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.

]]>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!

]]>Here’s my 7th grader’s solution to the problem:

Here’s my 9th grader’s completely different solution:

As always is is fun to hear kids working through problems – especially the amazing ones from Catriona Shearer!

]]>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.

]]>Today as sort of a unusual way to play around with fractions I thought it would be fun to try to write some fractions in binary. It has been a while since we talked about binary, though, so I had my son tell me what he knew about binary first:

Next we moved on to writing fractions in binary – we started with some simple cases:

Finally, we tried to write 1/3 in binary. This video shows what a kid thinking through a math problem can look like, and also shows why I thought this exercise would be a nice fraction review:

]]>Using Steven Strogatz’s Infinite Powers with a 7th grader

Today I wanted to show him a hand waving overview of two of the more well-known ideas from calculus – finding tangent lines and finding areas under a curve.

I started with the tangent line problem:

He was struggling to remember some of the basic ideas about lines, so I broke the talk about tangent lines into two pieces to let him take his time remembering how to describe lines. Here’s the second part of the discussion:

With the tangent line discussion finished, we moved on to finding the area under a curve. To keep things simple, I stuck with the same function:

]]>The book is terrific and the math explanations are so accessible that I thought it would be fun to ask my younger son to read the first chapter and get his reactions.

Here’s what he thought and a short list if things that he found interesting:

After that quick introduction we walked through the three things that caught his eye – the first was the proof that the area of a circle is :

Next up was the “riddle of the wall”:

Finally, we talked through a few of the Zeno’s Paradox examples discussed in chapter 1:

I think you can see in the video that Strogatz’s writing is both accessible and interesting to kids. I definitely think that many of the ideas in Infinite Powers will be fun for kids to explore!

]]>My older son was working on something else tonight, so I talked through the problem with my younger son (he’s in 7th grade). The aim of the project tonight was not to solve the problem, but just to have him play around with a few simple cases and see if he could take a guess at the answer.

I started by just sharing the problem and making sure he understood the ideas and the constraints:

Having looked at the 2×2 case in the prior video, we moved on to the 4×4 case in this video. He had some pretty interesting ideas about how to check if we’d found the maximum number of dominoes:

Now that we’d satisfied ourselves with the 4×4 case, we moved to the 6×6 case. This case is a little harder, but still accessible to kids. Here’s his first attempt at a solution – the trouble is that we weren’t sure if it used the maximum number of dominoes:

It took a bit more experimenting to see that we’d used the most dominoes we could in the last video, and you can see him starting to understand some of the patterns in the problem as he experiments.

By the end of this video he guessed that the maximum number for the 8×8 grid would be 10.

The final challenge was an 8×8 grid. In the first attempt at filling in the board with dominoes we kept getting stuck – but his thoughts about the problem are very interesting:

With one more try through the 8×8 board we were able to fit 10 – yay!

This is a great problem to share with kids. Again, even if they can’t get all the way to understanding the general solution, there are plenty of things they can play with and understand – and also tons of ways to approach the problem!

]]>I started by simply showing him the picture and asking him to describe what he saw and if it looked like what he expected a picture of a black hole to look like:

Next we watched Katie Bouman’s Ted Talk video (published in 2017) about the black hole imaging project. This video is fantastic and great to share with students because the explanation of the project, and especially why the project is so difficult, is done at a level that kids can appreciate even if they can’t understand all of the details. Here’s that video:

After we finished watching, I asked my son to tell me some things from Bouman’s talk that caught his eye:

Next I had my son read through two great twitter threads from when the announcement happened last week. Those threads are from two physics professors – Katie Mack from North Carolina State University and Chanda Prescod-Weinstein from the University of New Hampshire. Those two twitter threads are here:

Here are two of the tweets from those threads that caught my son’s eye and his explanation of why he thought those two tweets were interesting:

Now I had my son play with a on online program that Leo Stein made a few years ago that shows the amazingly beautiful paths light can take orbiting a black hole. You can find the program here:

Unfortunately our camera was having a strange fight with the computer screen at the beginning of this video, so I cut that part out. Because of that cut the video starts mid conversation, but you’ll still be able to see that Leo’s program is really great to use with kids. I love his comment at the end: “Black holes seem pretty mysterious and neat and have weird properties.”

Finally, I showed him a picture from one of the papers about the black hole image that was published last week. The link to the paper with the image is here:

Here’s our quick discussion about that image and what my son thought about it. Talking a little bit about this image can help younger students see and understand some of the statistical work that went into producing and checking the image:

It was really incredible to see the announcement of the black hole image last week. It is equally incredible that so many people in the physics community take time to share their ideas about discoveries like this with the public. I’m super grateful for the public-facing work those people do because it makes sharing new discoveries with kids possible (and fun!).

]]> Evelyn Lamb’s review of *Count like an Egyptian *

The goal for today was more for him to talk through what he learned as opposed to getting the math details right. We definitely had a few stumbles, but it was still fun and the multiplication and division ideas are really neat.

Here’s the introduction to the book and the arithmetic ideas:

Next I asked my son to talk through a few multiplication problems:

Finally we wrapped up a division problem.

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