I have taught 3D Pythagoras for several years now. Some students have such a hard time visualizing it on a 2D projection, so I’ve been trying to come up with new ways of introducing it. I also get very bored with doing the same slides and examples each year, because I teach it in both IGCSE and IB Math Studies. I was about to start my lesson for year 10 on it today, and remembered an introduction to it in the IB textbook that I have. I modified itÂ slightly to make it more applicable to my class.

“Imagine you are on your school Experiencing China trip and you come across a sword replica that you want to take home. Since you’re flying, you’ll be able to put the sword in your checked bag, but you need to know if it will fit. Your bag is the standard size of 56*35*23 cm of a carryon bag. What is the longest length of sword you can buy?”

Then I had them work with partners to see if they could figure it out. We’ve been doing Pythagoras for a few days now, and even started with The Taco Cart from Dan Meyer. I walked around the room to hear their thinking.

“What about all the clothes? It’ll have to lie flat because you have other things in there!” “But you can pack the clothes around it! You can go from this corner to this corner.”

“I know you have to go from here to here, but I don’t how to get that measurement.”

After a few minutes of productive struggle (I hope!), I pulled them all together around a table. I had found an empty box in my room that I could open the top of and make it sort of like a suitcase. I also have yarn in my room (I host knitting club), so we cut yarn to talk about how we might find the measurement we needed. First I had them tape on the diagonal from one corner to the opposite corner to symbolize the sword fitting just right. Then we talked about what else we needed to know in order to find that. Someone mentioned the diagonal on the bottom of the suitcase, so we taped some yarn there as well.

Then I held up the box and asked what they saw. “A right triangle!” We then talked about how to solve the measurements, and then they were able to work out the maximum length of their sword.

All in all, I felt that this method of introducing the concept went well, because they discovered it, and it helped erase the pseudocontext of the textbook examples. I’ll keep this idea in the future.

How do you teach 3D Pythagoras? Leave a note in the comments!