Recently my younger son and I have been study some introductory geometry in our Prealgebra book, and several of the questions in this chapter have sparked great conversations. Today I saw a fascinating question about a rectangle and couldn’t wait to hear his ideas. So, off we went.
We talk through the problem for a bit at the start – this is a challenging problem just to understand! Since he’s a little stuck, he decides to try out a few test cases and draws a 2×1 rectangle. This simple rectangle immediately gives him some new ideas – great start!
It is nice to hear all of his ideas, and especially “I think I see something . . . . ”
I broke the talk into two pieces when it seemed that his idea at the end of the last video didn’t quite work. The trouble he’s run into is that when the width increases by two units the area changes in a way that is different that when the length increased by one unit.
One bit of luck is that he’s working with some small numbers and that probably helped him notice that the change was twice what he was expecting. Upon noticing that fact, he’s able to write down the dimensions of the rectangle. The last part for him is just checking. At the end I show him a little bit of geometry that was hiding in the problem that helps us simplify the calculations a little.
When I think of what learning math looks like, a lot of the ideas in these two videos are what I think of. When I think of kids who are not enjoying learning math – or worse – I’d love to walk through a problem like this, or even just show them the ideas here so that they can see what learning math looks like. Questions, ideas, trying out a few things here and there, trying to find what works and what doesn’t, and maybe even finding the solution to a pretty challenging problem.
I visited the Museum of Math earlier this week and shot these two videos in one of their exhibits. The videos show two different ways to slice a cylinder into smaller pieces (sorry for the vertical video – feel free to mock me!)
The reason I shot these two videos is that my older son and I have just started a chapter on three dimensional shapes. I wanted to use these two videos to show two different ways to calculate the volume of a sphere – one seemingly easy, and one seemingly pretty hard.
There is no intention here for my son to understand every step. What I do want to show, however, is two different ways of looking at a problem, the idea of building up a solution to a complicated problem in steps, and also the surprise that the extremely ugly looking sum of rectangles actually converges to the right answer. I think this is a fun example of the power of math, and, as always, it is really fun to hear how kids talk through advanced math ideas.
Here’s my son’s reaction to the first video showing how to chop up the cylinder into circles. Before seeing the second video with the rectangles, it is pretty hard to see the rectangles.
Here’s his reaction to seeing the rectangles in the second video:
So, now we take a deeper dive into the rectangles to see what we can say about them. This video and the next one are really just writing down a Riemann sum (informally). Again, my aim here is not for my son to understand every step, but simply for him to start to see the ideas of building up the solution to a complicated problem in a few steps. At the end of the second video it is nowhere close to obvious that the sum we’ve discussed will converge to the correct volume of the cylinder.
Since I have no interest in actually evaluating this sum, we go to Wolfram Alpha to do it. What we see there is that the sum does indeed converge to the value we thought it would – amazing!
So, a fun little project with a nice surprise at the end. The ideas of looking at a problem two different ways – in this case slicing into circles and slicing into rectangles – and the idea of building up a solution to a complicated problem in several (hopefully) easier steps are what I’m trying to show here. I was lucky to have seen the MoMath exhibit at the same time we started studying 3D geometry. It will be fun to return to this project when we study calculus – many, many years from now 🙂