# A fun prime problem I saw in an Evelyn Lamb tweet

Saw this sequences of tweets because of Evelyn Lamb yesterday:

It was a fun problem to play around with and, frankly, solving it was the most impressive thing I did at the gym today so I’ve got that going for me . . . .

My younger son was home sick today, but he was feeling better this afternoon so I thought it would be interested to see what he thought about the problem. It turned into a really nice talk about numbers and arithmetic.

Here’s the introduction to the problem and his first 5 minutes of work:

and here’s his work up to the solution:

This is a really great problem to get kids talking about numbers, arithmetic, and primes. Testing whether the various multiples of 6 are near a prime is a good challenge for kids, too!

# Using Matt Parker’s Menger Sponge video to talk fractions with kids

Saw Matt Parker’s latest video via an Evelyn Lamb tweet yesterday:

Here’s the video:

Watching the video last night I thought that there’d be a lots of different ways to use this one with kids. I chose to use it as a way to talk about fractions and scaling. Kids, I think, will be surprised by the result involving $\latex \pi$ but diving into that part is probably too much for kids. They can appreciate the result, though, and the discussion of fractions and scaling leads right up to the Wallis formula.

I started by asking the kids what they thought about Parker’s video. We talked about their thoughts as well as a few other fractal shapes that the knew:

Next we talked about Sierpinski’s carpet and walked through the calculations for the area. The boys saw the area essentially as a subtraction problem, so I spent a lot of time trying to help them understand how to see it as a multiplication problem:

I think that the boys still didn’t really see the area change from one step to the next as a product, so we spent a little more time talking about that idea in the modified Sierpinski’s carpet that Parker talks about in the video. The way the area changes from step to step is a great fraction problem for kids.

I’m sorry that we got a little bogged down in the calculations here, but I really wanted to be sure that the kids saw the relationship between the subtraction and multiplication approaches to calculating the area.

Finally, we looked at the complete calculation for the area of this modified Sierpinski carpet. The boys noticed a pattern in the products, which was cool, and we were able to transform our product into the famous Wallis product.

So, a fun project for kids. Parker’s video is great and serves as a great motivation for diving into the calculations. The calculations themselves are a great exercise in fractions for kids.

I’d like to try to work through the 3d version, too, so maybe we’ll do that tomorrow. My back of the envelope calculation tells me that the 4d version doesn’t have the same property of having the “volume” of the 4-d sphere (which is $\pi^2 / 2$, though I may not have done the calculations correctly the first time around and will revisit them later this weekend, too.

# My fun interaction with prime numbers this week

Last week I saw a amazing new result about primes by two mathematicians at Stanford – Robert J. Lemke Oliver and Kanna Soundararajan – via an Evelyn Lamb article:

Peculiar Pattern found in “Random” Prime Numbers by Evelyn Lamb

Erica Klarreich at Quanta magazine also wrote a fantastic article about the result:

Mathematicians Discover Prime Conspiracy by Erica Klarreich

and there’s also a neat discussion of the result on Terry Tao’s blog:

Terry Tao’s blog post about the new result

After seeing the two articles (I only saw Tao’s blog post today) I thought it would be fun to play around with some similar ideas and chose to look at the last digits of triples of consecutive primes. Over the course of the week I was able to use a simple program in Mathematica to count how often the different triples of last digits occur in consecutive primes in the first 10 billion primes. Right from the start I found something I didn’t expect – counting the occurrence of the triples of last digits seemed to pair the sets of last digits quite naturally into groups of 2.

For example, for 3 consecutive primes in the first 10 billion prime numbers the last digits (3,7,1) occur 178,500,881 times and the last digits (9,3,7) occur 178,500,928 times. Another example of the strange grouping is that the triple (1,1,3) occurs 147,750,170 times and the triple (7,9,9) occurs 147,761,746 times. Weird – what’s causing this clustering?

All of my data is in the google doc linked below. I’m sorry that the data in the google doc isn’t organized very well – I was just playing around for myself, but thought that it might be fun to share anyway:

My google doc with all of the data I collected this week

I didn’t really study any number theory in college or graduate school, so I have essentially no way to know if something like the counts for the last digits of consecutive prime triples pairing up is an easy to prove fact or an impossible to prove fact. After thinking about the strange groups of two for a few days without having any decent ideas I sent an e-mail to authors of the new paper and asked them for help. They wrote back last night – which was super cool! – and provided a (possibly) easy way to think about it. I sort of can’t believe that they wrote back, but I’m really excited to spend a bit more time trying to understand their explanation.

