## Thursday, April 28, 2016

### Day 148: Pythagorean Theorem Proof #2 & Intersection Posters

My 2nd period class had the best estimates of the day in my opinion. Katie knew that from San Francisco to Florida it's about 2400 miles. She reasoned that from Pennsylvania to Florida it would be half that, so she estimated around 1200 miles. This was a pretty accurate estimate. Estefany reasoned that today's distance was 3 times longer than yesterday's distance of 271.
 First attempt at explaining the what converse meant... not so good.
 Gary did an awesome job showing his work when finding the distance between 2 coordinates using the pythagorean theorem and showing all his work.
The second proof we did today was now starting with two squares with a total area of a^2 + b^2. Instead of using a ruler to measure side b, I had students tear off a strip from their half sheet of paper to make marks for side length B. They then slid it to the right and marked a leg of length b, and then connected that with a dashed diagonal line to the top left to form a right triangle with hypotenuse of length c.

Students then reasoned that the rest of the base must be A, because the whole top is a+b and a+b-b is equal to a. Then they drew a diagonal from leg A to leg b to form a second right triangle.

To avoid any mistakes, I got a highlighter and outlined the border, and made sure to point out where I did not want them to cut (the line where the square a and b are adjacent to each other. They then cut this, after being sure to label side c. In this example I forgot to label the second side c.

After this I instructed them to cut the 2 triangles out by staying on the pink line. They then rearranged it to form a square. Some used the 2 triangles to make a rectangle, and I asked them if that was a square. I encouraged them to use all 3 pieces. After some rotations, it fit together. I asked them what the area of the new square was and they said c^2. They realized the hypotenuse was the side length. A lot of students appreciated the elegance of this proof.

 My best attempt at explaining the converse. In the top right, we stated the pythagorean theorem followed by its converse.