Teaching Laws of Exponents

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This year I made the change from teaching middle school to teaching high school. I still teach Algebra, and the difference is tremendous. All the methods I’ve used in the past really don’t apply anymore. I no longer look for ways to challenge students, but am now focused on how to get the students to understand the basics. It’s very discouraging.

Take teaching exponents. You either multiply, and therefore add the exponents; if you have a power, you multiply the power with the exponent(s); and if you divide, you subtract. Three simple “rules.” And, yes, I taught it that way, and regret doing that. I do have a handful of students that do get the laws of exponents – but most confuse all the different rules and do whichever operation they fancy at the moment with no rhyme or reason.

For students who don’t have a good handle on math, the above “rule” method is not a good one. I should have gone to the basic method of having them write it out: x^2(x^3) = x*x*x*x*x which is x^5. And (x^3)^2 = x^3*x^3 = x*x*x*x*x*x which is x^6. As for division, they can just cancel out top/bottom pairs after expanding them out. Even though this method is rather crude, it’s something the students can understand without relying on rules. And it’s probably true that those handful of students who got the “rules” would have seen the pattern of the rules after writing everything out anyways, and that self discovery of the rules would have been that much more meaningful to them.

My point is, you know the level of your students. If your students tend to struggle, then teach them to use the definition of exponents (i.e. writing everything out in expanded form) to evaluate exponential expressions. In fact, the next thing on my list is to create worksheets with smaller exponents so that expanding them out won’t be so torturous on the students. I think one of the reasons why I taught “using the rules” is because their workbook is full of problems with larger exponents like (x^12)^4. Who on earth wants to write out 48 x’s. You’d think after writing the first few, they’d get the point, but you can’t always assume those kinds of things.

So if you haven’t gotten to that unit yet, hopefully by the time you do, I will have several worksheets on expressions that have small enough exponents that are more ideal for writing out/expanding. I’m fairly fortunate in that I taught this unit to my students who are repeating second semester Algebra, and based on the mistake I made with them, I can make sure I don’t repeat that mistake with my one-year Algebra students. Towards the end of the year, I will reteach my second semester students this unit using expansion, since they probably won’t recall having done anything with exponents come March/April…

By the way, I had a student ask me why x + x is 2x and x * x is x^2. I explained that multiplication is “shorthand” for adding the same thing over and over again, or repeated addition. Instead of saying 7 + 7 + 7, it’s better to say 3 times 7. Likewise, exponents is “shorthand” for repeated multiplication. She did not get it. I used the visual, if you have an x in one hand, and an x in the other, all together you literally have two x’s, that’s what addition is. Adding does not change what you have, just how many of them you have. She didn’t get that. I expanded exponents to show her why only multiplication produces changes in the exponents, and she still didn’t get it. If you have a better way to explain this common error away, please let me know…

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