Receiving their e-mail got me even more interested in / excited about their paper, so I spent several hours today going through it one more time. The results and conjectures are general enough to apply to the problem of consecutive triples and that led me to try to see if the paper could help me get a better understanding of the data I’d collected. Happily, I was able to understand a bit more of the paper the 2nd time through,

With sort of an “I know enough to be dangerous” understanding I attempted to predict the number of various prime triples in the next set of 1 billion primes (so, last digits of three consecutive primes from the 10 billionth prime number to the 11 billionth prime number). My guesses are in column R and column U of the “Approximations” tab in my google doc. The results should be in tomorrow morning 🙂

One fun thing about the two sets of guesses is that the sum of the guesses for all of the triples adds up to almost exactly 1 billion! Since I’m looking at 1 billion primes the sum be 1 billion, but I didn’t take that constraint into account (not directly anyway) when I was playing with the numbers.

One other bit of structure I was able to notice in the data after re-reading the paper today was a different set of clustering. The triples with three of the same numbers have the lowest counts, triples with two of the same number in a row have (generally) the next lowest counts, triples with two numbers that are the same, but not in a row have (generally) the next lowest counts, and triples with three different numbers have (generally) the highest counts. *I think* their paper predicts this ordering.

So, a really fun week of playing around with prime numbers. There are still a few things to think about – the e-mail from the paper’s authors, and seeing if there’s any way to improve the predictions – but I’m extremely happy with how this little side project went this week. Haven’t had that much fun learning new math in a long time 🙂

# Looking at the new discovery about primes with kids

Earlier in the week Robert J. Lemke Oliver and Kannan Soundararajan of Stanford announced a totally surprising discovery about prime numbers. Evelyn Lamb has a fantastic article about the paper here:

Peculiar Pattern Found in “Random” Prime Numbers

The new paper suggests that the prime numbers are not distributed quite as randomly as mathematicians had previously expected. In particular, the last digit of consecutive prime numbers has a distribution that is different from what you’d expect if the distribution of primes was random.

I thought this would be a fun result to discuss with kids. One surprising thing about this result is that it is pretty easy to understand. In fact, you can double check the result on a computer really quickly.

Before jumping in to the result about primes, though, I wanted to spend a few minutes talking about probability and probability distributions. This part of the project turned out to have some extra fun when my older son asked a neat question about dice.

Here’s the introduction to today’s project and my son’s question – what is the probability of seeing at least one 5 when you roll two dice?

With the complimentary counting problem behind us, we moved on to talk a little bit about the difference between probability and probability distributions. Once again we had a little detour following a statement that my older son made. I was happy to have these extra little conversations about probability this morning:

The next part of the project involved talking about prime numbers. We talked a little bit about how mathemticians viewed the primes. Number theory and prime numbers are not my field – hopefully the details in this part are right. A great (and accessible) read about prime numbers and randomness can be found in Jordan Ellenberg’s How Not to be Wrong.

It was really fun to hear what the kids had to say about the last digit of consecutive prime numbers here. This problem is a great way to get kids thinking about math.

Now we moved to the computer and used a little Mathematica program to study what the distribution of the last digit of consecutive prime numbers looked like. We chose to look at prime numbers that ended in 1. It was neat to see the results. Loved hearing what my younger son observed: “it seems like there’s a lot less primes ending in 1”:

We wrapped up by looping through the first 25,000,000 primes and compared our results to the results that were given in Evelyn Lamb’s write up from above. Our results were really close – yay!

So, I think that talking about this new discovery makes for a really fun project for kids. I’m sure that our project could be improved quite a bit by any mathematician who had a good understanding of number theory (since my understanding of that subject is essentially zero!), but even the high level walk through that we did today was fun. It is pretty amazing to find a new discovery about prime numbers that can be understood by kids!

# Amazing math from mathematicians to share with kids

About two years ago I saw this Numberphile interview with Ed Frenkel:

One of the ideas that Frenkel mentions in the interview is that professional mathematicians haven’t done a good job sharing math with the general public. Although I’m not really the kind of professional mathematician Frenkel was talking about, I took his words to heart and have been on the lookout for math to share – especially with kids.

