Yet another busier-than-normal week has gone by. I am looking forward to our break next week to catch up on some sleep, get some work done, and even relax a little.

Last Friday, I taught the students how to do polynomial division, and it seemed to go ok. I had typed up most of the notes already on their note-taking template, just leaving the examples for them to do, and they were very excited by not having to write as much down. The algorithm is pretty straightforward, and the only student who really had trouble was one who had learned division in a different country, with a different algorithm. A lot of students who learn long division in Mexico use the same DMSB algorithm that we do in the US, except they do the multiplication and subtraction steps in their head and just write down the difference. But this student had a totally different format (the division sign is written upside-down, the numbers go beneath, etc.). I'd never seen it before, but after watching him use it to do a division, I got how it worked. I couldn't come up with an analog for polynomial division on the spot, however, so I just tried to work with him on that a little more. Maybe I can offer him some extra credit if he works out a way to base a polynomial division algorithm on his division method...

I decided to skip synthetic division this year since you don't really need it if you can do polynomial division, and I am also skipping the factor and remainder theorems. (I'm not holding them accountable for knowing this stuff, but I am offering it up as an extra credit assignment over the break.) I'd like to push those concepts into our pre-calc curriculum. It's the middle of February and I already feel the STAR test breathing down my neck. I need to get through Rational Functions (which is a long unit - I'm already thinking about what concepts I can trim and save for pre-calc) and well into Exponentials and Logarithms before the test, as it has an absurdly heavy focus on logs.

But I digress. This week, we've been working on the properties of exponents, and operations on rational monomial expressions. I have been putting a heavy focus on having students understand why the properties of exponents act as they do - especially when dealing with negative exponents. When kids just learn the rules (add/subtract/multiply the exponents), they constantly make mistakes, putting the result in the wrong place, multiplying instead of adding, and so forth. I've found that this year, so far, they are doing a lot better since I am not talking about the "rules" at all, and instead, having them reason through their work each time. We have explored how a negative exponent works, and the only "rule" I want them to use now is to move the factor from the numerator to the denominator (or vice versa) and make the exponent positive.

I have also been doing a lot of problems like 3^900 / 3^x = 9. These help push a deeper understanding about what is happening when you divide and multiply power expressions.

Tomorrow we will do a scientific notation review (hence the title of today's post), just to make sure they have this down before they move into chemistry next year. The end of the exponents unit seems like a good time to do it - especially now that they better understand what x 10^-5 actually means.

Making Sense of Arithmetic

1 day ago

## 10 comments:

This is funny, because jd had just talked about synthetic vs. long division of polynomials over at jd2718. Is there any way you could elaborate on how your student did his division? See, I come from a different country where the division algorithm is presented in a different way, and I am always confused at how kids do it here. Well, confused may be a too strong of a word, rather taken aback :)

What you said about reasoning out the rules for exponents was music to my ears. Just yesterday, I was eavesdropping on a tutoring session at a local bookstore that related to this. The tutor told the student that all numbers raised to the power of zero equal one. The student asked why. The reply was some variation of "It just is." I almost choked on my chocolate chunk cookie It's not like that fact is obvious. Mathematicians puzzled over what a power of zero would entail for quite some time.

Anyway, kudos to you.

e:

The student is from Peru. I've been trying to google an example, but haven't yet found one (I can't remember exactly how he did it). We're on break now, but when we get back, I'll ask him to show me a problem again, and then I'll post it. All I remember is that the division "L" he used was reflected horizontally from the one we use in the US, the quotient went underneath, and the work went on the left side.

Tony:

I'm glad you agree. I have a big problem with teaching the rules without explanation. Not only is it not interesting, it doesn't actually work for the majority of students. Working one on one with a student is great, because you can move at the exact right speed for that person, and it's a shame to waste that kind of time. I looked at your blog - good luck with it and with becoming a teacher. There is a severe shortage of good math and science teachers.

Nice to hear that explaining where the rules come from works. I do that too, but as a second-year I have little basis for saying that the students learn it better that way, and sometimes I wonder whether I would serve them better by hammering the rules. My natural inclination is to insist that the students can reason about these from scratch, but natural inclination is not a justification for anything (and I do think that I sometimes should emphasize "steps" and algorithms more, that I require too much "understanding" too early; it's hard to know.)

How do you go about explaining negative powers? I write 3^4, 3^3, 3^2, and so on in a column on the board, then ask the students to discuss what happens to the answer as the powers are reduced by one. Following the same pattern of dividing by 3 every time we reduce the exponent by one gives 3^0 and 3^-1. Then we pick pairs of values from the column, comparing 3^3 with 3^-3, and 3^2 with 3^-2 concluding that if 3^a means multiply 3 by itself a times, then 3^-a means divide by 3 a times. We repeat the procedure for some fraction, and see how the pattern dictates that the fraction gets flipped when the powers get negative. What exactly do you do by way of exploring/reasoning about negative powers?

H:

The way you described is pretty much how I teach the meaning of negative and zero exponents. I don't know of any better method, but it seems to work pretty well.

The problem is that the use of 0 and the negative sign in exponents is giving a totally different meaning to well-understood symbols, seemingly created so that there is closure among the properties of exponents. This confuses students to no end, and 5^-2 is often thought to be -25. After I get students to see this, I think they really just need to memorize that a negative exponent is just a notation for the reciprocal of the power with a positive exponent.

I do think that students should be taught the algorithms, but I don't think they should be taught first or taught exclusively. After they have practiced reasoning them through with carefully selected exercises and properly coached, then the algorithms will make a lot more sense to them - and they are more likely to remember and use them correctly. You're right - you can't expect all students to have as deep an understanding as you'd like them to in the time you have available. I think becoming a better teacher in math means learning to balance these opposing forces.

I guess this means I am not a "constructivist" per se, because I don't think students should discover everything and construct all mathematical meanings for themselves. I do think that discovery work, with proper scaffolding (discussion, coaching, homework, and assessment) is much more powerful than a simple direct instruction model.

Thanks - for cutting down a little on the trial-and-error of a beginner by writing about which approaches (don't) work!

No problem. But of course, what works for my students may not work for yours (just to keep things nice and complicated!)

I'm always interested in collaboration, so feel free to email me if you want to discuss how to teach a certain concept, implement a new classroom policy, etc.

I kept the factor and remainder theorems... I have an odd year and a half algebra II/precalc hybrid... so we are doing all the traditional algebra II stuff, but heavy on theorems and proofs.

e is right: I taught both long division and synthetic, and let the kids choose, but favor long division myself (it was a hard algorithm to learn in 4th or 5th or 6th grade, why throw it away for something that is only marginally more efficient?)

What do you do with rational functions? Do you do the graphing part, or just the operations? (I'm wondering because I have a way of teaching rational function graphing that I really like, and I'd be interested in how it compares to what you do. I'll check back to see how rational functions goes.

I know this is from forever ago, but I also do division that way! I don't think of it as an L, but a cross.

The number being divided goes in the top left quadrant. The divisor goes in the top right quadrant. The quotient is in the bottom right quadrant, and the work is in the bottom left quadrant.

I was also highly confused when I learned polynomial division. I just couldn't follow the teacher's work, since I never did it that way.

And yet I still managed to get a PhD in engineering. There is hope for your students yet!

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