Pythagorean Theorem in Sketchpad

1. Pythagorean Theorem in Sketchpad Jen Lamontagne Math 531

2. 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

3. 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.

4. 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

5. 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?

6. 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

7. 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.

8. 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.

9. 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

10. 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.

11. New to me this course? • How important it is for students to understand variance and invariance of a structure.