Kids Say the Darnedest Cubics

Celeste Ng, if you didn’t know, is a novelist. Her bestseller Little Fires Everywhere will soon become a Hulu miniseries starring Reese Witherspoon and Kerry Washington. (Meanwhile, my begging-to-be-filmed blog post on mathematical Britishisms remains in development limbo. That’s Hollywood for you.)

Anyway, Celeste Ng is also the parent of a future math teacher.

I, for one, warmly welcome our 9-year-old hero to the time-honored profession of foisting equations upon the reluctant.

As a kid, I made similar efforts to stump my dad. Mine were mile-long arithmetic problems: “13406824360 times 78645103465” and the like. Such multiplication is certainly hard. But it’s also tedious and shallow (as my dad’s taxed patience can attest).

These equations, in contrast, run deep. Deeper than their inventor could have known.

That’s the way it is with math. When you put a shovel in the dirt, sometimes you get a shovelful of dirt; but sometimes, a 200-foot geyser springs forth. Beneath the innocuous patch of surface you’re exploring, there may lurk terrific geological forces.

This is one of those times. (Beware: spoilers for the problem follow.)

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Our system involves four variables: f, a, c, and e (which pleasingly spell “face”). Before we solve algebraically, it’s worth investigating the matter geometrically.

Here’s the relationship between a and e:

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Between a and f:

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Between f and c:

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And between a and c (which I derive from the final equation by replacing e with 11 – a, an equivalence established by the first equation):

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Since each equation relates just two variables, it can be visualized in two dimensions. The results are two lines, a parabola, and a hyperbola.

But what if we want to combine them all? With four variables in play, we need four dimensions. I don’t want to speak for you, but when my brain tries to visualize four dimensions, it mostly achieves something like this:

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In context, each 2-variable relationship becomes a 3-dimensional solid, living in an ambient 4-dimensional space. The solution we seek is a point that belongs to all of the solids simultaneously.

Pretty heavy dinner-table conversation for a 9-year-old.

Thwarted by geometry, we turn to more algebraic methods. What do you do with 4 equations involving 4 unknowns?

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Turn it into 3 equations involving 3 unknowns.

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(In this case, I used the substitution mentioned above, then factored the left-hand-side.)

And what do you do from there? Why eliminate another variable, of course, boiling it down to 2 equations with 2 unknowns.

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(In this case, I added the first and second equations from before.)

And from there, you can guess our next move: dessert.

Then, with our sweet tooth sated: boil our pair of equations down to a single one, in a single unknown.

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(In this case, I solved the second equation for c, then substituted into the first equation.)

The result is a cubic. Potentially a nasty piece of work. Consult this table:

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In this case, fortunately, our 9-year-old teacher has thrown us a bone. One of the solutions to this cubic can be guessed without too much trouble: a = 4.

(Ben Dickman, my arch-rival in being a mathy Ben on the internet, has a nice explanation of why, if you’re curious and also a traitor who is forever dead to me.)

Knowing this solution, we can factor the cubic into a linear and a quadratic:

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Then, pick your favorite quadratic-solving method, and voila:

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All of these solutions check out, giving us a total of three solutions to the original system of equations:

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Back to the geometry: apparently those 3-dimensional surfaces, crisscrossing in 4-dimensional space, meet at exactly three points. To higher-dimensional aliens, this would perhaps be as obvious as noting that two lines cross at a single point. Or that a line and a parabola at a pair of points. It’d be elementary geometry, almost too trivial to speak of. But despite my siblings’ insistence, I am not an alien, and cannot see such things.

(Though if we retreat back to the step where we had two equations in two unknowns, Desmos can show us why three solutions emerge: the hyperbola and parabola intersect in three places.)

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This is, if I may reiterate, some hefty dinner-table chitchat. Even my wife and I (together comprising one math professor, one author of math books, and two incurable nerds) don’t delve this deep on your typical weeknight. Hats off to the Ng household.

Anyway, this all got me thinking about the vast pointlessness of algebra.

I know I’m not supposed to say algebra is pointless. But I can’t help it. The other day, as we moved into factoring quadratics, my Algebra I students raised the timeless chorus: “Why do we have to learn this?”

I had no good reply. I never do.

It’s not for lack of thought. I’ve spent years meditating on this very question, which is exactly why I find the standard answers so empty and unsatisfying.

If algebra class is to be of daily use to the citizenry, why not focus on probability, statistical literacy, and personal finance?

If algebra class to impart valuable professional skills, why not focus on spreadsheets (plus maybe a little Python)?

If algebra class is to teach us “how to think” (which was once my preferred answer), then what do we make of research revealing that lessons in reasoning don’t generalize? That algebra teaches us to think, not about life or logic as a whole, but about equations only?

Finally, if algebra class is simply a platform upon which to compete for college admissions… then please excuse me while I scream into the void.

So. Why do I teach this stuff?

Here’s one reason. It’s an idiosyncratic reason, inspired by the exploits of the Ng household, offered with the caveat that your mileage may vary. It goes like this.

Arithmetic raises questions that it cannot answer.

Arithmetic poses riddles that it cannot solve.

Arithmetic uncovers patterns that it cannot explain.

But algebra can.

Arithmetic is the solid ground, the numerical surface of things. You figure it must be dirt all the way down. But if you stick your shovel in the right spot, as the Ng family did, you’ll be greeted with a geyser.

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How can you not love those eruptions? How can you not want to know their source? How can you resist the urge to probe the depths?

So, three cheers for algebra, and for all the 9-year-olds at all the dinner tables tossing off questions so profound that, even on a well-caffeinated day, a half an answer is the very best I can muster.

A NOTE ON THE TITLE OF THIS POST: Apparently the TV show is spelled “Kids Say the Darndest Things,” with only one “e” in “Darndest.” This is an abomination and I will have no part of it.

EDIT: A cool animation from Adam in the comments, showing what the graphical approach looks like with three equations.



14 thoughts on “Kids Say the Darnedest Cubics

  1. Regarding the last portion of this post (“why Algebra?”), I am put in mind of a conversation I had with a good friend of mine when we were both Maths undergrads.
    We were trying to come up with a good answer to a common question to maths students – “What is ‘pure’ maths?” (See also, “how do mathematicians research”, “When is maths finished” etc).
    The answer we settled on is that pure maths is a game. You set out your basic rules and see how far you can take these. What fun ideas can you prove or disprove? How can the rules be broken or twisted? And if you break the rules, can you invent new rules to continue the game (eg imaginary numbers). And to come back to your question, as with any game you have to learn the rules to be able to play.
    So this may be an analogy you can put to your students in future (and this is a concept that applies to all areas and skills, not just maths, so it is useful to learn early)

    1. Excellent response. I esp like it because this is the explanation my math-grad-student sister once gave me when I was struggling with high school geometry and coming close to hating math. “It’s really just a game,” she said, “with made up rules that you have to learn to get good at the game. The fact that the rules actually produce useful results for the real world is just icing on the cake.” Turned my head completely around. I got my degree and have *actually used algebra* in my work life. It does happen!

  2. The thing that struck me about this problem was that finding A solution was pretty easy (I started with the assumption that 15+f was a perfect square, and things pretty much fell into place from there); but finding ALL solutions is considerably more complicated.

    1. I started with the assumption that all the variables were positive integers, and a solution fell out very quickly.

      Is that a good assumption? I can’t say, but start looking where the light is good before you venture too far into the darkness

  3. Why learn algebra? Speaking only for myself, I have to say that I used algebra several times daily in my non-math profession, and that, even retired, I find myself in need of it a couple of times a week or more. Perhaps most people don’t.
    In contrast to what the studies show, I find that that algebraic concepts generalize to other areas of my life and learning. Perhaps I see math in places that others might not.
    I think that statistical literacy should be an absolute requirement for all students at all levels, especially those in post-graduate work! (Personal finance, too, but that’s mostly arithmetic, not math.)

  4. Well, on the other hand, context matters. A 9-year-old that doesn’t even know how to use exponents (see second equation) is unlikely to have used anything but natural numbers to craft this problem. So, a has to be a perfect square between 15 and 21, and the problem becomes quite tractable.

  5. Without algebra, you can’t progress to any of the higher maths, like trig and calc. But then, you might ask, why do we have to learn these? Like you, I have no good answer unless you’re going to be a physicist.

    It wasn’t until I starting learning the “why” via abstract algebra and real analysis that I really got a hold of the “how” that we learned as kids.

  6. The graphing approach does kind of work for this problem. Substituting (11-a) for “e” in the final equation, the bottom three equations have only 3 total variables, so we can draw each of them as a surface in 3D space. The three solutions are where all three surfaces intersect. Actually finding those points is maybe best done with algebra, but there is a perfectly good graphical representation of this system and its solutions. Here’s a nice animation of this:

  7. One reason to learn algebra is without it you can’t grasp calculus, and calculus makes many real physics and engineering problems trivial, as in “solve in your head” trivial. Calculus also opens up methods of solution for really complicated stuff like electromagnetics and heat flow and “what part will break first?” Admittedly, not everyone will solve these problems, but for those of us who do, the sooner you can get to the good stuff. You can make similar arguments about “Why do I have to learn French/ History /Chemistry/ Gym / theology?” We all end up with a brain full of useless skills. I can re-time a car, recite the books of the Bible, and point out that my mothers cat is on the blue couch in Paris. I don’t regret any of the useless junk I learned.

  8. I’ll add one more reason to teach math: some people like it! And there’s no way to know if you like math until you actually start learning some of it. If one of the purposes of school is to help you decide what you want to do with your life, then of course we should teach math! Even by high school, I think there are a bunch of people who are unsure if they like math or not and if we replace algebra 2 with personal finance, how will those people ever find out? (Not saying that we shouldn’t teach personal finance–just that we shouldn’t replace algebra 2.)

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