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.

An experiment with a difficult algebra problem

Instead of working through some medium level AMC 10 problems today, I decided it would be fun to walk through a single really challenging problem with my older son.

The problem we chose was #23 from the 2015 AMC 10b:

Problem 23 from the 2015 AMC 10a

Here’s the problem:

The roots of f(x) = x^2 - ax + 2a are integers.  Find the sum of all values of all possible values of a.

Though this problem is difficult, I choose it because my son is able to understand the solution.  Over the course of about 20 minutes we worked through that solution – slowly.  There were a few misconceptions along the way, but I hope this was a productive exercise.

We began by talking about the problem itself and getting a few initial thoughts from him:

In the second section of the talk he began to try to play around with the values of the roots, but that didn’t go anywhere. Eventually he thinks to try the quadratic formula which leads to an interesting breakthrough:

Now he’s found a new equation that he knows needs to be a perfect square to make the original equation in the problem have integer solutions. Here we start down the path to finding when our new equation is a perfect square, but we hit a strange problem about dividing by zero. This problem causes a little detour:

So, with the division by 0 problem out of the way we returned to trying to find when our new equation could be a perfect square. In this part of the solution he checks a few cases by hand.

Finally, we wrap up by trying to see if we’ve found all of the solutions. There a little bit of number theory / number sense that helps us out here:

So, a tough problem for sure, but nice to see that each individual step is something that my son was able to understand. Learning to pull together all these pieces is an important part of problem solving.