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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Playing with Annie Perkins’s counting problems

I saw a neat tweet from Annie Perkins last week:

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!

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!

More intro number theory with my son inspired by Martin Weissman’s An Illustrated Theory of Numbers

I’ve been thinking about more ways to use Martin Weissman’s An Illustrated Theory of Numbers with the boys lately:

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.

Writing fractions in binary

I saw an interesting tweet last wee that got me thinking about fractions:

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:

Following up our conversation about Steven Strogatz’s Infinite Powers with some basic calculus ideas

Yesterday I had my younger son (in 7th grade) read chapter 1 in Steven Strogatz’s new book Infinite Powers and we had a fun time talking about what he learned:

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:

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!