In this lesson, students examine the meaning of negative exponents in context. In both cases, time \(t\) is the independent variable and \(t = 0\) corresponds to some particular moment when a quantity is first measured. In these situations, a value of \(t\) that is less than 0 corresponds to a time before the initial measurement, while a value of \(t\) that is greater than 0 refers to a time after the measurement. Students write equations given a table, and then use those equations to answer questions, which requires working with negative values representing time. In one case, students also produce a graph. As with many other activities in this unit, the mathematical work is grounded and interpreted in a context (MP2).
An optional activity here addresses the difficulties inherent in producing graphs representing exponential change, namely that the graphing window has to be selected carefully in order for the graph to be useful. The savvy setting of a graphing window is an important skill when using graphing technology, and intentionally addressing this skill puts students in a better position to use technology strategically in the future.
- Interpret a negative exponent in equations that represent exponential growth or decay.
- Write and graph an equation that represents exponential decay to solve problems.
Let’s look more closely at exponential graphs and equations.
Devices that can run Desmos (recommended) or other graphing technology and GeoGebra (recommended) or other spreadsheet technology should be avalable as an option for students to select during the lesson. Graphing technology is needed if doing the optional Graphing Windows activity.
The activity "Coral in the Sea" has a digital applet in the activty synthesis. Be prepared to display the applet for all to see.
- I can describe the meaning of a negative exponent in equations that represent exponential decay.
- I can write and graph an equation that represents exponential decay to solve problems.
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.