Using John Shonder’s amazing US temperature visualization with kids

The videos in this project are a bit longer than what we normally do. Also the 2nd one is badly out of focus even though I didn’t do anything that I know of (!!) with the camera between any of the videos. Oh well, don’t let the length or the focus issues distract from Shonder’s amazing piece of work.

So, last week I saw a really neat tweet about a blog post on Wolfram’s site:

I started the project by showing the boys Shonder’s visual and asking them what they thought about it and what they noticed. At the end I showed them the raw data and we talked about some of the difficulties that come when you are dealing with a big data set:

Next we walked through Shonder’s blog post. I wanted to show the boys that although some of the code looks a little complicated, for the most part Shonder was dealing with ideas that were reasonably easy to understand. So, almost all of the steps and ideas in this presentation were things that were accessible to kids.

Next we stepped through the individual lines of code using our home version of Mathematica. Here we go pretty slowly and carefully through most of the code and discuss (and show) what each command does to the data. I hoped that this slow walk would help the kids see that although the pieces of the code might have looked a little intimidating, it was mostly pretty simple stuff. Happily, the boys seemed to understand almost all of the steps, which was really fun!

Finally, I asked each of the boys to think (off camera) of a follow up project that they thought we could do.

My younger son thought about making a graph showing the percent change in the average temperature. That led to a short discussion of how we’d measure that percent change, which was nice. This idea seems like one that we can implement pretty easily and should be accessible for a 7th grader.

My older son wondered if we could make a prediction about future temperatures. This idea is obviously quite a bit more difficult, but hopefully we can find a way to explore it. One thing that might be fun would be to take the first 50 years of data, use that for a prediction of the next 50 years, and then compare that prediction to what actually happened.

Anyway, we’ll think about how to explore both of the ideas in the next week:

I really had a lot of fun prepping for this project and talking about the ideas (and the implementation in Mathematica) with the boys today. It is really amazing to me that data analysis ideas like the one Shonder is sharing here can be made accessible to kids.

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Using some temperature graphs from Deke Arndt to talk about probability distributions with kids

Via my friend Ed Adlerman, I saw some amazing temperature graphs made by Deke Arndt:

I thought these graphs could be used for a great introductory statistics talk for kids, so I decided to used them to talk about probability distributions today.

We started by looking a probability distributions in a relatively simple situation -> dice:

Next we moved to talking about a probability distribution in a more complicated situation -> a “Galton board”:

Now we moved on to discussing Arndt’s graphs. The conversation about these graphs went on for 11 minutes. As always it is fascinating to hear what kids see when they look at advanced math.

Here’s the first part of that 11 min discussion – here you’ll hear their initial observations and a bit about how to interpret the distributions on display here.

Here’s the second part of our conversation about the graphs. Here we talk about what all of these pictures are telling us about the temperature in Sydney during the last 60 years.

We wrap up by talking about whether or not they liked this presentation and different ways to present the data that would have been made the presentation harder to understand (I though this would be an easier question for kids than finding ways to make it better).

I think Arndt’s work here is amazing all by itself, but is also something that I think can be used really effectively to talk about probability and statistics with kids. Thanks to him for sharing this great work.

Sharing an e surprise with kids

Yesterday I saw a neat request from Sam Shah on twitter asking for ideas about how to “stumble upon” e with kids in Algebra 2 (other than compound interest). I shared an old project we did (and am doing again below) which I think is a terrific way to share a fun and surprising idea about e with kids.

Later in the thread, though, there was a tweet that surprised me:

Strogatz has done more math for the public that just about anyone, and he’s also taught a college course that shared beautiful and advanced ideas in math with students not intending to be math majors, so I was really caught off guard by his thoughts about e.

But rather than getting into an academic discussion about whether or not ideas about e can be shared with Algebra 2 students, I decided to revisit our old project with the boys today.

The idea we’ll take a look at today is this -> Take an NxN set of squares and place a random integer from 1 to N^2 in each of the squares. How many of the integers from 1 to $N^2$ do you expect to not appear in any of the boxes?

I introduced the idea with a 2×2 square and selecting random integers from 1 to 4 by rolling a 4-sided die:

Next we moved on to a 5×5 grid and talked about what we’d expect to happen:

Now we moved to a computer to help us look at the grids more quickly. In this video I explain the program using a few simple examples. The program itself is picking random numbers and counting how often each integer from 1 to N^2 appears in the list of numbers selected.

