The Open University
Maths Dept
Promoting Mathematical Thinking
University of Oxford
Dept of Education
Only Connect:
who makes connections when,
and how are they actually made?
John Mason
Poole
June 2010
1
Outline
 Topics
as Connections
 Themes as Connections
 Powers as Connections
2
Crossed Ladders
 In
an alleyway there is a ladder from the base of
one wall to somewhere on the opposite wall, and
another the other way, reaching to heights 3m
and 4m respectively on the opposite walls. Find
the height of the crossing point.
 Find the position of the crossing point.
3
Crossed Ladders Solution
4
3
h
h

h
4
3
1
1
4

3
Dimensions of
possible
variation
1 1
1
 
a b
h
1

1
h
At point dividing width in ratio
1 1
:  4 :3
3 4
4
Harmonic or
Parallel Sum
Fraction Folding
 Take
a rectangular piece of paper.
 Fold it in half parallel to one edge,
make a crease and unfold.
 Fold along a diagonal; make a crease.
 Now fold it along a line from one corner
to the midpoint of a side it is not already
on; make a crease.
 Note the point of intersection of the
diagonal crease and the last crease.
Fold along a line through this point
parallel to an edge.
In what fraction have the edges been divided
by this last crease?
What happens if you repeat this operation?
5
Couriers
A
courier sets out from one town to go to another
at a certain pace; a few hours later the message
is countermanded so a second courier is sent out
at a faster pace … will the second overtake the
first in time?
 Meeting Point
– Some people leave town A headed for town B
and at the same time some people leave town
B headed for town A on the same route. They
all meet at noon, eating lunch together for an
hour. They continue their journeys. One group
reaches their destination at 7:15 pm, while the
other group gets to their destination at 5pm.
When did they all start? [Arnold]
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Meeting Point Solution
 Draw
a graph!
B
Distance
from
A
Dimensions of
possible
variation?
A
time
7
Cistern Filling
A
cistern is filled by two spouts, which can fill the
cistern in a and b hours respectively working
alone. How long does it take with both working
together?
a
b
time
8
Dimensions of
possible
Crossed Planes
 Imagine
three towers not on a straight line
standing on a flat plain.
 Imagine a plane through the base of two towers
and the top of the third;
 and the other two similar planes.
– They meet in a point.
 Imagine a plane through the tops of two towers
and the base of the third;
 and the other two similar planes
– They meet in a point
 The first is the mid-point between the ground and
the second.
9
Tower Diagrams
10
Remainders & Polynomials
 Write
down a number that
leaves a remainder of 1
on dividing by 2
 and also a remainder of 2
on dividing by 3
 and also a remainder of 3
on dividing by 4
 Write
down a polynomial
that takes the value 1
when x = 2
 and also takes the value
2 when x = 3
 and also takes the value
5 when x = 4
1
1
1 + 2n
1 + 2(2 + 3n)
1 + 2(2 + 3(1 + 4n))
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1   x  2  p( x )
1   x  2 1   x  3  p ( x ) 


1   x  2  1   x  3 1   x  4  p ( x ) 
Combining Functions (animation)
Making mathematical sense of
phenomena
Using coordinates to read graphs
Getting a sense of composite functions
Generating further exploration
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Combining Functions (Dynamic
Geometry)
Making mathematical
sense of phenomena
Using coordinates to read
graphs
Getting a sense of composite
functions
Generating further exploration
13
Cobwebs (1)
14
Cobwebs (2)
15
Cobwebs (3)
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Generating Functions
f  x   2 x  3 g  x   3x  2
 What
functions can you
make by composing f and
g repeatedly?
f x   x  1
 What
g x   x
2
functions can you
make by composing f and
g repeatedly?
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2x+3, 3x+2, 4x+9, 6x+7,
6x+11, 8x+21, 9x+8,
12x+17,
12x+25, 12x+29, 16x+45,
18x+19, 18x+23, 18x+35,
24x+37, 24x+53, 24x+61,
24x+65, 27x+26, 36x+41,
36x+49, 36x+73, 36x+53,
36x+77, 36x+89, 54x+55,
54x+59, 54x+71, 54x+107,
81x+80
One More

What numbers are one more than the product of
four consecutive integers?
Specialising
in order to locate structural
relationships
and then to generalise
Let a and b be any two numbers, one of them even.
Then ab/2 more than the product of any number, a
more than it, b more than it and a+b more than it, is
a perfect square, of the number squared plus a+b
times the number plus ab/2 squared.
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Characterising
 How
many moves in Leapfrogs if there are a pegs
on one side and b pegs on the other?
 What is the largest postage you cannot make
with two stamps of value a and b (no common
divisor).
 What numbers can arise as
ab + a + b? as ab – a – b?
 Answer: numbers 1 less than the product of two
numbers
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Original Tangents
what point does the tangent to ex pass through
the origin?
 At what point does the tangent to e2x pass
through the origin?
 Generalise!
Dimensions of Possible
 At
Variation
Range of Permissible Change
What is the locus of points at
which the tangent to f(λx)
passes through the origin?
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What is the locus of points at
which the tangent to μf(x)
passes through the origin?
Vecten
 Imagine
a triangle
 Put squares on each side,
outwards
 Complete the outer hexagon
How do the triangles
compare?
Area by sines
Area by rotation
Extension
s?
21
Cosines
22
Mathematical Themes
 Invariance
in the midst of change
 Doing & Undoing
 Freedom & Constraint
 Extending & Restricting Meaning
23
Invariance in the Midst of Change
 Angle
sum of a triangle
 Circle Theorems
 Identities (a + b)2
– (a2 + b2)(c2 + d2) = (ac + bd)2 + (ad – bc)2;
 Scaling
& translating a distribution
 Sum of an AP or GP
 Area formulae
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Doing & Undoing
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Multiplying numbers
Factoring
Expanding brackets
Factoring
Differentiating
Integrating
Substituting
Solving
Adding fractions
Partial Fractions
Given triangle edge
lengths, find medians
Given median lengths,
find edge lengths
Natural Powers
 Imagining
& Expressing
 Specialising & Generalising
 Conjecturing & Convincing
 Organising & Characterising
 Stressing & Ignoring
 Distinguishing & Connecting
 Assenting & Asserting
26
Connections
 Who
makes them?
 When are they made?
 How are the made?
 How are they prompted or supported or
scaffolded?
27
After Thought
What fraction of the
unit square is shaded?
Anything come to
mind?
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Further Thoughts
 Website:
mcs.open.ac.uk/jhm3
 Thinking Mathematically (2nd edition) Pearson
 Questions & Prompts for Mathematical Thinking (ATM)
 Counter-Examples in Calculus College Press
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