Saturday, June 9, 2012

Pythagorean Triples Part 2: Teacher Learning

If you are not careful, teaching can become very boring, very quickly. Most teachers of specialized areas teach the same content arranged in the same manner numerous times throughout a career. It is no wonder teachers are constantly warned of burnout. Opening up space for student initiative serves a two-fold purpose:

First, the extra freedom allows students to create significance in memorable ways.
Second, the sheer variety of student queries can raise questions for teachers.

Essentially, surprises lead to great student and teacher learning. The Pythagorean triple exercise (detailed in a previous post) is a perfect example.

I left the class with two main questions:
  1. Why does multiplying a Pythagorean triple by a constant yield another?
  2. Does the method of squaring odd numbers and finding consecutive whole numbers always work?
The first was fairly straightforward. I had seen the number theory proof before; it is quite simple and relies on the use of a greatest common factor. Until this class, I had never stopped to think about how my students would make sense of this fact. I guess I wanted them to think of it as a happy coincidence. I began to ask myself if there were any curricular connections to the idea of non-primitive Pythagorean triple generation.

One student had put it this way:

"The numbers all work because each one is being altered by a constant amount."

It was almost as though it had been staring me right in the face; I had just found an excellent transition activity between the Pythagorean Theorem and Similar Polygons. Ironically, those two topics appear as consecutive units in my Workplace 10 course this year. By complete chance, I had manufactured an opportunity to combine two seemingly unrelated topics.

When we began similar triangles, I had students construct some of their triples and notice the connections. Until that moment, I had always ignored the geometric implications of the theorem because it is typically presented in an algebraic way throughout high school.

The second question needed to be solved with a piece of paper. I enjoy doing mathematics that relates to my job. Solving unique problems from the fields that I teach gives me a deeper understanding of the discipline. It allows me to approach the classroom from a different light--as someone who holds more than answers. Someone who also holds questions.

I wrote out the proof and digitized it later. Essentially, I define the conditions:
From here I used the facts that the new "b" and "c" summed to the square of the new "a" and the new "b" and "c" were consecutive to define both the new "b" and "c" in terms of "a".

I then had all three components of the new triple. I added the squares of the new "a" and "b"

and then squared the new "c" to check equality

The perfect match led to the conclusion that the three new components do indeed create a new triple. What's more is that because the new "b" and "c" differ by one, they cannot both be multiples of a number larger than one. So the new triple, if generated by a primitive triple, is also primitive.

I challenged the students to find an infinite number of Pythagorean triples thinking that they would come up with the fact that multiplying by a constant preserves the relationship. I had no idea that this particular student would find something much deeper.

If teachers aren't learning anything new themselves, they aren't looking hard enough.


Sunday, June 3, 2012

Pythagorean Triples Part 1: Student Strategies

The school year is winding down for me and my project-based grade ten classes. I have found myself looking at the curriculum more and more as the final day approaches. I was told by many that content coverage would be impossible in a project-based setting; this only made me more anxious. Compounding this problem, I needed a substitute teacher for a day and do not like throwing them into a project setting without any briefing. In order to accommodate them, I chose to photocopy a worksheet on the Pythagorean Theorem for my students while I was gone. When I alerted them of this, the response was clear:

Monday came and went, and I planned to review the work on Tuesday. The class came and I had numerous students ask if we were going to be doing the theorem again. It was obvious that they were not accustomed to the tedium of worksheets--this made me extremely happy. It had taken a mammoth effort, but students were starting to understand that the mathematics done in my room was supposed to be practical. They clearly distinguished the disconnect between medium and purpose.

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.