It turns out that there are some fantastic ideas that are out there for kids to see. Some surprising fun I had sharing Larry Guth’s “no rectangles” problem with kids earlier this week (see below) made me want to share some of the ideas I’ve found in the last couple of years, so here are a few examples:

(1) One of the most incredible lectures that you’ll ever see is Terry Tao’s “Cosmic Distance Ladder” lecture at the Museum of Mathematics in New York City:

I used Tao’s video for three projects with my kids – but there are probably 20 math projects for kids you could get out of it.

Part 1 of using Terry Tao’s MoMath lecture to talk about math with kids – the Moon and the Earth

Part 2 of using Terry Tao’s MoMath lecture to talk about math with kids – Clocks and Mars

Terry Tao’s MoMath lecture part 3 – the speed of light and paralax

(2) The Museum of Math’s public lectures are a great source beyond Tao’s lecture.

Here’s a project based on Bryna Kra’s lecture:

Angry Birds and Snap Cubes – Using Bryna Kra’s MoMath public lecture to talk math with kids

Eric Demaine’s lecture was part of our Fold and Cut theorem project:

Fold and Cut part 3

and I can’t say enough good thinks about Laura Taalman’s work – she’s inspired dozens of our projects.  Just search for her name on the blog:

(3) and Speaking of Fold and Cut . . .

Katie Steckles and Numberphile put together an incredible video about the Fold and Cut theorem. I used the video this week for project with 2nd and 3rd graders at my younger son’s school earlier this week.  Steckles’s presentation is so incredible – this is the kind of math that really inspires kids:

We used it for three projects (including the Eric Demaine one above):

Our One Cut Project

The Fold and Cut Theorem is Awesome!

In prepping for the grades 2 and 3 projects I also totally coincidentally ran across a “fold and punch” exercise that is a great activity to try with kids before trying out fold and cut:

(4) Another great success with the 2nd and 3rd graders was Larry Guth’s “no rectangles” problem. I had a great time playing around with this problem with my kids, but nothing prepared me for how enthusiastic the kids in the two programs were about this problem.

Larry Guth’s “No Rectangles” problem

After the 3rd grade night, Patrick Honner sent me this picture that I used to wrap things up with the 2nd graders.

(5) The Surreal Numbers

I’d seen John Conway’s surreal numbers previously via an amazing Jim Propp blog post:

and I wanted to revisit them after finally reading Donald Knuth’s book:

Revisiting the Surreal Numbers

Infinity + 1 and other Surreal Numbers

Playing with the surreal numbers via checker stacks is an incredibly engaging way for kids to learn about mathematical thinking.

(6) Speaking of John Conway –

In the 2014 edition of the Best Writing in Mathematics Conway had an article about variations on the Collatz conjecture. It was a fascinating article that even gave us the idea to translate some of the math into music.

The Collatz Conjecture and John Conway’s “Amusical” variation

I’ve also talked with the boys about the standard version of the Collatz conjecture:

It is a great way to introduce kids to an unsolved problem in math while also sneaking in a little bit of arithmetic practice!

(7) Occasional contest math problems

I happened to run across another MoMath lecture yesterday – this one by Po-Shen Loh. He was talking about “Massive Numbers.” I thought maybe he’d be talking about the book “Really Big Numbers” by Richard Evan Schwartz:

A few projects for kids from Richard Evan Schwartz’s “Really Big Numbers”

or maybe Graham’s Number:

An attempt to explain Graham’s number to kids

The last 4 digits of Graham’s number

His presentation is fascinating and I even talked through the first version of the problem with my younger son:

Show that any positive integer n has a (positive) multiple which has only the digits 1 and 0 when represented in base 10.

A challenging arithmetic / number theory problem

(8) Building off of popular books by mathematicians as well as public lectures

I was surprised at how much great math writing and speaking there has been for the general public in the last couple of years.

Jordan Ellenberg’s “How not to be Wrong” inspired several projects – probably my favorite was using his idea of “algebraic intimidation” to talk about the famous 1 + 2 + 3 + . . . = -1/12 video by Numberphile. :

Jordan Ellenberg’s Algebraic Intimidation

Jacob Lurie’s Breakthrough Prize public lecture inspired two projects about a year apart from each other:

Using Jacob Lurie’s Breakthrough Prize Lecture to Inspire Kids

Using Jacob Lurie’s Breakthrough Prize talk with kids

And, Ed Frenkel, who got me thinking about sharing advanced math with kids in the first place has inspired a few projects, too:

Fine Ed Frenkel – you convinced me

Ed Frenkel, the square root of 2, and i

and one of my all time favorites:

