Shading fractions

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Shading fractions
Question How many ways can you shade three eights
(3/8) of a 4x2 Rectangle
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Our systematic shading strategy

• We found four strategies useful to us in our
  investigation
• Strategy 1 involved manually shading all
  possibilities on grid paper
• Strategy 2 was to substitute letters eg
  a,b,c,d,etc to replace shading the squares as a
  formula to assist us in our investigation
• Our 3rd strategy involved a formula we found
• Our 4th strategy is using Pascals triangle

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Strategy 1

• To ensure all possibilities are discovered a
  strategy must be used
• We did this by first randomly shading any three
  separate squares

• We found that this strategy was unorganised and
  difficult to find all the possibilities for
  shading the squares
• We abandoned using this strategy as it was not
  orderly and we thought that an orderly strategy
  was required if we were going to find the maximum
  number of ways we could shade the three eights of
  this 4x2 square

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Strategy 2

• Our second method involved us finding all the
  squares using a slightly more organised strategy.
• We decided that we were going to group two
  squares together and shade a third in other
  random squares

• We decided that this was more orderly than our
  first attempt at the shading the fraction
• However it was necessary to try the first
  strategy to discover this one

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Strategy 3

• Our third method was to group all three squares
  together and place them in different positions on
  the table

• We found that we had to combine all these three
  methods in our first strategy to be able to
  maximise the amounts of ways we could shade the
  squares
• We found using this method 52 ways of shading 3/8
  of a 4x2 square
• We believe that this method is inaccurate and
  that we had not found all the possibilities

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Strategy 1

• I found that numbers were easier to come up with
  a more precise answer to this question
• I found 52 ways to shade the 3/8 of the rectangle

I concluded that this method is inaccurate
because there could have been human error
involved using this method It was necessary to
try this strategy and fail to reach a solution so
that I was able to try other strategies
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Strategy 1

• I found that letters were easier to use as a
  substitute for shading
• I found 51 options for shading the rectangle
• This method is inaccurate due to human error

I decided that we would use excel to find other
possibilities so that we found a solution We did
this and we came up with 56 possible ways which
we believe is the solution
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Magic Formula

• We found this formula from second year primary
  teachers. The equation can be used for this
  question all you have to do is substitute in the
  variables.
• The formula is as follows
• __c!__
• r! (c-r)!
• c Number of total cells
• r Being shaded
• ! Function on calculator

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Pascals Triangle
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Pascals Triangle

• Due to a tip from Professor Bob Dengate, Dr
  McIntyre and Professor Lazarou have discovered
  the relevance of the eighth row of Pascals
  triangle.
• It has come to our attention that our solution
  lies on the eighth row
• Our solution that we have been trying to discover
  is the magical number 56
• Our time of need had ended as the Doc and I
  discovered a true solution

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Depth and Range

• Due to the fact that we have the formula to work
  out all the possibilities for shading fractions
  we found that it was not necessary to explore in
  a great depth and range changing the dimensions
  of the rectangle

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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, however we have found the
  formula so therefore we did not need to explore
  all these possibilities.