1 Thinking Algebraically & Geometrically John Mason University of Iceland Reykjavik 2008

2 Remainders of the Day (1)  Write down a number which when you subtract 1 is divisible by 7  and another  Write down one which you think no-one else here will write down.

3 Remainders of the Day (2)  Write down a number which is 1 more than a multiple of 2  and 1 more than a multiple of 3  and 1 more than a multiple of 4 …………

4 Remainders of the Day (3)  Write down a number which is 1 more than a multiple of 2  and which is 2 more than a multiple of 3  and which is 3 more than a multiple of 4 …………

5 Remainders of the Day (4)  Write down a number which when you subtract 1 is divisible by 2  and when you subtract 1 from the quotient, the result is divisible by 3  and when you subtract 1 from that quotient the result is divisible by 4  Why must any such number be divisible by 3?

6 Some Sums = Generalise Justify Watch What You Do Say What You See = = =

7 Cubelets Say What You See

8 Differences Anticipating Generalising Rehearsing Checking Organising

9 Word Problems In 26 years I shall be twice as old as I was 19 years ago. How old am I? ?=?2( - 19) ?26 ? 19 ? =

10 Mid-Point  Where can the midpoint of the segment joining two points each on a separate circle, get to?

11 Scaling P Q M Imagine a circle C. Imagine also a point P. Now join P to a point Q on C. Now let M be the mid point of PQ. What is the locus of M as Q moves around the circle?

12 Map Drawing Problem  Two people both have a copy of the same map of Iceland.  One uses Reykjavik as the centre for a scaling by a factor of 1/3  One uses Akureyri as the centre for a scaling by a factor of 1/3  What is the same, and what is different about the maps they draw?

13 Difference Divisions 4 – 2 = 4 ÷ 2 4 – 3 = 4 ÷ – 4 = 5 ÷ – 5 = 6 ÷ – 6 = 7 ÷ – 2 = 3 ÷ – (-1) = 0 ÷ (-1) oops 1 – 0 = 1 ÷ oops 1 1 How does this fit in? Going with the grain Going across the grain

14 Four Consecutives  Write down four consecutive numbers and add them up  and another  Now be more extreme!  What is the same, and what is different about your answers?

15 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.

16 Gasket

17 Leibniz’s Triangle 1

18 How Much Information?  How few rectangles needed to compose it?  Design a rectilinear region requiring –3 lengths to find the perimeter and –8 lengths to find the area  How few rectangles needed to compose it?  How much information about lengths do you need in order to work out –the perimeter? –the area?

19 More Or Less Perimeter & Area moresameless more same less are a Perimeter same perim more area more perim same area more perim more area less perim more area less perim less area more perim less area same perim less area less perim same area Draw a rectilinear figure which requires at least 4 rectangles in any decomposition Dina Tirosh & Pessia Tsamir

20 Two-bit Perimeters 2a+2b What perimeters are possible using only 2 bits of information? a b

21 Two-bit Perimeters 4a+2b What perimeters are possible using only 2 bits of information? a b

22 Two-bit Perimeters 6a+2b What perimeters are possible using only 2 bits of information? a b

23 Two-bit Perimeters 6a+4b What perimeters are possible using only 2 bits of information? a b

24 Parallelism How many angles do you need to know to work out all the angles?

25 Kites

26 Seven Circles How many different angles can you discern, using only the red points? How do you know you have them all? How many different quadrilaterals?

27 Square Count

28 Ratios in Rectangles

29 Some Mathematical Powers  Imagining & Expressing  Specialising & Generalising  Conjecturing & Convincing  Stressing & Ignoring  Ordering & Characterising  Distinguishing & Connecting  Assenting & Asserting

30 Some Mathematical Themes  Doing and Undoing  Invariance in the midst of Change  Freedom & Constraint

31 Worlds of Experience Material World World of Symbol s Inner World of imagery enactiveiconicsymbolic

32 Structure of the Psyche Imagery Awareness (cognition) Will Body (enaction) Emotions (affect) Habits Practices

33 Structure of a Topic Language Patterns & prior Skills Techniques & Incantations Different Contexts in which likely to arise; dispositions Root Questions predispositions Imagery/Sense- of/Awareness; Connections Standard Confusions & Obstacles Only Behaviour is Trainable Only Emotion is Harnessable Only Awareness is Educable BehaviourBehaviour EmotionEmotion AwarenessAwareness

34 Mathematics & Creativity  Creativity is a type of energy  It is experienced briefly  It can be used productively or thrown away  Every opportunity to make a significant choice is an opportunity for creative energy to flow  It also promotes engagement and interest  For example –Constructing an object subject to constraints –Constructing an example on which to look for or try out a conjecture –Constructing a counter-example to someone’s assertion