This lesson has two aims. The first aim is to prompt students to write an equation of the form \(y = a \cdot b^x\) to represent an exponential function without a context from limited information.
A second optional activity encourages students to look more closely at how different equivalent expressions can be written to highlight different aspects of a quantity. For example, we can write an expression that shows the growth factor of a bacteria population each week, or one that shows the daily growth factor.
In the first activity, to find \(a\) and \(b\), students need to reason abstractly and make use of structure. Calculating the growth or decay factor, the \(b\) in \(a \cdot b^x\) , from the coordinates of two points requires students to understand that exponential functions change by equal factors over equal intervals (MP7). In addition, because the first activity uses the Information Gap routine, students first must decide what information they need to solve the problem and why they need it. Obtaining useful information may take multiple rounds of questioning (MP1) and the use of increasingly precise language (MP6).
Lesson overview
 6.1 Warmup: All Equivalent? (5 minutes)

6.2 Activity: Info Gap: Two Points (25 minutes)
 Includes "Are you Ready for More?" extension problem
 6.3 Optional Activity: Bacteria Growth Expressions (10 minutes)
 Lesson Synthesis
 6.4 Cooldown: Finding \(a\) and \(b\) (5minutes)
Learning goals:
 Create an equation for an exponential function from inputoutput pairs in which the inputs do not differ by 1.
 Determine what information is needed to write an exponential equation. Ask questions to elicit that information.
Learning goals (student facing):
 Let’s decide what information we need to write an equation for an exponential function.
Learning targets (student facing):
 I can write equations for exponential functions from two inputoutput pairs, even when the input pairs are not one unit apart.
Required Preparation
 Print the blackline master for the Info Gap activity so that there is 1 copy for every 2 students. Cut the cards into sets for students to use during the activity.
Required Materials
 Preprinted slips, cut from copies of the blackline master
IM Algebra 1, Geometry, Algebra 2 is copyright 2019 Illustrative Mathematics and licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.