By Toni Gardner

1
Maths Investigation

• By Toni Gardner
• Nicole Hancock

2
Investigation Question

• Make a strip of squares. Colour some whole
  squares black, leaving others white.

3
Formulas that can be applied

• Possible formulas that can be used to establish
  the amount of squares in the sequence are as
  follows
• Black squares white squares total squares
• B W T
• Total squares black squares white squares
• T B W
• Total squares white squares black squares
• T W B

4
Continuing the pattern

• The pattern presented here may continue
  indefinitely

This pattern can be made using the DRAPE program
with the following directions
2
10
(
)
DRAPE..\Desktop\DRAPE.exe
5

• There are a number of possibilities that can
  incorporate black and white squares.
• eg. using 2 black squares and 3 white squares
  the possibilities are done by adding 4 3 2 1

6
Possible arrangements

• We found that a pattern is formed when there is a
  constant variable of 2 black squares.
• When there is a total of 3 squares there are 3
  possible arrangements.
• When there is a total of 4 squares there are 6
  possible arrangements.
• When there is a total of 5 squares there are 10
  possible arrangements.
• When there is a total of 6 squares there are 15
  possible arrangements.
• The pattern is emerging here is that each time a
  square is added to the total the possible
  arrangements are increasing by the previous
  number of squares.

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This proves that the formula 4 3 2 1 10
combinations
8

• When all possibilities are explored with various
  black and white squares there are many
  possibilities, an example of this is on the
  following slide

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Possibilities
10
Further investigation

• With further investigation we discovered that the
  five squares presented could also be used as an
  introductory sample of tessellation the five
  squares could then be extended to further
  tessellations such as triangles and irregular
  shapes.
• Eg.

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Maths Games

• Games such as Battleships and Bingo can be
  played using the strips.
• Playing Battleships could be a great way to get
  younger students interested in investigating
  math's problems, learning about possibilities and
  realizing what can be done with strips.

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Web games on the net

• http//games.funschool.com/game.php?g1043
• http//www.primarygames.com/patterns/question1.htm
  
• http//www.learner.org/teacherslab/math/patterns/p
  eople/start.html
• http//nlvm.usu.edu/en/nav/frames_asid_184_g_1_t_1
  .html
• http//www.shodor.org/interactivate/activities/tes
  sellate/index.html

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Investigation Conclusion

• We have found out that
• Patterns can be continued indefinitely.
• Predictions of different colourings depends on
  the length of the strip. A clear pattern emerged
  as to how to obtain all possibilities using 2
  black squares.
• Patterns can be made a variety of combinations of
  black and white squares.
• This pattern can be used as a tool in the
  teaching of tessellation.

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• tgardn01_at_postoffice.csu.edu.au