Sharing the Coupon Collector problem with kids

We are working through Mosteller’s 50 Challenging Problems in Probability and today’s problem was a nice surprise -> the famous coupon collector problem.

Here’s how I introduced it and what the boys thought the answer would be:

Next the boys talked through how to think about the problem. The diagram they ended up drawing was such a great surprise! At the end of this video the boys were a little stuck on the problem of trying to figure out how many steps it would take, on average, to go from having one coupon to having two.

Now we dive into the calculation of the number of steps required to go from one coupon to having two coupons. This is a tricky calculation but we broke it into pieces and had a good guess at the answer by the end of this video:

Now that we had a good guess at the answer for the sum of the complicated series, my older son found a pretty clever way to evaluate the sum. That was a really nice break through!

By the end of this video we were wondering about how to find the expected number of steps it would take to go from 2 coupons to three coupons:

The next step was trying to sum the series that would give us the expected number of steps that we would need to go from 2 steps to 3 steps. Once we made that calculation the boys saw how to evaluate the remaining terms.

Finally, we went to a little matahematica program to check our results and to see what the maximum number of trials would look like.

This was a super project. I loved hearing the ideas that the boys had and was super proud that they were able to work through all the way to the solution of the original problem.

Sharing basic ideas about Cramer’s Rule and Fourier Analysis with my older son

My older son is studying linear algebra out of Gil Strang’s book this year. Currently he’s in the chapter on determinants and we’ve spent the last couple of days talking about Cramer’s Rule.

As we talked about the proof of Cramer’s Rule, I was struck by how similar the ideas were to the ideas used in Fourier analysis. This morning we had a fun discussion showing how the ideas are connected.

I started by asking him to talk about Cramer’s Rule. He did a nice job, especially since his knowledge about this rule is only a few days old:

Next we played around on Mathematica with a 4×4 example and found that the solutions you get from Cramer’s Rule do indeed match the solutions you get from other methods!

Next I gave a really short introduction to a problem that initially seems very different, but has a lot of the same mathematical ideas hiding in the background -> pulling a signal out of noise:

Finally, we went back to Mathematica to play with a few examples of signals hiding in noise. We saw how the ideas from Fourier Analysis could often pull out the signal even though it wasn’t obvious at all that a signal was hiding in our data in the first place!

Sharing the tiles from Cherry Arbor Design with kids

Last week I was really lucky to be able to visit ICERM in Providence and saw some amazing mathematical tiles made by Cherry Arbor Design. Their website is here:

When I got home from that trip I ordered 3 sets of tiles. They arrived today!

Tonight I asked the boys to play around with a set and see what they could make. My younger son chose the Twin Dragon Tiles and played with them for 45 min! Unluckily I had a call that came in at roughly the 2 minute mark of the video below, but we resumed after that call. You can see from the video that he really enjoyed creating all kinds of different shapes:

My older son chose to play around with the Penrose Tiles. These tiles are completely stunning. Here’s his creation and what he had to say about the tiles:

The mathematical tiles and puzzles from Cherry Arbor Design are absolutely beautiful. If you are looking for something fun and math-y to get for someone for a present, definitely check out their selections!

Revisiting NetLogo’s “Simple Economy” model

A few weeks ago I learned about the “Simple Economy” simulation on NetLogo thanks to a presentation by Bill Rand at the Santa Fe Institute. I shared the program with the boys when I got home from the conference – that project is here:

Sharing the NetLogo “Simple Economy” simulation with kids

and the program itself is here:

NetLogo’s “Simple Economy” simulation

Today I revisited the project with my younger son, though in Mathematica rather than in NetLogo.

First we explored the original problem -> 500 people start with \$100 and play a game. At each step of the game they give \$1 to one of the 500 people selected at random (assuming they have more than \$1). How does the distribution of the money evolve over time?

Then, as in the prior project, we talked about what happens when you give away 1% of your money at each step rather than \$1:

At the end of the last video I asked my son to design a new experiment. Here’s what he decided to do:

(i) If you are below \$50 you have to give only 50 cents
(ii) If you are above \$200, you have to give \$3, and
(iii) Otherwise you give \$1.

We talk through how to code his idea:

After we finished coding we discussed what he thought would happen, but in this video we didn’t yet see what would happen since the simulation was taking a bit longer to run that we’d guessed:

Here are the results of the simulation. My son explains where his guess was right and where it was wrong:

Finally, we wrapped up by looking at some of the individuals and how their money evolved over time:

This “simple economy” model is super fun to play with both on Netlogo and on your own. It still feels like such an unusual result to me, but I’m really enjoying hearing my kids talk about the results of the original model and the modifications that we are playing around with.

