# Using Po-Shen Loh’s quadratic formula idea to calculate the Cosine of 72 degrees

In December Po-Shen Loh made a video about a really neat approach to the quadratic formula:

We did a project using the ideas of sums and products of roots at the time, and I wanted to revisit the idea tonight now that my son is studying complex numbers. An idea I thought would be fun was to explore was how to use the sum and product of roots ideas for quadratic functions to calculate the cosine of 72 degrees.

I started with a quick review of the main ideas we’d be using in this project as it has been several months since we went through Po-Shen Loh’s idea:

Now we dove into the problem of finding the roots of the equation $x^5 - 1 = 0$.

Now we moved to the main idea in the project – how can we factor the polynomial we found in the last video – $x^4 + x^3 + x^2 + x + 1$ – into two quadratic polynomials?

The work here is a little tricky, but my son got through it really well. The ideas here are definitely accessible to students who have learned a little bit about polyomials and sums and products of roots.

Finally, we solved for the roots of the quadratic equation $x^2 + x - 1 = 0$ (I accidentally wrote this equation wrong, so we get off to a bad start. Luckily we caught the error after about a minute.)

Solving this equation gives us the value of cos(72)!

It was really fun to see that the combination of introductory ideas from complex numbers, polynomials, and sums / products of roots of quadratics could help us calculate the value of cos(72). I’m excited to play around with Po-Shen Loh’s idea a bit more and see where else we can find some fun applications!

# Working through some introductory problems on matrix powers

I saw a really neat problem in Strang’s Linear Algebra book earlier this week:

Tonight I had my son work through them on camera. These problems bring together ideas not just from linear algebra, but also from a high school algebra class.

Here’s his work on the first problem:

Here’s his work on the 2nd problem – this one is a fun surprise. The numbers don’t get big at all. In fact, this matrix has powers that are the identity matrix:

Here’s the third problem – a lot of the work in this problem is him remembering how to multiply complex numbers. I really like this problem because it brings in quite a bit of math from outside of linear algebra:

Here’s the last problem which is another fun surprise. We change one entry of the matrix in the previous problem by a tiny amount, and the powers of the matrix behave in a completely different way:

# Sharing Po-Shen Loh’s new idea about the quadratic formula with kids

Yesterday thanks to a tweet from Tina Cardone I saw a neat article about a new idea about the quodratic formula from Po-Shen Loh:

I thought it would be fun to see what the boys thought about this new idea. We haven’t looked at the quadratic formula in a long time – probably at least 2 years – so I started with a review of the ideas. I asked my younger son if he remembered the formula and then my older son was able to derive it using ideas about completing the square.

Next I wanted to show some ideas about the sum and product of roots of equations. Personally, these are some of my favorite ideas from algebra as they were my high school math teacher’s favorite ideas. But, again, we haven’t talked through these ideas in a while so I wanted to review the ideas about the sum and product of roots in a quadratic equation with the boys before they watched Po-Shen Loh’s video:

Next we watched Loh’s video that introduces his idea:

Having watched Loh’s video, I asked the boys to give me two ideas that they took away from that video. We then talked through the ideas with a relatively simple quadratic equation:

Finally, we solved a general quadratic equation using the ideas from Loh’s video – the general solution requires a fair amount of algebra, but really is a fascinating way to get to the general result!

I think this is a really neat approach to solving a quadratic equation. The ideas of sum and product of roots are neat ideas and were emphasized in the Algebra book from Art of Problem Solving that my kids learned from. It is fun to see those ideas coming up again in a slightly different context as my older son is studying eigenvalues and eignevectors in his linear algebra book now. Hopefully Loh’s ideas will help lots of kids see the quadratic formula in a new and interesting way!

# Revisiting The Cat in Numberland

Yesterday this tweet happened to appear in my feed:

It reminded me of an old project on infinity that I’d done with the boys:

Talking about The Cat in Numberland”

Last night I had the boys read the book again and write down three things that they found interesting. This morning we talked through some of those ideas. I’d forgotten that my older son had an appointment this morning, so we were unexpectedly pressed for time. So, sorry if parts of this project feel a bit rushed – I think I panicked a bit more about the time than I should have.

