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1 Generalisation: Fostering & Supporting Algebraic Thinking John Mason Trondheim Oct 2007

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2 Assumptions Generalisation lies at the very core of mathematics and mathematical thinking A lesson without the opportunity for learners to generalise … is not a mathematics lesson!

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3 What’s The Difference? What could be varied? –= First, add one to each First, add one to the larger and subtract one from the smaller What then would be the difference?

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4 Think Of A Number (Thoan) intrigues adolescents Displays power over numbers Introduces a device for dealing with as-yet- unknown numbers

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5 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? + 1 + 2 + 3 + 6 4

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6 Powers Imagining & Expressing Specialising & Generalising Conjecturing & Convincing Classifying & Characterising Fixing & Changing Stressing & Ignoring Attending & Intending

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7 Pattern Continuation … …

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8 Experiencing Generalisation Pleasure in use of powers; disposition: affective generalisation (Helen Drury) Going across the grain: cognitive generalisation Going with the grain: enactive generalisation

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9 Raise Your Hand When You See … Something which is 2/5 of something; 3/4 of something; 5/2 of something; 4/3 of something; 3/4 of 2/5 of something; 3/4 of 4/3 of something; 1 ÷ 2/5 of something; 1 ÷ 3/4 of something

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10 CopperPlate Multiplication

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11 Paper Folding Shape?

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12 What Would Happen If …? The tap wasn’t turned off It never rained The power went off A nearby stream flooded You kept on cutting a piece of paper in half …………

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13 One More What numbers are one more than the sum 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. What numbers are one more than the product of four consecutive integers?

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14 Perforations How many holes for a sheet of r rows and c columns of stamps? If someone claimed there were 228 perforations in a sheet, how could you check?

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16 Consecutive Sums Say What You See

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

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18 Remainders of the Day (1) Write down a number which when you subtract 1 is divisible by 5 and another Write down one which you think no-one else here will write down.

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19 Remainders of the Day (2) 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?

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20 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 … … … …

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21 Remainders of the Day (4) 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 … … … …

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22 Four Odd Sums

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23 Slope Reading

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24 Cutting Chocolate Bars In how few cuts can you separate the bar into its pieces? You can only cut one piece at a time!

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