I love the wide variety of resources available for exploring the Pythagorean theorem! I decided to go pretty low tech this year although I've seen some fun explorations using Geogebra and Sketchpad. Yesterday I started with a set of squares and had students try to form a right triangle by using 3 squares as the sides of the triangle. We recorded all of the trios of numbers they found, and then set that list aside.
Next, we explored a 3, 4, 5 right triangle, and built squares on the three sides and looked for patterns. Students pretty quickly identified that the sum of the areas of the squares on the two shorter sides was equal to the area of the square on the longest side. Using this information, we were able to generalize and "discover" the Pythagorean theorem.
Going back to our list of trios of numbers, students were able to check to see if their triangles were right triangles. We developed a list of Pythagorean Triples, and then just looked for other trios of numbers that would work.
Today I brought out Dan Meyer's Taco Cart Problem. Students did a great job of asking questions and figuring out what they needed to know, and we brought back in a lot of the vocabulary we had used yesterday when exploring right triangles. I supplied the information for Act Two and sent them on their way to work out the problem in groups of 3 or 4 on 2ft by 3ft whiteboards. This was the first problem they had solved with the Pythagorean Theorem, and there were some great misconceptions for us to discuss!
When I started the discussion, I asked what was the first piece of information that we needed to agree on. Students suggested the length of the hypotenuse, as all other calculations were based on that distance. Some misconceptions: Students squared the legs and found the sum, but left that value for the hypotenuse instead of taking the square root. Students added the legs and recorded that value as the hypotenuse. Students squared the legs and found the sum, but then thought that dividing by two was equivalent to taking the square root.
The next part of our discussion was how to use the rates. Also some misconceptions here, but I love that we were able to discuss if groups had a strategy that would have worked if their hypotenuse had been right. Common misconceptions here: multiplying the length by the rate of 2 or 5 ft per second instead of dividing. Forgetting that one person had part of the walk on sand and another part on the sidewalk.
Monday we'll be looking at a few more problems, including a version of the television 3 act problem from Timon Piccini. Also really liking revising the Wizard of Oz from Robert Kaplinsky. Also looking forward to using Tilted Squares from NCTM as an enrichment activity- I think it's from an article that you have to be a member to see, but there is an illuminations activity that is pretty similar.
What are your favorite Pythagorean Theorem tasks or explorations?