Elvis Impersonators
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Lesson Title – Applications of Exponential Models
Course Title – Honors Pre-Calculus X
Time Frame – One 50 minute period
Topic/Concept – Modeling real-world problems using exponential functions
Prerequisite/Prior Knowledge – The students must be able to solve a 2x2 system of equations.
The students must understand the concept and behavior of exponential growth functions. The
students must understand how to use the exponential regression tool on their graphing calculators.
The students also must be able to graph exponential functions and interpret the meaning of these
graphs.
Lesson Overview – In this lesson, students will explore exponential models in a real world
setting. In the launch section, the students will be introduced to radioactive decay and the concept
of half-life. Students will also watch a music video sung by Elvis Presley, grabbing their attention
for the exploration. In the exploration section, the students will use exponential modeling to
answer questions about population growth involving Elvis impersonators in the United States. In
the summary section, we will examine the real-world accuracy of this exponential model and any
outside factors that would have a significant effect on the numbers produced in the exponential
model. I will also have the students complete a problem of the day about bacterial growth, giving
me a quick and easy way to assess the students in a formative way and evaluate the effectiveness
of my teaching methods.
Purpose and/or Goal(s) – The purpose of this lesson is for students to be able to use the
principles of exponential modeling to answer real-world problems about population growth,
radioactive decay, and compound interest.
Objectives – The students will be able to use exponential growth and decay functions to model
and solve real-world problems. The students will be able to solve story problems about compound
interest. The students will be able to solve real-world problems about carbon-dating and
radioactive decay using exponential models.
Standards –
Linear, Quadratic, and Exponential Models: Construct and compare linear, quadratic, and
exponential models and solve problems
F-LE.1: Distinguish between situations that can be modeled with linear functions and with
exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals, and that
exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval
relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per
unit interval relative to another.
F-LE.2: Construct linear and exponential functions, including arithmetic and geometric
sequences, given a graph, a description of a relationship, or two input-output pairs (include
reading these from a table).
F-LE.3: Observe using graphs and tables that a quantity increasing exponentially eventually
exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial
function.
Interpreting Functions: Analyze functions using different representations
F-IF.8: Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the function.
b. Use the properties of exponents to interpret expressions for exponential functions. For
example, identify percent rate of change in functions and classify them as representing
exponential growth or decay.
Interpreting Functions: Interpret functions that arise in applications in terms of the context
F-IF.4: For a function that models a relationship between two quantities, interpret key features of
graphs and tables in terms of the quantities, and sketch graphs showing key features given a
verbal description of the relationship. Key features include: intercepts; intervals where the
function is increasing, decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.
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Materials Needed – Laptop, ActivInspire software, EInstruction Mobi wireless tablet, pencils,
paper, graphing calculators, Elmo Visual Presenter, Elvis Exploration handouts, problem of the
day answer sheets
Launch – The purpose of the launch section is to introduce the problem to the class and grab their
attention with the music video “Jailhouse Rock” sung by Elvis Presley. The launch section will
take approximately 10-15 minutes. When the bell rings, I will begin going around the classroom
taking attendance and checking the homework that was assigned on Friday. While I am checking
the homework from Friday, I will have the students complete their “bellringer” for the day. The
“bellringer” will ask the students to explain, using sentences or a bulleted list, how to graph the
function 𝑦 = 𝑙𝑜𝑔2 (𝑥 − 2) + 1 starting with the parent function 𝑦 = 2𝑥 . This will be a great
formative assessment and review of the concepts taught last week, especially because many of the
students struggled with this concept. Once the students complete this, I will briefly introduce one
of today’s new concepts, the idea of radioactive decay and half-life. I will show students the
model for continuous growth, 𝑃 = 𝑃0 𝑒 𝑟𝑡 , and then I will guide the students through two
examples, giving them an opportunity to see the process, understand the concept, and ask
questions if anything is unclear. After this segment, I will grab the students’ attention by playing
the “Jailhouse Rock” music video (about 3 minutes long) performed by Elvis Presley and some of
his look-alikes. This will lead perfectly into our exploration today which focuses on the concepts
behind exponential modeling using the real-world context of Elvis impersonators. After the video
is over, I will pass out the Elvis Exploration handout (attached) and give the students their
instructions.
