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.
I started by asking him what he already knew:
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.
I saw a tweet from Amy Hogan this morning that reminded me of an old project:
Two thieves steal a necklace w/ 10 rubies and 14 emeralds, fixed in some arbitrary order on a loop of golden string. Show they can cut the necklace in 2 places so when each thief takes one of the resulting pieces, they get ½ the rubies and ½ the emeralds. From @MoMath1#mathchat
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!
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.
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.
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:
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.