# Revisiting one of Annie Perkins’s drawings

Back in 2018 Annie Perkins posted this drawing:

We did a project talking about it back then, but I decided to revisit it today with my younger son.

First we recreated part of the drawing using our Zometool set, and then I had my son share his thoughts about the shapes – it is always so fun to hear what kids see when they look at shapes:

# An introdution to sin(x) and cos(x) with my younger son

With school nearly over for the year I was looking for some ideas to explore with my younger son over the summer.  I thought some introductory trig ideas might be fun since he saw some basic right triangle trig inn his math class at school this year.

The first thing that came to mind for me was a short exploration of what the functions sin(x) and cos(x) look like.  It was fun to hear his ideas about these functions evolve over the course of our discussion this morning.

After that introduction, I introduced the unit circle and asking him to make a guess as to what the graph of y = cos(x) would look like:

In the last video we looked only at the interval 0 to 90 degrees. Here we made a sketch of y = cos(x) and y = sin(x) from 0 to 360 degrees. It was fun to hear what he thought of these graphs as he was drawing them:

The discussion we had today was really fun and even had a few nice surprises. I’m excited to continue this discussion a bit more over the next couple of weeks.

# Revisiting Grant Sanderson’s “fair division” video thanks to an Amy Hogan tweet

I saw a tweet from Amy Hogan this morning that reminded me of an old project:

The old project is here:

And Grant’s video is here:

Before we started the project today, I had my son watch Grant’s video. Here are some of his takeaways:

Now we tried to see if we could find a fair division of a string with two colors using only two cuts:

Finally, to wrap up we tried to see if we could do a fair division of a string with three colors using three cuts:

This is a fun and relatively easy project for kids, and also a way for them to see some really fascinating advanced math!

# Having my younger son try out Matt Enlow’s algebra inequality problem

I saw an interesting tweet from Matt Enlow earlier today and thought it would be fun to have my younger son try out the problem:

One bit of hesitation I had was that I didn’t know if my son had seen rational functions in school or not, so we started with an overview of the problem just to make sure that he understood it:

In the first video he made some progress on the problem in the case when x > -4. Now we finished up that piece of the problem:

Now he tackled the case when x < -4. This part was not as difficult since we’d done most of the necessary work already:

Finally, we went to the computer to look at a graph of both equations. Here he talks about how these graphs help us see the solution to the original equation:

I liked this problem and was happy that my son was able to work through it. Definitely a nice problem for students learning algebra to think through.

# Introducing my younger son to the Poisson distribution – part 2

Last week I did a brief introduction to the Poisson distribution with my younger son. That project is here:

Today I dove in a little more to see if he could see some of the patterns that emerge in the distribution. We started with a quick review and a look at data from a few simulations I ran:

Next we looked at the data from four simulations with an averages of 1, 2, 3, and 4 events expected per year. It was a little hard for him to see the overall pattern, but after a few hints he was able to see what was going on:

To wrap up today, we looked at the pattern from the simulations and tried to write down the pattern that we’d expect to see for an event that happens 5 times per year on average. At the end of this video he was able to write down a formula for the general pattern!

# Introducing some basic ideas about the Poisson distribution to my younger son

For our project today, I though it would be fun to talk about the Poisson distribution. For me it is one of the most interesting and important ideas in probability. This question, for instance, is fascinating -> If a random event happens on average once per time period, what is the probability that it happens twice?

I started the introduction with a version of the idea I mentioned above and asked my son for some estimates of what he thought the answer would be:

Then we looked at some simulations. Here I’m looking at the idea of a random event that happens on average once per year and chopping the year up into 52 weeks:

Next I chopped the year up into 365 days – would we get different answers?

This project turned out to be a little more interesting to my son than I was expecting – I’m looking forward to exploring Poisson distributions a bit more next week.

# Having my son talk through an interesting probability problem from Pasquale Cirillo

Last week I learned about an really interesting probability problem from Pasquale Cirillo:

Today I asked him to think about the problem while I was out and when I got back home he walked me through his solution:

The last video shows his general approach – now he calculated the asnwer:

To wrap up I showed him how to modify his original argument just a bit to avoid the infinite series calculation. This is a much shorter way to solve the problem, but does require a bit more mathematical sophistication:

I really love the problem and think that it is a great one to share with kids. Even if kids can’t quite solve it, it would be really fun to hear their thought process and how they might estimate the probability of winning.

# 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:

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.