In this lesson, students study a situation characterized by exponential change and learn the term growth factor. They represent this relationship using a table, an expression, and a graph. They also explain the meaning of the numbers \(a\) and \(b\) in an exponential expression \(a \cdot b^x\), identifying their meaning in terms of a context (\(a\) is the initial amount and \(b\) is the multiplier or growth factor) and also in terms of a graph (where \(a\) is the vertical intercept and \(b\) determines how quickly the graph increases). Students interpret the different representations of growth in terms of a bacteria population (MP2).
In this and following lessons, students will often work with properties of exponents, a topic developed in grade 8. There is an optional activity intended to remind students of the convention that \(a^0 = 1\) for a nonzero number \(a\).
Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
Lesson overview
 3.1 Warmup: Math Talk: Exponent Rules (5 minutes)

3.2 Optional Activity: What Does \(x^0\) Mean? (10 minutes)
 Includes "Are you Ready for More?" extension problem
 3.3 Activity: Multiplying Microbes (15 minutes)
 3.4 Activity: Graphing the Microbes (15 minutes)
 Lesson Synthesis
 3.5 Cooldown: Mice in the Forest (5 minutes)
Learning goals:
 Explain (in writing) how to see \(a\) and \(b\) on the graph of an equation of the form \(y=a \cdot b^x\).
 Interpret \(a\) and \(b\) given equations of the form \(y=a \cdot b^x\) and a context of exponential growth.
 Write an equation of the form of \(y =a \cdot b^x\) to represent a quantity that changes by a growth factor \(b\).
Learning goals (student facing):
 Let’s explore exponential growth.
Learning targets (student facing):
 I can explain the connections between an equation and a graph that represents exponential growth.
 I can write and interpret an equation that represents exponential growth.
Required preparation:
 Acquire devices that can run Desmos (recommended) or other graphing technology as an optional tool for students.
Glossary:
 growth factor  In an exponential function, the output is multiplied by the same factor every time the input increases by one. The multiplier is called the growth factor.
 Access the complete Algebra 1 Course glossary.
Standards:
 This lesson builds on the standard:CCSS.8.EE.A.1MS.8.EE.1MO.8.EEI.A.1
 This lesson builds towards the standard:CCSS.HSFIF.C.7MS.FIF.7MO.A1.IF.C.7
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