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.

Responding to Dan Meyer’s Quadratic question

Interesting question in Dan Meyer’s blog today:

Dan Meyer’s blog post from June 17, 2015

Here’s the question:

Instead, ask yourself, “Why did mathematicians think this skill [ factoring quadratics with integer roots ] was worth even a little bit of our time? If the ability to factor that trinomial is aspirin for a mathematician, then how do we create the headache?”

I happened to see this post right before leaving the office. On my 45-ish minute bike ride home I spent some time thinking about why I thought this skill was worth some time. Here’s where my thoughts took me . . . .

The first thing I thought of was a passage from Jordan Ellenberg’s How not to be Wrong. On page 323 in the section “The unreasonable effectiveness of classical geometry” he makes this point about why ellipses seem to show up all over the place in math:

In math there are many, many complicated objects, but only a few simple ones. So if you have a problem whose solution admits a simple mathematical description, there are only a few possibilities for the solution. The simpleset mathematical entities are thus ubiquitous, forced into multiple duty as solutions to all kinds of scientific problems.

I think quadratic equations fall into the bucket of simple mathematical entities that Ellenberg is talking about here, and I’m not surprised that math folks would think they are worth of study. The simplest case is probably quadratics where the solutions are integers, so that’s a natural place to start.

But there are other important mathematical ideas that you see – maybe for the first time – when you start to think about factoring quadratic equations. Here are three ideas that I thought of specifically:

(1) Factoring integers

Mike Sipser, the former head of MIT’s math department, has a nice public lecture which includes a discussion of difficulty of factoring integers in the first 10 minutes:


By the time a student encounters quadratic equations, he or she will have had a lot of practice multiplying numbers but probably less practice factoring them. When factoring x^2 + bx + c you have to grapple with the problem of finding two numbers that multiply to be c and sum to be -b. As Sipser’s lecture shows, this is by no means an easy question – particularly when you are seeing it for the first time.

I asked my kids to take a crack at finding two integers whose product was 120 and whose sum was 26 as an illustration:



One more advanced project that I have done with my older son that involved factoring polynomials was based on a neat post by University of Colorado math professor Richard Green that Patrick Honner pointed out on Twitter:

A “new to me” proof that there are infinitely many primes

(2) Exploring properties of numbers in depth

Prior to encountering quadratic equations, students will (hopefully!) have studied and solved linear equations like 3x - 6 = 0. My kids, at least, will solve equations like this by moving the 6 to the “other side” and then dividing by 3. Great technique for linear equations, but not so great for quadratics. You need a new idea and that idea is pretty deep – if two numbers multiply to be 0, then one (or both) of the numbers has to be zero.

You’ll can hear Julie Rehmeyer talk about struggling with a similar idea in this Inspired by Math interview. The part I’m referencing begins around 31:30 when she talks about her time at Wellesley and trying to prove that 0 + 0 = 0:

July Rehmeyer interviewed by Inspired by Math

One other pretty profound idea that you encounter for the first time with quadratics is finding that equations can have multiple solutions. My 3x - 6 = 0 example has the solution x = 2. With quadratics you can see 2 solutions – but what does that even mean? It has to be very confusing seeing multiple solutions for the first time. Sometimes those second solutions are fun to explore, though:

Dan Meyer’s Geometry Problem

Also, going beyond two solutions can be interesting to kids:

Quadratic and Cubic equations

(3) Finding new types of numbers

This part isn’t so much about factoring as it is about quadratic equations in general. Or maybe just quadratic equations with integer coefficients rather than ones that factor into integers. In any case, with some simple quadratic equations that will not factor easily you are able to talk about some new kinds of numbers:

I saw two interesting pieces from prominent mathematicians talking generally about numbers. The first piece was Ed Frenkel’s book Love and Math. I used some of Frenkel’s ideas about quadratic equations to talk about some surprising similarities between \sqrt{2} and i with the boys:

Ed Frenkel, the square root of 2, and i

The next piece was the public lecture that Jacob Lurie’s gave after winning the Breakthrough Prize. His lecture is an absolutely wonderful talk about math from a mathematicians point of view, and it has a couple of great ideas that you can use with kids. In the first three minutes of the lecture you can see some of the important ideas about irrational and imaginary numbers that come into play with quadratic equations:

Using Jacob Lurie’s Breakthrough Prize lecture with kids

All of this was a long way of saying that I think quadratic equations serve as a gateway to some interesting and advanced mathematical ideas as illustrated in the three points above. They also come up in enough places in math and physics (hinting at Ellenberg’s idea) that I’m not really surprised to see that mathematicians think they are important to study.

Though a subjective feeling, obviously, I feel that kids will find many of the ideas related to quadratic equations to be fascinating, which is why I’ve tried them out with my own kids.

Oh, and just as I was finishing writing up this post I remembered another fun project with the boys that involved a quadratic that can factor over the integers – probably the most internet famous math problem so far in 2015 🙂

Talking with the boys about Hannah and her sweets

So, I guess that’s sort of the complete set of thoughts from the bike ride home 🙂

*The quadratic formula and Fibonacci numbers

I knew ahead of time that I was going to have a busy week of work this week and was looking for something fun to cover with my older son in the limited amount of time that we would have.  We were supposed to be covering properties of functions so I was looking for a topic that would at least be tangentally related to that, but I also wanted to get him a little review with solving quadratic equations.  Diving into the formula for the Fibonacci numbers seemed to fit the bill quite nicely . . .

We started with a short talk about the Fibonacci numbers that focused on thinking about the usual recurrence relation definition:

I didn’t do a great job with the graph at the end, and we spent another 10 minutes after the video talking through the graph and comparing it to y = x^2 and y = 2^x.  I was pretty happy with how that talk about the graphs went after the first video and wanted to reinforce some of those ideas the next day.  With that in mind, the next talk begins by discussing how the Fibonacci numbers grow and then considers what happens if the Fibonacci numbers could be written in the form F(n) = C * \lambda^n.

I’ve always found this technique for finding the closed form for the Fibonacci numbers to be really beautiful.  Turns out it is a great little tool for algebra review, too!

After finding that the Fibonacci numbers somehow related to the numbers \frac{1 \pm \sqrt{5}}{2} at the end of the last video, in this one we finish up the calculation and write down the closed form for each Fibonacci number.  Lots of great algebra review for kids in these calculations, too!

With the formula for the Fibonacci numbers now in hand, I wanted to play around with the formula so we jumped over to Wolfram Alpha.  The first neat thing I wanted to go through was how we could now easily calculate each Fibonacci number without reference to the prior two numbers.  Fun!  The second thing I wanted to show was that the second term in the formula doesn’t play much of a role for the larger Fibonacci numbers.  It is pretty amazing to see how well \frac{1}{\sqrt{5}} * (\frac{ 1 + \sqrt{5}}{2})^n approximates the larger Fibonacci numbers.  The meat of the formula is all in the first term!  I thought this would be an especially fun fact to show him since the two terms look so similar when you write them down.

We’ve talked a little bit about the Fibonacci numbers previously, but not with this level of math.  I’d chosen this topic because I thought the math is really interesting – which it was – but all of the algebra review turned this in to a nice learning opportunity, too.  For what was supposed to be just a  little diversion way from the book during a busy week of work for me, this topic turned out to be really fun.