This problem (#22 from the 2014 AMC 10a) gave my son some trouble this morning:
We ended up having a nice talk about the problem this morning. To see if the ideas really sunk in, I asked him to talk through the solution tonight, and he did a nice job:
After we finished, I wanted to go back to the 2014 AMC 10 and just happened to notice that google was also showing that Art of Problem Solving had a video about the problem. So, I thought it would be fun to watch Richard Rusczyk’s solution. Turned out to be a lucky decision since his solution was totally different than the one we found:
It was neat to see this second solution – I learned a lot about 15-75-90 triangles today!
Moon Duchin gave a talk about math and gerrymandering in San Diego yesterday that generated an enormous amount of excitement. One lucky bit of that excitement for me was that Francesca Bernardi shared the teaching resources from a math and gerrymandering conference in Madison, Wisconsin organized by Moon Duchin and Jordan Ellenberg:
This morning I decided to try out some of those ideas with my kids. The boys are in 6th and 8th grade and really enjoyed working through the project this morning. Overall, my impressions are that:
(i) The math all by itself is both interesting and accessible for middle school and high school kids.
(ii) Working with a larger group would produce some fascinating discussions about the tetris-like shapes involved in this project. For example, what sorts of shapes do kids consider natural and what sorts of shapes seem unnatural when dividing up a square?
(iii) The project is great for showing why gerrymandering is a difficult math problem. I think that students will see quickly that creating 6 “winning” regions out of 10 for a group that has only 40% of the population seems unfair. However, they’ll also see quickly that it isn’t as easy as they might think for the math to flush out that unfairness.
So, here’s how things went with my kids today. I started by trying to give a simple explanation of gerrymandering – a concept that they’d not heard of before:
Now I had them each work on one of the exercises from the materials that Bernardi shared yesterday. In this exercise you start with 10×10 grid that has 40 orange squares and 60 purple squares. The first challenge is to divide the large square into 10 connected regions of 10 small squares each in which exactly 4 regions have majority orange squares. The next challenge is to try for exactly 6 majority orange regions.
Here’s how the boys explained their approaches to the two exercises. You’ll see that this problem is a great way to get kids to talk and think about some basic ideas in geometry.
Now we moved on to the part of the exercise that tries to use geometric ideas to identify gerrymandering. Again, working through these different math ideas in this part of the exercise is a fantastic exercise for kids.
Before diving into this part of the project I explained three of the geometric ideas just to make sure they boys understood them prior to diving into the calculations:
The boys did their calculating work off camera. I had them pick 3 regions from each of the two shapes and work through 3 of the different metrics.
Here’s what my older son had to say after he finished:
And here’s what my younger son had to say (if you look really carefully you’ll see that he was confused on some of the calculations, but that shows, I think, why this exercise can be a great activity for kids – this could easily be a week long activity in a 6th grade math class):
Wow is this a great project for kids – and we barely scratched the surface!
One surprise for me was that the ideas of “packing and cracking” didn’t come up in the conversation with the boys. Maybe looking at the different shapes while simultaneously noting the different colors inside of those shapes is a harder exercise for kids than I guessed. Introducing the “packing and cracking” ideas would make a good follow up project.
Anyway, I think the educational project from Wisconsin’s math and gerrymandering conference is absolutely fantastic. Huge thanks to Francesca Bernardi for sharing these resources. The exercises and ideas will make a great addition to just about any middle school or high school math class – I hope they are shared widely!
If you are looking for additional resources, here a few that I’ve found to be helpful from the last year:
(1) Erica Klarreich’s article in Quanta Magazine last year
I thought that the boys would love reading the book and asked them to each read it twice prior to today’s project. Here are some of the things that they thought were interesting (ugh, sorry for the focus problems . . . .) :
The first thing the boys wanted to talk about was the “smallest” infinity -> . Here we talked about that infinity and other sets of integers that were the same size.
Next we moved on to talk about the rational numbers – we had a good time talking through the argument that the “size” of the rational numbers was the same as the positive integers.
This argument is represented in the book by a painting of a shark!
