Transcript Match Stick Puzzles
Situation: Match Stick Stairs By
(Cor) 2 an 1
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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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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Problem • How many match sticks would you need to create a track of 500 square match stick units?
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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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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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Make a Chart 2 3 Squares 1 4 Match Sticks 4 7 10 13 7
Look for a Pattern: Deconstruct the Information in the Chart Squares Match Sticks 1 2 3 4 4=4 7=4+3 =4+3(1) 10=4+3+3 =4+3(2) 13=4+3+3+3 =4+3(3) 8
3 4 Make a Generalization Squares 1 2 n Match Sticks 4=4 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) 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.
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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.
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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.
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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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Maybe Draw a Table: Base Length (b) Number of Squares (s) 1 2 3 4 1 3 6 10 16
1 2 3 4 Look for patterns: Base Length (n) Number of Squares (s) 1 3 (1 + 2) 6 (1 + 2 + 3) 10 (1 + 2 + 3 + 4) 17
1 2 3 n Steps to follow: THREE Conjecture/Generalisation Base Length (n) Number of Squares (s) 1 (1 x 2)
2 3 (2 x 3)
2 6 (3 x 4)
2 n(n+1)
2 or (n 2 +n)/2 18
1 2 3 10 Steps to follow: FOUR Justify or Prove for all cases.
Base Length (n) Number of Squares (s) s(n) = (n 2 +n)/2 1 (1 + 1)
2 3 (4 + 2)
2 6 (9 + 3)
2 55 (100 + 10)
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Steps to follow: FOUR Justify or Prove for all cases.
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Another Table: Base Length (b) Number of Matches (m) 1 2 3 4 4 10 18 28 21
Look for patterns: Base Length (b) Number of Matches (m) 1 2 3 4 4 (1 + 3 x 1) 10 (4 + 3 x 2) 18 (9 + 3 x 3) 28 (16 + 3 x 4) 22
1 2 3 4 b Steps to follow: THREE Conjecture/Generalisation Base Length (b) Number of Matches (m) 4 (1 + 3 x 1) 10 (4 + 3 x 2) 18 (9 + 3 x 3) 28 (16 + 3 x 4) b 2 + 3b 23
2 3 4 6 Steps to follow: FOUR Justify or Prove for all cases.
Base Length (b) 1 Number of Matches (m) m(b) = b 2 + 3b 1 2 + 3 = 4 2 2 + 6 = 10 3 2 + 9 = 18 4 2 + 12 = 28 6 2 + 18 = 54 24
Steps to follow: FOUR Justify or Prove for all cases.
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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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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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