A list Ed Frenkel will love

(9) Finally, it would be impossible to write a post like this one without mentioning the work that Evelyn Lamb is doing writing math articles for the general public. I’ve lost count of how many projects she’s inspired, but it is probably well over 20. I’m especially grateful for her talk about topology which have generated really fun conversations with the boys. For example:

Using Evelyn Lamb’s Infinite Earring with kids

Evelyn Lamb’s fun torus tweet

and

Henry Segerman’s Flat Torus

which arose after Lamb pointed out this video:

So, I’m really happy that mathematicians are sharing so many amazing ideas. I think this is the sort of math promotion that Frenkel had in mind. Hopefully it continues for many years to come 🙂

# Henry Segerman’s Flat Torus

[Sorry for writing this one up at one million miles per hour. We are seeing Star Wars at 9:30 am today and I wanted to get this one published before we left. Hopefully the fun of this amazing shape outweighs the quick and unedited writing.]

Saw another amazing post from Evelyn Lamb a few weeks ago:

In that post she showed this video by Henry Segerman:

I decided to buy this shape from Shapeways and do a project with the boys. Here’s where you can order it:

Henry Segerman’s Flat Torus on Shapeways

Before starting the actual project this morning I had the kids explore the shape on their own. Although these individual explorations were a little more difficult than I was expecting I’m glad that we did them because it gave the kids a bit more familiarity with the shape. The videos from that part of the project are here:

Background for our project on Henry Segerman’s Flat Torus

So, with that background out of the way, we began today’s project by talking about how to make Möbius strips. The discussion to a little detour when I asked them to describe how to make the shape, but it was a cool and important detour. It is neat to hear kids talk about complicated shapes!

Next we talked about how to make a torus from a single sheet of paper. The kids had a hard time imagining exactly how a square would fold up into a torus, but that’s sort of the point of the project! At the end we looked at how the shape was sort of like a bagel, but sort of not like a bagel!

For the 3rd part of our project we looked more carefully at Henry’s shape an how a flat torus was different than the bagel.

The last part of the project was a quick discussion of the Klein Bottle and the projective plane, which we can also understand from a single square piece of paper. The boys were really interested in the ideas here, and especially how difficult it was to make the Klein bottle in three dimensions! They did make a connection between the Klein bottle and the Möbius strip, which was really cool to hear.

So, a super fun project exploring some really complicated shapes. Infinite thanks to Evelyn Lamb and Henry Segerman for inspiring this project!

# Frank Farris’s patterns

A couple of weeks ago Evelyn Lamb’s article Impossible Wallpaper and Mystery Curves introduced me to Frank Farris’s work. On Saturday I stumbled on his book at Barnes and Noble:

I was excited to try out some of his ideas with the boys even though they use complex numbers and exponentials which are over their heads. We did the whole exploration this morning using Mathematica.

To start, we just explored the exponential function.

Next we moved to looking at sums of two exponential functions. The boys were surprised by the graphs and we played around with a few more examples. They had some interesting ideas about what the pictures looked like, and I’m glad that the pictures also reminded them a little of Anna Weltman’s loop-de-loops.

Next we moved on to sums of three exponential functions motivated by the idea of trying to produce another kink in the loop. There was a little discussion at the beginning of this part of the talk about complex variables. I thought going down this path was going to be too difficult to explain, so I tried to bring the conversation back to the sums. I love the ideas they had about symmetry here.

Next we looked at Farris’s “mystery” shape and played around a bit more with the ideas. These shapes also led to fun conversations about symmetry:

Finally, I let the kids just play around. As I was writing up this project I got a “hey dad, come here and look at this cool shape” call:

So, despite the math underlying these shapes being a little over their heads, the kids seemed to really enjoy seeing these shapes. I loved hearing their ideas and I loved seeing them play around with the ideas for a long time after we turned off the camera.

Also, Farris’s book is absolutely amazing – you’ll love the ideas and the presentation, and probably most of all the incredible pictures he creates from the ideas!

# Geometric Constructions with Origami

Yesterday I saw a great tweet from Evelyn Lamb:

I’d never see constructions using origami before, so the idea that you could trisect an angle using paper folding was brand new to me. One of Zsuzsanna Dancso’s comments in the Numberphile video made me believe that you could also solve the “doubling a cube” problem using paper folding. Sure enough that construction is here:

How to double a cube with origami from Cutoutfoldup.com

The combination of that construction and the construction in the Numberphile video made for a great Family Math topic for today.

