Download Pythagorean Theorem in Sketchpad

Pythagorean Theorem in
Sketchpad
Jen Lamontagne
Math 531
Goals and Objective
• To learn the ratios of the sides for some special
angle triangles, namely the 45-45-90 and 30-6090 triangles to solve problems involving special
right triangles.
• To have students understand the Pythagorean
Theorem of 90 degree triangle
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.
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
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?
I. 45-45-90
C
hypotenuse
leg
A
leg
mCBA = 45
mBAC = 90
mACB = 45
AC = 1.94 in.
CB = 2.74 in.
BA = 1.94 in.
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
II. 30-60-90
•
•
•
C
Hypotenuse
long leg
B
A
short leg
mCAB = 90
mBCA = 30
mABC = 60
CB = 2.05 in.
BA = 1.03 in.
AC = 1.78 in.
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.
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 a^2 + b^2 = C^2. They should also
see from their exploration that this rule only
works when using right triangles.
Further exploration
• Have students form squares with each triangles
side’s 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
In Summary
• Now we can see why the 30 60 90 triangle’s 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.
New to me this course?
• How important it is for students to understand
variance and invariance of a structure.