Key Stage 3: age 11-13
Links to the National
Curriculum and Numeracy Strategy
- Represent problems and solutions in algebraic or graphical forms.
- Describe the general term of a simple sequence in words, then using symbols.
Duration
Two hours
Learning Objectives
Pupils will:
- select appropriate stategies to use for a numerical and algebraic problem.
- represent problems and solutions in algebraic form.
- find and describe in symbols the next term or nth term of a sequencewhere the rule is linear.
Resources required
Matchsticks (straws cut into matchstick size) and/or access to Matchstick
Patterns
Squared
paper
Learning activity
Triangular matchstick patterns.
Using matchsticks or straws pupils make the following growing patterns
How many matchsticks would be needed to make 20 triangles?......100 triangles?
A table of results can be drawn
|
Number
of triangles
t |
Number
of matchsticks
m |
|
1
|
3
|
|
2
|
5
|
|
3
|
7
|
|
4
|
9
|
The pupils might notice
the following.
Two more matchsticks are needed for each triangle. This sequence is generated
3, 5, 7, 9, 11.......
Each triangle needs two more matchsticks except the first triangle which
needs an extra one.
A general rule for the number of matchsticks can be generated:m = t x 2 + 1 or m = 2t + 1
Therefore 20 triangles
will need 20 x 2 + 1 = 41 matchsticks.
Therefore 100 triangles
will need 100 x 2 + 1 = 201 matchsticks.
Inverse functions
Use function machines to generate the inverse function:
t = (m-1)/2
Extension
Investigate other growing
matchstick patterns: growing matchstick squares:
|
Number
of squares
s |
Number
of matchsticks
m |
|
1
|
4
|
|
2
|
7
|
|
3
|
10
|
|
4
|
13
|
The pupils might notice
the following.
Three more matchsticks are needed for each square. This sequence is generated
4, 7, 10, 13, 16.......
Each square needs three more matchsticks except the first square which
needs an extra one.
A general rule for the number of matchsticks can be generated:m = s x 3 + 1 or m = 3s + 1
Therefore 20 squares
will need 20 x 3 + 1 = 61 matchsticks.
Therefore 100 squares
will need 100 x 3 + 1 = 301 matchsticks.
Use function machines
to generate the inverse function:
s = (m-1)/3
Move on to pentagons and hexagons. The formulae are m = 4p + 1 and m = 5h + 1
It is now possible to generate a rule for any growing polygon matchstick pattern.
If s is the number of sides of the polygon, n is the number of polygons and m is the number of matchsticks:
m =(s-1) x n +1 or m =n(s-1) +1