Exponential and Logarithmic Models (Lesson 3.8)
Unit 3  Day 9
Learning Objectives

Create models for halflife or double life problems and use solving techniques to answer questions based on the model.

Use exponential functions to model population growth, disease growth, and compounded interest.

Interpret answers to exponential and logarithmic modeling problems
Quick Lesson Plan
Experience First
In this lesson students use halflife to reason about recommended doses of DayQuil™, and specifically the active ingredient of Dextromethorphan. Students do not need prior knowledge of halflife or halflife formulas to be successful in this lesson.
Students first think about how much of the chemical remains in the body after certain time intervals in order to arrive at the formula for D(t). On question 1c, ask students how many 3hour periods are in 10 hours or how many times the drug has halved in 10 hours. If students are really struggling with this, have them first estimate that the amount after 10 hours would be somewhere between 2.5 mg and 1.25 mg, since the initial quantity of 20 mg has halved more than 3 times but less than 4 times. As you monitor groups, expect to hear students say that after one hour, the body will eliminate ⅓ of ½ , or ⅙ of the quantity. Push students to explain and defend their conjectures and to have other students critique their reasoning (MP3).
In question 4, students must use logs to work backwards and find the time at which the drug becomes ineffective. Although the exact answer is a decimal, ask students whether the label would likely give this very specific answer or round instead.
Question 5 can be tricky for students. The table goes up in 2 hour increments, so students must determine how much remains after 2 hours or simply plug in the appropriate tvalue into their equation from question 2. Note that at 12 PM, the person will have taken their next dose, so 20 mg is added to the amount remaining from their first dose. This is also true at 4 PM and at 8 PM. At 8 PM students must decide whether the person did or did not yet take their new dose. This is a defining characteristic of some of the ambiguities that come with realworld modeling tasks.
Formalize Later
New ideas that are presented in this lesson are the ideas of halflife and mathematical modeling. Students should be able to generate the halflife formula on their own by reasoning about how many halflife periods have occurred in a certain interval of time. Additionally students are formally introduced to the idea of a mathematical model, even though they have been interpreting such models throughout the whole year. I use the example of a model train or a model of the Eiffel Tower to explain the idea that models are a convenient representation of the reality, but they have limitations. We use models because they are useful, not because they are perfectly accurate.
Many strategies are possible for question 3 and this is worth discussing in the debrief. Most students do not immediately jump to the algebraic generalization that every hour there would be (½)^(⅓) or 79.37% remaining of the chemical. Many students will calculate the amount left in the body after 1 hour, and then divide this by the initial amount to see the ratio. This is a great approach, and leads to a discussion about why the initial amount was actually irrelevant to the calculation.