Intro to Linear Algebra

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.

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Sharing a new problem from Catriona Shearer with the boys

Saw this problem from Catriona Shearer today and just had to share it with the boys when they got home:

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!

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

My copy of Steven Strogatz’s new book arrived a few weeks ago:

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 \pi r^2:

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!

Exploring a fascinating idea from special relativity – Penrose-Terrell rotation – with kids

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

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!):

Revisiting a connection between arithmetic and geometry

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:

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.

Sharing James Tanton’s area models with my younger son

I saw a really neat tweet from James Tanton yesterday:

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:

James Tanton’s area model lecture

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:

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.