Compare exponential and linear functions
Understand that over time a quantity increasing exponentially will eventually exceed a quantity increasing linearly (F-LE.A.3$^\star$).
In this section, students will compare linear and exponential functions. Students are familiar with linear functions and linear growth so students will use this base understanding to develop the notion that increasing exponential functions will eventually exceed increasing linear functions.
WHAT: This task gives verbal descriptions of five situations that can be modeled by a linear or exponential function; students must choose which model is most appropriate (MP4).
WHY: Now that students have a good understanding of the difference in the underlying growth laws for linear and exponential functions, this task is a good way of assessing that understanding. Students must decide in each case which growth law is being described. This task serves as a good lead in to the section, where students start comparing linear and exponential functions in earnest. It could also serve as introductory activity in the next unit.
WHAT: In this task students construct and compare linear and exponential functions and find where the two functions intersect. They do so by comparing the exponential growth of a population with the linear growth of a food supply, and answer questions regarding when food will run out.
WHY: One purpose of this task is to demonstrate that exponential functions grow faster than linear functions even if the linear function has a higher initial value and even if we increase the slope of the line. By comparing the functions and their graphs, students can see that the slope of an exponential function continues to increase, while the slope of a linear function stays the same.
WHAT: Students are asked to compare with two different payment schemes for completing the chore of raking leaves; one is an exponential model and the other is linear.
WHY: The purpose of this task it to have students discover how (and how quickly) an exponentially increasing quantity eventually surpasses a linearly increasing quantity, in a context where naïve intuition might prefer linear growth (MP2). The task serves as a reminder of the power of exponential growth discovered in the hook lesson at the beginning of this unit, and segues into a more complex comparisons of linear and exponential growth.
WHAT: Students are again asked to compare a linear function and an exponential function. This time, however, the base of the exponential function is greater than but very close to 1 so the exponential function does not grow as quickly as seen in Exponential Growth Versus Linear Growth I.
WHY: The purpose of this task is to provide students with a linear and exponential model where, initially, it does not look as if the exponential model will ever surpass the linear model. Although a calculator can and should be used on this problem, more value will be gained from the task by thinking about the function values than by simply having a calculator plot graphs (MP2, MP5). This aspect of the task is emphasized in the second solution.
WHAT: In this task, students investigate and construct a model for how memory loses its fidelity as the number of remembrances increases. Students create and examine both linear and exponential models (MP4) and use these to determine when a memory becomes semi- or unreliable (F-LE.A.1$^\star$, F-LE.A.2$^\star$).
WHY: The purpose of this culminating task in the section is to provide students a complex context where it is not obvious whether a linear or an exponential model fits best, and give them an opportunity to explore the consequences of choosing either model (MP4).
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