I started the project today by asking my son to tell me some things he knew about primes. He gave the definition of a prime numbers, explained how we know that there are infinitely many primes, and talked about twin primes, though he apologized for not knowing how to prove that there were infinitely many twin primes:

Next I showed him the polynomial and we talked about this equation producing a lot of primes.

Now we went to the “prime time” explorable and my son talked about what he saw in the first two examples -> the Ulam spiral and the Sack spiral.

Finally we looked at the last two patterns -> the Klauber triangle and the Witch’s spiral.

I saw some really neat tweets from John Carlos Baez and Greg Egan on Penrose-Terrell rotation last week:

A cube whizzes past at nearly the speed of light, c. What do you see? Lorentz contraction squashes the cube. But you *see* it as rotated and bent, since light from different parts of the cube take different amounts of time to reach you – and it's moved a lot by then!

[1/3] At @johncarlosbaez’s suggestion, I made some animations showing how Terrell Rotation [the distortion in the *appearance* of an object moving sideways https://t.co/vyt148Znut ] looks for both relativistic physics and Galilean physics with a finite speed of light. pic.twitter.com/jptwIJey5K

[Correction] I missed some of the true weirdness in the faster-than-light non-relativistic version. In that case, an object’s world-line can cross the incoming light cone twice, so parts of the cube can have two distinct images at the same time. pic.twitter.com/7RaXu7NAsd

Even though even the most basic ideas from relativity are far outside of what kids can grasp, I thought it would be fun to share these animations with my younger son. The animations in the above tweets are definitely something that kids can appreciate, and I was excited to hear what my son would have to say.

So, I started out the project today asking my son to describe anything he knew about relativity and then what he thought he’d see if a cube passed by him really fast:

Next we talked about some simple ideas from relativity and what impact those ideas might have on a cube passing by. Also, since he’s just starting to learn about square roots and quadratics in school, I showed him the Lorentz contraction formula and we did one simple calculation:

Finally, we went to the computer to look at the tweets and animations from John Carlos Baez and Greg Egan that I linked above. As always, it is really fun to hear a kid react to and describe ideas from advanced math (and physics!):

Though it would be fun to try . . . and the game arrived today!

Tonight I had the chance to play with my younger son – he didn’t know what gerrymandering was, but we talked a little bit about the idea as we opened the box:

Here’s a quick peek at the game set up:

Finally – here’s what the game looked like at the end and then a short discussion of some of the mathematical ideas for kids that come through in the game. What ideas can help you win? What can you do to cause your opponent some problems?

I really like this game a lot. It is easy to play right out of the box – the directions probably took under 5 min to read and understand. The game is a great way to teach important ideas about gerrymandering and also has great math ideas in it for kids. I highly recommend it – thanks to Jordan Ellenberg for the recommendation.

I saw a really great thread on twitter this week and wanted to share some of the ideas with the boys for our Family Math project today:

My observation that you can sum the first n cubes to get 1^3 + … + n^3 = [(n*(n+1)]^2 by counting rectangles in an nxn square is of course well-known (as is everything simpler and clever).

We started off looking at the sum 1 + 2 + 3 + . . . .

Next we looked at the sum of squares and searched for a geometric connection:

Now I showed them the fantastic way of looking at the sum of squares in the Jeremy Kun blog post. This method is a terrific way to share an advanced idea in math with kids – it is totally accessible to them and gives them a chance to talk through a fairly complicated idea:

Finally, I showed how the ideas we were just talking about extend to some of the basic ideas is calculus. It was neat to hear my younger son talking through the ideas here, too:

Definitely a neat morning – it is always amazing to see the connections between arithmetic and geometry.

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.

About a week ago I saw a great tweet from Dave Richeson about some journal articles that were being made freely available for the month of March:

Editors from Math Horizons (me), the American Mathematical Monthly, Mathematics Magazine, and College Mathematics Journal each chose five articles about π, which will freely available from now until the end of March. Enjoy! https://t.co/tcuSxxykBbpic.twitter.com/LXyWMVJ71K

The paper from the The College Mathematics Journal by Susan Jane Colley caught my attention for being both an really neat result and being explained at a level a student taking calculus could understand.

So, this morning to celebrate Pi day I decided to use the paper to talk a bit of calculus with my son. Pulling all of the different ideas together was challenging for him, so we went slow but still made it through the main points in about 30 min.

We took a quick look at the paper and then started digging into the math by looking at the famous alternating series for .

I should say for clarification that I forgot to look up Susan Jane Colley’s current position before we started the project and wasn’t sure if she was still at Oberlin or had moved to a different university since the paper was published in 2003. But to be clear, she is the Andrew & Pauline Delaney Professor of Mathematics at Oberlin.

