# Pythagorean Triples For The Whole Class!

After introducing my geometry students to the Pythagorean Theorem, I introduced them to the idea of Pythagorean Triples. I showed them that the numbers 3, 4, and 5 will work in the equation: 32+42=52. Then I had students pick a couple of numbers, and we generated a brand new Pythagorean Triple with some fun algebra. At least, I thought it was fun, and students were curious where I was going with my algebraic manipulations. There are several methods, but the one I used went like this.

Pick two interesting numbers. Make them integers, not too big, but not too small. Did I hear 7 and 19? Perfect.

2 ∙ 7 ∙ 19 = 266

Next, find the (absolute value of the) difference of the squares of those two numbers:

192 – 72 = 312

Now, find the sum of the squares of those two numbers:

192 + 72 = 410

And just like that, we have a brand new Pythagorean triple:

2662 + 3122 = 4102

Go ahead – check it on your calculator if you’d like. Pretty nice, huh? I’ll leave it to you to do the algebra to see that it will always work. The most amazing thing about this little task is that students were immediately convinced (and rightly so) that even though it may be hard to find Pythagorean triples, there are an infinite number of them.

And then the game started. I gave each group of 3-4 students a stack of 24 numbers that can be turned into 8 sets of Pythagorean triples. (Of course, there’s only one way to get all 8 sets, but since there are some doubles, it’s possible to match up some that will throw you off).

For both of my geometry classes, all students (no exaggeration!) were engaged bell to bell, furiously testing and playing with the numbers, discussing strategies, and trying to find the eight correct sets. Not only that, although students were in separate groups, they soon realized that they’d be more successful working together. I heard so many great ideas about how to attack the problem that I wasn’t surprised when, even though it came down to the wire, one of the classes found all 8 sets!

This, to me, was a great low floor, high ceiling activity. Every student was able to work on calculations – calculating squares, calculating sums, etc. And some students really worked through some great strategies – what units digits can sum to the biggest number’s units digit? Can we work backwards from that? And of course, the little bit of algebra review in the beginning was helpful as I found later, when I had students repeat the process to develop their own new Pythagorean triples. It was easy for them to check their answers at the end and then look for any mistakes they may have made, and there’s something about big numbers that makes students truly proud (and rightfully so, even when they are using a calculator to do the calculating work).

# Student Generated Problems Are Best!

Around this time of my geometry course each year, students have learned a whole lot of geometry – parallel lines, congruent triangles, basic proof techniques. It is the perfect time for them to come up with challenge problems for me. Believe it or not, classes never disappoint in this fashion. I’ve had some great questions in the past. (One of my favorites – can you fold an 8.5 by 11 piece of paper into an isosceles triangle where the entire triangle has the same number of layers of paper. I’m still not sure about that one, but I really enjoy watching students try every year.)

I tell my students that there are two things that make a challenge problem great:

1. It is relatively easy to explain to anyone.
2. It is challenging to find a solution (or even figure out if a solution is possible).

This year, a student came up with a problem that fit those requirements perfectly: Is it possible to take a square, and cut it into isosceles triangles that aren’t congruent? We then restated it to require a finite number of non-isosceles triangles (though the fractal version is a pretty cool visual and a nice approach).

We spent a few minutes on this at the end of class, but didn’t get anywhere right away. I sent this out via twitter, and almost immediately got a tweet back from Henri Picciotto:

What? That brought out some resolve in me to find a solution, and later that day I did. The next morning, before school, I had a student rush in to show me her solution, which was pretty similar to mine. I suggested to both my geometry classes that there may be more solutions – how many unique solutions or approaches may there be? Can we find an even number? Fewer than seven? I ended up waking up in the middle of the night thinking about this, and came up with even more solutions. So far, I’ve got solutions for 6, 7, 8, 11, 13, and 17 triangles. But the best part of this whole story is that a week later, I have students talking about this before and after school, at lunch, outside my classroom, totally invested in a problem that they made on their own.

When you are a teacher, these are the moments that you live for. Students of varying backgrounds and supposedly different ability levels collaborating and arguing with passion about squares and triangles. Sometimes, that question of “When am I going to use this?” or “How does this apply in the real world?” never come up because the math, by itself, is just fun.

So…can anyone find a solution with fewer than 6 triangles?

# Reflections on a Busy Year

Yes, I’ve been busy before, and yes, all teachers are perennially busy, but this year has been different. The summer before the school year, I was asked to teach a second section of Algebra 1. The enrollment at our school increased (yay!) but not enough to hire another math teacher (boo!). I agreed, and am receiving a nice stipend to do it (which we’re pretending we don’t have and putting straight into savings). What I wasn’t prepared for was just how little time I would have. Teaching six periods, with no prep period at school, and spending most of my time before school, during lunch, and after school working with students has left me no time during the school day to do those other parts of my job that are important (if sometimes menial) – reviewing lessons from years past, preparing and updating lessons, reflecting on daily lessons, communications with parents, and oh so much grading and student feedback. I had a schedule like this in the past, but there were a few things that were very different.

1. I was much younger, and had much more energy.
2. My commute was much shorter (15 minutes vs. 45-75 minutes).
3. I didn’t have any kids at the time. Man, having kids really eats up your time. Making breakfast, giving baths, story time, preparing lunches, putting the girls to bed. (Make no mistake about it – my wife does as much as I do, sometimes more, but we do try to split the duties evenly).
4. I decided to implement standards based grading (which I really need to blog about), an endeavor that is mostly working really well, but needs some tweaks to push forward.
5. For my Algebra 1 class, I finally decided to move forward with CPM, which has meant a lot of extra work and a very different approach to my teaching style.

I fully chose to do this, but it’s led to working from 8 to 11 or later most nights, while waking up at 5:30 every morning, working most of the time on weekends, mostly not blogging this semester, and committing to myself to not do this to myself again. I ‘m well past the point of being in danger of burning out (I think) in this career, but next year I want some more time for myself and my family, and maybe really think about doing that Desmos fellowship that I decided not to apply for this year (but instead followed longingly on Twitter).

So, for all you teachers in a similar position, I’d love to know what strategies you have. What do you cut back on? What do you just cut out? How do you survive? Any thoughts on how to be productive during commutes besides listening to podcasts, etc.?

Also, if you are interested, blog posts should be coming, in no particular order, on:

• Some Thoughts on Implementing Standards Based Grading (So Far)
• Debate #2: Discrete vs. Continuous Graphs
• Debate #3: Do We Need to Prove Things in Class That People Already Proved?
• A Great Geometry Question from a Student
• When a Pythagorean Triples Warm-up Becomes a Class Long Activity

I’ve got a nice long train ride from LA to San Jose after Thanksgiving, and maybe after finishing my Asilomar Talk (11:00, Oak Shelter – Get Your Students Talking: Introducing Debate to Math Class, with Noirin Foy), sub plans, and grading, I’ll put some thoughts down. Thankfully, the in-laws will be coming with us on the train, and should occupy the girls while my wife and I get some valuable time to work.