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.

The Square of a Sum

An arithmetic and geometric approach to Algebra

A very common mistake test takers make when they have been out of school for a while is that they automatically try to expand the square of a sum as if it was the same as the plain sum of the squares of the individual terms. For example, sometimes some students, when presented with say, (x + 5)², they wrongly expand it as x² + 25, leaving out the middle term 10x.
They forget they have to use the foil method to multiply the given sum times itself. Since typographically the sum of the squares looks like something that could equal the square of the sum, they guess they can expand the square that way, as if it was a real math rule, and then they get the problem wrong.
I try to preemptively address the possibility of this mistake, because both sums of squares and squares of sums show up a lot in the math section of standardized tests. For a long time I have used a drawing, with two little squares and two congruent rectangles filling up a larger square, to show the geometric interpretation (in terms of area) for the algebraic formula (a + b)² = a² + 2ab + b². I emphasize the middle term, 2ab, in this expansion, and I point to the corresponding two rectangles in the drawing.
The above approach is useful but many students forget the formula when it comes to actually applying it. Most do not realize it is a universal pattern where to plug any other expressions in.
So, lately I have been experimenting with a slightly different approach: I present two or three numerical examples first, before drawing the squares and rectangles. I ask the student to choose two numbers, and then I guide them through a sequence of calculations that allows them to actually compute and see the numerical difference between the square of the sum, and the sum of the squares. Then I ask them to multiply the original numbers, then to double this product. By asking all the questions in the right order, I have gotten comments from them like: “Wow!” or “How weird!” or “That is so funny!” or “This is very interesting!” or “Does it always work like that?” or “How can that be?” or “What is going on?” or “I’m sure there is a pattern here!” Then I draw the squares and rectangles with all the numbers in their proper places. So, creating the perception of a “mystery” with the numbers, and then explaining it away with the picture seems to work well. It makes sense for the students. They usually say: “Oh, right! Now I get it.”

Integration by Parts in Calculus 2

It is really neat when students make their own discoveries.

This morning I had a Calculus 2 tutoring session about Integration by Parts.
My student was having some trouble with those integrals that are calculated by applying the integration-by-parts formula multiple times.
In this type of problem you get somewhat complicated expressions with parenthesis nested inside parenthesis, often with multiplying coefficients and/or negative signs in front of each parenthesis.
Students usually get confused when doing these integrals for the first time. The main reason is because they do not expect so much complexity. They believe the problems are going to be shorter than they actually turn out. So one key for getting these integrals right is to keep track of every step separately, identifying each new integral, and labeling it with a new variable of its own (using I, I1, I2, I3, ... works well). Then you keep working all the way down until you finally reach an integral you can actually solve, without any new “left-over” integrals. Then you re-trace your steps back one at a time, substituting each (ever longer) expression into the corresponding place for the previous partially solved integral, until you get to the original one.
We were working with the function (x^3)(e^x), where you start by deriving x^3, and integrating e^x.
My student understood everything but he wanted to make sure he would remember it later, so he started all over again. I suggested this time to start from the bottom up, so he integrated x(e^x), then (x^2)(e^x), and then (x^3)(e^x). I suggested for him to keep going so he integrated (x^4)(e^x).
At this point I said he could even memorize all the resulting formulas just for the test but it did not seem a good idea to him. Then I asked: “Well, maybe there is a pattern here. Do you see after factoring out e^x the last number (the constant term in the polynomial factor) is a factorial? And the polynomial always starts with the power of x we have in the original integral, does it not?”
I started looking for a way to factor out the polynomials. It turned out to be not always possible (the second-degree one already had complex roots) but all of a sudden my student exclaimed: “They are all derivatives!” So the pattern was not apparently multiplicative, but one more closely related to calculus, because it involves successive derivatives.
The best part here is my student saw it himself without my help. He realized what the pattern was before me.
That really made my day. I enjoy watching students making their own discoveries. This was not a textbook exercise, but a question I came up with after we went beyond the examples in the book, and my student was able to come up with the answer faster than me, showing he really understood the question. And now he has a very effective reference point to remember all those integrals if they show up in his exam! He won’t forget it because he discovered it on his own.