Sharing a intro calculus idea with my younger son inspired by Steven Strogatz’s Infinite Series appendix

Last week Steven Strogatz released two previously unpublished appendicies for his book Infinite Powers:

My older son and I did a fun project with Fermat’s idea. He’d taken calculus last year and the ideas Strogatz shared made for a really nice calculus review:

Sharing Appendix 1 to Steven Strogatz’s Infinite powers with my son

My younger son is in 8th grade and has not taken calculus. I thought some of the ideas about finding areas under simple curves would be interesting, so I tried sharing some of those ideas this morning.

We started by taking a look at the first page of Strogatz’s appendix and then talked about finding the area under $y = x^n$ for small values of $n$

Now we moved on to the case $n = 2$. He had the really neat idea of thinking that this piece of the parabola might be a quarter circle. That idea made for a great little exploration:

I asked for another idea had he decided to chop the parabola up into rectangles. This isn’t an idea that came out of the blue because we have talked about some intro calculus ideas before. I was still happy to have this idea jump to the front of his mind, though:

Finally, I shared the full Riemann sum calculation with him so that he could see how to arrive at the exact answer of 1/3. This part was not as much an exploration for him as it was just me showing him now to do the sum. I was ok with this approach as there is plenty of time after 8th grade to dive into the details of Riemann sums:

I’m very happy that Strogatz shared these unpublished appendixes. They are yet another great way for kids to see some introductory ideas from Calculus.

Sharing Appendix 1 to Steven Strogatz’s Infinite Powers with my son

Yesterday Steven Strogatz shared an unpublished appendix to his book Infinite Powers:

I read it and thought it would be terrific to share with my older son who took calculus last year. This year we’ve been working on Linear Algebra – so not a lot of polynomial calculations (yet!) – so I also thought Strogatz’s appendix would be a terrific review.

I had him read the note first and when he was ready to discuss it we began:

At the end of the last video my son had drawn the picture showing Fermat’s approach to calculating the area under the curve $y = x^n$. Now we began calculating. He was able to write down the expression for the approximate area without too much difficulty:

The next step in working through the problem involved some work with a geometric series. Here my son was a little rusty, but I let him spend some time trying to get unstuck:

I just turned the camera off and on at the end of the last video and he continued to struggle with how to manipulate the geometric series into the form we wanted. After a few more minutes of struggle he found the idea, which was really nice to see.

Once he understood the simplification, the rest of Fermat’s proof was easy!

I’m really happy that Strogatz shared his unpublished note yesterday. It is terrific to share with kids who have already had calculus, and would, I think, also be terrific to share with kids studying Riemann sums.

Exploring trig and 2d geometry with 3d printing

This week I’ve been doing a fun 3d printing project with my younger son who is learning trig (from Art of Problem Solving’s Prealgebra book). We have used 3d printing to explore 2d geometry before – see some of the projects here, for example:

3d Printing ideas to explore math with kids

Exploring Annie Perkins’s Cairo Pentagons with kids

Evelyn Lamb’s Pentagons are Everything!

This week I had my son create, code, and then print some simple 2d shapes – the project combines ideas from trig, geometry, and algebra.

Here’s his description of the first shape -> a 3-4-5 triangle:

Here’s the 2nd shape – a 7-6-3 triangle. Creating this shape shows how ideas from introductory trigonometry come into play:

Finally, here’s a regular pentagon that we made yesterday. Unfortunately we made a mistake in the code for the print – mixing up a Sin() and a Cos(), but here is explanation of how to make the shape is correct:

I’d forgotten how useful 3d printing can be as a tool to explore 2d geometry – this week was a happy reminder of how fun those activities can be!

Sharing problems 11 and 12 from Mosteller’s 50 Challenging Problems in Probability with kids

We’ve taken a break from going through Mosteller’s 50 Challenging Problems in Probability, but are back at it today. Problems 11 and 12 are not calculation problems, but rather are discussion problems.

Honestly, at first I wasn’t sure whether or not to even use them, but the discussion turned out to be pretty interesting.

Here’s the first problem:

And the 2nd problem which is similar:

It would be fun to share either of these problems with a classroom – I’m sure the discussion would be great!

Sharing Laura Taalman’s slices with my younger son

I had an opportunity to visit Laura Taalman at ICERM today who made me a copy of her latest creation. I couldn’t wait to get home to share it with my younger son:

He had some interesting ideas about what the shape was in the last video. Now I shared where the slices came from and had him explain how Taalman’s creation worked:

This is such a fun way for kids to experience shapes from a different point of view. I’m really excited to see if we can create some similar objects to play with!

What a kid learning trig can look like

My younger son is working through Art of Problem Solving’s Precalculus book this year. Today we were working through some problems in chapter 4 on right triangle trig. He had been working through this section on his own while I was traveling last week.

One of the problems struck me as one that would make a nice project – the problem gives two lengths of a triangle plus an angle and asks you to calculate the area.

Since the angle is 30 degrees, his solution to the problem did not really use trig:

Now I changed the angle to 40 degrees. My expectation was that this change would not produce a solution that was all that different – wow was I wrong.