Task
DDT is a toxic agricultural chemical that was used in the United States before it was banned in 1972. DDT has a halflife of 15 years. That means it takes 15 years for one half of a quantity of DDT to degrade into a different, harmless chemical. Suppose an environmental scientist in 2015 measured 9g of DDT in a soil sample taken from land where DDT was once heavily used. The scientist modeled the amount of DDT in the soil, a, with the function \(a(t)=9(0.5)^t\). She indicated in her notes that t represented "time."

Find a(0). What might this value represent in this context?

Find a(1) and a(1). What might these values represent in this context?

Explain, with more specificity, what you think t represents in the function a(t).
IM Commentary
The purpose of this task is for students to encounter negative exponents in a natural way in the course of learning about exponential functions. If students are permitted calculators for this task, they may miss out on needing to make sense of evaluating a negative exponent. Without a calculator, they might reason that \({9(0.5)^{1}=9(}{1\over2})^{1}=9(2)^1\). Calculator use is a judgment call at the teacher's discretion, but the teacher could still call attention to the result of raising 1/2 to a negative power. The interpretation of negative values for t as referring to periods of time in the past will still be elicited, even if calculators are used for evaluating.
This task could be used as a foundational context to ask some related questions. For example, a teacher may need stretch questions for students who finish early. Here are some suggestions:
"The scientist has historical records that show the original amount of DDT in a soil sample of the same size taken from the same area was 72g. Figure out (by any method you devise) the value of t when a(t) = 72. What does this value tell you about when the DDT was originally introduced to the soil sample?" Students might count the number of doublings backward to 72g to find t = 3 it is not necessary that students know how to use a logarithm in order to solve \(9(0.5)^t=72\). Rather, they should use appropriate familiar tools strategically (MP.5), for example, they could create a table to organize their backwardscounting, or they might graph the function a(t) with technology and look at the point (t, 72).
As an alternative to the previous question, we could ask about a less convenient amount of DDT (less convenient than 72g which works out to exactly t = 3). For example, "In a certain year, a scientist found that the amount of DDT in a soil same of the same size taken from the same area was 50g. About how many years before 2015 was that soil sample taken?" Students could again use appropriate tools strategically to find that this occurred around 2.5 halflives before 2015, which would be around 37.5 years before 2015, or some time in 1977. However, perhaps it would meet the teacher's goals for the class for students to determine that the sample was taken between 2 and 3 halflives ago.
"Rewrite the function in a different form so that someone could just plug in number of years from today without having to find how many halflives first." If we let u = number of years from now (recall that t = number of halflives from now), we can reason since there are 15 years in one halflife that u = 15t. So in the function, we could replace the t with u/15 and come up with \(a(t)=9(0.5)^{u\over15}\).
Note: the halflife of DDT is, in reality, approximately 15 years, but for simplicity's sake we are assuming in this task that the halflife is exactly 15 years.