Pythagorean Spiral Activity
The Pythagorean Theorem states that “The square on the hypotenuse (c) of a right triangle is equal to the squares on the two legs (a and b).” The theorem is talking about the area of the squares that are on each side of the right triangle. The green and blue squares represent the legs, and the red square represents the hypotenuse. The area of the blue square is a^, the area of the green square is b^, and the area of the red square is c^. From the theorem, the relationship is: area of blue square + area of green square = area of red square, or a^+b^=c^. So, this proves the Pythagorean Theorem.
Throughout this project, we started out by making our spiral. We did this by plotting a point 9 centimeters down and 10 centimeters in on a regular piece of paper. From here, we made a line 3 centimeters north of the point, and 3 centimeters east. We then connected those lines to make a right triangle. After that, we made another 3 centimeter line from point a/b, using a protractor. So, the hypotenuse for the first triangle became one of the legs for the second triangle. We kept repeating these steps until we finished our 17th triangle, thus, making our Pythagorean Spiral. Afterwards, we had to fill out a chart explaining the length of leg a, leg b, and the hypotenuse. We found out the exact answers for them, using the Pythagorean Theorem. The pattern that I encountered was that by plugging the numbers into the Pythagorean Theorem, the first answer was 3√2. As I completed the equations for each triangle, the pattern became clearer, 3√3, 3√4, 3√5, and so on.
What went well was that in the beginning, I thought I was going to have to redo my spiral, because I messed up, but apparently, the last triangle was supposed to overlap the first one by exactly 4 degrees. So in the end, I didn't have to redo my entire spiral.
One challenge that I came across was that when I was doing the equations for each of the triangles, I put my answers into decimal form, when I was supposed to put them in radical form. When I talked to Ms. X, she said that I didn't need to do all of the equations over again, and that I could just convert them in radical form, using a calculator.
One Habit of a Mathematician that I practiced throughout this project is "be systematic." I practiced being systematic because when I was having to perform all of the equations for the triangles using the Pythagorean Theorem I did the exact same operation, but with different numbers. Another Habit of a Mathematician that I used was "look for patterns." I used looking for patterns, because one of the reasons for doing this project was to find the specific pattern when using the Pythagorean Theorem.
What went well was that in the beginning, I thought I was going to have to redo my spiral, because I messed up, but apparently, the last triangle was supposed to overlap the first one by exactly 4 degrees. So in the end, I didn't have to redo my entire spiral.
One challenge that I came across was that when I was doing the equations for each of the triangles, I put my answers into decimal form, when I was supposed to put them in radical form. When I talked to Ms. X, she said that I didn't need to do all of the equations over again, and that I could just convert them in radical form, using a calculator.
One Habit of a Mathematician that I practiced throughout this project is "be systematic." I practiced being systematic because when I was having to perform all of the equations for the triangles using the Pythagorean Theorem I did the exact same operation, but with different numbers. Another Habit of a Mathematician that I used was "look for patterns." I used looking for patterns, because one of the reasons for doing this project was to find the specific pattern when using the Pythagorean Theorem.