answer
Team
1
28/07/2011
Question 3
Question 4
1 2
 ab 2
3 2
A cubic polynomial is given by
f(x) = x3 + x2 – 10x + 8.
Find the value of a and the value of b.
Factorise f(x) fully.
Answer
Team
Answer
Question 5
Question 6
Two circles of radius 10 overlap so that each passes through the
centre of the other. Find the exact area of the rhombus formed
by joining the centres and the points of intersection.
What is the value of 1.14?
Answer
Team
Answer
Team
Team
2
28/07/2011
Bonus
There are four pairs of positive integers
(x,y), such that
x2 - y2 = 105
Find them.
Answer
Team
3
28/07/2011
The Mathematical Marine Mobile Company
Manufacture - MECHANICS
Make a semicircular-tailed fish and a triangular-tailed fish.
THE FISH MUST BALANCE HORIZONTALLY WHEN SUSPENDED
Making a Profit – DECISION MATHEMATICS
Use the information to decide how many of each type of fish
to make for your test manufacture in order to maximise your
profits when you actually go into business.
Quality Control - STATISTICS
Lord Sugar (formerly Sir Alan Sugar), has
kindly given you the opportunity to go into
business with him. All you have to do is to
set up a test company for making the latest
in children’s groovy fish mobiles….
Dodgy fish = money lost. Set up a quality control system
to enable you to estimate how much of your production
run will be wasted due to poor manufacture. Then DO
SOMETHING ABOUT IT!
Presentation
Hired or Fired? Lord Sugar will decide …on the basis of your final presentation!
Mechanics – Moments and Centre of Mass
Manufacture - MECHANICS
Make a semicircular-tailed fish and a
triangular-tailed fish.
THE FISH MUST BALANCE
HORIZONTALLY WHEN SUSPENDED
1
28/07/2011
Finding the centre of mass of your fish tail… hang it from a
point and use a plumb line:
Centre of mass much closer to the bar than the athlete’s body, due
to the shape of the body as it passes over the bar: the “Fosbury
Flop”. [The centre of mass can even pass UNDER the bar!]
Some theory ….
BALANCING FISH
Magic Moments ….
I need to balance horizontally when suspended from point X.
X
Distance
Distance
? cm
? cm
Weight
Weight
A moment is a product. For see‐saw calculations, it is the product of weight and distance of centre of mass from the pivot axis.
To balance, the moments must be equal.
.. but my head is TOO BIG!
2
28/07/2011
A little more theory ….
BALANCING FISH
For flat shapes, the weight is represented by the AREA.
So the MOMENT is the product of area and distance of centre of mass from the pivot axis.
I also need to balance horizontally when suspended from point X…
Pivot axis
Centre of Mass
Centre of Mass
X
Distance T
? cm
? cm
Area
Distance C
Area
T
(Weight of triangle)
.. but my head also is TOO BIG!
A little more theory ….
For equal moments:
C
(Weight of circle)
Area
T
x Distance T = Area
C
x Distance C
BALANCING FISH
For flat shapes, the weight is represented by the AREA.
So the MOMENT is the product of area and distance of centre of mass from the pivot axis.
For Printing
Pivot axis
Centre of Mass
Centre of Mass
Distance C
Distance S
Area
Area
S
C
(Weight of circle)
(Weight of semicircle)
For equal moments:
Area
S
x Distance S = Area
C
x Distance C
3
28/07/2011
BALANCING FISH
HELPSHEET FOR CALCULATIONS:
TRIANGLE
For printing
Distance of Centre of Mass of TRIANGLE from join
Area of TRIANGLE
Moment of TRIANGLE from join
Distance of Centre of Mass of NEW CIRCULAR HEAD from join
Area of NEW CIRCULAR HEAD
Moment of NEW CIRCULAR HEAD
Moments Equation
Radius of NEW CIRCULAR HEAD
HELPSHEET FOR CALCULATIONS:
SEMICIRCLE
SOLUTIONS
Distance of Centre of Mass of SEMICIRCLE from join
Area of SEMICIRCLE
Moment of SEMICIRCLE from join
Distance of Centre of Mass of NEW CIRCULAR HEAD from join
Area of NEW CIRCULAR HEAD
Moment of NEW CIRCULAR HEAD
Moments Equation
Radius of NEW CIRCULAR HEAD
4
28/07/2011
The Mathematical Marine Mobile Company
Manufacture - MECHANICS
Make a semicircular-tailed fish and a triangular-tailed fish.
