LOGO - PULSE



Take Your Medicine

Author: Brink Harrison
Adapted from Section 9.1 in Hughes-Hawlett, Deborah, et.al; Single Variable Calculus; John Wiley & Sons, Inc.; New York; 2002


Time: 1 to 2 days
Preparation Time: none
Materials: none

Abstract

Students learn to change from the expanded form of a finite geometric series to the closed form of a finite geometric series and calculate the sum of the series. They use the closed form to calculate the amount of a specific antibiotic remaining in a body after a given number of doses have been taken and graph the amount of the drug in the body in between doses over time. From this graph students will infer the cumulative amount of the drug in the body as the number of doses taken increases and the see the importance of taking the full regimen of the antibiotic to combat antibiotic-resistance infections.



Objectives


(Day 1) Students will able to:

• Change from the expanded form of a finite geometric series to the closed form to facilitate calculating the sum of the series.

(Day 2) Students will able to:

• Use the sum of a finite geometric series to calculate the amount of antibiotic in the body just after the nth dose is taken, given the percentage of the original given amount of the antibiotic is present at the end of a given time period.

• Use the sum of a finite geometric series to calculate the amount of antibiotic in the body just before the nth dose is taken, given the percentage of the original given amount of the antibiotic is present at the end of a given time period.

• Draw a graph that shows the quantity of the antibiotic in the body between doses as a function of time.

• Draw a graph that shows the cumulative amount of the antibiotic in the body over time.

• Explain why the graph approaches a constant amount of antibiotic in the body.

Math Standards
Numbers and Operations
Compute fluently and make reasonable estimates
Algebra
Use mathematical models to represent and understand quantitative relationships
Problem Solving
Apply and adapt a variety of appropriate strategies to solve problems
Connections
Recognize and apply mathematics in context outside of mathematics


Teacher Background
This lesson should follow an introductory lesson where the students are asked to identify geometric series as well as determining the value of a and being able to calculate the value of r. They have not found the sum of a geometric series yet.

A geometric series is a series in which each consecutive term is a multiple of the one before. If the first term of the series is a and the constant multiplier, or common ratio of successive terms is r, then a finite geometric series with n terms has the form:

Notice that the exponent of the last term is (n-1). This is because when you include the first term a, you want to have a total of n terms,

We will use the symbol,
                                     

to represent the sum of the first n terms of a finite geometric series. The expanded form of the sum of the first n terms of a finite geometric series is written as the equation,

The purpose of the first part of the lesson is to be able to convert the expanded form of a finite geometric series to the closed form to make it easier to calculate the sum of a series with many terms. The closed form formula for the sum of the first n terms of a finite geometric series is:

where a is the first term of the series, r is the common ratio between consecutive terms, and n is the number of terms in the series.

 

 


Activity (
Day 1)

Have the students to discuss what they know about geometric series. (See teacher background)

Tell the students that they will use the symbol,
                                           
to represent the sum of the first n terms of a finite geometric series. Have the students write the expanded form for the following series:

Ask them how they would find the sum of the following series:

If there are not many terms, all they need to do is add the terms together. Have them discuss what they would do if there were 40 terms or 100 terms or n terms. The following procedure will lead the students to the formula for the closed form of any finite geometric series.

To find the sum of any finite geometric series, we write the sum of the first n terms as:

Multiply this equation by r to get:

Now subtract the second equation from the first:

We now have the equation:

Factoring out common terms, we have the equation:

Solving, we end up with the equation:

This is called the closed form for the first n terms of a finite geometric series, and it’s much easier to use to calculate the sum than to add all of the terms together. All the students need to know are the value of a (the first term), the value of r (the common ratio between successive terms) and the value of n (the number of terms in the finite series).

Examples: For the following finite geometric series, determine the value of a, the value of r, the value of n, and calculate the sum of the series by using the closed form.


-Top-


Homework

For the following finite geometric series, determine the value of a, the value of r, the
value of n, and the value of the last term in the series. Then calculate the sum of the series by using the closed form of the series.


8) Calculate the following quotients:

The answers to these problems are the same as the answers to problems 2 – 6 above. Why do you think this happens?

9) Think of some real-life application where you would need to calculate the sum of a finite geometric series.

