Students often misinterpret negative exponent notation and if this is cleared up, in the beginning, a lot of frustration and time can be saved.
Common Mistakes with Negative Exponents
There are often a lot of misconceptions about understanding and teaching negative exponents. Before we begin looking at the solutions, let's take a look at the common problems.
Students love to create or rework rules for negative exponents. Be aware that you may find negative signs in the strangest of places.
For example, you may find mistakes like this:
Just take the reciprocal can be a real answer later, but first students must understand the why. Why does this work and how can I know this rule? We have all seen students get done with the homework super quick, but then bomb the test. Simple may make the lesson go faster, but an easy gain is an easy loss.
Students, to retain these rules, must have an understanding of why and how they work in a way that works for them.
Using Common Language and Vocabulary
Most students will want to see several different ways of looking at this concept. But one thing is for sure, if we can connect this learning for them between the examples, they will develop a solid understanding of the concepts with negative exponents.
Some of the most common vocabulary: Exponents, exponent rules, repeating functions, reciprocals,
Activities to Help Understanding
I like to give the students this table and ask some questions. What do exponents mean? What is happening? Can this pattern continue? How do you know it can/cannot? Are we still multiplying?
I like to do this a few times, to allow all of the students to have a chance to see the patterns.
I like the idea of giving the students something they can picture to work within their minds. For example, thinking of cake, as we go into the negative exponents we get a fractional part of the cake, we do not get a negative cake.
And while seeing the pattern does help, they also need to understand the rules. At this time we are ready to put it all together.
Depending on your class and their level of understanding, you may need to review the exponential rules. Specifically, they need a good understanding of the product rule and the quotient rule.
I find it best to plug in positive numbers and explore the possibilities. The students will begin to come up with "shortcuts." This discovery is useful! We can then introduce the rule of, But now they understand why this works and we can move forward.
I found the following challenge on the website, http://math.stackexchange.com/questions/629740/how-would-you-explain-to-a-9th-grader-the-negative-exponent-rule and I loved the potential of teaching it this way. It builds on the exponent rules. I would stop talking at the arrow and see where the students take the conversation. I bet it would give you a lot of insight into their understanding.
Video - How I Feel About Logarithms - by Vi Hart
This video moves very fast, but for your students that are intrigued, this video will motivate and challenge them. The negative exponents begin around minute 7, but you need the whole video to understand the vocabulary. I can see this working for some and not so much for others, but I enjoyed it, and I hope you get some additional ideas from it as I did. I suggest watching this to be sure it is a good fit for your class.
"Sometimes to make the harder things simple, first you have to make the simple things harder"
Rewards of a Job Well Done
While it may take more time up front to ensure that students understand this concept, in the long run, it will save you time. The ability to see patterns and complete the calculations without help will be priceless. When you give students an in-depth understanding and allow time for processing the concepts go into the long-term memory, students gain quite an advantage as they continue their math education.