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:

Problem

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!

Seeing ideas about substitution for the first time

My son had an interesting problem on his enrichment math homework this week, and it gave him a lot of trouble this morning:

Tonight I thought it would be good to talk through the problem since I think the main idea he needed to solve it was new to him.

Here’s the introduction and some of the ideas he tried this morning:

Next we took a look at the equations on the computer and talked about some of the ideas we saw:

After looking at the graphs of the equations on the computer we came back to the whiteboard to talk about substitution.

Finally, having worked through the introductory part of u-substitution in the last video, I let him finish off the project on his own.

I can’t remember talking through this topic previously, but it was fun. It is always neat to be there when a kid is seeing a math topic for the first time.

What learning math can look like: Sums and products of roots

My older son has been studying properties of quadratic equations in his algebra book. Yesterday we talked a little bit about sums and products of roots and he was a little confused:

 

The confusion doesn’t bother or worry me – learning math is hardly ever a perfect, straight line process.

Today we revisited the topic and took a look at one easier and one more difficult problem.

First – find two numbers whose sum is -3 and have a product equal to -18:

 

Second – find two numbers whose sum is 2 and whose product is also 2:

 

So, hopefully a little bit of a gain in understanding from yesterday to today. Once again, learning math isn’t always a straight line.

An interesting introduction to completing the square

My older son started a new section in Art of Problem Solving’s Introduction to Algebra book today. The chapter begins to look at quadratics in a bit more depth. Given the question my son picked for today’s movie I assume that one of the topics will be completing the square. It is always interesting to see how a kid approaches an advanced topic before really knowing much about that topic.

The question asks you to find the minimum value of x^2 + 10x - 7. Here’s his work:

 

After he finished his work I gave a really basic introduction to completing the square using this problem as an example:

 

Moving beyond the quadratic formula

It has been interesting watching my son learn a bit more algebra this year. He seems to have made quite a bit of progress studying linear equations, but he’s still at a point where the quadratic formula is the first thing he thinks about with any sort of non-linear equations (which, given that he’s just learning algebra are mostly quadratic equations).

Today he ran across a problem that was probably designed quite specifically to help kids see beyond the quadratic formula. The problem is #20 from the 2007 AMC 10 a

Here’s the problem:

Suppose that the number $a$ satisfies the equation 4 = a + a^{ - 1}. What is the value of a^{4} + a^{ - 4}?

This problem gave him some difficulty and I asked him to explain his original approach using the quadratic formula first:

 

Next we talked about how to approach this problem without solving for a first. We had briefly talked through this approach in the morning but this was his first time trying to explain it.

 

Next we went to Wolfram Alpha to see that the solution he’d found with the quadratic formula actually produced the answer of 194. After that we talked about the graph of y = x + 1/x. It was a little hard for him to see that the minimum value on the graph occurred at x = 1, but zooming in a little helped him see it.

 

Finally, I wrapped up by showing him one way that we could use the quadratic formula to help us see where that minimum occurred. I took this approach to help him see that even though the quadratic formula wasn’t so helpful in solving the original problem, it still could be helpful as a way to learn a little bit about x + 1/x.

 

So, a nice little problem that provides a good example of a situation where the quadratic formula isn’t so helpful. Hopefully examples like this one will help him see that there are lots of to approach non-linear equations, and the quadratic formula is just one of them.