EXAMPLE #1
x/2 + 5
Student: "7!"
Me: "Ok. I heard 7. We are going to put 7 in for x in the expression on the board. Then, whatever we get as the result, we are going to put in for x. And then again...and again....and again. But we aren't EVER going to stop. What do you think is going to happen? Tell your partner."
They had various ideas, we tested them out, and made some cool observations. After that, I encouraged them to experiment with anything they were curious about. What happens if we change the expression? What happens if we change the starting value? What happens if we use two rules instead of one and alternate? There were lots of options.
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EXAMPLE #2
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EXAMPLE #3
Eventually, we got a few questions with potential. Will it always hit the corner? How many times will it hit the sides before it hits a corner? Does it matter if the side lengths are odd or even? Is it possible to hit every grid line on the side BEFORE it hits the corner? Is it possible to end up in the same corner that you started in?
I suggested that each group: 1) pick a question they were interested in, 2) set their own rules/parameters and 3) get to work. It was interesting. Groups worked for a couple days and then things really stalled out. Because there were only four people (or so) working on a problem, there wasn't the same opportunity for them to bounce ideas off of other groups, for us to work through difficult things together as a class, or the same chance that someone might have an insight that led to progress for the whole class. We eventually proved that 1) it would ALWAYS end in a corner and 2) that corner would NEVER be the starting corner (assuming you launch at a 45 degree angle from a corner). So, from there, I suggested we all work towards finding a way to predict exactly which corner it would land in based on the rectangle size.
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Help Me Out...
- I don't love the idea of tricking students into asking the question you want them to ask, but I also have trouble when students are all working on different things.
- I'm curious about the "initial event" that prompts student questions. Should I start small and well-defined and then move to open exploration (example #1) or should I start wide open and leave it wide open (example #3)?
- In most cases, I found it hard to facilitate student work. How can I get students to share work, challenge each other, and challenge themselves?