Pythagorean Theorem in Sketchpad

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Pythagorean Theorem in Sketchpad

• Jen Lamontagne
• Math 531

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Goals and Objective

• To learn the ratios of the sides for some special
  angle triangles, namely the 45-45-90 and 30-60-90
  triangles to solve problems involving special
  right triangles.
• To have students understand the Pythagorean
  Theorem of 90 degree triangle

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MA standards

• 8.G.2 Classify figures in terms of congruence
  and similarity, and apply these relationships to
  the solution of problems.
• 8.G.4 Demonstrate an understanding of the
  Pythagorean theorem. Apply the theorem to the
  solution of problems.

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Prior Knowledge Learning Styles

• Students should have an understanding of algebra
  topics such as taking the square root of a
  number.
• Sketchpad reaches more Visual Spatial Learners

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Exploration

• It is believed that the Egyptians were able to
  use triangles for land surveying. Some believe
  that they also used it to help design their
  pyramids. Today, surveyors, carpenters and
  woodworkers also use specific triangles.
• What is it about these triangles that assist
  workers in these professions?

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I. 45-45-90

• Activity
• 1. Measure the angles of the triangle. How can
  you classify the triangle by its angles?
• 45, 45, 90 triangle is an isosceles right
  triangle because one angle is 90 degrees and the
  other two angles are equal.
• 2. Measure the lengths of the two legs of the
  triangle. What do you notice about these lengths?
  Move the points on the triangle around and see if
  your conjecture always works.
• The leg lengths are equal. As the length of 1
  leg is changed there will be a corresponding
  equal change to the other. Also the ratio of the
  leg and hypotenuse is constant.
• 3. How can you classify the triangle by its
  sides?
• Two sides equal is an isosceles triangle
• 4. What is the relationship between the legs and
  the hypotenuse?
• The sum of the legs are always greater than the
  Hypotenuse. If you square each side the sum of
  the two legs squared is equal to the sum of the
  hypotenuse squared, leg squared leg squared
  Hypotenuse squared
• 5. Is this relationship always true? Move the
  points on the triangle around to test your
  conjecture.
• As students shrink and pull the side length they
  should see that the angles measurements do not
  change. However, as the lengths change, the
  relationship between the side lengths in
  questions 4 proves true.
• Students observe variance and invariance

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II. 30-60-90

• Open sketch
• Stretch and shirk the triangle using point B
• 1. Measure the angles of the triangle. How can
  you classify the triangle by its angles?
• The angles measures are 30, 60, 90 degrees, this
  is a right triangle
• 2. Measure the lengths of the sides of the
  triangle. What could you do to the short leg to
  get the length of the hypotenuse? What could you
  do to the short and long leg to get the length of
  the hypotenuse?
• After exploring the relationships found in 45,
  45, 90 triangles, students may test the equation
  they developed with the 30-60-90 triangle. They
  should observe a relationship, that as one of the
  leg lengths changes, the hypotenuse and other leg
  change. The sum of the short and long leg
  squared is equal to the length of the hypotenuse
  squared. Leading students to find the
  Pythagorean Theorem. Some students may or may
  not discover the sides relationship of short leg
  x, long leg 2x, and the hypotenuse length x 3.
• 3. Move the points around on the triangle. Does
  your conjecture always work?
• Yes, as students view the squared lengths sum it
  always equals the hypotenuse lengths squared.

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III.

• Using sketch one and two, can you formulate a
  rule to determine the length of the hypotenuse?
  Does it work with both sketches? Try this with
  non-right triangles, does your rule still work?
• Students should formulate the Pythagorean theorem
  a2 b2 C2. They should also see from their
  exploration that this rule only works when using
  right triangles.

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Further exploration

• Have students form squares with each triangles
  sides length. Have students measure the area of
  the squares, and again look for a relationship.
  This should further confirm their conjectures of
  the Pythagorean Theorem

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In Summary

• Now we can see why the 30 60 90 triangles 3-4-5
  triangle is frequently used by surveyors,
  carpenters and woodworkers to make their corners
  square.
• We can now see that the Pythagorean Theorem works
  with any right triangle
• Through sketchpad students are able to make
  conjectures. They are left to explore their
  ideas in the controlled environment of sketchpad.
  Students are able to prove their conjectures,
  this aids in the retention of the information.
  Students are also reinforcing the invariant and
  variant attributes of the Pythagorean theorem.
  Taking notice that while the angles and side
  length proportion are invariant, the lengths
  themselves are variant.

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New to me this course?

• How important it is for students to understand
  variance and invariance of a structure.