### I Notice...I Wonder...The Pythagorean Theorem in 3D

I am pretty tired and close to brain-dead, but I wanted to post while today's activities were fresh in my mind.

I had a brainstorm Saturday morning. Algebra 1 was finishing up the Pythagorean Theorem this week, and I wanted to take a look at the Pythagorean Theorem in 3D. I tried it last year with not a lot of luck. A few saw it, many did not.

My "A-Ha" came at the strangest time, but I went with it and began planning. What if I approach it with an "I Notice...I Wonder..." angle? What if I can get kids to see where we're headed without telling them?

I decided to show kids a diagram of a rectangular prism with the interior diagonal in blue and ask what they noticed and wondered. Then I showed them the same diagram with the dimensions of the prism and the question, "What is the length of the blue line segment?"

I split kids into groups of three, gave them a copy of the diagram they could write on, and I gave each group a box (various sizes) with string as the interior diagonal so they could have something to touch and see.

And then I turned them loose.

I was scared. What if they all stared at their papers and said, "We don't know what to do!"? What if I had to walk everybody through step by step by step? What if it was a total flop of a lesson?

My fears were unfounded.

Students noticed and wondered some good things. They were advised to give serious, math-related observations before they began. :)

As I distributed diagrams and boxes, everyone got to work. Almost everyone tried SOMEthing. Maybe two groups all day said, "We don't even know where to start."

I tried to not give too much away as I helped groups. "You're on the right track." "Tell me why you approached it that way." "Does what you found have anything to do with the length of the string?" I'd ask a question or two, smile, and walk away.

I would say by the end of the activity, most students saw the two triangles they needed to see to find the length of the string/blue line segment.

It was interesting to see their incorrect approaches and misconceptions. Many wanted to start with area of the rectangular faces or the volume of the box. Some would find the diagonals of the rectangular faces and add them. A few had visualization difficulties (interestingly, some of my strongest math-y students had the most trouble with the visualization of the string in the box).

A few went right for what is the formula/shortcut for finding the length of a box's diagonal, extending what they knew about the Pythagorean Theorem, but since they couldn't explain to me why it worked, they had to find another way (the "8th grade way") to get the correct answer. Those groups got mad at me. :)

The cardboard boxes were a MUST. I didn't start the day with enough boxes for every group, but after some searching by a dear friend, I ended up with enough, and I'm so glad I did. Being able to hold, touch, and outline what they were finding on a real box helped make the lesson a success. One kid even said, "I'm glad we had this box; I can't tell anything on the picture!"

I'm wondering if I needed some triangles of the proper sizes cut out so that students could SEE the triangles they were trying to see. I would need the same size boxes for everybody to make that process easier. It's something I'll think about before I do this activity with my Pre-Algebra kids.

I wish I had added one thing to the notice/wonder start to the lesson. After their initial observations and questions, I wish I had said, "If I told you we were dealing with the Pythagorean Theorem in 3 dimensions, does that add anything to what you notice or wonder?" I wanted to tell the students as little as possible, but it took a few of them a little bit to realize this activity was tied to what we've been doing in class for the last week.

The next part of this lesson I need to tweak is the wrap up. All the groups in my 3rd Period class finished, saw both triangles they were supposed to see, and found the length of the string/segment. But I'm pretty sure a couple in my 5th Period and a few more in my 7th Period never got there, even with prompting and hints. I hate feeling like some students leave a lesson thinking, "That was nice. What on earth were we doing?!?"

So...how to get them all "there" and tie up the lesson with a nice little bow without just giving it all away?

I'll keep thinking on it...sleeping on it...and waiting for the next Saturday morning "A-Ha!".