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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.
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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.
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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
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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.
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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.
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