1 Only Connect: who makes connections when, and how are they actually made? John Mason Poole June 2010 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

2 Outline  Topics as Connections  Themes as Connections  Powers as Connections

3 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.

4 Crossed Ladders Solution Dimensions of possible variation Harmonic or Parallel Sum 4 3 h At point dividing width in ratio

5 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. In what fraction have the edges been divided by this last crease? What happens if you repeat this operation? In what fraction have the edges been divided by this last crease? What happens if you repeat this operation?  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.

6 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]

7 Meeting Point Solution  Draw a graph! B A time Distance from A Dimensions of possible variation?

8 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? time a b Dimensions of possible variation?

9 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.

10 Tower Diagrams

11 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 = n 1 + 2(2 + 3n) 1 + 2(2 + 3(1 + 4n))

12 Generating further exploration Getting a sense of composite functions Using coordinates to read graphs Combining Functions (animation) Making mathematical sense of phenomena

13 Combining Functions (Dynamic Geometry) Generating further exploration Getting a sense of composite functions Using coordinates to read graphs Making mathematical sense of phenomena

14 Cobwebs (1)

15 Cobwebs (2)

16 Cobwebs (3)

17 Generating Functions  What functions can you make by composing f and g repeatedly? 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

18 One More  What numbers are one more than the product of four consecutive integers? 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. Specialising in order to locate structural relationships and then to generalise

19 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

20 Original Tangents  At what point does the tangent to e x pass through the origin?  At what point does the tangent to e 2x pass through the origin?  Generalise! Dimensions of Possible Variation Range of Permissible Change What is the locus of points at which the tangent to f( λ x) passes through the origin? What is the locus of points at which the tangent to μf(x) passes through the origin?

21 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?

22 Cosines

23 Mathematical Themes  Invariance in the midst of change  Doing & Undoing  Freedom & Constraint  Extending & Restricting Meaning

24 Invariance in the Midst of Change  Angle sum of a triangle  Circle Theorems  Identities (a + b) 2 –(a 2 + b 2 )(c 2 + d 2 ) = (ac + bd) 2 + (ad – bc) 2 ;  Scaling & translating a distribution  Sum of an AP or GP  Area formulae

25 Doing & Undoing Multiplying numbers Factoring Expanding brackets Factoring Differentiating Integrating Substituting Solving Given triangle edge lengths, find medians Given median lengths, find edge lengths Adding fractions Partial Fractions

26 Natural Powers  Imagining & Expressing  Specialising & Generalising  Conjecturing & Convincing  Organising & Characterising  Stressing & Ignoring  Distinguishing & Connecting  Assenting & Asserting

27 Connections  Who makes them?  When are they made?  How are the made?  How are they prompted or supported or scaffolded?

28 After Thought What fraction of the unit square is shaded? Anything come to mind?

29 Further Thoughts  Website: mcs.open.ac.uk/jhm3  Thinking Mathematically (2 nd edition) Pearson  Questions & Prompts for Mathematical Thinking (ATM)  Counter-Examples in Calculus College Press