I started by talking through some basic ideas of compass and straight edge constructions from Euclidean geometry. My older son and I have touched on this topic in our geometry work, but my younger son has never seen it. Because this topic was new to my younger son I didn’t want to go that much into detail. The main idea for today was to introduce the three famous impossible constructions: (1) doubling a cube, (2) trisecting an angle, and (3) squaring a circle and then show how to solve (1) and (2) in the last two videos.

Our first origami construction was solving the “doubling a cube” problem by constructing $\sqrt[3]{2}.$ The directions I linked above are really easy to follow, so each of us made our own version of this one:

The last thing we did was attempt the “trisect an angle” origami construction that Zsuzsanna Dancso demonstrates in the Numberphile video. This construction is a little bit more difficult than the doubling a cube one, so we worked on this one together rather than making three separate constructions. I also used a ruler to draw in some of the lines just to speed things up, but it is easy to see that we could make the same lines by folding.

So, not as much mathematical detail in this one, but some fun history and some fun constructions. I wasn’t aware of the idea of origami constructions before seeing the Numberphile video, so this project had a little bit of extra fun for me because the kids and I got to learn something new together!

# 10 fun math things from 2014

I’ve been paying attention to math a little more in 2014 than I have in previous years and thought it would be fun to put together a list of fun math-related things I’ll remember from this year:

(10) Dan Anderson’s “My Favorite” post

Dan asks his students to talk about things they would like to learn more about in math class, and the students talked about subjects ranging from topology to diving scoring. I was really happy to see the incredibly wide range of topics that the kids thought would be interesting. Beautiful post by Dan and a fantastic list of topics chosen by his students – this one made a big impression on me:

Dan Anderson’s “My Favorite” post

My initial reaction to Dan’s post is here:

A list Ed Frenkel will love

(9) Laura Taalman’s Makerhome blog:

We bought a 3D printer early in the year and it allowed us to do a bunch of math projects that wouldn’t have occurred to me in a million years. Most of those projects came either directly or indirectly from reading Laura Taalman’s 3D printing blog. As 3D printing becomes cheaper and hopefully more available in schools, Taalman’s blog is going to become the go to resource for math and 3D printing. It is an absolute treasure:

Laura Taalman’s Makerhome blog

An early post of mine about the possibilities of 3D printing in education is here:

Learning from 3D Printing

and one of our later projects is here:

Klein Bottles and Möbius Strips

(8) Numberphile

It has been nearly a year since Numberphile’s fun infinite series video hit the web. I know people had mixed feelings about it, but I loved seeing a math video spark so many discussions:

I’ve used so many of their videos to talk math with my kids, I’m not even sure which of them to pick for examples. Here are two:

Using Numberphile’s “All Triangles are Equilateral” video to talk about constructions

Some fun with Numberphile’s Pythagorean Theorem video

(7) Fields Medals and the Breakthrough Prizes

Erica Klarreich’s coverage of the Fields Medals over at Quanta Magazine was absolutely amazing. Two of her articles are below, but all of them (including the videos) are must reads. Her work her made it possible for anyone to meet the four 2014 Fields medal winners:

Erica Klarreich on Manjul Bhargava

Erica Klarreich on Maryam Mirzakhan

A really cool opportunity to understand the work of one of the Fields Medal winners came when the Mathematical Association of America made an old Manjul Bhargava’s paper available to the public. I had a lot of fun playing around with this paper (that he wrote as an undergraduate, btw). It made me feel sort of connected to math research again:

A fun surprised with Euler’s identity coming from Manjul Bhargava’s generalized factorials

The Breakthrough Prizes in math didn’t seem to get as much attention as the Fields Medals did, which is too bad. The Breakthrough Prize winners each gave a public lecture about math. Jacob Lurie’s lecture was absolutely wonderful and a great opportunity to show kids a little bit of fun math and a little bit about the kinds of problems that mathematicians think about:

Using Jacob Lurie’s Breakthrough Prize talk with kids

I’m glad to see more and more opportunities for the general public to see and appreciate the work of the mathematical community. Speaking of which . . . .

(6) Jordan Ellenberg’s “How Not to be Wrong”

Jordan Ellenberg’s book How not to be Wrong is one of the best books about math for the general public I’ve ever read. I have it on audiobook and have been through it probably 3 times in various trips back and forth to Boston. My kids even enjoy listening to it – “consider the set of all integers plus a pig” always gets a laugh.

