I found a neat problem in Mosteller’s Fifty Challenging Problems in Probability to share with the boys this morning:
In terms of “what kids learning math look like” this is one of the most interesting projects we’ve done. The problem turned out to be basically accessible to the boys, but pretty close to the end of what they could solve. Eventually they got there, though. I feel like this post could be useful for both the math and for seeing how kids think about math as they are learning it.
I started with a quick introduction to the problem and asked for their initial thoughts:
The first thing the boys wanted to do is look at the replacement and no replacement strategies with the urn with three blocks:
Next they looked at the two strategies in the urn with 201 blocks. We pretended the chances of each block were 50/50 to simplify the math a little:
Now things started going in a direction that was a bit different from what I was expecting. I thought we were just about to wrap up, but it turned out that the right way forward wasn’t yet clear in their minds. But even though things aren’t yet clear to them, you’ll see that they are making progress towards the answer:
Next we were able to combine the two decisions trees and start looking at how often our decision rule would lead us to picking the correct urn. Towards the end of this video suddenly the ideas in the problem become clear to the boys:
Here we finish the calculation of guessing right with our strategy – with the problem now making sense to them the calculation goes pretty quickly:
My older son is studying a bit of introductory statistics right now. I was a little surprised to see this statement in his book:
“. . . you will learn that if you repeat an experiment a large number of times, the graph of the average outcome is approximately the shape of a bell curve.”
I certainly don’t expect middle school / high school textbooks to be 100% mathematically precise, but a little more precision here would have been nice.
For today’s project I decided to show them one example where the statement was true and one where it wasn’t.
For the first example I chose an exercise from the book -> the situation here is a basketball player taking 164 shots and having 64.2% chance of making each of those shots.
Here’s our discussion of that problem (sorry that we were a little clumsy with the camera):
Next we revisited the archery problem that we studied previously. Here’s the problem:
Sharing an advanced expected value problem from Nassim Taleb with kids
Here’s our discussion of this problem today. It is fascinating to see that even with 100,000 trials both the mean and standard deviation of the outcomes jump all over the place.
I think it is really important to understand the difference between these two different types of experiments. Both situations are really important for understand the world we live in!
This problem gave my son a lot of trouble this morning. It is from one of the challenge problem sections in Art of Problem Solving’s Introduction to Geometry book:
A triangle has side lengths of 10, 0, and 12, find the lengths of the radius of the inscribed and circumscribed circles.
This problem looks like a pretty plain vanilla math contest problem, but he’s not studying for math contests. He drew an amazing picture of the situation:
After talking about the problem a bit this morning we moved on to other things. When he got home from school we revisited it. Here are his thoughts on the incircle part of the problem:
Here are his thoughts on the circumcircle part of the problem:
Finally – and unfortunately interrupted by a phone call – here’s his explanation of how the project from today relates to a 3d printing project that we’d done previously
Definitely a fun problem to talk through, and really nice to see that he was able to explain the ideas in a problem that he’d really struggled with in the morning.
My younger son has been studying Art of Problem Solving’s Introduction to Geometry book this year. He’s been doing most of the work on his own, but I check in every now and then.
Today he was working in the section about medians, so I thought I’d ask him what he’s learned.
Here were his initial thoughts.
In the last video he was struggling to remember how to prove that the 6 small triangles formed when you draw in the medians of a triangle all have equal area. Here I gave him a hint and he was able to finish off the proof:
We wrapped up with a short discussion about the lengths of the medians.
Overall a fun discussion. I’m a big fan of the Art of Problem Solving books and am happy that my son is enjoying working through their geometry book.
My younger son is working his way through Art of Problem Solving’s Introduction to Geometry book. He’s been doing almost all of the work on his own – I just check in every now and then.
Today we had a little extra time so we did a project on the section he’s currently studying -> angle bisectors.
We started with some of the basic properties -> why is the intersection point of the angle bisectors the same distance away from every side:
Next we moved on to a slightly harder problem -> what is that distance?
This problem gave him a little trouble. BUT, after a hint to think about how the 1/2 base * height formula for the area of a triangle might help, he made some nice progress:
Finally, I had him work put the ideas we talked about to work in a specific triangle. Here he finds the radius of the inscribed circle in a 3-4-5 triangle:
It was interesting to see him pull some old ideas from geometry in to help understand some of the new ideas he’s learning here. I haven’t looked ahead in the book, but assume that the angle bisector theorem is coming soon. That theorem was really difficult for my older son to grasp, so I’m going to try to work a little more carefully with my younger son in the coming weeks to help him with any difficulty he might have there.
This is another problem from the Iowa State problem collection:
Yesterday I looked at one of the number theory problems with boys:
Some of the Iowa State “problems of the week” are great to share with kids
Tonight I tried another terrific (though very challenging) problem with my older son:
Here’s are his initial thoughts about the problem:
Now we rolled up our sleeves a bit and started to solve the problem. His first thought about what to do was to try to solve the problem with one inscribed circle and then with three inscribed circles:
The problem with three inscribed circles was giving him trouble so we moved on to a new movie and sort of started over on the three circle problem. While he was re-drawing the picture he was able to see how to make some progress:
Finally, having solved the problem with three circles, he moved on to solving the problem in general and found the surprising answer:
I really like these problems. Obviously not all of them are going to be accessible to kids, but the ones that are accessible are really amazing treasures!
Saw a neat tweet from Joel David Hamkins at the end of last week:
I thought it would be fun to talk through the knight’s tour problems with the boys today and end by showing them the infinite problem. I ran into trouble almost immediately when we began to talk about the tours on the 3×3 and 4×4 boards. The difficulty they had explaining was a big surprise to me. We ended up talking about the 4×4 problem for almost 30 min.
Tonight I sat down with each of them and asked them to talk me through the problem and explain why the knight’s tour on the 4×4 board was impossible. You can see that my older son (in 8th grade) was able to explain the problem pretty well, but my younger son (in 6th grade) still really struggled.
Here’s what my older son had to say:
Here’s what my younger son had to say:
Definitely a much harder problem for kids than I thought. Hopefully will have some time during the week to explore this and maybe a few other tour problems with them.