Interesting question in Dan Meyer’s blog today:
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 you have to grapple with the problem of finding two numbers that multiply to be and sum to be . 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:
(2) Exploring properties of numbers in depth
Prior to encountering quadratic equations, students will (hopefully!) have studied and solved linear equations like . 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:
One other pretty profound idea that you encounter for the first time with quadratics is finding that equations can have multiple solutions. My example has the solution 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:
Also, going beyond two solutions can be interesting to kids:
(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 and with the boys:
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:
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 🙂
So, I guess that’s sort of the complete set of thoughts from the bike ride home 🙂