Although I struggled a little bit with the output of the program (the joy of filming these things live . . . ) we eventually found our way and the kids noticed some potentially interesting patterns in the number counts:

Now we moved up to some larger grids and the kids began to notice more and more patterns in the number counts – :

Finally, we looked at a few very large grids – starting with a 50×50 grid – and the boys began to notice the pattern emerging in the number counts that allowed you to take a guess at each number in the list. It was fun to see them begin to understand these patterns more and more throughout this project:

I guess I’ll conclude by saying that my view differs from Strogatz’s view. I think this project would be appropriate for Algebra 2 kids. It shows them a pretty advanced idea but also gives them a chance to explore that idea using things they’ve learned in K-12 math ranging from simple arithmetic, to a bit of geometry and algebra, and also elementary statistics. I’m happy that we were able to go through this project again today.

Sharing a great Random Walk program with kids

I saw a fun random walk program shared by Steven Strogatz yesterday:

Today I shared the program with the boys. It has 4 different types of random walks to explore. For each one I asked the boys what they thought would happen. At the end we looked at all 4 simultaneously.

Sorry that the starting videos are so blue – I didn’t notice that while we were filming (and didn’t do anything to fix it, so I don’t know why the last two vides are better . . . .)

Also, following publication, I learned the author of the program we were playing with:

Here’s the introduction and the first random walk – in the walk we study here, the steps are restricted to points on a triangular lattice:

In the next random walk, the steps were chosen from a 2d Gaussian distribution. It is interesting to hear what the boys thought would be different:

Now we studied a random walk where the steps all have the same length, but the direction of the steps was chosen at random:

The last one is a walk in which the steps are restricted to left/right/up/down. They think this walk will look very different than the prior ones:

Finally, we looked at the 4 walks on the screen at the same time. They were surprised at how similar they were to each other:

Definitely a fun project, and a really neat way for kids to explore some basic ideas (and surprises!) in random walks.

Using a John Allen Paulos problem to discuss probability and expected value with my younger son

I saw a neat problem from John Allen Paulos earlier in the week:

Today my older son was working on a different math project, so I thought I’d use Paulos’s problem for a nice project with my younger son.

I started by introducing the problem (and forgetting to zoom out after introducing it – sorry about the middle 3 min of this video . . . .). Despite the filming goof, you’ll see my son head down a path that illustrates a common counting mistake.

Now we found that our probabilities didn’t add up to 1, so we tried to found out where we went wrong. Fortunately, my son was able to track down the error.

The next part of the project was discussing the expected amount we’d win playing this game. I didn’t define “expected value” but my son was able to come up with a good way of thinking about the concept.

Finally, we went to the computer to write a little program in Mathematica. This part of the project turned out to be a nice lesson in both simulations and in statistics.

Sharing Ben Orlin’s Math with Bad Drawings with kids -> talking dice and p-values

A few weeks ago I ran across a copy of  Ben Orlin’s Math with Bad Drawings at a book store:

Last night I asked each kid to pick a chapter in the book to read so that we could talk about those chapters for a project today.

My younger son picked the chapter about dice – hardly a surprise as he’s been fascinated with dice forever! Most of the dice you see in this video can be found at the Dice Lab’s website if you are interested in more information about them. Here’s what my younger son had to say about the chapter and about dice this morning:

My older son picked the chapter on p-values – gulp! This topic is pretty advanced and once that isn’t super easy to explore with kids. But I gave it a shot.

First, here’s what he found interesting:

Next I designed a little experiment on Mathematica. For this experiment I wasn’t using p-values but rather confidence intervals – this was just for simplicity, but was still also not super easy for the boys to understand.

In my experiment, I picked 30 numbers from a normal distribution with mean of 5 and standard deviation of 10, and we looked to see if we could tell (statistically) if the mean of the numbers was greater than 0.

What we found was that roughly 25% of the time, 0 was in the 95% confidence interval of the mean. Also, roughly 2.5% of the time, the lower end of that confidence interval was greater than 5 (so we excluded 5 from the confidence interval roughly 5% of the time!).

Hopefully this little experiment helped the kids understand how you could find “wrong” results every now and then:

I love Ben’s book – definitely a fun read and although it isn’t specifically meant for kids, there are plenty of ideas in the book that can be shared with kids.

*Helping kids understand when the Central Limit Theorem applies and when it doesn’t

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:

Sharing an advanced expected value problem from Nassim Taleb with kids

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!