Finishing up our project on Pasquale Cirillo’s introductory probability / expected value problem

Yesterday my younger son and I talked through this fun problem I learned from Pasquale Cirillo:

Our project is here:

https://mikesmathpage.wordpress.com/2021/05/01/talking-through-a-neat-introductory-probability-expected-value-problem-from-pasquale-cirillo-with-my-younger-son/

and Cirillo’s discussion of the problem is here:



Yesterday we did not get to the optimal solution, but rather looked at the strategy of stopping when you get a 6 on the first or second roll, and then at stopping when you get a 4 or higher on the first or second roll. I asked my son to think about the problem a bit more this morning while I was out and he was able to find the optimal solution.

Here’s what he did in his own words:



Next he showed how he used Mathematica to help him find the best solution:



Finally, I showed him an alternation approach to finding the optimal solution that comes from working backwards. This is the approach that Cirillo takes in his discussion of the problem:

Talking through a neat introductory probability / expected value problem from Pasquale Cirillo with my younger son.

This problem was posted by Pasquale Cirillo last week and I thought it would be great to talk through with my son:

This is the video where Cirillo talks through the problem if you want to see his full solution:

I’d asked my son to think about it a bit ahead of us sitting down to talk. Here are some of his ideas:

Here we find the expected winnings if we use my son’s first rule – stop in either of the first 2 rounds if you get a 6:



We ended the day today by talking through the strategy of stopping in either of the first 2 rounds if you see a 4, 5, or 6.



At the end of the last video I let my son know that we hadn’t quite found the best strategy, yet. Tomorrow we’ll finish up and find that best strategy.

The paradox of the 2nd ace part 2

Yesterday we did a project on this fun problem from Futility Closet:

Today we finished the project by talking about the 2nd part of the problem and then having a discussion about why the answers to the two questions were different. Unfortunately there were two camera goofs by me filming this project – forgetting to zoom out in part 1 and running out of memory in part 4 – but if you go through all 4 videos you’ll still get the main idea.

Here’s the introduction to the problem and my son’s solution to the 2nd part of the problem. Again, sorry for the poor camera work.

Next we went to the computer to verify that the calculations were correct – happily, we agreed with the answer given by Futility Closet.

In the last video my son was struggling to see why the answers to the two questions were so different. I’d written two simulations to show the difference. In this part we talked about the difference, but he was still confused.

Here we try to finish the conversation about the difference, and we did get most of the way to the end. Probably just needed 30 extra seconds of recording time 😦 But, at least my son was able to see why the answers to the two questions are different and the outputs from the simulations finally made sense to him.

So, not the best project from the technical side, but still a fun problem and a really interesting idea to talk through with kids.

Revisiting (again!) the Paradox of the Second Ace

Today my younger son and I are going back to a problem that we’ve looked at before – the “Paradox of the Second Ace.” This is a problem I learned about from Futility Closet:

Here’s the problem as described on their site

This problem teaches a couple of good counting lessons. Today we focused on the first part – if you have at least one ace, what is the probability that you have more than one. First, though, we talked through the problem to make sure my son understood it:



Next I asked my son to work through the calculation for the number of hands that have “at least one ace.” He made a pretty common error in that calculation, and we discussed why his calculation wasn’t quite correct:



Now we talked about how to correct the error from the last video via complementary counting:

Now that we had the number of hands that had at least one ace, we wanted to count the number of hands with more than one ace. My son was able to work through this complementary counting problem, which was really nice to see:



Finally, since we had all of our numbers written down as binomial coefficients and these numbers were going to be difficult to compute directly, we went to Mathematica for a final calculation:



Excited to continue this project tomorrow and hear my son’s explanation for the seeming paradox.

Going back to James Tantons’ amazing Möbius Strip cutting project

James Tanton’s Solve This book is full of incredible math projects to do with kids:



Today we went back to do a project that we’ve looked at a few times before – cutting various versions of a Möbius Strip.

We started with a cylinder just to get going with an easy shape:

Next we moved on to a Möbius strip – but when we made this one we didn’t notice that we’d made a full twist rather than a half twist. So we sort of got two surprises. I’ve included this mistake to show that you do have to be a little careful when you do this project – it is easy to accidentally make the wrong shape:

Here’s the correction and the actual Möbius Strip:

Next was a shape where we had a 2nd little error – this time the tape held the final shape together a bit too much. In our video you can sort of guess what the actual shape will be, but I definitely encourage everyone to try this one out for yourself!

The 4th shape is where things really get interesting. For this and the next two shapes we are making a circular cut through a cylinder with a hole cut out of it. Each of the shapes has a interest set of twists in it – the shape in this video has a single half twist on one of the “arms”:

The 5th shape has half twists (going the same direction) in each of the arms:

The 6th shape has half twists going in different directions in each of the arms:

This really is an amazing project for kids. As our version of the project here shows (accidentally!), you do have to be careful with the preparation, but even with the couple of errors we had, it was still an incredibly fun morning.

