We start with patterns. This works as a great intro for a variety of reasons:
- it's an easy way to figure out where kids are and get their brains back in math mode
- patterns are naturally low threshold and high ceiling
- my Geometry course is all about investigation, conjecture and (rather informal) proof: patterns hit all of these
- they're on the state test (which students take this year, so it does have to always be in the back of my mind)
The Massachusetts state test usually has some good open response pattern questions, so I pulled the questions I used from past tests and from a variety of other problems I've saved over the years.
Day 1 (a short class) I gave students a questionnaire, to be fair, I filled it out too. One of the questions asks about your experience with math, this was my response:
"In my experience math is a fun challenge. I enjoy solving puzzles and math is often puzzling."
We start the next class with a visual pattern, where the prompt is "What questions could you ask about this pattern?"
After letting them stare at it for as long as it took me to take attendance, I suggested if they hadn't asked it yet, a good question would be "what comes next?" As students answered that and proclaimed they didn't have any questions, I asked for observations instead. When I asked the class to share we pretended it was Jeopardy and found a question that each answer could have come from. We noticed patterns (re: left, right and total faces, blank space) and pondered what happened when you got to row 6 or 7.
Then, I gave each student 1 of 4 patterns to work on. They were welcome to talk to their neighbor (who had a different pattern). This time, if they ran out of questions or observations I directed them to hint cards at the front of the room. I saw this model in a video of a Japanese classroom while in grad school and have always wanted to use it. I thought ahead of time what I would say to students who were stuck, then wrote those questions/phrases on index cards. Then students had ownership of when they got a hint and no one had to wait for me. Most students thought walking all the way to the front of the room was effort, so they didn't go until they had exhausted other options.
After a while of this, I directed students with the same pattern to gather at a white board (I have one or two empty boards on each wall, which is awesome). They drew their pattern, shared questions, answered ones the original asker hadn't been able to answer and discussed interpretations. Then, I brought everyone together for a discussion of hint card 4 (about multiple representations). By now most every group had a list of numbers somewhere on their board, so I asked them to think back to Algebra and other ways they knew to represent data. It took various amounts of prompting, but each class got to the point of "table, graph and equation." I sent them back to the boards to develop those three models for one of their questions.
The results were pretty cool.
In the last 10 minutes I had students present their findings and reflect on the class. Homework was to make up their own pattern and ask, then answer at least 3 questions.
This process (plus a quick read through of the syllabus) took 90 minutes. I was shocked that Fundamentals and Standard Geometry students could work on one pattern for an entire block! Especially on a Friday, the first week of school, at our second class meeting. I had prepped number patterns to do as well but they had enough to say about these patterns that I could push all groups into multiple representations and reach some good depth on just one pattern. Students came up with ideas, discussed, got stuck, made mistakes and helped each other. Not everyone was engaged all the time (mastering group work is a long term goal that I have yet to tackle) but I did convince some students to share ideas with each other since I'd worked with each of them on different parts of the problem (next year I should do this on purpose!).
Next up we studied Pascal's triangle and looked for patterns in there. The discussion led to an impromptu proof that odd + odd = even!
Homework was to make pascal's triangle starting with 2's on the ends rather than 1's.
Overall I'm happy with how this unit went. It may have been worth spending an extra day on the numeric patterns I prepped that we never used, but it being geometry doing all visual patterns is okay.
How do you start your Geometry course?