Yesterday, and today, I taught negative exponents to my Algebra 1 class. Students started out with a “discovery” activity where they filled out two tables, one for and another for , for n = 5 to -5 where they eventually figured out that each successive result was half of the prior result, for the table, and one-third of the prior result for the table, which made the answer for n=0 and any negative n more understandable. I say “eventually” since most students did not recognize the pattern in the table until we covered it explicitly.
This activity led into an introduction of the following definition for a negative exponent.
For this definition, I mentioned that when any number a, a≠0, is raised to a negative power n, you took the reciprocal of the number a raised to the positive power n. Students learned what reciprocal meant earlier in the year, as well as in prior math classes, so I did not explicitly mention how to write the reciprocal. I also did not provide an explanation for why the definition was true.
If I had to explain why the definition is valid, I would show the following where any number a, a≠0, which is raised to a negative power n can be multiplied by its multiplicative inverse to yield 1. The first step in the following started by multiplying the denominator in the definition above with the left hand side of the equation.
Likewise, when using fractions instead of whole numbers, the definition above is written as follows.
Since there is a fraction in the denominator in this case, we invert the fraction in the denominator and multiply it by the fraction in the numerator, 1 divided by 1. The result is the reciprocal of the fraction raised to a positive power n.
The “proof” for this is the same as shown above for whole numbers as follows.
Hence, in my instruction, I took care to show students the first couple of steps each time, as in the following example, instead of directly proceeding to the solution using a slight of hand (e.g. just flipping the fraction and making the exponent positive).
My CT mentioned after class that she felt my method might be too confusing for students. She mentioned that she simply taught students to write the reciprocal of a fraction to a negative exponent by flipping the fraction and making the exponent positive. In essence, she skipped my intermediate steps two and three, above, and jumped straight to my step four. While I mentioned to students that they will be able to do that in time, simply by recall, I wanted them to see the mathematical reason behind that “trick.”
After hearing my CT’s suggestion, however, I am not so sure my approach was the best. Did I confuse students when I intended to help them? Was my use of mathematical procedure for dividing fractions more confusing to students versus simply telling them they flip the numerator and denominator?
Did I overcomplicate my instruction? I care deeply about helping my students learn math. I will always show them tricks or simpler ways to solve problems. However, I do not prefer to show the “memorize this” approach first. I want to show them the mathematically correct way first, at least mostly correct, then show them shortcuts, tricks and the like.
What do others think?