Yesterday I was at a curriculum meeting. My Algebra students reviewed the formula for finding the slope of a line when given two points. In my 9th period class, I have one student who likes to try to trick me with logic puzzles. Today he had the diagram below drawn on the board when I got to class. He pointed out that the bottom 5.5 feet of the boat was under water and asked me how many feet of the boat would be under water if the water level rose by 2 feet. He was trying to trick me into saying 7.5, but I didn’t fall for it.

I said to the class, “Ok now I have a puzzle for you.” I drew two boxes on the boat and explained that they were very heavy and caused the boat to sink down to 5.5 feet underwater. I explained that when I add a third box, the boat will be 6 feet under water. “How far under is the boat when there are no boxes on board??” I was surprised by how quickly student started calling out, “Easy – 4.5 feet!” I also noted that these were the higher achieving students in the class.

I currently have my students seated homogeneously, with the higher achieving students sitting in the back of the room. I checked in quickly with them to see how they felt about yesterday’s lesson and they said they felt good. I gave them an online practice assignment to do during this class period. My plan was to review the homework assignment with the lower achieving students. I threw those plans out the window and told them to gather around the whiteboard. At first they didn’t want to stand, but I think it mattered. I said “No paper, no pencils. Just calculators in your hands.”

I drew something like this on the board and asked them to see if they could figure out the weight of each bar.

I asked them **not** to answer out loud. I asked if the bars could be 5 pounds each.

Sheldon: “No.”

Me: “Why not?”

Sheldon: “That would be too heavy.”

Me: “Maybe they’re bricks.”

Sheldon: “Ok but then the bag on the left would be 20 pounds without counting the weight of the bag.”

Me: “Good point. Ok then I’m ready to hear your ideas. Who knows the weight of each bar?”

Silence.

Me: “Ryan, you seemed like you had an answer earlier but now you’re not volunteering. Did you change your mind?”

Ryan: “Yeah, I was going to say that they would be 0.75 pounds each because I did 3 pounds divided by 4 bars for the one on the left, but then I saw that it wouldn’t work because the one on the right would only add up to 4.5 pounds.”

Me: Very impressed with Ryan’s thinking. Remained calm. “Yeah good point – because you didn’t consider the weight of the bag.”

Students: “Ohhhh…..”

Mason: “I don’t know. Aren’t they one pound each? Because like, you add two bars and it weighs two pounds more”.

Ryan: “I was thinking that but then the one on the left would weigh more than four pounds so that doesn’t work.”

Me: “Oops! That’s my bad. Good idea though Mason!”

We did two more problems like this before class was over. One of them with this group of students but the last one with the whole class. I felt really good about how the students were **thinking** rather than following a memorized procedure. I loved how they articulated their ideas without prefacing every statement with “*I think this is wrong, but…*” I liked that everyone was talking about slope/rate of change but no one was using those words or any formulas. I’m feeling really encouraged (for the moment) and now need to figure out how to keep this going tomorrow.

This is a great story, Kate. I was wondering what you would do about the bag that weighs -1 lb, and I really like that you can just say “Whoops!” and your kids will accept that. It speaks to a high level of trust in you. You can find several similar problems in _Common Sense Math_, and I imagine that you already know about dy/dan (at http://blog.mrmeyer.com/ ) , but his blog is a focal point of great discussion about teaching algebra. Let’s talk more sometime!