-Top-

Day 2

Teacher Background

One of the objectives of the lesson is to write the expanded form of a finite geometric series to represent the amount of the antibiotic ampicillin in the body as the patient continues to take the prescribed doses. The students will then use the closed form for the sum of a finite geometric series to calculate how much of a specific antibiotic remains in the body over time. The formula for the sum of the first n terms of a finite geometric series is:

where a is the first term of the series, r is the common ratio between consecutive terms, and n is the number of terms in the series. But this time each term of the series,

represents a dose of the drug and the sum of the series represents the drug level in the body in the long run. In this situation the value of a is the first dose of the drug, n represents the nth dose of the drug, and r is equal to the percentage of the original given amount of the antibiotic that is present at the end of a given time period.

**** Optional: You can make the process more complicated if you wish by having the students calculate the value of r instead of giving it to them. To calculate what percentage of the original given amount of the antibiotic is present at the end of a given time period, the students must use the half-life of the antibiotic along with exponential decay function

For example, the antibiotic doxycycline has a half-life of 16.3 hours. We want to find what percent of the original given amount of the antibiotic is present at the end of a 24-hour period. First you need to find the value of k in the above equation. It is easier to assume that you have 100mg in the beginning. Since 50mg is what is left of the original 100mg after 16.3 hours, the equation then becomes:

Solving for k, the equation becomes

After 24 hours,

Thus, the percentage of the original amount of doxycycline that remains after 24 hours is approximately 35.6%. To calculate the sum of a finite number of doses, 35.6% is the number used as the value of r, the common ratio between the terms, in the closed form of the finite geometric series. For most of the problems in the homework section, the value of r will be provided.

By calculating the amount of the drug in the body over time, the students should see that as the number of doses increases, the amount of the drug in the body right after each dose is taken approaches a constant value. This “steady state” amount is high enough to kill the bacteria causing the infection. However, if this level of medication is not maintained for long enough, it can lead to the development of bacterial resistance.

Antibiotic Resistant Bacteria

Many infectious diseases, including bacterial pneumonia, tuberculosis, and gonorrhea, are caused by bacteria. Antibiotics work by a variety of mechanisms to kill bacteria and have proven vital in curing infectious diseases. However, bacteria have the capacity to evolve defense mechanisms against antibiotics and can become resistant to their effects. When such resistance develops, bacteria are no longer killed by the antibiotic and, thus, the antibiotic is no longer capable of treating or curing the disease. The more an antibiotic is used, the more likely that bacteria will "learn" to evade it.Many infectious diseases, including bacterial pneumonia, tuberculosis, and gonorrhea, are caused by bacteria. Antibiotics work by a variety of mechanisms to kill bacteria and have proven vital in curing infectious diseases. However, bacteria have the capacity to evolve defense mechanisms against antibiotics and can become resistant to their effects. When such resistance develops, bacteria are no longer killed by the antibiotic and, thus, the antibiotic is no longer capable of treating or curing the disease. The more an antibiotic is used, the more likely that bacteria will "learn" to evade it.

Natural selection plays a key role in the development of antibiotic resistance. Most bacteria die when exposed to antibiotics to which they are sensitive. That leaves more space and available nutrients for surviving bacteria (i.e., for antibiotic-resistant bacteria). As a result, the resistant bacteria can reproduce and multiply freely and pass on the antibiotic-resistant genes to the next generation.

Not only can resistant bacteria proliferate after other bacteria are killed off by an antibiotic, but they also can transfer that resistance to other bacteria that have never been exposed to the antibiotic. Bacterial cells can join briefly and exchange loops of DNA (called plasmids) that contain genes that confer antibiotic resistance. For example, if one bacterial species becomes resistant to a broad-spectrum antibiotic, it could transfer its resistance genes to other bacteria that have never encountered those antibiotics.

The genes that cause antibiotic resistance function in a number of ways. Some do not permit the antibiotic to get into the bacteria; others actively pump it out of the bacteria; some produce enzymes that inactivate the antibiotic; and others modify the antibiotic’s target site in the bacteria. To make matters worse, many bacteria have become resistant to multiple antibiotics. That property can result from the cumulative effect of treating stubborn infections with multiple types of antibiotics (or acquiring a plasmid with numerous resistance genes).

