I always try to plan every lesson to death. I don't plan out minute by minute, but I do try to figure out exactly what I'm going to say, how I'm going to introduce an activity, exactly how the students will be organized for this or that problem, but sometimes I forget that the best activities or lessons can spring organically out of our collective energy. I had an idea for a warm-up problem today about 2 minutes before class was going to begin, and I decided to run with it.
Our school is in Sheridan- which is a very small town, so most of the teachers and the students live in McMinnville which is a much larger town 20 minutes away. The topic I was planning to introduce was graphing absolute value functions as the culmination of our algebra 1 unit on linear relationships.
When the students came in I asked who I should make part of the warm-up problem and everyone volunteered. I picked two students- one who is 15 and one who is 13. I said that the 15 year old just got his license but because he's so nervous, he drives to McMinnville at an agonizingly slow 10 mph. Sheridan and McMinnville are about 20 miles away from each other. At this point, my students- on their own I might add- started making x-y tables to figure out how far the two students are from McMinnville at any given time. We made a simple table on the board and realized that it took the two students two hours to get to McMinnville from Sheridan. At this point the students start poking fun at the 15 year old because he's driving so slowly, and he starts defending himself saying he doesn't want to kill anyone and however long it takes doesn't matter. I asked the students if this relationship is linear and if we can model it with the equation of a line and immediately they start finding the equation for the line. At this point I interjected with a new piece of information. The 13 year old navigator wasn't paying attention and forgot to tell the 15 year old to stop and that they made it to McMinnville safely. My 13 year old student says that actually, she fell asleep. So the 15 year old driver drives right on through McMinnville and out the other side. So we continue the table for distance vs. time, but now they're getting further and further away from McMinnville. So here's what our table looked like:
Hours driving 0 1 2 3 4
Distance from McMinnville 20 10 0 10 20
It made perfect sense to the students why we didn't go into the negatives, why the values started to "bounce" back up. The students were having a lot of fun poking fun at the 15 year old and the 13 year old for driving so slowly and for spacing out and missing McMinnville. One student even made his own table asserting that, since McMinnville is 4 miles across, the second half of the table should go 6, then 14 instead of 10 then 20.
After this warm up problem I had no trouble at all showing the students that an absolute value equation "bounces" but that on either side of the vertex, we have linear relationships. I had complete engagement with the rest of the lesson.
I think maybe why it worked so well was that the problem was almost a "make your own adventure". The students were able to add details for fun or change the numbers to fit in with their understandings of the real world. I always start lessons with a warm-up that tries to pull the material we're learning that day into a real-world context to see what kinds of intuition the students already have for the subject material, but rarely does it work so well. Sometimes no planning is the best planning- but I wish there were a linear relationship between how much time I spend on a lesson and how well it goes because then every day would be awesome.