Exponent Sprint 3 and Exponent Sprint 4 today. Students will still be getting used to the new format, so their scores may still be a little lower than they were with the Power Sprints. However, you're training them to think in logarithmic terms, so it's worth the practice.
In between the two sprints, briefly discuss strategies for efficiency. This might be a good point at which to quickly review the properties of exponents. Everyone needs to be totally comfortable with these so that they can understand the properties of logarithms when they're introduced.
Now, I'll ask the class if they were able to find the linear equation that fits the given points in Two Points Determine a Function. I don't expect anyone to have had trouble with this, so we should be able to agree on the correct equation very quickly.
Next, I'll ask for a student volunteer to go to the board to explain how the exponential equation can be created, and I'll do my best to leave the explaining to the students. What I will do is ask questions in order to draw out the following points:
1. Given the general equation, y = a*b^x, we can use the two given points to create a system of equations. It's clear from this that two points determine a unique exponential equation.
2. In the general equation, a is the initial value or y-intercept.
3. In the general equation, b is the constant factor that we recognized in yesterday's class as the characteristic of exponential growth or decay.
Once we're all in agreement on the correct exponential equation, it's time to turn our attention to the third function type.
Finally, it's time to examine the nature of quadratic growth in a new light. I'll begin by reminding the class of our discussion of linear & exponential functions in the previous lesson. Then I simply ask, "How can we characterize quadratic growth in terms of its graph, equation, and data?" (Again, check out this video for the motivation and rationale of this lesson.)
To spur this conversation along, I'll go ahead and quickly jot down a simple data table to illustrate quadratic growth.
Students will see that the function values do not increase with either a constant difference or a constant ratio. Soon some will certainly notice that the differences themselves increase with a constant difference. I call these values the "2nd differences" for lack of a better name and point out that this constancy is characteristic of quadratic growth. (It might be worth recalling our models of projectile motion from a previous lesson.)
Here, you can see here what my whiteboard looked like when this conversation was finished.
Some students almost always ask about third degree polynomials, so I also take some time to quickly investigate. A very simple table of cubic data shows that the 3rd differences are constant, and from this I tell the class that this is characteristic of all polynomial functions. The degree of the function determines which column of differences will be constant. It's an interesting aside and reveals a little more clearly just how different exponential functions are from polynomial.
Now, given this conversation, it's time for everyone to tackle the problem of creating a quadratic equation that fits the two given points. As students work individually or in small groups, I'll move around the class offering encouragement, asking questions, and giving explanations.
Once again, they should begin by using the given points to create a system of two equations. The difference is that this time there is no unique solution. This is what makes this lesson so awesome!
Students may explain this in one of two ways. First, they may note that since a quadratic equation has three coefficients, it generally takes three data points to create a system of three equations in order to determine them. Second, they may point out that with only two data points, the constant "2nd difference" isn't fixed. In this case, we're free to choose any value we please, which will result in any number of different quadratic equations.
For a challenge, you might consider asking students to identify the quadratic equation that "best" fits either the linear or exponential equation. Of course, "best" is open to interpretation, and I would leave it up to the student to determine what is best along with some justification. Some technology like GeoGebra can be really useful for an investigative approach to this problem.
Homework for today is to complete the remainder of this problem. We'll discuss the solution tomorrow.