Antibiotic resistance has increased rapidly in the U.S. and abroad in recent decades. Streptococcus pneumoniae, a bacterium that can cause ear infections, pneumonia, blood infections, and meningitis, is becoming increasingly resistant to antibiotic treatment. In 1987, antibiotic-resistant pneumococci were unknown. By 1997, as many as 40 percent of pneumococcus isolates were resistant to penicillin and other commonly used antibiotics.
(Source: http://www.cspinet.org/reports/abiotic.htm)

Related Websites
http://www.cspinet.org/reports/abiotic.htm

-Top-

Activity (Day 2)

1) Go over the first three problems in yesterday’s homework. Ask the students if they needed to know the last term of the finite series to be able to calculate the sum. (They did not need it.)
Ask if anybody came up with a real-life application where they need to be able to find the sum of a finite geometric series. Discuss any answers you might get.

Tell the students that one application is in taking antibiotics when they have an infection. Let’s suppose that they have strep throat and the doctor tells them to take the antibiotic ampicillin regularly for two weeks. Ampicillin is usually taken in 250 mg doses four times a day, that is exactly every six hours. It is known that at the end of six hours, due to excretion, about 4% of the drug is still in the body. How much ampicillin is in the body right after taking the eighth tablet? What about right after the 48th tablet?


2) Ask the students how they can set up a finite geometric series to represent the situation. Tell them to let the symbol
                     

represent the quantity in milligrams of ampicillin in the blood right after having taken the nth tablet. This means that right after taking the first tablet,
                    
Six hours later, a second tablet is taken, and the amount of the drug in the body becomes
                    
because you took a new tablet (250 mg), plus you have 4% of the first tablet as well.

Six hours later, after the third tablet is taken, the amount becomes
                    
because you took a new tablet (250 mg), plus you have 4% of the second tablet, and (4%)(4%) of the first tablet as well.

3) The students will hopefully see the pattern. Ask them to give you the next three quantities.

Teacher Cheat sheet:

Point out that the equations are the expanded form of a finite geometric series with a = 250, r = 0.04, and the highest exponent used is (n –1). There are n terms in the series, one for each tablet.

4) Ask the students to write the expanded form of the finite geometric series that represents the amount of ampicillin in the blood right after having taken the eighth tablet and the 48th tablet, using “+…+”, instead of writing out all of the terms.


5) Ask the students if they can think of a short cut for calculating the amount of ampicillin in the blood right after having taken the 48th tablet instead of writing out all 48 terms and adding them together. Hopefully somebody will remember how to calculate the sum using the closed form:


6) Have the students use the closed form to calculate the following amounts of the drug

                    


7) Ask them to describe, as the number of doses increases, what is happening to the amount of the drug in the body right after a tablet is taken. Have them discuss why this might be happening.

Teacher cheat sheet: Below are answers to the respective amounts.

Notice that all of these values would round to 260.42mg, the same amount (to two decimal places) as the amount of the drug in the body right after the 48th tablet. This means that by the end of the first day (after taking four tablets), the amount of the drug appears to have stabilized at approximately 260.42mg.


8) Ask the students write the closed form if n tablets are taken. What does this closed form predict about the long-run level of ampicillin in the body? In other words, what happens to the level of the drug right after a tablet is taken if a very large number of tablets are taken?

Teacher cheat sheet: The closed form of the series for n tablets is:

If you allow the value of n to get very large,

Thus, assuming that 250mg continue to be taken every six hours, in the long run the level of the drug in the body, right after a tablet is taken, appears to be approaching a constant level of 260.42mg.

9) Ask the students how they can set up a finite geometric series to represent the amount of the drug in the body six hours after having taken a tablet. Remind them that about 4% of the original amount is left in the body six hours after the tablet is taken. Let the symbol represent the quantity in milligrams of ampicillin in the blood six hours after having taken the nth tablet. This means that

six hours after taking the first tablet, or just before taking the 2nd tablet,

Six hours after the second tablet is taken, and the amount of the drug in the body becomes

as 4% of the second tablet and (4%)(4%) of the first tablet remain in the body.

