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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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 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.

Having the boys work through some of Kate Owens’s math contest problems

Yesterday I saw a fun tweet from my friend Kate Owens who is a math professor at the College of Charleston.

These problems from yesterday’s math contest looked like they would make a fun project, so had the boys work through the first 6 this morning.

Here’s problem #1 – this problem lets kids get in some nice arithmetic practice:

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Here’s problem #2 – the challenge here is to turn a repeating decimal into a fraction:

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Here’s problem #3 – this is a “last digit” problem and provides a nice opportunity to review some introductory ideas in number theory. The boys were a bit rusty on this topic, but did manage to work through the problem to the end:

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Problem #4 is a neat problem about sums, so some good arithmetic practice and also a nice opportunity to remember some basic ideas about sums:

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Next up is the classic math contest problem about finding the number of zeros at the end of a large factorial. My older son knew how to solve this problem quickly, so I let my younger son puzzle through it. The ideas in this problem are really nice introductory ideas about prime numbers:

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The last problem gave the boys some trouble. BUT, by happy coincidence I’m about to start covering partial fractions with my older son, so the timing for this problem was lucky. It was interesting to see the approach they took initially. When they were stuck I had the spend some time thinking about what was making the problem difficult for them.

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Continued Fractions and the quadratic formula day 2

Yesterday my younger son and I did a fun project on continued fractions and the quadratic formula. He really seemed to enjoy it so we stayed on the same topic today.

I started by asking him to recall the relationship that we talked about yesterday and then to make up his own (repeating) continued fraction:

After he’d chosen the continued fraction to study, we looked at the first few approximations to get a feel for what the

Finally he used the quadratic formula to solve for the value of the new continued fraction – it turned out to be (3 + \sqrt{17}) / 2!

A fun quadratic formula project with continued fractions and Fibonacci numbers

My son had a 1/2 day of school today due to the snow storm. Instead of having him work through problems form Art of Problem Solving’s Algebra book, I thought it would be fun to do a quadratic formula project since we had more time than usual.

I was a little brain dead as the spelling in the project will show, but still this was a nice project showing a neat application of the quadratic formula.

We started by looking at a common continued fraction and seeing how the Fibonacci numbers emerged:

Next we tried to see how to find the exact value of this continued fraction – here is where the quadratic formula made a surprise appearance:

Finally, we tried to decide which of the two roots of our quadratic equation were likely to be the value of the continued fraction. We had a slight detour here when my son thought that \sqrt{5} was less than 1, but we got back on track after that:

It was definitely fun to show my son how the quadratic formula can appear outside of the problems in his textbook 🙂

Sharing a problem from Gil Strang’s new linear algebra book with kids

I was fortunate to get Gil Strang’s new linear algebra book last week – it really is terrific:

Though it is hardly a book for kids, one problem from the first section jumped out as ones kids could actually work through, so I shared it with the boys today.

Here’s my younger son (in 7th grade) talking through the problem – he struggles a bit, but eventually finds the right idea to get to the end of the problem:

Next my older son (in 9th grade) talked through the problem. He was able to work through the problem without too much difficulty, so I asked him to explain his solution in more detail at the end:

I’m really excited to go through this book on my own over the next few months – hopefully will find a few more problems here and there to pull out and share with kids.