One of the more mathy takeaways for me was his discussion of infinite series and what he calls “algebraic intimidation.” Both led to fun (and overlapping) discussions with my kids:

Jordan Ellenberg’s “Algebraic Intimidation”

(5) The Mega Menger Project

The Mega Menger project was a world wide project that involved building a “level 4” Menger sponge out of special business cards. We participated in the project at the Museum of Math in NYC. The kids had such a good time that they asked to go down again the following weekend to help finish the build.

It was nice to see so many kids involved with the build in New York. It also made for another fun opportunity to explore the math behind the project a little more deeply:

The Museum of Math and Mega Menger

(4) People having a little fun with math and math results

For some family fun, check out the new game Prime Climb:

our review is here:

A review of Prime Climb by Math for Love

Also, don’t forget to have a little fun when tweeting about new and important math results. Like Jordan Ellenberg tweeting about the solution of an old Paul Erdos conjecture:

Erica Klarreich’s Quanta Magazine article on the same result was just published yesterday by coincidence:

Erica Klarreich on prime gaps

For me the math laugh of the year was Aperiodical announcing the results of an 8 year search confirming the 44th Mersenne Prime:

(3) Evelyn Lamb’s writing

Evelyn Lamb’s blog is a must read for me. I love the wide range of topics and am pretty jealous of her incredible ability to communicate abstract math ideas with ease. Her coverage of the Heidelberg Laureate Forum was sensational (ahem Breakthrough Prize folks, take note!). This post, in particular, gave me quite a bit to think about:

A Computer Scientist Tells Mathematicians How To Write Proofs

My thoughts on proof in math are here:

Proof in math

Away from her blog, if you want a constant source of fun and interesting math ideas just follow her on Twitter. For instance this tweet:

led to a great little project with the boys:

Irrationality of the Square root of 2

(2) Terry Tao’s public lecture at the Museum of Math

On of the most amazing lectures that I’ve ever seen is Terry Tao’s public lecture at the Museum of Math. I don’t know how it had escaped my attention previously, but I finally ran across it about a month ago. What an incredible – probably unparalleled – opportunity to learn from one of the greatest mathematicians alive today:

Explaining a few bits of his talk in more detail led to three super fun projects with the boys:

Part 1 of using Terry Tao’s MoMath lecture to talk about math with kids – the Moon and the Earth

Part 2 of using Terry Tao’s MoMath lecture to talk about math with kids – Clocks and Mars

Part 3 of using Terry Tao’s MoMath lecture to talk about math with kids – the speed of light and paralax

(1) Fawn Nguyen’s work

When one of the top mathematicians around is tweeting about projects going on in a 6th grade classroom 2000 miles away, the world is working the right way!

Fawn is producing and sharing some of the most interesting math projects for kids that I have ever seen, and I’m super happy that her work is getting recognized. She’s probably inspired more than 20 projects with the boys, and I can’t wait for the next 20 in 2015. Here are two from this year:

Fawn Nguyen’s Geometry Problem

A 3d Geometry proof without words courtesy of Fawn Nguyen

If you have even a passing interest in fun, exciting, and generally kick-ass math projects for kids – you have to follow Fawn.

# Popular math

I’m sitting in our room in the Great Wolf lodge outside of Philadelphia. It is just after 8:00 and both kids are already sacked out! Catching up on the news of the day I saw this tweet from Evelyn Lamb:

Read her post, btw, it is great!

It has always been a little funny to me what videos or blog posts become popular. My two most popular videos couldn’t be more different.

The most popular one is actually the 2nd math video I ever made. When I decided to start teaching my kids I’d not been in a classroom for about 15 years and was worried that I’d be a little rusty talking about math. To get back in the habit I made some practice lectures that followed the sections of of “Geometry Revisited.” The second chapter in the book is on Ceva’s theorem, so that was the 2nd video. There must not be too many resources on Ceva’s theorem online because this is what the second day of talking about math for the first time in 15 years looked like:

As a little comedy extra, I’m wearing the same shirt right now as I was wearing in the video 🙂

The popularity of the second video makes a little more sense as it is a question that people ask about all the time – why is a negative number times a negative number equal to a positive number? Pretty sure it was Dan Meyer who asked about this back in June of 2013. I saw the post during the day and although it wasn’t something that I was talking about with either kid at the time, it seemed like there was a neat project in there somewhere. I sort of daydreamed up the idea of using blocks and the principle of inclusion / exclusion. It turned into a fun little discussion that evening with my older son.