Explore – The purpose of the exploration section is for students to explore the use of exponential
models in real-world situations. This section will take approximately 20-25 minutes. The students
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will be instructed to complete this packet in their groups. The groups will be allowed to work
through the packet at their own pace, but I will remind them to ask questions if they get stuck,
because the questions build off of each other. The Elvis Exploration packet has students using
exponential models, exponential regression, and real-world data to answer questions and predict
the future number of Elvis impersonators. Using two data points, the students will create an
exponential model that represents the number and growth of Elvis impersonators in the United
States in any given year. They will then use those same two points and use the exponential
regression tool on their calculators to determine the regression equation and then compare these
two methods and their hypothetical results as well. The students will then have to analyze their
findings and synthesize answers to open-ended questions based on their findings and the current
population of the United States. While the students are working on this handout, I will be
circulating the classroom, listening to the discussions and seeing how each group is approaching
the problem. If students are having difficult or appear to be on the wrong track, I will provide
guidance through careful questioning of their processes and the results that came from those
processes. However, if the students become severely stuck, I will ask them to converse with their
classmates, and see if other students and/or groups have a better understanding of the problem.
This exploration will serve as great preparation for the homework assignment and all complex
story problems in the future. Students will also build confidence in themselves after successfully
conquering such a multifaceted problem as this, making other story problems seem less
intimidating.
Summary – The purpose of the summary section of this lesson is to review the concepts
introduced in today’s class, as well as give the students an opportunity to share and discuss their
findings, and more importantly, the real-world implications of their findings. After today’s lesson,
students will be able to use exponential growth and decay functions to model and solve real-world
problems about compound interest, radioactive decay, and population growth. The summary
section will take approximately 10-15 minutes. After the students have completed their
exploration packets, we will come together as a class and discuss the findings of each group. I
will ask students to explain their answers and demonstrate their work, if applicable, on the
eInstruction Mobi tablet. I will guide the discussion with the following questions:
o Is the growth of Elvis impersonators in this exploration realistic? Why or why not?
o What other, outside factors could affect our results?
o What type of growth model would be more appropriate for this example? Why?
o What else in the real world can be modeled by this type of growth?
After the discussion, I will pass out the homework for tonight, and give the students the
instructions for completing this assignment. After that, with approximately 5 minutes left in class,
I will give the students the problem of the day. This problem will be a real-world problem about
bacteria growth, which is modeled by the continuous growth function 𝑁 = 250𝑒 𝑘𝑡 . Students will
have to employ the same problem solving techniques used in the exploration in order to arrive at
the proper solution, making this problem an excellent formative assessment and it will also serve
as an evaluation of my teaching methods today.
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Assessment – The students will be primarily assessed informally throughout the Elvis
exploration. The students will be working in small groups, allowing the teacher to circulate the
classroom, assessing the students understanding of the material and success level after completing
each portion of the exploration section. Furthermore, the students will be required to discuss their
solutions with the rest of the class, explaining their answers using mathematical reasoning, giving
the teacher even more opportunities to assess their understanding at a conceptual level. The
students will also be assessed through the collection and grading of today’s problem of the day,
giving the teacher an opportunity to evaluate the students’ understanding of the material through
mathematical calculations and written explanations. The students will also be given a homework
assignment (worksheet attached) that includes two exponential model story problems, two
continuous growth model story problems, and two radioactive decay (half-life) story problems.
This homework assignment will be an excellent application of mathematics in a real-world
context. It will connect the math to several other disciplines and contexts.
Accommodations – Accommodation will be provided as required by an IEP (individualized
education program). For students who struggle with reading or have a reading disorder, or are
auditory learners, they will be placed into groups in which a strong reader will accompany them.
