My younger son has a water rocket project for his 6th grade science class. I thought that playing around with his rocket would make for a fun little project this morning because there was an interesting (sort of) math-related question he wanted to answer -> what amount of water would propel the rocket to the maximum height?

What I ended up learning was that being a good 6th grade science teacher is probably a bit out of reach for me . . . .

Here are a few trials (and fails) as we tested various amounts of water:

Here’s our final attempt with 500 ml of water in the rocket and some extra help from me to prevent the rocket from falling over.

Two wrap up we went inside and looked at what data we had.

So, maybe not our best project ever in terms of getting results, but a fun one anyway. Definitely a fun way to spend an hour on Father’s Day.

My older son is studying a bit of introductory statistics right now. I was a little surprised to see this statement in his book:

“. . . you will learn that if you repeat an experiment a large number of times, the graph of the average outcome is approximately the shape of a bell curve.”

I certainly don’t expect middle school / high school textbooks to be 100% mathematically precise, but a little more precision here would have been nice.

For today’s project I decided to show them one example where the statement was true and one where it wasn’t.

For the first example I chose an exercise from the book -> the situation here is a basketball player taking 164 shots and having 64.2% chance of making each of those shots.

Here’s our discussion of that problem (sorry that we were a little clumsy with the camera):

Next we revisited the archery problem that we studied previously. Here’s the problem:

Here’s our discussion of this problem today. It is fascinating to see that even with 100,000 trials both the mean and standard deviation of the outcomes jump all over the place.

I think it is really important to understand the difference between these two different types of experiments. Both situations are really important for understand the world we live in!

We stumbled on this problem in the book my older son is studying over the summer:

A game involves flipping a fair coin up to 10 times. For each “head” you get 1 point, but if you ever get two “tails” in a row the game ends and you get no points.

(i) What is the probability of finishing the game with a positive score?

(ii) What is the expected win when you play this game?

The problem gave my son some trouble. It took a few days for us to get to working through the problem as a project, but we finally talked through it last night.

Here’s how the conversation went:

(1) First I introduced the problem and my son talked about what he knew. There is a mistake in this part of the project that carries all the way through until the end. The number of winning sequences with 5 “heads” is 6 rather than 2. Sorry for not catching this mistake live.

(2) Next we tried to tackle the part where my son was stuck. His thinking here is a great example of how a kid struggling with a tough math problem thinks.

(3) Now that we made progress on one of the tough cases, we tackled the other two:

(4) Now that we had all of the cases worked out, we moved on to trying to answer the original questions in the problem. He got a little stuck for a minute here, but was able to work through the difficulty. This part, too, is a nice example about how a kid thinks through a tough math problem.

(5) Now we wrote a little Mathematica program to check our answers. We noticed that we were slightly off and found the mistake in the 5 heads case after this video.

I really like this problem. There’s even a secret way that the Fibonacci numbers are hiding in it. I haven’t shown that solution to my son yet, though.

Part of the talk was on how algorithms – and black box algorithms, in particular – can create unfair outcomes. O’Neil goes into this topic in much more detail (but also in a very easy to read and understand way) in her book Weapons of Math Destruction.

The talk was part of a conference honoring Harvard math professor Barry Mazur who was O’Neil’s PhD advisor. At the end of the talk one of the questions from the audience was (essentially): What can someone who has a focus on academic math do to help the public understand some of the problems inherent in the algorithms that shape our lives?

O’Neil said (again, essentially) that a good approach would be to find ways to communicate the mathematical ideas to the public in ways that were not “gobbledygook.”

Although I’m not an academic mathematician, this exchange was on my mind and I decided to try out a simple idea that I hoped would help the boys understand how small changes lead can lead to very unequal outcomes. There are no equations in this project, just our new ball dropping machine.

First I asked to boys to look at the result of several trials of the machine dropping balls and tell me what they saw. As always, it is really interesting to hear how kids describe mathematical ideas:

Next I tilted the board a bit by putting a thin piece of plastic under one side. I asked the boys to guess what would happen to the ball distribution now. They gave their guesses and we looked at what happened.

One nice thing was that my younger son noticed that the tails of the distribution were changed quite a bit, but the overall distribution changed less than he was expecting:

I’m sorry this part ran long, but hopefully it was a good conversation.

To finish up the project I tried to connect the changes in the tails of the distribution with some of the ideas that O’Neil talked about on Thursday. One thing that I really wanted to illustrate how small changes in our machine (a small tilt) led to large changes in the tails of our distribution.

I hope this project is a useful way to illustrate one of O’Neil’s main points to kids. Algorithms can create unfairness in ways that are hard to detect. Even a small “tilt” that doesn’t appear to impact the overall distribution very much can lead to big changes in the tails. If we are making decisions in the tails – admitting the “top” 10% of kids into a school, firing the “bottom” 10% of employees, or trying to predict future behavior of a portion of a population, say – that small tilt can be magnified tremendously.

It may not be so easy for kids to understand the math behind the distributions or the ways the distributions change, but they can understand the idea when they see the balls dropping in this little machine.

