Investigation Seminar Bounces

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Investigation SeminarBounces

• Adam Simpson Scott Holliday

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Bounces

• Imagine a rectangle on dot paper.
• Investigate the path of a ball which starts at
  one corner of the table, is pushed to an edge,
  bounces off that edge to another, and so on, as
  shown in the diagram. When the ball finally
  reaches a corner it drops off the table.

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Introduction

• Our investigation was to follow the path of a
  ball which starts at one corner of the pool table
  , is pushed to an edge, bounces off that edge to
  another, and so on.
• When the ball finally reaches a corner, it falls
  off the table.

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Assumptions

• At first look we assumed that this pool table had
  no middle pockets for the ball to enter as it
  only exited the table from the corner.
• We also assumed that we could make the pool table
  any size not just the traditional shape of the
  rectangle.
• We decided to keep the path of the ball as it
  hits each side of the table at 45 degrees.
• It was assumed that the speed of the ball was
  keep constant so the ball can hit the sides
  without slowing down until it exits the table.

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Initial Theories

• Some of our initial thoughts were
• That the ball would never hit the middle of the
  table
• If a side has a prime number then the ball would
  exit the opposite corner to where the ball
  started.
• Through looking at the problem in this way we
  were able to come up with many other patterns for
  the path of the ball.

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Where to Start ?

• At first we focused on the example we were given
  and were having some trouble grasping the
  concept. It was not until varied the size of the
  table that we started to see different patterns.

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Prime Numbers

• From this we experimented with using prime
  numbers as the length of the sides of the pool
  table. The prime numbers can not be the same as
  it forms a box and the ball will go straight out
  the other side. This is an example of a table
  with the sides of 5 and 3.
• From this we found that for tables with two
  different prime numbers as its side the ball
  always made the same pattern and the ball will
  always exit from the opposite corner of the
  table. This is true for all prime numbers except
  for 2 and primes that are the same. (e.g. 3 and
  3)

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Examples of Prime number tables
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11 by 7
11
13
2
19
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3 by 2 (an example of how 2 as a prime number
does not work)
19 by 13
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Old school to New school

• We found a internet site that enabled us draw
  these tables at high speed and made our
  investigation easier and quicker.
• The internet site was from the National Council
  of Teachers of Mathematics (http//www.nctm.org/)
• This internet site allowed us to put in any
  numbers up to 21 and it would plot the path of
  the ball for us.
• This allowed us to look at more aspects of the
  path without having to draw every size.

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The New School Eliminating the Graph Paper
Taken from www.nctm.org
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After Clicking the Button
Taken from www.nctm.org
This took 2 seconds and a steady hand
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Hints for Bounces

• At the same time we found this site we were also
  given our clue sheets in class. We took some of
  the questions asked on the sheets and looked at
  them for our investigation.
• Some of the questions we looked at were
• What happens for rectangles in which one
  dimension is a factor of the other?
• We considered the length of the lines as it hits
  the vertical side.

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Factors

• What happens if one side of the table is a factor
  of the other?
• We choose the number three and looked at its
  factors up until 18.

3 and 6
3 and 9
3 and 12
3 and 15
3 and 18
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Factors Findings

• As we can see from the examples with three and
  its factors the exit corner changes every factor
  that we go up. The exit corner alternates from
  the bottom right corner to the top right corner
  assuming that the ball starts at the bottom left
  corner.
• We can also see that there is a pattern that
  forms. The number of triangles in the table goes
  up by one for every factor that we increase. We
  can see this in the examples above.
• As we all know you can not just focus on one
  aspect to draw a conclusion, so we took some
  random numbers and their factors to prove our
  findings.

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Factors Findings cont.
Factors of five
5 and 15
5 and 10
5 and 20
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Path Lengths
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Path Findings

• From the example above we can see that the length
  of any line hitting the vertical side will be the
  same length as the horizontal dimension of the
  table.
• This will be true for all rectangles no matter
  what the size of either dimension.
• 
  
  

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Conclusion

• In our concluding thoughts were
• There is no overall conclusion for this
  investigation, as there is many different aspects
  of investigating for the path of the ball.
• When we started this investigation we were sure
  we would find some formula for the table but we
  were unable to find one due to the many variables
  that are possible in this investigation.

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E-folio

• This Maths investigation will be included in our
  Curriculum and Knowledge section of our e-folios
  as it is an example of the curriculum and
  knowledge we have learnt this semester.