Next we dove in to the connection between the alternating series and . I thought I’d try to introduce the connection in a sneaky way, but it was sort of a dead end. Eventually, though, he thought about arctangent.

At the end of the last video the formula for an infinite geometric series came up, but that formula wasn’t quite at the top of his head. So we took a little detour to re-derive that formula. Once we had that formula we could see that the alternating series we were looking at converged to :

Now we looked at the main result of the paper – a different series for that converges really fast.

Here we look briefly at the formula for this series (sorry for the reading typos by me – trying to read the paper and stay behind the tripod and not cast a shadow was hard . . . . )

Finally, we went to Mathematica to evaluate the integral and look at the speed of convergence of the two series we’ve been studying:

I think that Colley’s paper is absolutely terrific and a great resource to use to show calculus students some advanced math. It is an extra terrific resource to use on Pi day 🙂

My son is in the middle of a long review for the BC Calculus exam in May. Right now he’s doing some integration practice and came across this integral today:

Although the discussion hasn’t actually happened yet because he’s currently evaluating the integral, this integral will lead to a very good discussion this morning pic.twitter.com/LDlyq2ZKcx

We didn’t get a chance to talk about it too much this morning, but we did review the problem when he got home tonight.

First we I asked him to approach the problem as he did this morning -> by trying to evaluate the integral. The practice integrating rational functions turned out to be useful:

Next we took a more geometric approach:

Finally, I wanted to show him how you could see that the integral was zero from the u-substitutions that he made in the first video. Even though this method wasn’t really the best one for this particular integral, I still wanted him to see how the algebra worked out:

The excel random seat generator placed all 3 Matts in the same group today. Some are calling shenanigans…randomness is a goofy thing! pic.twitter.com/Km34q4ZgNi

For today’s Family Math problem I thought I’d talk through the math behind this situation with my younger son.

First I showed him what had happened and asked for a few ideas about how to approach finding the probability that all of the Matt’s would end up in group d. It has been a while since we’ve done a probability and counting project, but the ideas came back to him as we talked:

Next we talked about how to calculation the number of different ways to put the students into the the groups of 5 and 4. Then I asked my son to estimate what the number was without calculating. His estimate was off and we returned to why in part 4 of the project:

Now we calculated the number of ways to arrange the students with all 3 Matt’s landing in group D. Once we had that number we found the probability of the outcome that happened. I also asked my son if he thought the outcome happened by random chance, or was Bob possibly playing a little joke:

Finally, we revisited the calculation from part 2 of the project. It turned out that his estimate actually was nearly right – investigating where it went wrong was a good use of time. Once we had the exact value by hand, we also computed the exact probability for the original problem by hand.

Thanks to Bob Lochel for sharing this fun outcome. If you’d like to see a similar probability / counting problem check out our exploration of the “Snapchat problem” from a few years ago:

We had a snow day today and I was able to use some extra time with my son to talk about numerical integration techniques. I’d guess that I hadn’t seen the topic in at least 25 years, so it was fun to teach and review at the same time!

This evening I had my son review the ideas using an integration problem from the 2012 BC Calculus exam. I started by having him evaluate the integral exactly with no numerical integration techniques:

Next we used the trapezoidal rule and compared the answer to the exact answer –

The one thing I learned today was that the midpoint rule was roughly twice as accurate as the trapezoidal rule. Here my son used the midpoint rule and we do find that the answer is closer to the true value than the answer we found in the last video:

Finally, we used Simpson’s rule. Despite one little mistake in the middle, we ended up finding an answer that was surprisingly close to the exact asnwer.

I saw a really neat tweet from James Tanton yesterday:

I've started a new project on quadratics – simple videos of me chatting in my home and matching PDF essays (perhaps better). The first three are up, all about the story of COMPLETING THE SQUARE. What do folk think? https://t.co/NxsCdZnMRR@cheesemonkeysf @mythemathics @k8nowak

The video in the “setting the scene” link is terrific and I had my younger son watch it this morning. I can’t figure out how to get the video to embed, so here’s a direct link to the page it is on:

One reason that I was extra excited to see Tanton’s video is that by total coincidence I’d used essentially the same idea with my older son to explain why a negative times a negative is a positive:

But I’d never gone through the same ideas with my younger son – at least not that I can remember. So, for our Family Math project today I had him watch Tanton’s video and then we talked about it.

Here’s what he thought about the video and area models in general – I was really happy to hear that my son liked Tanton’s area models and thought they were a really great way to think about multiplication and quadratics:

Next I had my son walk through the negative / positive area ideas that Tanton used to talk about multiplication. He did a really nice job replicating Tanton’s process. I think this is a great way for kids to think about multiplication: