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Q: Is there a way we can teach polynomials, avoiding the "watch me do it & now you do it" training method, that will allow students to anticipate and predict the existence or formulation of polynomials? (as it applies to, say, middle school mathematics)

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There are some properties of polynomials that may not be fully realized or proved at the grade 7 - 9 level. This is fine. I am more concerned about the introduction of polynomials to students. What I want to avoid is the particularly robotic and soulless routine of (some) math teaching which goes something like this:

  1. here is a thing that's unfamiliar or you've never seen before
  2. here is what we call some of the parts of this thing
  3. here are some things you can do with them
  4. now watch me do it and you do it until you get it right
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  • $\begingroup$ I assume this is after they have extensive experience with quadratic equations. $\endgroup$ – Gerald Edgar Jun 9 '16 at 21:42
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    $\begingroup$ There's not so much wrong with the scenario of your 1-4 except for the very end, "until you do it right". Maybe better "Now you try", anyway. How do people learn except by imitation? Just leave out the judgemental or adversarial part. One point is that no mandate for introduction of polynomials will be discovered by kids living a normal life... rather, I'd think one would want to persuade them that there are surprises, and good surprises, ... and that it's not hard, just practice "like so". "Now you try..." $\endgroup$ – paul garrett Jun 9 '16 at 22:14
  • $\begingroup$ @GeraldEdgar No experience with quadratic equations. Linear equations only. $\endgroup$ – Marian Minar Jun 9 '16 at 22:48
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    $\begingroup$ For me it is very hard to answer the question of how to teach something without at least reflecting on why it is being taught. Why are students of these ages being taught to manipulate polynomials? Except for the few who will become engineers and scientists, it is unlikely that they will need to use polynomials in the future. The motivations for teaching the rest polynomials seem to be nonutilitarian and noninstrumental. What are they? $\endgroup$ – Dan Fox Jun 10 '16 at 5:37
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    $\begingroup$ Echoing @DanFox's comment: it is hard to make a case that most people will "need" polynomials, ever. The people who will need them will not need them as contrived analogues of decimal expansions, etc. (In reality, they'd need them as simple examples of functions.) The catchiest appeal (not "need") is, as mentioned in passing in the answer below, is as explanation of certain quasi-magic tricks: "Think of a number, but don't tell me. Add 6. Multiply by 3. Subtract the original number. Add 5. ...Tell me what you get. Your original number was ..." That was my original motivation. $\endgroup$ – paul garrett Jun 12 '16 at 16:48
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One of the things I discovered with polynomials when working with them as a high-school student is that they generalize the idea of a place value system. This is a very cogent way to introduce the subject to middle-school students, who have probably been exposed to working in multiple bases.

For example, $11$ times $11$ is $121$ in every base except binary where the 2 in the middle gets carried out. This isn't a coincidence — it's because $(x + 1)(x + 1) = x^2 + 2x + 1$. Same with $12 \times 13 = 156$ for bases 7 and higher — it's just saying that $(x + 2)(x + 3) = x^2 + 5x + 6$, and evaluating it for $x = 10$.

So what you could do is to do a series of classes about working in different bases, and then ask — what if we work in a base with an unknown number? What happens then? Can we make arithmetic statements that work in all bases?


If you use this place value paradigm, teaching polynomial arithmetic becomes easy as well — it's simply the algorithms they're used to, but you never carry anything like you would in a positional number system with a fixed base.

To multiply, say, $2x + 3$ by $x + 6$, you simply treat it as multiplying $23$ by $16$ in "base $x$", in which case you get $2\ 15\ 18$ as the "digits", or $2x^2 + 15x + 18$ in standard notation. They can verify this for themselves by plugging $x = 10$ and seeing that $23 \times 16 = 368$, which you would expect if you were to carry everything out. They can also plug in $x = 100$ to see that $203 \times 106 = 21518$, and that the carrying only happens when $x = 10$ because the digits are all scrunched together.

It will also help explain why certain mental tricks with multiplication work, such as the trick where you multiply a 2-digit number by $99$ by subtracting 1 from it, then appending the new number's difference from $99$ to the end (e.g. $43 \times 99 = 4257$). It works because multiplying a monomial $ax + b$ by $x^2 - 1$ gives you $ax^3 + bx^2 - ax - b$, which you can rewrite as $(ax + b - 1)x^2 + x^2 - ax - b$, and then $(ax + b - 1)x^2 + (x^2 - 1) - (ax + b - 1)$.


Afterwards, you can discuss further differences between polynomials and regular place values, such as what happens when you use fractional coefficients. Why can't you represent, say, $\frac12 x^2$ as a polynomial with lower degree? What happens when you divide 100 by 2 in several different bases? Do they ever return the same thing, or even something similar?

