Negative exponents

Using powers of two for explaining the concept of negative exponents

Negative numbers confuse many students. Usually students tend to struggle with almost everything related to negative numbers. From the very concept of using the (-) sign to refer to conventional spatial directions (left, down, back), to the different rules for adding, subtracting, or multiplying positive and negative numbers; there are plenty of instances where the (-) sign is overlooked, or misinterpreted, resulting in a wrong answer. Given that a high number of wrong answers produce a low score, it is only natural for many students to react defensively whenever a new concept involving negative numbers shows up in their radar. This is the case for negative exponents. Remember that the first definition of exponent (positive) most students are introduced to is: “the number of times you multiply a number times itself.” When one tries to apply this definition to the case of negative exponents it does not make sense because, how do you multiply a number times itself “minus once,” or “minus twice”? One possible answer is that, when it comes to negative exponents, you do not multiply but you divide instead. Even with that interpretation the rule still needs some adjustment because, if you start with a number a, and you divide it by itself once, you get 1 as a result, so you would think a^-1 was 1 but that would be wrong because 1 = a⁰, whereas a^-1 = 1/a.
In order to avoid this kind of confusion, I follow another approach: By asking very simple questions I help the student build a list of powers of 2 from 2 to 1024, each related to their corresponding positive exponent. Then we notice the patterns how the numbers change, and how the exponents change. Then we move backwards all the way to the beginning of the list, and beyond, continuing into fractions, on one column, and into negative exponents, on the other column. In this way it becomes crystal clear to the student how the exact same pattern correlates the negative exponents to the fractions, and even the general formula a^-n = 1/aⁿ becomes apparent. I have successfully used this method many, many times, and so far it has worked wonderfully, clearing away all confusion and anxiety students of all ages had around negative exponents. The key to this approach is being very thorough, and asking the right questions, in the right way, at the right time.

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A Number Divided By Another Number

One more instance of Math not being English

Lately I had a few students who had some trouble with fractions. Part of the problem was a very specific type of confusion at the time of setting up a division calculation. When asked “How much is 5 divided by 12?” for example, they would sometimes correctly calculate the result but most other times they would set up the division as 12 divided by 5. One of them asked me a few times if the result would be the same. When I tried to explain that division is not commutative, using some visual representations of fractions, he was not totally convinced. So I just pulled the calculator, I asked him to give me a couple numbers, and I did both divisions (let’s say, 17/4 and 4/17) with the calculator, showing him the results. We repeated the “experiment” with two other examples, and then he was convinced that division is not commutative. However, such discovery created some anxiety in him because he doubted he would choose the right order of calculation in any given problem. Actually, it took multiple repetitions on my part for him to finally learn how to set up the right calculation when a word problem involves the phrase “divided by.” At first he wanted to transliterate the written phrase (“105 divided by 15” for example) word by word, and number by number, in the exact same order into the division calculation by writing, from left to right, the 105 first, outside the division symbol, then the division symbol (the one that looks like a rotated “L”), and finally the 15, inside the division symbol. I would tell him that the order of the numbers in the phrase “105 divided by 15” is backwards to the order of the same numbers in the actual calculation but this only seemed to surprise him, and confuse him. Once more, I resorted to the calculator. I said: “O.K. just do the division.” When he asked: “In what order?” I said: “Do them both.” Once he had calculated both results by long-hand, I gave him the calculator, and I said, “Now do them both with the calculator.” As he punched the calculator keys, I directed his attention to what sequence in the calculator was giving him the same result as he had calculated long-hand. I said “Do you see when you enter into the calculator: ‘105, division symbol, 15, enter’ that gives you the same result as when you did ’15, division symbol, 105’ by long-hand?” When he saw this evidence he still said “It is confusing.” Then I said: “Yes, I know, but that is just the way it is, so you are going to have to remember it that way. In the calculator division the numbers go in the same order as they are in the phrase ‘105 divided by 15,’ whereas in the long-hand division, to get the same result, the numbers have to go backwards.” It took some repetition over a few tutoring sessions but he finally got it consistently right.