Now my older son wanted to talk about Cantor’s diagonal argument. He was a little confused about the arguments presented in the book, but we (hopefully) got things straightened out. I think this shows kids can find ideas about infinity to be really interesting.
Finally, we wrapped up by talking about the implications of the infinity of binary strings being larger than the infinity of counting numbers.
Definitely a fun project. I love the content of the book and so do the kids. The only problem is that the quality of the binding is awful and although we’ve only had the book for a few days, it is falling to pieces. Boo 😦
My son was working on a few old AMC 10 problems yesterday and problem 17 from the 2016 AMC 10a gave him some trouble:
I thought this would be a nice problem to go through with him. We started by talking through the problem to make sure that he understood it:
In the last video he had the idea to check the cases with 10 and 15 balls in the bucket, so we went through those cases:
Now we tried to figure out what was happening. He was having some difficulty seeing the pattern, so I spent this video trying to help him see the pattern. The trouble for me was that the pattern was 0, 1, 2, . . ., so it was hard to find a good hint.
Finally he worked through the algebraic expression he found in the last video:
This isn’t one of the “wow, this is a great problem” AMC problems, but I still like it. To solve it you have to bring in a few different ideas, and combining those different ideas is what seemed to give my son some trouble. Hopefully going through this problem was valuable for him.
Today we moved on to a really neat surprise, and what makes the math behind this problem incredibly fun -> the “ABRACADABRA” problem.
First, we reviewed the ideas from yesterday:
After that review, we though through a few of the states and the transition probabilities in the new word. The transition probabilities are subtly different than in the “COVFEFE” problem:
Now we went to Mathematica to code in the ideas we discussed in part 2. We did about half of the coding on camera and did the other half off camera:
Finally, having finished the code we discussed what results we expected. I don’t see how anyone could get the right intuition here seeing the problem for the first time, so what do you expect here is almost an unfair question. Still, the boys had some nice ideas and then we checked out the results:
There are other approaches to these problems – the approach via Martingales, for example:
What that approach is also interesting (and incredible – you can solve the stopping time in your head!) I think the Markov chain approach is a bit more accessible to kidsd. Well . . . maybe because the math is buried in the background.
Anyway – super fun project, and an great piece of math to share with kids.
I’d forgotten about that project, but when I mentioned to my younger son that we’d be looking at Markov chains today he told me he already knew about them!
So, I started today by having the boys watch the PBS Infinite Series video again. Here’s what they thought:
Next I introduced the “COVFEFE” problem. I was really happy how quickly the boys were able to pick up on how Markov chains could be used to solve this problem.
Next we looked at Nassim Taleb’s Mathematica code – that code is so clear that the problem becomes instantly accessible to kids, which is pretty amazing.
Finally, since things were going so well this morning, I introduced the word that we’ll study tomorrow -> ABRACADABRA. The kids were able to see why the transitions in this word were different. I’m excited to see how they think through the “ABRACADABRA” problem tomorrow!
The math behind this problem really was the most interesting math that I learned in 2017. It is really important math, too, and I’m excited that the Mathematica code makes some of the ideas accessible to kids. This was a fun one!
This is probably my favorite 3d printing project that we’ve done on our own. I didn’t do a specific project with the boys using the shape because it is really fragile (in fact, I have 3 other broken ones . . . ).
The problem is -> can you cut a hole in a cube large enough so that you can pass another cube of the same size through the first cube?
An old project where we talk about the problem (without 3d printing) is here:
Here’s a really fun shape to play with – the rattleback. It wants to rotate one way, but not the other way. There’s very little indication when you look at it that it would have such an odd property:
(10) James Tanton’s tetrahedron problem
This one has a special place in my heart because it was one of the first times we used 3d printing to solve a “new to us” problem. I loved how these shapes came together. The problem involved understanding the locus of points that were 1 unit away from a tetrahedron:
Another one of my all time favorites projects came from Laura Taalman. Right after the discovery of a 15th type of pentagon that tiles the plane, Taalman created 3d print models of all 15 of the pentagons so that anyone could explore this new discovery: