## IM Commentary

When students first start studying exponential functions they evaluate them at integer inputs. After they have learned about rational exponents they can evaluate them at other inputs. The purpose of the task is to help students become accustomed to evaluating exponential functions at non-integer inputs and interpreting the values. They do this by constructing a model to interpret a real-world situation with an emphasis on evaluating at non-integer values.

There are several opportunities to attend to precision MP.6 in this task. First of all, all of the outputs of the function used to model the amount of the drug left in the patient's bloodstream are approximations. (How quickly a body processes chemicals can be affected by a range of factors, including the amount of exercise it gets and the presence of other chemicals in its system.) This is indicated by the presence of $\approx$'s in the solution, but the issue might bear a few moments of attention. Secondly, the task does not include rounding instructions and calculated answers aren't "nice" numbers, so the teacher can decide whether to prescribe that students round to the tenths place, as shown in the solution, or decide for themselves what an appropriate level of precision would be.

Although part (c) doesn't elicit this observation explicitly, a nice thing to notice is that the percentage of the drug that leaves the bloodstream in the first half an hour, 8%, is not half of 15%, because the decrease won't behave "linearly" each hour -- the drug leaves faster at first, so more than half of 15% leaves in the first half-hour. A teacher could opt to prompt students to explain why more of the drug leaves in the first half-hour than the second half-hour. A teacher could push students further to think in terms of exponential functions decreasing by equal factors over equal intervals, so if the factor for an hour is 0.85, the factor for a half hour is $\sqrt{0.85}$.