After attending the math conference, I was inspired by David Coffey to create this project. And since then I’ve had several teachers ask me for it so I thought I may as well post it on my blog and you can feel free to share it, use it, tweak it, ignore it……pretty much whatever you want to do with it!
Pythagorean Triangles…..Math 8, Alberta Program of Studies
The Set Up
I began by simply presenting my class with this statement: It can be said that a square is a rectangle but it cannot be said that a rectangle is a square. Then I asked them what the heck that means. They came up with a really great list of criteria for what makes a square, a square.
At this point I gave them a handout with several “perfect” squares drawn onto a grid. I then told them that not only were these squares, they were perfect squares. And I asked them to come up with a criteria list for what makes a “perfect square”. Long story short….after about twenty minutes the class came to the decision that a perfect square has all the same requirements as a regular square but also has four equal sides that are whole numbers with no decimals.
The next step was to give them two right angled triangles and six perfect squares and ask them to see what kind of connections or relationships they could make between these objects. Warning….if all good teachers have bite marks on their tongues….I was drawing blood.
It was really really hard not to simply give them the answer or even a hint. I watched them play around and stack them, make nice creative designs, all the while asking them, “Do you see any connections? Do you see any relationships?”
Because I held off on giving them an answer, it was all the more powerful when one of the kids FINALLY figured out that the squares fit onto the side of the triangles. And as soon as I confirmed it for one kid, it spread like wildfire around the room. One kid showed another, who showed another, and so on…..
To cut this down….the kids began to figure out that the side of the square was the length of the triangle side and that the areas of the squares were all connected. In their words, small square plus medium square equals large square.
So now we have our understanding of right angled triangles (note: there were more classes that focused on maneuvering back and forth between squares and the side lengths of the triangle). It was at this point that I attended the math conference and met David Coffey. He challenged us to ask questions of our students that have them make their thinking visible. Questions that will require students to make decisions and empower them with the ability to have choice.
So here was the question I presented them with: If we assume a “well built” building has 90 degree corners, is this school a well built building?
The Project Begins
It became obvious to the kids that they were about to do some measuring of some corners. And that’s when they proceeded to ask:
How many corners should we measure?
Where should we measure them?
What corners count?
What do we use to measure them?
And I told them…..I don’t know! You’re going to have to decide this for yourself. How many corners will you need to measure before you are satisfied you will be able to answer this question?
This was a big departure for me. In the past I would have given them an assignment that would have outlined exactly how many corners I wanted and where in the school I wanted them to measure.
David Coffey is very much about choice though. And when I thought about it, by giving them the parameters of the project I was really limiting their creative control. Some of my kids have been all over the school. Some focused mostly on one area. Another student decided he only wanted to do the structure of the original school and went so far as to find the blue prints from 1952 so he knew what corners to look at! This would never have happened if I had told them what to do.
In the end they will present their answer to the question in either video format, written response, or through a teacher/student interview. They are expected to explain exactly how they arrived at their answer. This includes reasoning through how many corners they measured, which corners they picked, and ultimate weather the math proves or disproves their 90 degree angles.
So far many of the students are finding that the corners are NOT 90 degrees. Some students have found that the corners are close according to the math, which brought up an interesting discussion point. What about margin of error? Is close enough, good enough?
What I’ve Learned
It’s hard. It’s really hard to give up control of the project to the kids and trust them to do what you hope they will do. I do prod them along with questions the whole way and they do come to me to touch base. It’s important to keep in mind that the real point of the project is to see if they understand the relationship between squares and how they are used to create a Pythagorean triangle. If their measurements are off? If they only use 4 corners while someone else uses 20? In the end they are responsible for justifying the decision they made to solve their project and it’s that thinking that I’m interested in.
If you have any ideas on ways to tweak this project or if you try this with your own class I would love feedback on how it went! This is based on the Grade 8 Program of Studies Curriculum for Alberta, Canada.