THE FISH MUST BALANCE HORIZONTALLY WHEN SUSPENDED
The first stage of your task, the manufacture of prototype fish, will be
explained to you. But here are the key points to remember:
- Find the centre of mass (COM) of the semicircle and triangle using the
paper clip, string and weight.
- Measure how far the COM is from “join” of the fish for both the triangle and
semicircle.
- Calculate the AREA of the triangular tail and semicircular tail.
- Use MOMENTS to work out the radius of the “head” of the fish.
- Using compasses and scissors, cut out the correct size of head for both fish,
making sure you keep the head attached to the tail.
- Check using the paper clip that your fish balances.
SUMMARY
The Mathematical Marine Mobile Company
Making a Profit – DECISION MATHEMATICS
Use the information you have been given to
decide how many of each type of fish to
make in order to maximise your profits
Making a Profit – DECISION MATHEMATICS
Use the information to decide how many of each type of fish
to make for your test manufacture in order to maximise your
profits when you actually go into business.
You are given the following information:
S = number of semicircular tailed fish to make
T = number of triangular tailed fish to make
.
Each S fish uses 0.1 kg of wood. Each T fish uses 0.05kg of wood. You can buy up to
20 kg of wood to make the fish. [“Wood Constraint”]
It costs £0.40 to make each S fish. It costs £0.50 to make each T fish. Lord Sugar has
given you up to £150 to spend on materials. [“Cost Constraint”]
It takes on average 8 minutes to make each S fish. It takes on average 5 minutes to
make each T fish. You have up to 3600 minutes of labour time available.
[“Time Constraint”]
5
28/07/2011
A graph showing the “Wood Constraint” and the “Time Constraint” is given on the
next page. But the “Cost Constraint” has not yet been included.
400
Draw an extra line on the graph which represents the “Cost Constraint”.
T
WOOD
It has been estimated that each S fish used in the final mobile will contribute
£0.65 of profit, and each T fish will contribute £0.50 of profit.
300
Use your graph to decide how many of each type of fish your company
should make.
200
What is the maximum profit you can make?
100
TIME
S
100
400 T
200
300
400
Quality Control - STATISTICS
SOLUTION:
P = S x £0.65 + T x £0.50
D
A
B
C
D
300
C
B
Dodgy fish = money lost. Set up a quality control system
to enable you to estimate how much of your production
run will be wasted due to poor manufacture. Then DO
SOMETHING ABOUT IT!
(200, 0) P = £130
(100, 200) P = £165
(75, 240) P = £168.75 MAX PROFI T
(0, 300) P = £150
So make 75 semicircular- tailed fish and
240 triangular- tailed fish
to get maximum profit of £168.75
200
100
A
100
200
S
300
400
6
28/07/2011
A “Normal Distribution” curve, from which probabilities can be calculated.
Quality Control - STATISTICS
NOTE: It is the AREA under the curve which gives the probability for a given
range of degrees
Dodgy fish = money lost. Set up a quality control system
to enable you to estimate how much of your production
run will be wasted due to poor manufacture. Then DO
SOMETHING ABOUT IT!
0.2
The graph on the next page is the very well known Normal Distribution. It shows
how likely it is that the fish will tilted from the horizontal (in degrees: positive
numbers are “clockwise” tilts; negative are “anticlockwise tilts.) The key to
calculating using a Normal Distribution is that the AREA under the graph gives the
probability of a fish having a tilt in a given range.
0.15
0.1
When manufacturing your mobile, any fish which has a tilt of more than 5
degrees is rejected.
Use the graph to ESTIMATE the probability that a fish will be rejected. What
is the probability that a fish will NOT be rejected?
Each finished mobile in fact uses 8 fish. What is the probability that ALL the
fish in the completed mobile will pass the quality control test?
0.05
degrees
-8
-6
-4
-2
2
4
6
8
The highlighted region is magnified on the next page
SOLUTIONS
A = (0.0088 + 0.0022) x 1 / 2
0.02
0.02
B = (0.0022 + 0.0004) x 1 / 2
0.015
C = (0.0004 x 1)/2
0.015
0.01
0.005
0.01
A
0
6
B
degrees
8
C
Total in each tail = 0.007 (This could be done by counting “squares”, or
using one big triangle)
0.005
5
7
degrees
0
5
6
7
8
so probability tilt > 50 = 0.014
Probability tilt < +/- 50 = 0.986
Probability 8 good fish = 0.9868 = 0.89 or 89%
7