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.

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 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?

6

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.

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 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)



2 19

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

20

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.

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