Shading Fractions

1
Shading Fractions

• By Steph Walters and
• Phillipa Rogers

2
Question

• How many ways can you shade in (3/8) three
  eighths of a 2 x 4 rectangle?

3
Is there a systematic shading strategy?

• We used a few different ways to calculate all the
  possibilities.

• Method 1 was to shade in the different options on
  grid paper.

• Method 2 was to apply transformation geometry to
  various shapes and find all possibilities from
  one shape.

• Method 3 was to substitute numbers into the
  shaded boxes and find the different options
  systematically.

• Method 4 was to use a formula to find the answer.

4
Method 1

• To do this successfully a strategy must be used
  to ensure all possibilities are found.

• We did this by grouping 3 boxes together as
  shown here.

• Then grouping 2 boxes together.

• Finally having all separate.

5
Method 1

• Using this method in Excel we found 54 options.

• This method proved to be difficult to follow and
  difficult to determine if all of the variables
  have been found. This is due to possible human
  errors.

• So to determine if all were found other methods
  were used.

6
Method 2

• Method 2 involved taking one shape and applying
  transformation geometry, which involves
  reflection and rotation.

• This is an example of one shape and all of its
  possibilities

• These options could be considered as one shape.

7
Method 2

• 13 options were found by using transformation
  geometry. However with this method again it is
  uncertain that all options are found.

• This however was unsuccessful as when rotating
  and reflecting the process became mixed up and a
  few of the same patterns resulted. This is also
  a result of human error.

8
Method 3

• Method 3 uses the coding system as follows

• Using these numbers we calculated all the
  possible combinations of 3 numbers from 1 8.

• The result showed that each number appears in the
  combination 21 times. So 21 must be multiplied by
  8 to get a total. However because there is a
  combination of 3 numbers it must be divided by 3.

• This gives a total of 56.

9
Method 3

• Because this method is the most consistent we
  presumed this to be correct, we also checked the
  other methods with this formula and realised that
  the other methods were not consistent and some
  patterns had been repeated while other not used
  at all.

• The sheets being passed around show all of the
  possible options for method 3.

10
Method 4

• We found this formula because the question can be
  linked to probability. The equation can be used
  for this question and also for questions relating
  to probability.

• The formula is as follows
• __c!__
• r! (c-r)!
• c Number of total cells
• r Being shaded
• ! Function on calculator

11
Method 4

• Using this formula results in 56 options for
  shading in 3/8 of a 2 x 4 rectangle.

• This is consistent with the result from method 4,
  so it could be concluded that there is 56 ways to
  shade in 3/8 of a 2 x 4 rectangle.

• This method is more reliable than method 3
  because it is more flexible, any size and any
  fraction can be accommodated into the equation.

12
Conclusion

• Through the use of four different methods it can
  be concluded that a purely mathematical formula
  produces the most accurate result. There are
  endless combinations for shading options and
  rectangle sizes but due to time constraints it
  was only possible to look at four different
  options for the one sized rectangle.