I saw an incredible tweet from Francis Su yesterday:

Any polygon can be cut into finite pieces and rearranged to form any other polygon of equal area. This beautiful visualization of the Wallace-Bolyai-Gerwien Thm just won a 2018 @NSF@PopSci Vizzies award. From a @harveymudd class taught by Satyan Devadoss https://t.co/RSXZinf8tV

After exploring the program a little bit last night I thought it would be really fun for the boys to play with it this morning. So, I showed them the basics of how the program works and had them each play around for 10 min. Here are their thoughts:

Younger son (in 7th grade):

Older son next (in 9th grade):

I am really happy that this program won an NSF award – what an incredibly fun way to share an advanced math topic with everyone!

Having finished a single variable calculus class with my son this school year, I’ve been thinking about what to do next. Probably the next step is going to be linear algebra and we’ve been watching a few of Grant Sanderson’s “Essence of Linear Algebra” videos to get a feel for the subject.

Today I wanted to have a short and introductory talk about vectors with my son, and I had two goals in mind. The first was to show some ideas about (for lack of a better phrase) thinking in vectors rather than thinking in coordinates. The seconds was just sort of a fun introduction to the dot product.

So, I started with a simple introduction to vectors that he’s seen a bit of via the Grant Sanderson series:

Finding a vector representation for the 2nd diagonal of the parallelogram we’d drawn was giving him some trouble, so we took a deeper dive here. I’ve always thought that the equation for the 2nd diagonal was non-intuitive, so I gave him plenty of time to make mistakes and work through the ideas until he found the answer:

Finally, I did a simple introduction to the dot product and we calculated the angle – or the cosine of the angle – between a couple of vectors as a way to show how some ideas from linear algebra help solve seemingly complicated problems:

So, next week I’m having him watch a few more of Grant’s videos while I’m away on a work trip. We’ll get going on linear algebra the week after that.

HUGE debates in class today over this question: 4 people sit at a round table, and they only care about who sits next to them. How many ways can they sit at the table? https://t.co/Wox7Nm8BPl

Today I thought it would be fun to play around with this idea with my younger son. First I introduced the 4-person problem and let him think through it. His thought process is a great example of what a kid learning math can look like:

At the end of the last video he’d determined that there were 3 different arrangements of the 4 people sitting around the table. In this video I asked him to find those arrangements:

Next we moved to the 5 person problem:

Finally, having decided that there were 12 different arrangements with the 5 person problem, I asked him to try to write down all 12. This is a good exercise in using counting techniques to make an organized list:

Definitely a fun problem for students, and also a really nice introduction to counting and symmetry. Thanks to Annie for sharing!

Today I was looking for a project with my son and flipped open to the chapter on quadratic reciprocity. It had a few introductory ideas that I thought would be fun to share with my younger son.

We first looked at Wilson’s Theorem:

After Wilson’s theorem, we moved on to talking about perfect squares mod a prime. After a fairly long discussion here my son noticed that half of the non-zero number mod a prime are perfect squares:

Finally, I asked him to make a mod 11 multiplication table and we talked through some of the patterns in the table – including that the non-zero numbers had multiplicative inverses:

It was a really fun discussion today. I know next to nothing about number theory, but I really would like to use Weissman’s book more to explore some advanced ideas with the boys.