This warm up activity will serve as a check for understanding from a previous lesson. In the first slide, students will be solving for the exponent value without graphing. First, students can isolate the base/exponent by multiplying or dividing. Then students can determine the exponent value by inspection. If students cannot tell by looking at the equation they can make a list of powers of the particular base (ex. 2^x = 32; 2, 4, 8, 16, 32; This shows that x = 5) (MP2).
In the second slide, students will first try to reason about the value of the exponent by estimating. Students with a greater number sense will be able to make a more accurate measurement. Allow these students to share their reasoning with the class so that students can following their thinking (MP3).
Possible student excerpt for 4^x = 14: Since 4^1 = 4 and 4^2 = 16 I know the exponent must be between 1 and 2. Then I noticed that 14 is closer to 16 than it is to 4. This made me estimate the value of the exponent to be about 1.9.
Students will then verify their estimation by graphing. In the above example, students could view the equation 4^x=14 as two separate functions. The first is f(x) = 4^x the second is g(x) = 14.
Students can graph these functions on their graphing calculator and use the "calculate intersect" feature to determine the point of intersection. The x-value of the point of intersection will be the solution to the original equation. (See image in the resources for this section)
Have students work on this investigation with a partner. Students should take 3-4 minutes to read and understand the question and develop a plan for solving it (MP1). Once students have taken a few minutes to develop a plan, survey a few partnerships to determine how students are planning on solving the problem.
Let students work on the problem for about 10-15 minutes. While students are working make note of the various approaches that you see from the class. Some students will most likely organize their work into a table to help them see the payment based on the number of bags of leaves (MP2). Listen for students who are beginning to compare the different growth rates. They may say that the linear growth ($2 per bag) is growing at a constant rate while the exponential growth starts out more slowly but grows quickly later on. Some students may only look at the first 10 bags while others will continue to investigate the pattern (MP8). If students have their tables set up correctly, the second payment method does not become better than the first until the 11th bag. After this, the second payment method will always be better than the first. Listen for students who continue the pattern who realize that the scenario becomes somewhat unrealistic. For example, if 15 bags are filled the payment is $328 (MP2).
In the last 5-10 minutes have various partnerships share their approach through use of a document camera or by displaying their reasoning in some other way. Begin with students who were starting to notice the pattern but did not have a sophisticated explanation. Then, move on to other groups who were beginning to compare the two types of functions. Find a group that has a nicely organized table and display it for the class to see (first column = bags (x), second column = method 1 (f(x)), third column = method 2 (g(x))). Ask the class, "How can you distinguish the exponential function from the linear function by looking at this table?" Let students turn and talk to discuss this and then share out their ideas with the class. Students may talk about the common ratio/difference respectively. They can also note that the linear grows at the same rate while the exponential grows slowly and then more quickly (MP3).
This closing activity will most likely take longer than 5 minutes and can be investigated for a homework assignment. That said, the idea is that students investigate linear and exponential decay models. Students should see how the exponential model decays almost a quickly as the linear model at first but then decays more slowly after that. The linear model decays at the same rate over time. Students who are able to reason about the output values (MP2) will see that Easton will never have a smaller population than Centerville. This is because 3% of 2500 is only 75 (which is smaller than 80). Moving forward in time 3% of smaller and smaller numbers will always be less than 80 and so Easton will always have the larger population.