Lesson 3: Understanding Rational Inputs
About this lesson
The goal of this lesson is to examine rational inputs of exponential functions and make sense of them in context, focusing on nonwhole number inputs. Prior to this point, students have studied the meaning of expressions such as \(5^{\frac{1}{2}}\) and \(10^{\frac{7}{3}}\). They have also graphed exponential functions with real numbers for the domain and used these graphs to estimate the values of functions, focusing mainly on the outputs when inputs were integers. Here, students use equations and graphs to determine outputs at rational inputs and make sense of what these inputs mean in context. In the next lesson, students will calculate the growth and decay factors for different rational input intervals.
Students construct exponential functions to model the growth of a population and the decay of the amount of medicine in the body. In both cases, they solve problems by interpreting functions (represented both graphically and with expressions) in context, working across different representations of the situations (MP2).
Lesson overview
 3.1 Warmup: Keeping Equations True (5 minutes)

3.2 Activity: Florida in the 1800’s (20 minutes)
 Includes "Are you Ready for More?" extension problem

3.3 Activity: Disappearing Medicine (10 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 3.4 Cooldown: Flea Treatment (5 minutes)
Learning goals:
 Understand the meaning of a rational input, particularly positive nonwhole number inputs, to an exponential function in context.
 Use equations and graphs to identify the value of an exponential function at a rational input.
Learning goals (student facing):
 Let’s look at exponential functions where the input values are not whole numbers.
Learning targets (student facing):
 I can determine the value of exponential functions at nonwhole number inputs.
Required materials:
 Graphing technology
Required preparation:
 Acquire devices that can run Desmos (recommended) or other graphing technology.
 It is ideal if each student has their own device. (Desmos is available under Math Tools.)
Standards:
 This lesson builds on the standard: CCSS.HSFLE.A.1.aMS.FLE.1aMO.A1.LQE.A.1aMO.A1.LQE.A.1b
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