"Exponents Sprints" 5 and 6 today.
Briefly discuss strategies for efficiency.
[Check out this video for an overview of how this lesson fits in with the previous two.]
Again, I'll ask for a student to volunteer to explain how a quadratic equation can be created to pass through the given points. (MP 3) My goals with this presentation - and what my questions will focus on - are the following:
1. We again use the general form of the equation and the given points to create a system of two equations. This is the same strategy we used for the linear & exponential models.
2. This time there is no unique solution. The best we can do is determine the necessary relationship between the coefficients. This means that there can be any number of parabolas that pass though the given points. (MP 7)
3. With only two given points, we cannot determine the "2nd difference" that is supposed to be constant for a quadratic equation. This explains in another way why there is no unique solution.
Finally, a quick look at the graphs on the indicated domain will wrap up this comparison of the three different function types.
At this point, my students have seen a single example clearly distinguishing linear, quadratic, and exponential functions. They were already very familiar with the graphical differences, and somewhat familiar with the differences in the equations. What is new is the characteristics of the data. The next problem set will help solidify this new understanding as students examine one data set after another to determine which type of function it exemplifies.
I hand out Types of Growth and briefly explain the task. My students are free to work individually or in small groups, but they must do two things. First, they must correctly identify each function type. Second, they must justify their answer by demonstrating that the data exhibits either a constant ratio, a constant difference, or a constant "2nd difference". (MP 6)
While they work, I'll move around the room to check for understanding.
With about 5 minutes left, I like to call a halt and quickly review the solutions to the first page. Just about everyone should be finished with these 8 data tables, and this immediate feedback will help them gain some confidence to complete the rest at home tonight.
I'll also point out that in the final three data sets, they may find that the numbers do not fit any of the models perfectly (due to rounding). In this case, they should determine which function represents the best fit.