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Thinking Algebraically & Geometrically John Mason University of Iceland Reykjavik 2008 1 Remainders of the Day (1) Write down a number which when you subtract 1 is divisible by 7 and another and another Write down one which you think no-one else here will write down. 2 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 … 3 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 … 4 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? 5 Some Sums 1+2= 3 4+5+6= 7+8 = 13 + 14 + 15 9 + 10 + 11 + 12 16 + 17 + 18 + 19 + 20 = 21 + 22 + 23 + 24 Generalise Say What You See Justify Watch What You Do 6 Cubelets Say What You See 7 Differences 8 1 1 1 1 1 1 7 6 42 2 1 2 1 11 1 1 1 1 1 11 3 2 6 8 7 56 6 24 4 8 Anticipating 1 1 1 1 1 Generalising 4 3 12 2 4 Rehearsing 1 1 1 5 4 20 Checking 1 1 1 1 1 1 1 1 1 Organising 6 5 30 2 3 3 6 4 12 Word Problems In 26 years I shall be twice as old as I was 19 years ago. How old am I? 40 + 26 ?=? = 2( 40- 19) ? ? 26 19 ? 19 9 Mid-Point Where can the midpoint of the segment joining two points each on a separate circle, get to? 10 Scaling Q Imagine a circle C. Imagine also a point P. M 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? 11 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? 12 0 Difference Divisions 1 2 4–2=4÷2 1 2 –3=4 5 1 3 –4=5 ÷1 4 6 1 4 –5=6 1 ÷ 4 7 1 5 –6=7 ÷1 6 4 13 1 ÷ 2 3 1 – (-1) = -2 1 –0=1 -1 1 oops 1 –2=3 1 1 0 ÷-2(-1) 1 ÷-1 oops 1 ÷ 1 2 3 How does this fit in? 3 5 5 Going with the grain Going across the grain Four Consecutives Write down four consecutive numbers and add them up and another and another Now be more extreme! What is the same, and what is different about your answers? +1 +2 +3 4 14 +6 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. 15 Gasket 16 Leibniz’s Triangle 1 1 2 1 3 1 4 1 5 1 6 17 1 6 1 12 1 20 1 30 1 2 1 3 1 12 1 30 1 60 1 4 1 20 1 60 1 5 1 30 1 6 How Much Information? How much information about lengths do you need in order to work out –the perimeter? –the area? How few rectangles needed to compose it? Design a rectilinear region requiring 18 – 3 lengths to find the perimeter and – 8 lengths to find the area More Or Less Perimeter & Area Draw a rectilinear figure which requires at least 4 rectangles in any decomposition are Perimetera more 19 more more perim more area same same perim more area less less perim more area same less more perim more perim same area less area same perim less area less perim same area less perim less area Dina Tirosh & Pessia Tsamir Two-bit Perimeters What perimeters are possible using only 2 bits of information? 2a+2b a 20 b Two-bit Perimeters What perimeters are possible using only 2 bits of information? 4a+2b a 21 b Two-bit Perimeters What perimeters are possible using only 2 bits of information? 6a+2b 22 a b Two-bit Perimeters What perimeters are possible using only 2 bits of information? 6a+4b 23 a b Parallelism How many angles do you need to know to work out all the angles? 24 Kites 25 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? 26 Square Count 27 Ratios in Rectangles 28 Some Mathematical Powers Imagining & Expressing Specialising & Generalising Conjecturing & Convincing Stressing & Ignoring Ordering & Characterising Distinguishing & Connecting Assenting & Asserting 29 Some Mathematical Themes Doing and Undoing Invariance in the midst of Change Freedom & Constraint 30 Worlds of Experience Inner World of imagery World of Symbol s Material World enactive 31 iconic symbolic Structure of the Psyche Awareness (cognition) Imagery Will Emotions (affect) Body (enaction) Habits Practices 32 Structure of a Topic Language Patterns & prior Skills Imagery/Senseof/Awareness; Connections Root Questions predispositions Different Contexts in which likely to arise; dispositions Standard Confusions & Obstacles Techniques & Incantations Emotion Only Emotion is Harnessable Only Awareness is Educable 33 Only Behaviour is Trainable 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 34