The Pythagorean Theorem Puzzles would be very beneficial to my students because, not only do they provide students with a visual of how and why the theorem works, but they also incorporate technology and transformations (rotations), and they require students to engage in critical thinking. I like how the puzzles diagram the dimensions of each side; thus, meaningful discussions could be generated about the theorem, in addition to using the activity to make a connection to finding the area of an irregular polygons. Furthermore, most of my students wouldn’t necessarily consider the activity a lesson in geometry. Rather, they would merely consider it a game or puzzle; therefore, I think they would exert more effort and thinking in an attempt to “win” and complete the puzzle.

When I solved puzzle #1, I found the puzzle on the right to be easier. For example, when I analyzed the white area, prior to rotating the figures, I knew that the square couldn’t fit in the lower right-hand corner. Thus, once I inserted the two triangles, everything else fell into place rather easily, because there was only one option left for placing the square. To the contrary, however, the puzzle on the right took a little bit more time; it was more difficult than the other. For example, since the square could fit into any one of the four corners, I spent more time deciding where to place the square. I realized that, when I placed the square in one of the four corners, the triangles didn’t extend the full length of the square box. Thus, I quickly realized that I needed to somehow rotate the square and place it somewhere other than one of the corners. This process took trial and error.

Fortunately, puzzle #2 didn’t give me any trouble at all. For example, the puzzle on the right was a carbon copy of the previous puzzle, and the one on the left was also pretty much the same. I enjoyed using the virtual manipulative; however, as a teacher, I prefer having my students use hands-on manipulatives. For instance, hands-on manipulatives allow students to create multiple examples at one time. Therefore, students can visualize more than one example/option at a time. In addition to this, students are given hands-on manipulatives on their standardized test. Thus, I think it’s best that they have more exposure and experience with the hands-on manipulatives, than the virtual manipulatives. Besides, hands-on manipulatives are more accessible; every teacher doesn’t have easy access to technology for an entire class.

Virtual manipulatives, on the other hand, break up the monotony of having students sit at a desk for the entire class, and they do a create job of incorporating technology into the lesson. Unlike hands-on manipulatives, you can often alter the shape, size and color of virtual manipulatives, in addition to having access to a wider array of manipulative options. However, as mentioned above, without access to technology, all of the advantages associated with virtual manipulatives are a mute point, although another added benefit is that they extend the class-time, since there’s no clean-up process.

mdougherty821

Jul 12, 2012@ 16:20:03I also prefer using hands-on manipulatives. You made a good point about being able to create multiple examples at one time with hands-on manipulatives. They also just give you a bit more freedom to go in whatever direction the lesson takes you. For example, if during the course of using hands-on manipulatvies you realize the class needs a refresher in perimeter, you can back-track and do a couple quick activities to reinforce the concept.