Pythagorean Path

1
Pythagorean Path

• Who was Pythagoras?
• What is the Pythagorean Theorem?
• What can it be used for?

• How can we use it today?
• Ramps
• Stairs
• Roofs
• Baseball
• Football
• Extensions
• Teacher Pages

2
Who was Pythagoras?

• Click on a button to find out about his life

Where He Lived
Who He Was
When He Lived
3
The Pythagorean Theorem

• The Pythagorean Theorem is one of the most
  well-known mathematical theorems.
• It has been proven by many different methods and
  entire books have been devoted to investigating
  its properties.
• Pythagoras is thought to be the first person to
  actually prove the theorem, although the
  Babylonians are believed to have discovered it
  about 1000 years before Pythagoras.

4
What is the Pythagorean Theorem?
Pythagorean Theorem
Proving the Theorem

• Another Visual Proof

5
What can it be used for?

• Find the hypotenuse of a right triangle
• Find a missing leg of a right triangle
• Determine if a triangle is a right triangle

6
How can we use this theorem today?

• If there is a right triangle, the Pythagorean
  Theorem can be used.
• There are many real applications of right
  triangles.
• Lets look at some of these uses.

7
Some real ideas

• Click on a button to find out how the Pythagorean
  Theorem is used for each of these things.

Ramps
Roofs
Stairs
Baseball
Football
8
Ramps

• Ramps for buildings are actually right triangles.
  
• Read about the regulations that are required for
  accessibility

Code for Ramps
9
Lets see how this would work

• We know the amount of incline that is allowed,
  the rise.
• We know the amount of distance required to cover,
  the run.
• We can calculate the ramp distance needed to
  accommodate these dimensions.
• a2 b2 c2

10
Stairs

• Staircases are basically right triangles. If
  you know how far a door is from the ground, you
  can find out how far your steps will
  need to come out from the wall.

door
height
Ground
11
Stair information
Stair Regulations
Rise
Chapter 3, Amendment R314.2
Run
12
Another use - Roofs

• Roofs also use the Pythagorean Theorem. They
  usually have a pitch (or slope) and can use the
  theorem to determine how much material will be
  needed to complete a roof project.
• This is important as this is a large expense for
  many homeowners.

13
Lets look at some roofs
From Wagner Rooflines Summer 1999
http//www.wagnerroofing.com/
14
Types of roofs

• Look at the roof types available on many houses.
  Decide which one you think would take the least
  amount of material to build or repair.

Roof Types
15
Another Use Baseball

• How difficult is it to hit a homerun in say
  Fenway park, where they have the Green Monster in
  left field?
• Lets check it out.

16
Baseball

• To find out how far a baseball must be hit to
  clear the Green Monster in left field at Fenway
  Park in Boston, you can use the Pythagorean
  Theorem.
• Find out how tall the wall is and how far it is
  away from home plate.
• These measurements create a right triangle and
  you can find out how far a ball needs to be hit.

17
Fenway Park

• Click on the Statistics button to find the height
  of the left field wall and the distance from home
  plate.

Fenway Park Statistics
18
Football

• Field goals are scored in football when a team
  kicks the football through the uprights of the
  goalpost.
• The team receives 3 points for a field goal if
  they are successful.
• Click on the Field Information button to find out
  about the field.

Field Information
19
Field Goals

• Now that you know how far the goalpost is off the
  ground, and you can find out how far the kicker
  is from the goal line, you can determine how far
  the football must be kicked to just clear the
  goalpost cross bar.

20

• Thank you for taking this tour of the Pythagorean
  Theorem. There are some additional sites you can
  go to and some other places you can go to explore
  other uses of the Pythagorean Theorem in the real
  world on the Extensions Page.

21
Extensions

• If you are interested in learning more, go to
  these sites
• To learn more about Pythagoras, go to
• http//www-history.mcs.st-andrews.ac.uk/history/M
  athematicians/Pythagoras.html
• To learn more about other proofs, go to
  http//www.cut-the-knot.org/pythagoras/index.shtml
  
• To learn more about ramps and accessibility at
  Kansas University, go to http//www.digitaljayh
  awk.org/kuedge/j415/415_projects/attig_h/full_stor
  y.html
• To learn more about other ballparks, go to
  http//www.ballparks.com/baseball
• To solve more Pythagorean theorem problems, go to
  http//www.pbs.org/wgbh/nova/proof/puzzle

22
Teacher Notes

• This is designed for 8th grade Algebra students
  as an individual trip or as a classroom
  excursion.
• Students will be able to use the Pythagorean
  Theorem to solve real world problems.
• This trip should take no more than one 90-minute
  class period.
• This can be used following the introduction of
  the Pythagorean Theorem, as students will be
  completing calculations.
• See the Field Trip Guide worksheet provided for
  Algebra.

23
References

• These are the websites that I used to complete
  this field trip.
• NOVA Online, The Pythagorean Puzzle. (2000).
  Retrieved October 27, 2003, from
  http//www.pbs.org/wgbh/nova/proof/puzzle.
• OConner, J. J. and Robertson, E. F. (1999).
  Pythagoras. Retrieved October 20, 2003, from
  http//www-history.mcs.st-andrews.ac.uk/history/Ma
  thematicians/Pythagoras.html
• Realtors Monthly Online. (05/01/2001). Retrieved
  October 27, 2003, from http//www.realtor.org/rmo
  daily.nsf.