A geometry puzzle from Catriona Agg that is also great for showing students some ideas from algebra

Last week I saw another terrific puzzle from Catriona Agg:



I left two copies of the puzzle for my son to work through while I was out this morning. For the first run through I asked him to solve the puzzle as it was stated. Here’s his work and his explanation:



For the second run through I asked him to solve the problem assuming that the radius of the circle was X rather than 5. This was first step in what I was hoping would be an interesting algebra exercise. Here he was successfully able to use the quadratic formula even though the equation he found had 2 variables:

For the last part of the project I wanted to see if he could factor the equation he found in the last video. This turned out to be a significantly more difficult challenge, but he figured out how to do it just as we ran out of space on the memory card!



I suspected that the factoring challenge would be more difficult than simply using the quadratic formula, though I didn’t realize how much different it would be. I might try to find some more challenges that involve multiple variables just to get a bit more practice with these ideas.

Why I love sharing ideas from research mathematicians with my kids

Yesterday we did a project inspired by the great podcast conversation between Steven Strogatz and Federico Ardila here:



That project is here:

https://mikesmathpage.wordpress.com/2021/04/03/revisiting-the-permutahedron-with-my-younger-son-after-listening-to-steven-strogatzs-interview-with-federico-ardila/

Last night I asked my younger son what he wanted to do today for a project and he said that he wanted to talk about the permutahedron a bit more. In yesterday’s project we talked about the permutations of the set (1, 2, 3, 4), so today we started by going down to some simpler sets of permutations:

Next we looked at the shape made by the permutations of the set (1, 2, 3). The way my son thinks through this problem shows why I love sharing ideas from math research to my kids.

To wrap up today we dove a little deeper into one of the ideas we talked about yesterday – in the permutations of the set (1, 2, 3, 4) is there a permutation that requires 4 or more flips to get back to the starting point of (1, 2, 3, 4)?

The permutahedron is a really neat shape to explore with kids, and hearing them talk about and think through the shape itself is incredibly fun.



Revisiting the permutahedron with my younger son after listening to Steven Strogatz’s interview with Federico Ardila

This week I listened a great conversation between Steven Strogatz and Federico Ardila on the Joy of X podcast:



We have played with the shape a few times before – see these blog posts of you are interested in seeing other ways that kids can explore the shape:

https://mikesmathpage.wordpress.com/2017/03/09/a-fun-shape-for-kids-to-explore-the-permutohedron/

https://mikesmathpage.wordpress.com/2019/10/27/labeling-each-vertex-of-a-permutahedron-is-a-terrific-mathematical-exercise-for-kids/

Today we explored the and idea that Ardila discussed in the podcast – finding paths on the permutahedron.

We started by just reviewing what the shape is and what it represents:



Next we tried an easy example of finding a path on the permutahedron going from a random permutation back to the correct order:

For the last part of the project we tried a more complicated scramble of the cards and found that walking back to the unscrambled state would take a minimum of 3 steps:



I love playing with this shape with kids. It is a great way to get them talking about fairly advanced mathematical ideas and also allows them to see a really neat 3d shape that research mathematicians find interesting!

Working through a neat circle / rectangle puzzle from Catriona Agg

Earlier this week I saw a really neat puzzle from Catriona Agg:



We’ve done lots of projects with Catriona’s puzzles in the past, so just search for “Catriona” and you’ll find them.

My younger son spent some time off camera solving the puzzle and then I asked him to walk through his solution. His solution gets the main idea about tangents and circles, and then computes the radius of the semicircles using the Pythagorean theorem:



Typically when we play with one of Catriona’s puzzles I have my son look through the twitter thread afterwards and find a neat solution. I took a different approach today and showed him how to use similar triangles to get to the answer with slightly less computation:



I really like Catriona’s puzzle. I also think that my son’s explanation is a great example of what kids doing math looks like.

Showing that 1/40 is in the Cantor set is a great arithmetic exercise for kids

Yesterday we did a fun project on the Cantor set inspired by an amazing tweet from Zachary Schutzman:



That project is here:

https://mikesmathpage.wordpress.com/2021/03/19/a-fun-fact-about-the-cantor-set-and-a-great-arithmetic-exercise-for-kids/

Today we extended some of the ideas from that project by showing that the number 1/40 is in the Cantor set. Here’s how my son approached the problem – the idea he uses builds on the idea we talked about with the number 1/10 in yesterday’s project. I was happy to see that those ideas had stuck with him!

Now that we knew 1/40 was in the Cantor set, we talked about what other numbers of the same form must be in it. Although we don’t prove it (that’s what the paper in Schutzman’s tweet does), he’s now found all of the numbers with finite decimal expansions that are in the Cantor set



Finally, I wanted to go down a path relating these base 3 expansions to infinite series, but my son’s ideas took this last part in a slightly different direction. Which was fine and also fun. It really shows that kids can have fun exploring – and also have the capacity to have some great ideas about – infinite series.

These two projects have been really fun. I think the ideas about the Cantor set are great for kids to play around with!