Transcript Situation: Match Stick Stairs  By

Situation:
Match Stick Stairs
By
(Cor)2an
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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
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
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Make a Chart
Squares
Match Sticks
1
4
2
7
3
10
4
13
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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
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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
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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
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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:
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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
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)
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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
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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
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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
4
2
10
3
18
4
28
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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)
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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
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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
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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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