Here’s what my younger son had to say about the book – he was interested in a few of the twists and turns that happened in Hilbert’s hotel:

My older son interpreted my instructions in a different way and came up with a few conjectures instead. In this video we talk about whether or not the complex numbers (with only integer coefficients) could fit inside of Hilbert’s hotel.

This was a pretty lucky break as my younger son had wondered about the rationals, which is essentially the same problem:

Finally, we discussed the real numbers and the boys both guessed that they wouldn’t fit in. At the end I showed them Cantor’s diagonal argument . . . just in time for my older son to head out!

# Sharing Appendix 1 to Steven Strogatz’s Infinite Powers with my son

Yesterday Steven Strogatz shared an unpublished appendix to his book Infinite Powers:

I read it and thought it would be terrific to share with my older son who took calculus last year. This year we’ve been working on Linear Algebra – so not a lot of polynomial calculations (yet!) – so I also thought Strogatz’s appendix would be a terrific review.

I had him read the note first and when he was ready to discuss it we began:

At the end of the last video my son had drawn the picture showing Fermat’s approach to calculating the area under the curve $y = x^n$. Now we began calculating. He was able to write down the expression for the approximate area without too much difficulty:

The next step in working through the problem involved some work with a geometric series. Here my son was a little rusty, but I let him spend some time trying to get unstuck:

I just turned the camera off and on at the end of the last video and he continued to struggle with how to manipulate the geometric series into the form we wanted. After a few more minutes of struggle he found the idea, which was really nice to see.

Once he understood the simplification, the rest of Fermat’s proof was easy!

I’m really happy that Strogatz shared his unpublished note yesterday. It is terrific to share with kids who have already had calculus, and would, I think, also be terrific to share with kids studying Riemann sums.

# Following up yesterday’s probability problem with one of my favorite ideas involving high school algebra

Yesterday we did a neat project on problem #1 from Frederick Mosteller’s probability challenge book:

Going through problem #1 from Frederick Mosteller’s probability challenge book with kids

Today my older son is off mountain biking so the follow up project is with my younger son who is in 8th grade. I thought it would be fun to look at additional solutions to yesterday’s puzzle and show him how we could write down a formula for those solutions.

We started by look at some of the small solutions to yesterday’s problem and looking for patters. My son noticed a connection to $\sqrt{2}$ that made me really happy!

Next I showed him the Internet Sequence Database. I wanted to show him that sometimes when you are looking at a sequence of integers, it is something that other people have studied before:

Now we returned to the whiteboard to study the sequence more carefully. Our starting point was the recurrence relation that we learned about in the last video:

Finally, and this is one of my favorite high school algebra examples, we took a first step at solving the recurrence relation. This step is a nice application of factoring and using the quadratic formula:

# Using the pre-test for the new Calculus course from Outlier.org with my younger son

Yesterday a new online calculus course taught by John Urschel, Hannah Fry, and Tim Chartier made its debut:

I think online learning has a lot of potential and even tried to put together a calculus lecture video library way back in graduate school. So I hope this one has success.

One thing that caught my eye browsing through the course information was the course’s pre-test. My younger son has been studying algebra this summer and the pre-test seemed like it might be a good challenge for him.

It is 10 questions – I think the work below is a nice example of how a kid thinks through ideas in algebra. Here are the questions and his work on them:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

# Sharing Numberphile’s “Amazing Graphs” video with kids

Last week Numberphile put out a fantastic video featuring Neil Sloane:

For today’s project we explored the sequence described in the first half of the video. Namely, the sequence that begins with $a_1 = 1$ and then continues with $a_{n+1}$ depending on the greatest common divisor of $n$ and $a_{n}$. See either the Numberphile video or the first video below for the full formula.

To introduce the boys to the sequence, I had them calculate the first 10 or so terms by hand:

Next we wrote (off camera) a Mathematica program to calculate many terms of the sequence, and studied what the graph of those terms looked like:

Finally, I asked the boys to watch the Numberphile video and the describe what they learned. They were both able to give a nice explanation of why the sequence eventually repeated:

I love math projects that allow kids to play with really interesting math and also sneak in some k-12 math practice. The first sequence in the Numberphile video is a perfect example of this kind of project!

# 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.

# 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.