​Learning Targets​

• Construct exponential models from an initial value and ratio or from two input-output pairs.

• Use an exponential model to make predictions about the dependent variable.

• Understand how equivalent forms of an exponential function can reveal different properties about its growth rate.

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Quick Lesson Plan

Experience First:

Although students have been working with modeling situations all unit, this lesson is specifically focused on constructing a model, interpreting the model, and understanding how an equivalent form of the model can reveal different attributes of the function’s growth.

 

In this lesson students use half-life to reason about recommended doses of DayQuil™, and specifically the active ingredient of Dextromethorphan. Students do not need prior knowledge of half-life or half-life formulas to be successful in this lesson.

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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 3-hour 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).

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In question 4, students must write an alternate model for D(t) that features an hourly rate. This is not specifically told to students, but the model only has t in the exponent, so students should realize that for integer values of t, as t increases by 1, the output is multiplied by b, so the b value needs to represent the constant proportion from one hour to the next. Flexibly manipulating exponential expressions is critical for being able to interpret exponential growth from multiple perspectives.

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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 t-value 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 real-world modeling tasks.

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Monitoring Questions:

• If we were to halve 4 times, what fraction of the original amount do we have? (Be prepared for students to say 1/8 instead of 1/16!)

• How many half-lives have occurred in 10 hours? Can you give an estimate? Can you give an exact answer?

• How do we calculate 3 ⅓  half-life periods?

• What are the advantages of your model in question 4 compared to your model in question 2? What are the disadvantages?

• What’s the decay rate for 1 minute?

• What’s the decay rate for 1 day?

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Formalize Later:

The debrief should include a discussion of the many strategies that are possible for answering question 3. 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.

 

Part of the process of writing an equivalent exponential form is rewriting growth factors for a different time interval. This can be done when you want the growth factor for a longer time period or for a shorter time period. If you have the growth factor, b, for 1 unit of time and you want the growth factor for k units of time, you can calculate b^k. If you have the growth factor, b, for k units of time and you want the growth factor for 1 unit of time, you can calculate b^(1/k). This should not be memorized but understood from a conceptual and algebraic perspective.

 

Students should understand that exponential models can be constructed from an initial value and ratio (something they’ve been doing the whole unit), other verbal descriptions (the quantity doubles every 5 days), as well as two input-output pairs. Question 2 on the Check Your Understanding has students work through this process of solving for the parameters of an equation given two input-output pairs.