Situation: Match Stick Stairs By (Cor)2an 1 A Square Match Stick Unit • Suppose a square match stick unit is defined to be a square with one match stick per side. 2 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. 3 Problem • How many match sticks would you need to create a track of 500 square match stick units? 4 What Problem Solving Strategies Can You Try? • • • • • Simplify the problem. Make a table. Look for a pattern. Make a generalization. Describe a function. 5 Simplify the Problem • How take • How take • How take • How take many match sticks does to make 1 square? many match sticks does to make 2 squares? many match sticks does to make 3 squares? many match sticks does to make 4 squares? it it it it 6 Make a Chart Squares Match Sticks 1 4 2 7 3 10 4 13 7 Look for a Pattern: Deconstruct the Information in the Chart Squares Match Sticks 1 4=4 2 7=4+3 =4+3(1) 10=4+3+3 =4+3(2) 13=4+3+3+3 =4+3(3) 3 4 8 Make a Generalization Squares Match Sticks 1 4=4 2 7=4+3 =4+3(1) 10=4+3+3 =4+3(2) 13=4+3+3+3 =4+3(3) 4+3(n-1) 3 4 n 9 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. 10 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 11 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. 12 Situation • Stairs made with matches: 13 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. 14 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 ? 15 Maybe Draw a Table: Base Length (b) Number of Squares (s) 1 1 2 3 3 6 4 10 16 Look for patterns: Base Length (n) Number of Squares (s) 1 1 2 3 (1 + 2) 3 6 (1 + 2 + 3) 4 10 (1 + 2 + 3 + 4) 17 Steps to follow: THREE Conjecture/Generalisation Base Length (n) Number of Squares (s) 1 1 (1 x 2) 2 2 3 (2 x 3) 2 3 6 (3 x 4) 2 n n(n+1)2 or (n2+n)/2 18 Steps to follow: FOUR Justify or Prove for all cases. Base Length (n) 1 Number of Squares (s) s(n) = (n2+n)/2 1 (1 + 1) 2 2 3 (4 + 2) 2 3 6 (9 + 3) 2 10 55 (100 + 10) 2 19 Steps to follow: FOUR Justify or Prove for all cases. 20 Another Table: Base Length (b) Number of Matches (m) 1 4 2 10 3 18 4 28 21 Look for patterns: Base Length (b) Number of Matches (m) 1 4 (1 + 3 x 1) 2 10 (4 + 3 x 2) 3 18 (9 + 3 x 3) 4 28 (16 + 3 x 4) 22 Steps to follow: THREE Conjecture/Generalisation Base Length (b) Number of Matches (m) 1 4 (1 + 3 x 1) 2 10 (4 + 3 x 2) 3 18 (9 + 3 x 3) 4 28 (16 + 3 x 4) b b2 + 3b 23 Steps to follow: FOUR Justify or Prove for all cases. Base Length (b) 1 Number of Matches (m) m(b) = b2 + 3b 12 + 3 = 4 2 22 + 6 = 10 3 32 + 9 = 18 4 42 + 12 = 28 6 62 + 18 = 54 24 Steps to follow: FOUR Justify or Prove for all cases. 25 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. 26 Steps to follow: FIVE Written Report • Strategies explored. • Data representation. Tables, graphs, diagrams. • Generalisations and/or mathematical formulae. • Justification. • Logical, Neat, Clear & Concise. 27 28
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