A challenging counting problem for kids learning algebra

My son is in a weekend enrichment math program and that program has been great for him. It comes to an end this week. The last problem on this week’s homework assignment gave him some trouble, so I thought it would be fun to see if we could work through it together.

I was a little worried because I’d not seen the problem until just before the project, but luckily things went ok.

Here’s the problem:

a, b, c, and d are positive integers less than 10. How many solutions are there to the equation a + bcd = ab + cd?

[post publication note: Originally the text presented the problem incorrectly. It is correct in the videos. Karen Carlson pointed out the typo to me. Sorry about that.]

Here’s how we got started – my son had found several cases, but not quite all of them:

After the introduction to the problem and my son’s work so far, we moved on to try to find more solutions. The main idea I gave my son involved writing the equation in a slightly different form:

Now that we had a plan, we moved on to counting the rest of the cases that we found in the last video:

Finally, we went to Mathematica to write a little program to count the solutions for us. This part of our project turned out to be more interesting than I was expecting. It was interesting to compare the brute force solution of the computer to the case by case counting technique that we’d just gone through.

So, a fun problem that definitely made my son think this week. It is

Sharing Grant Sanderson’s “derivative paradox” video with kids is really fun

Grant Sanderson’s new video series on calculus is incredible. We’ve done one project with the introductory video:

Sharing Grant Sanderson’s Calculus ideas video with kids

Today we looked at the second video in the series. This one introduces the idea of derivatives. The videos are not aimed at kids – not eve close. But watching it with them to explain (or just skip) some of the difficult parts was really fun.

The video below shows what they took away from the video when we talked about it roughly 2 hours after they watched it. I was pleasantly surprised by how much they remembered. They remembered the discussion about the moving car, they remembered the paradox of “instantaneous velocity”, and they remembered the rise over run idea of the derivative. Some of the main ideas stuck with them and those ideas definitely made them think!

Next we talked about the car and the graph of distance versus time. This discussion was really fun. The idea is not something that they’ve seen before but this short little discussion allowed us to explore the ideas in several different ways. It was a wonderful accidental moment when my younger son drew the graph heading back down to the x-axis – Sanderson does not touch on this idea in his movie!

I missed the chance to explore my younger son’s idea about the graph representing the velocity instead of distance – we’ll come back to that tomorrow.

Next we worked through one of the derivative calculations that Sanderson used to end his video. I picked the graph of $y = x^2$ just to keep the algebra on the easy side (he does $y = x^3$ in the video.

We didn’t have as much excitement here, but they were able to follow along pretty well. Most importantly, they were able to understand that “dx” was a variable rather than, say, “d” times “x” or something like that:

I’m not intending to teach my kids calculus right now. However, working through Grant Sanderson’s derivative video with them was really fun – and we have one more idea from this project still to study! I think I will go through some more of Sanderson’s videos with them – it is so great that he’s made something that makes some of the main ideas in calculus accessible to kids.