Lesson 1

This lesson builds on students’ experience with exponential functions in a previous course and with geometric sequences from earlier in this course. The goal is to recall some features of exponential change, such as:

• Exponential change involves repeatedly multiplying a quantity by the same factor, rather than adding the same amount.
• Exponential growth happens when the factor is greater than 1, and exponential decay happens when the factor is less than 1.
• A quantity that grows exponentially may appear to increase slowly at first but then increases very rapidly later.

In addition, students briefly explore the meaning of an exponential function at a non-whole number input and how they could determine the value of the function for that input. This exploration is done in the context of a pond whose surface is being covered by algae that doubles in size each day and wondering how much of the surface is covered half a day before 100% coverage. In future lessons, students will focus on making sense of the meaning of rational inputs in other contexts before using the principle that exponential functions change by equal amounts over equal intervals to calculate things like growth factors over different intervals of time.

Students may represent exponential changes in different ways. They reason abstractly and quantitatively by using descriptions to write expressions, create a table, or make a graph in order to answer questions about a situation (MP2). They may also use expressions to capture regularity in repeated reasoning (MP8). For instance, after being shrunk \(n\) times by a factor of \(\frac{4}{5}\), the height of a passport picture gets multiplied by \(\left(\frac{4}{5}\right)^n\). This work will support students throughout the unit, as they deepen their knowledge of exponential functions and extend it to include any type of rational input, with an emphasis on non-whole number input, later in the unit.

Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology, such as spreadsheets, to solve problems. We recommend making technology available (MP5). In particular, provide students access to calculators that can process exponential expressions for all lessons in this unit.

Rulers should be made available for the activity Shrinking a Passport Photo, but won’t necessarily be used by all students.