This activity/post is a little different than are usual posts. We look at a particular math concept and consider how to use context, models and CCSS math practices to increase student understanding. This activity focuses on negative exponents. Even if you aren't teaching negative exponents, this is a great opportunity to look at non-linear growth or positive exponents.
We use examples of a plant and a beanstalk growing exponentially and ask kids to determine the height of these plants in the future (growing at the same rate). Students skip ahead several inputs and are forced to consider a rule to model the situation. Next we ask students to think about how they might work backwards. If you know the height of a plant during its 9th month, how can you find it's height at the 8th and 7th months. Finally we ask them to consider how tall these plants were before they were purchased (again assuming that they were growing at the same rate). Students use this context to see the pattern of powers and develop an understanding of why positive whole numbers raised to a negative exponent result in a fraction (not a negative number).
For a suggested teaching approach check out Brian's video below. You will notice that there is an opportunity to attend to CCSS math practice #7, "Look for and make use of structure" by creating a middle column (in the video) and purposefully complicating the math. This allows us to see the math in a different way and to get a better understanding of what is happening, mathematically, between the variables, time and height. At this point you will notice that students can engage in math practice #8, "Express regularity in repeated reasoning". When Brian generalizes the math that he sees happening over and over again in the table, he is abstracting from repeated reasoning.
For these reasons MP.7 and MP.8 really jump out in this activity (as they do in many of our activities). Make sure to engage students in this process. The process should not be completely teacher directed. Students might try the problems themselves and then follow up with guided, teacher directed, whole class discussion.
The Activity: Fast-Growing-Plant.pdf
For members we have created solutions.