Six hours after the third tablet is taken, the amount becomes

as there is 4% of the third tablet, (4%)(4%) of the second tablet, and (4%)(4%)(4%) of the first tablet remain in the body.

10) The students will hopefully see the pattern. Ask them to give you the next three quantities.

Teacher Cheat sheet:

Point out that the equations are the expanded form of a finite geometric series with a = 250(0.04), r = 0.04, and the highest exponent used is n. There are n terms in the series, one for what remains of each tablet six hours after it was taken.


11) Ask the students to write the expanded form of the finite geometric series that represents the amount of ampicillin in the blood six hours after having taken the eighth tablet or the 48th tablet, using “+…+”, instead of writing out all of the terms.


12) Ask the students if they can think of a short cut for calculating the amount of ampicillin in the blood six hours after having taken the 48th tablet instead of writing out all 48 terms and adding them together. Hopefully somebody will remember how to calculate the sum using the closed form:


13) Have the students use the closed form to calculate the following amounts of the drug


14) Ask them to describe, as the number of doses increases, what is happening to the amount of the drug in the body right after a tablet is taken. Have them discuss why this might be happening.

Teacher cheat sheet: Below are answers to the respective amounts.

Notice that all of these values would round to 10.42mg, the same amount (to two decimal places) as the amount of the drug in the body six hours after the 48th tablet.


15) Ask the students write the closed form six hours after the nth tablet is taken. What does this closed form predict about the long-run level of ampicillin in the body six hours after each tablet is taken?
Teacher cheat sheet: The closed form of the series for six hours after the nth tablet is:

If you allow the value of n to get very large,

Thus, assuming that 250mg continue to be taken every six hours, in the long run the level of the drug in the body, six hours after a tablet is taken, appears to be approaching a constant level of 10.42mg.

16) Have the students make a graph of amount of drug in the body versus time (in six hours intervals). Assume that the first dose is taken at time t = 0.

Teacher Cheat Sheet: the first seven points would be as follows

All the rest of the y-values for Q will be 260.42mg and all the rest of the y-values for P will be 10.42mg.


17) Have the students make a graph of amount of drug in the body versus number of doses. Assume that no drug is in the body at time t = 0.

Teacher Cheat Sheet: the first seven points would be as follows


18) Looking at the graph of amount of drug versus dose number, ask the students what would happen if they forgot to take a tablet and went 12 hours in between doses? What would happen if they stopped taking the antibiotic before completing the full regimen they were given?

Teacher Cheat Sheet: See teacher background.

-Top-

Homework

The drug erythromycin may also be taken for streptococcal infections. Erythromycin is given in 400mg doses every six hours. At the end of six hours, there is 6.25% of the amount taken left in the body.

1) Write the expanded finite geometric series for the amount of the drug in the body right after having taken the nth tablet.

2) Write the closed finite geometric series for the amount of the drug in the body right after having taken the nth tablet.

3) Write the expanded finite geometric series for the amount of the drug in the body six hours after having taken the nth tablet.

4) Write the closed finite geometric series for the amount of the drug in the body right after having taken the nth tablet.

5) Calculate the amount of erythromycin in the body right after taking a tablet over the long-run.

6) Calculate the amount of erythromycin in the body six hours after taking a tablet over the long-run.

7) What is the difference between the numbers calculated in problems 5 and 6? Why does this occur?

Answer: 426.67-26.67=400mg, which is the amount of each individual dose of erythromycin taken. This occurs because once the level of the drug in the body right after a dose is taken reaches 426.67mg, the steady state amount, the body excretes 400mg over six hours, (426.67)(.9375) = 400, because taking one more tablet raises the level back to the steady state amount.

-Top-

Embedded Assessment

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


PULSE is a project of the Community Outreach and Education Program of the Southwest Environmental Health Sciences Center and is funded by:


an
NIH/NCRR award #16260-01A1
The Community Outreach and Education Program is part of the Southwest Environmental Health Sciences Center: an NIEHS Award

LOGO - SWEHSC
LOGO - NIEHS Center LOGO - NIEHS

Supported by NIEHS grant # ES06694


1996-2007, The University of Arizona
Last update: March 7, 2007
  Page Content: Rachel Hughes
Web Master: Travis Biazo