# Pythagorean Theorem and the Distance Formula: Live 3 Act

I was reading twitter when I found Mr. Orr’s 3 Act Task Corner to Corner task. I had just taught the Pythagorean Theorem the day before and the distance formula was on tap for the day. I had a giant thing of string from Algebra’s battleship task, so I thought….why not recreate the scenario in class.

I taped a piece of string from one corner on the floor to the kitty corner  one on the ceiling.  I have tables and a relatively small class ~15 students so I was able to push tables to the sides for the day. When I greeted the class at the door, I asked them to watch their heads. That got a few chuckles until they saw the giant string. Instead of giggles, I got excited chatter. Many were variations on”What is Ms Micaela up to now?”,  but many students were asking each other math-y questions as well. The bell hadn’t rung, I was still greeting in the hall, and already I overheard the question I wanted.

When class did start, one student asked the why question, so I asked them right back. “Why do you think I put this here?” and hinted I’d already heard some great math talk. We quickly settled on “How long is the string?” I told them I’d hung it right before class (in fact a few students were trickling in as I finished taping) and I didn’t measure it as I was doing it. In fact, the string was still attached to the yarn ball, but that I would give them other information if they wanted it and I knew it.

But first, I had them each guess/estimate the length. Afterwards I held up a meter stick and asked if anyone wanted to re-estimate. One student did.

I didn’t have nice pictures set up like Mr. Orr since it was last minute, but I did give them the height, width, and length of the room. One asked for the diagonal length and I honestly answered that I hadn’t measured that, but ensured them they could figure it out. (I though about having the students measure the room dimensions, but I also liked the double Pythagorean and visual understanding needed if the diagonal wasn’t known so I didn’t.)

Each time I gave a piece of information (height, width, length) at least one student would shout “I want to change my guess!” So we started collecting all the changes in estimates on the board.

They turned to their table partners and started in on trying to solve. One table quickly figured out that Pythagorean Theorem would work, but didn’t “see” the problem in 3-D quite right. Other groups saw the problem, but didn’t jump to Pythagorean theorem. I had them conference will a different group and then go back to their seats and try to solve. I also asked a group to come up and draw their thinking on the board.

They were excited and engaged and worked hard to explain their thinking to their tables.

It was a lot of fun. I know its not feasible for everyone, so I’m glad people like Mr. Orr make the videos, but I loved having the actual string in the room.   Instead of revealing the answer. We cut down the string and measured right there in class. It was so satisfying for them! We had a quick discussion on why our answers might have been a bit off the actual answer even if the math was done correctly.

Afterwards, we started in on the progression Mr. Orr shows here for the distance formula. He links to the Desmos files at the end. I adapted a bit and stuck them on a Power Point if that is easier: distance-formula-lead-up

I have always taught slope as as “change in x” over “change in y” and used the delta symbols, so we did the same thing in the distance formula. I have found this reduces sign errors and makes kids think about what it means. A few students who had seen the formula before asked it is related to “the y2 y1 thingy” so I added the traditional formula up as well and we discussed how they said the same thing. They seemed to leave feeling confident and I even overheard two students debating which way (distance or Pythagorean theorem) was better on the way to their next class. That is a teaching win.