The Pythagorean Theorem
The Pythagorean Theorem
- My name is Jacob Hampson.
- The class I envision using this sketch is primarily Algebra I and Geometry while studying right triangles.
- The topic being studied is: The Pythagorean Theorem.
- The mathematics behind this sketch involves basing a construction on properties of right triangles and squares so that when the figure is dragged the properties are maintained even though the lengths differ. The numerical values for a,b,c are shown on the right illustrate the relationship that the sides of the triangle have when using the Pythagorean theorem. This mathematics is also extended to other polynomials showing that the sum of the area of any two similar polygons a+b will equal a third similar polygon c.
- The usual way this is taught is by giving the students the formula and then proving the formula by completing a proof. Students are then shown many example problems using 2 sides to find the remaining sides.
- This sketch will enhance learning by allowing the students to understand why the theorem works rather than just that it works. With a deeper understanding of why it works the students will be able to figure out the problem even if they don’t remember the formula.
- The way I imagine using this sketch is by first creating the sketch on the overhead that way they can see how the sketch is made and the ideas used that allow the sketch to work. Then I would allow the students to play around with the dimensions on their own devices and maybe have a few tasks for them to complete to make sure they are grasping the concepts.
- “Okay class, today we are going to continue our discussion on right triangles. So we are starting with a right triangle, then we are going to sketch a square onto each side of the right triangle. Now notice that as you change the dimensions of the triangle the sizes of the squares change proportionally. Take a few minutes and play around with the dimensions of the triangle and take with your neighbor about trends you are seeing. So can anyone give me a trend they saw? Good, if you look on your right you will see the pythagorean theorem and values of a^2 +b^2 = c^2 adjusting as you change the dimensions of the triangle.”
- This activity covers the Common Core Standards for Mathematical Practice: Standard 2: reason abstractly and quantitatively, and Standard 5: Use appropriate tools strategically.
- The new functions of geogebra I used in this sketch were the semicircle function which allowed me to change the dimensions of the triangle but always preserve the right angle. Another new function I used was the condition to show object function which allowed me to show regular polygons with different numbers of sides using a slider.
- One thing I learned about mathematics from making this is that this is how the Pythagorean theorem was derived, I just knew it as a memorized formula and did not really think about why it worked while doing it. I also learned that the area of regular and similar polygons can also be found using these methods and that the area of the polygon with side a and plus the area of the polygon with side b is always equal to the area of the polygon with side c as long as the polygons are have the same number of sides and are regular or similar.
- One thing that I learned about pedagogy from this is that there are many ways to teach something but probably the most productive way is to teach it in such a way that gets the students interested and involved. When I started this project I truly did not know how to prove this the Pythagorean theorem. Something that I see while tutoring high school students is the lack of the ability to remember a lot of the formulas that they are supposed to know. I think that if we can help our students understand why the formula works in a way that is fun and visual they will be more able to remember the formula or figure it out if they forget and thus be more successful.