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1 Situation: Match Stick Stairs By (Cor) 2 an

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2 A Square Match Stick Unit Suppose a square match stick unit is defined to be a square with one match stick per side.

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3 A Track of Square Match Stick Units A track of two square match stick units would look like this. A track of three square match stick units would look like this.

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4 Problem How many match sticks would you need to create a track of 500 square match stick units?

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5 What Problem Solving Strategies Can You Try? Simplify the problem. Make a table. Look for a pattern. Make a generalization. Describe a function.

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6 Simplify the Problem How many match sticks does it take to make 1 square? How many match sticks does it take to make 2 squares? How many match sticks does it take to make 3 squares? How many match sticks does it take to make 4 squares?

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7 Make a Chart SquaresMatch Sticks 14 27 310 413

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8 Look for a Pattern: Deconstruct the Information in the Chart SquaresMatch Sticks 14=4 27=4+3 =4+3(1) 310=4+3+3 =4+3(2) 413=4+3+3+3 =4+3(3)

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9 Make a Generalization SquaresMatch Sticks 14=4 27=4+3 =4+3(1) 310=4+3+3 =4+3(2) 413=4+3+3+3 =4+3(3) n4+3(n-1)

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10 Create a Function f(n)=4+3(n-1) or f(n)=3n+1 where n represents the number of squares and f(n) represents the number of match sticks.

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11 Solution to Problem It would take 1501 match sticks to create a track of 500 squares. f(n) = 3n+1 f(500) = 3x500+1 = 1501

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12 Steps in Investigating Understanding. What is the investigation asking? Strategies that lead to mathematical conjecture. How? Generalisations. What I have discovered? Justification. Prove it! Communication. Tell the world.

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13 Situation Stairs made with matches:

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14 Steps to follow: ONE Understanding. What things could we consider? Number of matches. Length of stair case. Height of stair case. Number of squares. Etc.

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15 Steps to follow: TWO Explore and begin to develop strategy. Let’s examine more closely links or patterns with numbers of matches. length of stair case, height of stair case, number of squares, Etc. Conjecture ?

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16 Maybe Draw a Table: Base Length (b) Number of Squares (s) 11 23 36 410

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17 Look for patterns: Base Length (n) Number of Squares (s) 11 23 (1 + 2) 36 (1 + 2 + 3) 410 (1 + 2 + 3 + 4)

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18 Steps to follow: THREE Conjecture/Generalisation Base Length (n) Number of Squares (s) 11 (1 x 2) 2 23 (2 x 3) 2 36 (3 x 4) 2 nn(n+1) 2 or (n 2 +n)/2

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19 Steps to follow: FOUR Justify or Prove for all cases. Base Length (n) Number of Squares (s) s(n) = (n 2 +n)/2 11 (1 + 1) 2 23 (4 + 2) 2 36 (9 + 3) 2 1055 (100 + 10) 2

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20 Steps to follow: FOUR Justify or Prove for all cases.

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21 Another Table: Base Length (b) Number of Matches (m) 14 210 318 428

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22 Look for patterns: Base Length (b) Number of Matches (m) 14 (1 + 3 x 1) 210 (4 + 3 x 2) 318 (9 + 3 x 3) 428 (16 + 3 x 4)

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23 Steps to follow: THREE Conjecture/Generalisation Base Length (b) Number of Matches (m) 14 (1 + 3 x 1) 210 (4 + 3 x 2) 318 (9 + 3 x 3) 428 (16 + 3 x 4) bb 2 + 3b

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24 Steps to follow: FOUR Justify or Prove for all cases. Base Length (b) Number of Matches (m) m(b) = b 2 + 3b 11 2 + 3 = 4 22 2 + 6 = 10 33 2 + 9 = 18 44 2 + 12 = 28 66 2 + 18 = 54

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25 Steps to follow: FOUR Justify or Prove for all cases.

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26 Match Stick Triangles ? Here is a triangle made of 3 match sticks. A track of two triangles looks like this. A track of three triangles looks like this.

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27 Steps to follow: FIVE Written Report Strategies explored. Data representation. Tables, graphs, diagrams. Generalisations and/or mathematical formulae. Justification. Logical, Neat, Clear & Concise.

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