— Alexander Bogomolny (@CutTheKnotMath) June 6, 2018

I remember this problem from way back when I was studying for the AHSME back in the mid 1980s. I thought it would be fun to talk through this problem with my older son – it has some great lessons. One lesson in particular is that there is a difference between counting paths and calculating probabilities. It was most likely this problem that taught me that lesson 30+ years ago!

So, here’s my son’s initial reaction to the problem:

Next we talked through how to calculate the probabilities. This calculation gave him more trouble than I was expecting. He really was searching for a rule for the probabilities that would work in all situations – but the situations are different depending on where you are in the grid!

Despite the difficulty, I’m glad we talked through the problem.

(also, sorry about the phone ringing in the middle of the video!)

So, definitely a challenging problem, but also a good one to help kids begin to understand some ideas about probability.

My older son is doing a some review this summer in the Integrated CME Mathematics III book this summer. The topic in the 2nd chapter of the book is sequences and series. I thought it would be fun to show him where (at least some of) this math leads. So tonight we talked about some basic ideas in calculus.

First I introduced the topic and reviewed some of the basic ideas of sequences and series:

Now we used the ideas from the first part to find the area under the curve y = x by approximating with rectangles:

To wrap up we extended the idea to find the are under the curve y = x^2 from x = 0 to x = 2. It was fun to see that the basic ideas seemed to makes sense to him.

I was really happy with how this project went. Putting these ideas together to calculate the area under a curve – even a simple curve – is a big step. It might be fun to try a few more examples like these before moving on to the next chapter.

We had only a short time this morning for a project. At this point I should know better than to rush things, but I don’t!

Based on some twitter conversations this week I thought it would be fun to revisit talking about Graham’s number. We’ve done several projects on Graham’s number in the past, but not in at least a few years.

To get started, I asked the boys what they remembered:

Next we talked about one of the simple properties of Graham’s number (and power towers) -> they get big really quickly!

Here we talked about why 3^3^3^3^3 is already as about as large as one of the usual “large” properties listed for Graham’s number. Namely, you couldn’t write down all the digits of this number if you put 1 digit in each Planck volume of the universe:

Next we talked through how to find the last couple of digits of Graham’s number. This part of the project is something that I thought would go quickly, but didn’t at all. Still, it is pretty amazing that you can find the last few digits even though there’s next to nothing you can say about Graham’s number.

If you search for “Graham’s Number” in my blog or in google, you’ll find some other ideas that are fun to explore with kids. I highly recommend Evelyn Lamb’s article, too:

Last week we bought two packs of skew dice from the dice labs:

Today we decided to try a simple statistical test of the disc. Before we dove into that test, though, I asked the boys to tell me how they thought you might go about testing these dice to see how random they were.

We rolled the dice off camera. We rolled them in groups of 10 with each of us shaking up the dice in a container before each roll. After 12 rounds we had a total of 120 numbers -> here are the results:

Although we didn’t really have enough rolls to make a definitely statement about the dice, I think this was a nice way for the kids to see how a simple statistical test would work. I hope the kids are interested in playing around with these dice a bit more.

My older son is working thorugh the Integrated CME Project Mathematics III book this summer. Last week he came across a pretty interesting problem in the first chapter of the book.

That chapter is about polynomials and the question was to find a polynomial with integer coefficients having a root of . The follow up to that question was to find a polynomial with integer coefficients having a root of .

His original solution to the problem as actually terrific. His first thought was to guess that the solution would be a quadratic with second root . That didn’t work but it gave him some new ideas and he found his way to the solution.

Following his solution, we talked about several different ways to solve the problem. Earlier this week we revisited the problem – I wanted to make sure the ideas hadn’t slipped out of his mind.

Here’s how he approached the first part:

Here’s the second part:

Finally, we went to Mathematica to check that the polynomials that he found do, indeed, have the correct numbers as roots.

I like this problem a lot. It is a great way for kids learning algebra to see polynomials in a slightly different light. They also learn that solutions with square roots are not automatically associated with quadratics!

My younger son has been studying Art of Problem Solving’s Introduction to Geometry book this year. He’s been doing most of the work on his own. Every now and then, though, I decided to check in and see how things are going. Today the topic he was looking at when I checked in was constructions. The two questions on the table when I stopped by were:

(i) Construct the incircle of a triangle, and

(ii) Construct the circumcircle of a triangle

Here’s how things went.

(i) The incircle

He has the basic idea for how the construction works, but misses one important idea. That idea is that the tangents of the incircle are not the feet of the angle bisectors. My guess is that this is a fairly common point of confusion for kids learning this topic:

(ii) the circumcircle

Here his solution was completely correct. He had a tiny bit of trouble in the beginning figuring out to construct the perpendicular bisectors, but he worked through that trouble fairly quickly.

It is always interesting to hear the ideas that kids have when they talk through a mathematical process. For me it was especially nice to hear that most of the ideas he’s learning as he works through this geometry book are sinking in pretty well.