“What’s a real-life situation when you’d have no slope?”
This week, mathematicians in 8th grade who are taking Upper School Math 1 presented their work on a set of Exeter problems if someone drives for infinite hours at 5mph? Or, Teleporting?). Their teacher reminded them of the distance:time graph of “The Tortoise and the Hare,” prompting them to think about when in the story the hare’s graph becomes a horizontal line. What about a real-life situation for a vertical line? Students discussed why it is that you can’t divide by zero.:
“It’s not because it generates some huge unending number, or because it’s too complicated or long to do the arithmetic. We can think of the reason by considering what other way we can ask the question ‘What’s 12 divided by three.’” (“What number can I multiply by 3 to get 12?” is the converse. Try this for the 0 case.)
Exeter problems build off students’ earlier learning in Illustrative Math because the problems spiral through topics (revisiting content in many contexts over a long period of time), because they can each fundamentally be approached in more than one way (algebraic solutions, graphical solutions, coding solutions). The rich challenge creates genuine opportunities to practice our Math Process Skills (like showing your thinking, asking questions, critiquing reasoning).
I realized that the side that’s greater will always tell you where the point is.
Another student presented their problem: “My short answer to this question is that it’s below the line. And the long answer is that if you convert this to slope intercept form (y=-3x-2) and then put in the values, I noticed the right side of the equation is larger. So then I wondered how I could test a bunch of these and get a rule that worked. My first thought was that we just put the values of that in: 23 for y and 8.4 for x. Then you get 23=23.2, which, of course, it doesn’t. I tested this with a few different lines. And I realized that the side that’s greater will always tell you where the point is. If the greater number is on the right, and the slope is positive, then that point will be on the right side of the line. But it’s reversed if the slope is negative…” As students invent strategies and solutions based on prior knowledge, they present for peer conversation, and their teacher facilitates the synthesis of more efficient and elegant approaches.
Their teacher also supports naming and noticing specific strategies students are using more intuitively, “Oh, so you’re picturing the graph in your head. Or you can use this quick formula. I think you have to do one of those two things: you need to either sketch it or do the arithmetic. Did anyone come up with a more efficient way to do this?”
These students are preparing not only a strong foundation in algebraic and geometric reasoning, but also the ability to navigate novel problems and to build math understanding in a community where learning comes not just from the adult or a book, but from the reasoning and discussion of peers, which everyone can practice!