## Monday, April 2, 2007

### Motivating Zero, Negative and Fractional Exponents in Middle School and 1st year Algebra

Algebra teachers, like myself, are always looking for ways to help students make sense of exponents. We look through copies of the Mathematics Teacher, we go to the Math Forum and now we Google, Google ad infinitum (or some other search engine to be fair!). Here's an approach that I have found helpful. I assume the student has had some basic introduction to exponents and their properties. I call it the exponential function approach which sounds too challenging for middle schoolers but you decide if they can handle this. Students will use pattern-based thinking and graphs to make conjectures about extending powers of 2 to include zero, negative and even fractional exponents. Properties of exponents will then be used to 'justify' the conjectures. The juxtaposition of the numerical, symbolic, graphical and verbal descriptions are consistent with the Rule of Four that is now regarded as the most powerful heuristic in teaching mathematics.

Begin by making an x-y table - this is the critical piece.

Exponent (x)...........................Power (y = 2x)
3 ..................................................23 = 8
2 ..................................................22 = 4
1 ..................................................21 = 2
0 .................................................20 = ??

The instructor of course is prompting the students for the powers while they are taking careful notes. At the same time the instructor is plotting these results as ordered pairs and the students do likewise. It might be helpful to let 2 or 4 boxes represent one unit on the y-axis since, at some point, the y-values will be fractional. Similarly for the x-axis (play with it first).
At this point, the instructor asks a key verbal question (you may phrase it much differently depending on the level of the group and your preference):
[While pointing to the left and right columns]
"When the exponent decreases from 3 to 2, the corresponding power of 2 is divided by ___.
Repeat this phrase a couple of more times until you reach an exponent of 0, then -1 and voila! Keep going until x = -3, plot the points and the students are seeing an exponential curve in grade 7? 8? 9?
Motivating zero and negative exponents using a function model (tables!) seems to make sense to me because it begins to create a 'function' mind-set that can be carried through all subsequent math courses. It may also help students to 'see' that the range of the function consists of positive real numbers. If you're wondering why I didn't mention turning on the graphing calculators to make the TABLE and GRAPH, I hope you can guess why. It was important for me to have students do this by hand first, then I will turn on the overhead viewscreen and we can explore with technology. Just my opinion of course but students in my classes seem to make sense of this. Of course, I don't kid myself that this approach will lead to better grades on tests of this unit! Facility with the properties of exponents only comes from considerable skill practice with paper and pencil.
For fractional exponents, I'll begin the discussion but I will have to explore further on another post or leave it to your imagination. "Ok, boys and girls, if mathematicians believed exponents could be zero or negative integers, would you be surprised if they wondered about 21/2? From the table and the graph, 21/2 should fall between ___ and ___? Do you think it will be exactly 1.5? Why or why not?
I know many of you use the exponent properties to develop this topic, but I wanted to suggest an alternative. I usually follow this discussion with arguments like: " Hmmm, I wonder what
21/2 times 21/2 would be?" etc...