One of the major sources of confusion for my students is around the idea of a negative exponent. Today, we look at a series of problems that help to illustrate the meaning of a negative exponent and show it in several different contexts.
I ask students to answer these questions and to support their reasoning with a picture. The goal is to solidify their confidence around algorithms for negative exponents by asking them to generate concrete images that make sense to them.
It often takes several iterations to take the students as far as I want to, so we continue the conversation with other scenarios (e.g., comparing multiplying a third and dividing by 3 and so forth). I try as many combinations as I can until I am convinced that they understand multiply ways to represent negative exponents (MP8).
Here I ask students to spend 15 minutes filling out this chart:
I ask students to complete all the problems by hand. I say,"But, you can leave numbers in exponential form if the calculations are too tedious." For example, I would never ask a student to solve 9^6 power, but I do want them to realize that 9^6 = 9^6 and that 9^-6 = 1/9^6. So I ask them to leave those tougher exponents in positive exponential form. By this I literally mean that no exponents can be negative.
Then we review the chart and compare observations. I like to follow up with a question that taps into their intuitions (or misconceptions as the case may be), something like, "what number is 3 times greater than 3^3?" The answer of course is 3^4, but many students don't make this connection immediately. We continue with other examples until I think they've got it (MP8).
I like to finish this fluency lesson by returning to context. The topic is our ancestors and we are approaching it from a sort of flipped model. We often think of the connection to future generations and note that our children will have children of their own and so forth. I show what I mean in a simple tree diagram. The notion is that the population grows over time. However, when we look back we don't often see how this pattern grows as well.
Our parents each had their own parents who each had their own parents and so forth. Each step here is called a generation. So the question is, how many our ancestors would be on Earth at a particular time?
I ask them to consider a few years:
1929 (stock market crash)
1861 (USA civil war)
1776 (USA Independence)
1669 (Newton Explains Calculus)
1450 (Printing Press Invented)
Here students must consider the average length of a generation and apply the exponential function 2^x, however this number grows to such a height, that we talk about the limits of our answer and the implication that we share ancestors. For example, go back 1000 years and that is about 40 generations assuming a generation is about 25 years). But can we accept that we had 2^40 ancestors on the planet at that time? I ask students to consider what this means about our function and our past.