## Drug Filtering

- Lesson

In this lesson, students observe a model of exponential decay, and how kidneys filter their blood. They will calculate the amount of a drug in the body over a period of time. Then, they will make and analyze the graphical representation of this exponential function.

Announce that today you will be modeling the how the body's kidneys filter blood. Pour 1 liter of water into a pitcher, and explain that the water represents some of the blood in your body. To start the discussion, ask, "About how much blood do you have in your body?" [5 liters.]

Put several drops of food coloring into the water. Explain that this food coloring represents 1000 mg of a drug (such as acetaminophen or ibuprofen) that you have taken. Mix the food coloring well with the water. Tell students that after four hours the kidneys will filter out about 25% of this drug. Ask, "How can this be modeled?" Allow students to offer suggestions, and then remove 250 ml of the mixture and replace it with 250 ml of clear water. Then ask, "How many milligrams of the drug remain in my blood?" [750 mg.]

Before removing any more colored water, ask, "How could the kidney's work be modeled after another four hours?" Many students will respond that you should remove another 250 ml of colored water and replace it with another 250 ml of clear water. Then ask, "If we did this, how many milligrams of the drug would remain in my blood?" Allow students to make a prediction, often 500 mg, without correction. Ask again for one more four‑hour period. Most often students will say that again 250 mg of the drug are removed. Finally, ask, "So, if I repeat this process four times, will the drug be completely out of my system?" Most often students will answer, "Yes."

Now, remove 250 mg of the colored water a second time, and replace with 250 ml of clear water. Repeat and ask, "If I do this once more, will all the color will be gone, leaving clear water?" The students can now see that this will not happen. Have students debate whether it was the model or the prediction that was incorrect. Students may realize that the second time you removed 250 mg of colored water there was only 750 mg of the drug in the blood, so replacing a fourth of it only removed 1/4 × 750 = 187.5 mg.

Distribute the Drug Filtering Activity Sheet. Students may work alone or with a partner. Circulate around the room to make sure everyone is engaged in the activity. Help students connect their data to the demonstration. Make sure that students are graphing the model correctly. Check that their data is correct and that they are reading the scale of the graph correctly.

Ask students if the graph representing this situation is a linear graph. [No. It is exponential.] Explain that drug filtering is not linear, because the same amount of the drug is not removed during each four‑hour period. Have students look at their graphs. Explain that this is an exponential decay model. Wrap up the class by asking students to suggest other scenarios that can be represented by exponential decay models.

- Clear 2 liter pitcher
- 250 ml beaker
- Spoon for mixing
- Water
- Food coloring
- Container for discarded colored water
- Drug Filtering Activity Sheet
- Drug Filtering Answer Key

**Assessment Options**

- Have students work similar problems with different initial conditions such as a 200 mg dose and 10% filtering every 6 hours.
- Have students research drug and alcohol test methods and report their findings to the class. Have them compare their findings to the results of this activity.

**Extensions**

- Have students model this activity with a spreadsheet and investigate different rates of filtering.
- Have students calculate the amount of drug remaining in the blood, but instead of taking just one dose of the drug, now take a new dose of 1000 mg every four hours. Assume the kidneys can still filter out 25% of the drug in your blood every four hours. Have students make a complete a table and graph of this situation. How does the results differ from the situation explored during the main less? Ask, "Does the amount of drugs in your system continue to grow as long as you keep taking the drug?" Have students use their data table and graph to justify their response.
- Have students write an equation for the situation modeled in this lesson.

**Questions for Students**

1. Do you think the drug will ever be completely out of your blood?

[No. There is always a small amount of drug in the blood. However, it may be so little that it is undetectable.]

2. How would you describe the amount of drug in your blood as time goes on?

[The amount of drug decreases by a smaller and smaller amount, approaching 0, but never quite reaching 0.]

3. Do you think all things leave your body this way?

[There are some substances that that are filtered so slowly that repeated exposure can cause a toxic build up. These include lead and mercury.]

**Teacher Reflection**

- Would you do this activity again? If yes, what would you change? If no, what did not work for you or your students?
- Were all students engaged in the activity? If not, what could you do to engage these others?
- Did students recognize the limiting behavior for both the single dose and the repeated dose?
- Did you find it necessary to make adjustments while teaching the lesson?

### Learning Objectives

- Predict the behavior of a situation represented by an exponential decay model.
- Graph an exponential set of data.

### NCTM Standards and Expectations

- Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.

- Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships.

- Draw reasonable conclusions about a situation being modeled.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP5

Use appropriate tools strategically.

- CCSS.Math.Practice.MP7

Look for and make use of structure.