# Try bringing the problem to physical reality

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.

## 5 thoughts on “Try bringing the problem to physical reality”

1. hm: you can solve this problem with only the pythagorean theorem. Dangit. How can we emphasize trig here?

Or maybe it’s an opportunity to explore the glories in the connections between circles and triangles?

2. This seems much more involved than cutting corners out of a piece of copier paper. Unless I’m missing something.

1. Sorry; is it clear that the pyramid thing is separate from the box thing?

The problems are similar in that you have to cut in different ways to achieve different results. With the boxes, you have to cut 3 cm to get a volume of 360 cm^3, for example. With the pyramid, you have to cut at 53 degrees to get a height of 10 cm.

Both activities require a lot of leading questions. The pyramid is more involved in that it requires triangles instead of squares, and they’re harder to think about, but otherwise I’m not sure what you mean.

3. Have you tried it? (I just asked over at Kate’s place, but here’s the right place to ask.) I love this.

>With the pyramid, you have to cut at 53 degrees to get a height of 10 cm.

Hmm, I thought the problem Kate gave us showed a triangle standing perpendicular to the ground, with the triangle’s base running to the center of the base of the pyramid, and the hypotenuse running up the slanty side of the pyramid. If your pyramid has a 10cm square base and a height of 10 cm, then I get Kate’s angle to be 63 degrees, and the angle you need to cut to be 66 degrees.

I’m wondering if we’re seeing different 3d objects from these 2d pictures and diagrams.

Cool stuff!

4. No, I’ve never tried it – and before it can be tried, some way to disqualify the pythagorean theorem needs to be found (or it can be tried as an awesome pythagorean theorem practice ;)). This is very much an unfinished idea.

I didn’t calculate 53 degrees, I was just giving an example. I don’t even know how wide the base of the pyramid is so I don’t think it’s possible to determine the angle yet

You definitely found a 4th interpretation of the original triangle diagram that I didn’t even think of, with the lower-right vertex of the triangle in the MIDDLE of the left side of the pyramid. I was assuming it had to be at the vertex of the pyramid (running up the slanty edge of the pyramid instead of the slanty side). But I think we’re probably thinking of the same 3D shape.

Thanks Sue!