This activity is one of my favourites to do in a group, either with pre-service maths teachers or with students. There are millions of possible verifications of the famous a2 + b2 = c2 relationship between the lengths of sides in a right angled triangle, with some more pedagogically useful than others. There are hundreds of different proofs of the Pythagorean theorem, and I like to discuss with trainee teachers the usefulness of showing some of these in a maths class. Overall, I think that this is my favourite verification activity as it provides a concrete, visual representation of the meaning of the symbols with a hands-on activity which is fun and engaging. I first saw this idea in the article from Flores and Yun, published in the NCTM publication Mathematics Teaching in the Middle School in 2008.
The way I like to begin, is with each student having a large sheet of paper or card, and constructing a right angled triangle with any dimensions they like. (note: I do give some guidance to ensure that all of their 'fences' will fit later, but don't give away the punch line too early). It's nice that they all have different triangles to show that it doesn't just work for one special case. (note: this also leads into a nice discussion of verification vs proof - generally with the pre-service teachers). They then measure each of the sides of their triangle, and then cut 2cm width strips, the same length as each of the sides, from a separate piece of card. They create 4 strips for each side of the triangle - so 12 altogether.
I then get them to make cuts half way through each strip (so 1cm), about 1cm apart. This then allows them to fold alternate flaps created in opposite directions, making a firm base for their "fence" which can then be glued on top of the sides of the triangle.
With the result being a square 'paddock' fenced off on each edge of the right angled-triangle.
Rather than jellybeans, I tend to use Smarties or M&Ms (or round plastic counters) which tend to give a nice fit, and students then explore how many M&Ms will fit in each paddock. It always comes very, very close to the expected
number of M&Ms in paddock a, plus number of M&Ms in paddock b is equal to the number which fit in the largest paddock, c.
Kind thanks to student Sarah, for allowing me to include the photo of her beautiful work.
The understanding of Pythagoras' theorem in terms of areas is very common in textbooks and school examples, but I find that the process of constructing these fenced areas, and calculating the area in terms of the number of "animals" each can hold gives a concrete, and extremely memorable learning experience. Not to mention being able to eat the sweets afterwards...
References:
Yun, Jeong & Flores, Alfinio. (2008). The Pythagorean Theorem with Jelly Beans. Mathematics Teaching in the Middle School. 14. 202-207. 10.5951/MTMS.14.4.0202.