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  • $\begingroup$ Thank you for taking the time to write this out. Indeed, we are on the same page as this is something that I've done as well. I think this is one type of a great start in a classroom. Haven't done this yet, but this leads to other great questions like, could we write numbers with negative digits? Silly, but really cool: e.g. 49 could be \5\-1\. Perhaps, as educators, we need to teach about a variety of bases earlier on, as well as modular arithmetic (which I see was recently asked on this forum)? $\endgroup$ – Marian Minar Jun 10 '16 at 17:15
  • $\begingroup$ I think so too. The place value model just makes sense. Meanwhile we're putting things like synthetic division in the curriculum that don't make any sense and that don't intuitively generalize to or from anything else. $\endgroup$ – Joe Z. Jun 10 '16 at 17:17
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    $\begingroup$ And yes, negative digits are a similar result of the lack of carrying — there's no borrowing either, so if you subtract $342 - 173$ in base $x$ (as you do in the Tom Lehrer song New Math), you actually get $2\ {-3}\ {-1}$, which evaluates to $169$ when we carry everything out (or even $147$ in base 8, as the song then goes into). $\endgroup$ – Joe Z. Jun 10 '16 at 17:20
  • $\begingroup$ Nice answer. On the same theme, I'll point out that the Common Core standards for these grades include explication that polynomials are to integers (closed under add/subtract/multiply and factorable) as rational expressions are to rational numbers (closed under add/subtract/multiply/and nonzero division). $\endgroup$ – Daniel R. Collins Jun 11 '16 at 22:35
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What the OP left aside was:

Who cares? Why would we ever need to manipulate polynomials?

Permit me to answer this unasked question. Every* piece of text you see rendered in PDF on your laptop screen, or printed out on a printer, draws each letter either with Adobe Postscript cubic Bézier polynomials, or Apple/Microsoft's TrueType, which uses quadratic Bézier polynomials.


          Fonts
          (Images from Mark Kilgard's presentaion. The dots are control points for the Bézier curves.)
We can hardly get through a day without seeing polynomials in action! :-)

As an advanced tanget, one could discuss the advantages/disadvantages of cubic vs. quadratic Bézier curves: continuity, control points, evaluation complexity, etc.


*Nearly every.

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  • $\begingroup$ While this is interesting, it's not quite what the question is about. Introducing kids to parametric equations like Bézier curves as an example of polynomials is a good way to get them even more intimidated about the subject than they already were, because you're already going four or five steps ahead of what they're used to. The question is asking about how to gently introduce kids to how polynomials work in general, before introducing them to the harder topics that they can be applied to. $\endgroup$ – Joe Z. Jun 14 '16 at 6:03
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    $\begingroup$ I really like your example @Joseph O'Rourke. This is an awesome applied math project that students can investigate - I'm thinking grade 10 or 11? $\endgroup$ – Marian Minar Jun 15 '16 at 21:38
  • $\begingroup$ @MarianMinar: You could have students design a fancy version of the first letter of their name. There are interactive web drawing tools all over the web. E.g., here is one at Princeton. $\endgroup$ – Joseph O'Rourke Jun 16 '16 at 1:58
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I like Joe Z's answer. It's how I introduce polynomials as well. In the interest of other approaches and since the idea is to introduce students to polynomials, here is another thought:

Consider a geometric approach and start with squares and cubes. If we have a square with side length $3$ units, then the area is $3^{2} = 9 \mbox{ units}^{2}$. A cube with side length $3$ would have volume $3^{3} = 27 \mbox{ units}^{3}$. A next step would be to use $x$ in place of the side length to obtain $x^{2}$ and $x^{3}$ for the area and volume, respectively.

Now, we can start to tease out some rules about polynomials if we keep the units hanging around. For example, we could ask what in the world it would mean to have $x^{3} + x^{2}$. How can we add volume with area? We can't in any sensible way, and hence we must maintain $x^{3} + x^{2}$, or analagously, $27 \mbox{ units}^{3} + 9 \mbox{ units}^{2}$ --- these units don't mix.

On the other hand, if I had two squares, I could join them and have their areas add to $2x^{2}$. Or have two cubes and have their volumes add to $2x^{3}$. All the while, we would carry our units along. Then we could really do some mind bending with some computation. Ask the question, "If I had three squares, each with side length three, what would my total area be?" The answer would be $27 \mbox{ units}^{2}$. Then we compare this result with one cube of side length three. The only difference is that, while in value, both quantities are equal in dimensionality they are not --- apples and oranges.

Multiplying polynomials can be brought back to finding the area of a square or the volume of a sphere. $x$ is a polynomial of degree 1. Thus, $x\times x = x^{2}$ and $x\times x\times x = x^{3}$, but $x\times x\times x = x^{2} \times x = x^{3}$ or $x\times x\times x = x\times x^{2} = x^{3}$.

I haven't really had much occasion to offer this type of explanation, but I keep it in reserve in case the approach as outlined by Joe Z just doesn't resonate. One distinct "advantage", is that it gets students to think about dimension and units.

This allows us to bring the conversation back to simple numerical substitution. What if we had $x^{3} + x^{2}$ and were asked to substitute $x = 3$? Why is the answer $27 + 9 = 36$ a valid answer, but in the case of our square and cube, we were forced to leave it as the nonsensical $27\mbox { units}^{3} + 9\mbox{ units}^{2}$? The answer is that $x = 3$ is dimensionless, whilst $x = 3 \mbox{ units}$ is not.

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