It takes a lot of class time to have students build models, but here’s what I’m almost ready to call a pro tip: people love building models. I wanted to bring Algebra kids from an understanding of the typical calculus box (cut 4 squares out of the corners of a rectangular piece of…) to a function relating height and volume. I spent 100 minutes of class doing this – a huge amount of time. But you should have seen how proud kids were when they got their equation to hit all the points that represented the various boxes they made. I literally gave them 6 or 7 sheets of paper of the same size, had them cut corners out of them, and record, plot, and draw diagrams of their results.
To them, they got to spend 100 minutes on a single, rewarding task. To them I’m going slow.
To me, they’re spending 100 minutes practicing and learning the benefits of switching between multiple representations of a problem, and finally coming up with the ultimate representation, a function, about which we are just starting to learn <shocked gasp!/>.
Kate Nowak is writing about a trig problem. She just wants to review, so she doesn’t have 100 minutes, but literally making a pyramid is such a good idea, you guys. Kids can start from the design below, and in a series of careful questions, you eventually ask them at what angle they should cut the triangles (what angle should that little arc be?) to get a height of 10 cm.
To me this is the ultimate “be less helpful,” right? You have, verbally, given them literally zero information about the problem except the question. They have to go get a ruler from the ruler drawer (that is always accessible) after realizing they need a ruler. They have to come up with a way to check if the pyramid is actually 10 cm high (kind of hard!). And, to boot, they have an intrinsically rewarding experience at the end: a sweet little pyramid. They should probably color a diagram RIGHT ON IT for awesome mathiness that they can be proud of, and that you can glue to the door and have a sweet spikey door of trigonometry. Make sure to give different groups different goal sizes so that as a class you can confirm any patterns kids are seeing. And so that your spikey door is cooler.