We reviewed answers on Tuesday, but time fell well short. I had nothing planned. Every teacher knows that feeling. In pure desperation, I described what a Pythagorean Triple was. I told a story about my German university professor who called them "Try-Pulls" and then posed them a problem:
3-4-5 and 5-12-13 are both triples. There are an infinite amount of such triples.
Can you find me more?
The result was mixed. Personally, I felt like this would be a perfect way to kill twenty minutes. Some students sensed complexity and gave a token effort. Others, bless their hearts, ran with it. The results were amazing.
Students began by adding numbers to each triple and testing to see whether they developed another. Adding 'one' to each entry yielded no results. Two was the same story. Some tried only inserting perfect square numbers into the relationship. No dice.
These two activities may be thought of as traditionally incorrect, but just take a second to ponder what excellent thought is going on in these students' minds. First, they recognize that adding an identical number creates a pattern. They are essentially pattern matching. Second, they recognize what a perfect square number is, and that the theorem has perfect numbers at its heart. Thirdly, they clearly demonstrate that they know how to use the Theorem.
Some students started to multiply and light bulbs began to go off around the room. One student kept doubling each triple and receiving a new one. He was essentially multiplying by two, then four, then eight, etc. I asked him if that pattern would work for three or five--he got right to testing. Others just multiplied 3-4-5 all by two, then three, then four etc. They got the same result. The whole time, students were using the theorem!
One student created a master list of squares on a piece of paper. It looked something like this:
I asked her how that would help. She said that now she didn't have to worry about multiplying. She just added any two of the squares together and check to see if the result was on the list. She had essentially taken the task complexity down a notch. I asked her if it was a lot of work. She said:
"It was at first, but then patterns started to happen."
One student pulled me aside and said that she had a way to estimate Pythagorean Triples. She said that she looked at the patterns between the original two that I gave them (3-4-5 & 5-12-13) and created a linear relation for each variable:
" 'a' goes up by 2, 'b' goes up by 8, and 'c' goes up by 8 as well. I looked at the pattern, then tested it on the side "
Sure enough, she had the first few tested out.
Another student saw a different pattern. He called me over to explain that if you squared any odd number and then found the two consecutive numbers that added to that square, you have just created a new Pythagorean triple.
I took a step back to think, and then asked him how he knew. He said he had noticed that 3-4-5 and 5-12-13 both had the last two numbers that were one apart, and that the sum was the first square. He just expanded this with all numbers. Evens didn't work because you can't have two consecutive integers sum to an even number.
I was floored. The level of pattern insight from these kids was amazing. All of this thinking coming from a "time waster". Needless to say, they worked with the theorem in much deeper ways than a worksheet could ever elicit.
The class ended with deep though all around. The students were wondering if their pattern or method would always be true, and I was left amazed at their initiative. I was also puzzled as to why the last method worked. That is the topic for another post. The whole experience reiterated an important point: teachers should always provide space to be surprised by their students.
Keep reading: Pythagorean Triples Part 2: Teaching Learning