References –
Jan McNulty, Mathematics Teacher, North Central High School, Indianapolis, Indiana. September
2012
Common Core State Standards Initiative: Mathematics. (2012). Retrieved from
http://www.corestandards.org/the-standards/mathematics/
Appendices –
Appendix A: FlipChart Presentation (attached)
Appendix B: Elvis Exploration handout (attached)
Appendix C: Exponential Modeling Worksheet (attached)
Reflections – After teaching this lesson on modeling exponential growth and decay, I can pick
out many things that worked and things that didn’t. As I have come to notice, we ran out of time
in class without being able to finish our lesson. I don’t necessarily see this is an issue because I
always tend to over-plan. It is much better, in my opinion, to not get to everything you prepared
than to finish your lesson with 10-15 minutes left in class. This is typically when students are
unproductive and discipline issues seem to arise. Furthermore, some of the directions in the
exploration packet were unclear and needed to have more guided descriptions. If I were to teach
this lesson again, I would add the proper instructions, and also begin the exploration much
differently. Instead of handing out the packet and letting students jump right in, I would read all
the directions out loud with the class, and then I would launch them into the packet by giving a
few guided hints about #1 and asking if any of the students had any questions. This would cut
down on time spent by the students at the beginning in which they were stuck on the first question
where they were asked to solve a 2x2 system of exponential equations. The students just needed a
reminder that you have to divide one equation by the other. This helpful hint gave the students the
push in the right direction that they needed without taking anything away from the conceptual
goals of the exploration activity. Additionally, the examples I modeled about radioactive decay
and half-life could’ve been cleaned up a bit in my opinion. Math has always come easy to me, and
I have found that one of the biggest struggles to being a math teacher is finding an easy,
understandable way to explain something, that comes to you so easily, to a student who is
completely lost. I should’ve explained the concept of half-life in more detail and provided
students with real-life examples of how it works. While most of the students already had a solid
idea of what half-life was, some would’ve benefited much more if they would’ve understood the
concept first before attempting the math behind it.
On the other hand, there were several aspects of the lesson that worked very well. For
example, the music video of Elvis Presley served as a perfect “recharge” moment for the students.
It came after the launch and lecture portion of the class, and it allowed the students to take a short
break from math and enjoy some music. It also served perfectly as a launch into the exploration
about Elvis impersonators. I will definitely use this activity for years to come. Also, the
homework assignment was an exponential applications worksheet. I think this assignment, while
very challenging, was excellent practice for the students in solving story problems about
exponential growth and decay. Another form of assessment that I have started to use is the
problem of the day. I begin and end each class with a short question for the students to answer. At
the beginning of class, I have a problem that requires the students to write a short paragraph,
usually related to the previous day’s lesson. This gives me excellent feedback about the
effectiveness of my previous lesson. If students can accurately write about it, then they have
learned it pretty well. Furthermore, the problem at the end of the class period serves as a quick
formative assessment about the effectiveness of my lesson on that day. Usually the last 5 minutes
of class are devoted to this problem, but since we ran out of time today due to the length of the
exploration, the students were given an additional day to do it. After grading and returning them, I
am very pleased with the results. All but 5 students in the class got a perfect 5 out of 5 on the
problem of the day. This shows me that the students are grasping the most important concepts
being taught. All in all, this lesson was pretty successful, although it would’ve been beneficial if
we had an additional 10 minutes at the end of the period to complete the summary discussion and
the problem of the day.
Appendix A: FlipChart Presentation
Appendix B: Elvis Exploration Handout
Pre-Calc X
Name:____________________
Exponential Applications
In 1977, there were 37 Elvis Impersonators in the United States.
In 1993, there were 48,000
1. Assuming this growth to be exponential, (i.e. one Elvis Impersonator "inspires"
others), how many Elvis Impersonators were there to celebrate the new millennium
(the year 2000)?
a) Using the facts above and 1950 for a "benchmark" year, write 2 ordered pairs
in the form of (t, E) (Where t is number of years since 1950) then use these to write a
2x2 system of equations in the form E(t) = a*bt
b) Solve for a & b. (4 sig digs).
a=
;b=
c) Substitute the values and write the final equation below.
E(t) =
2.
a) Put the data into 2 lists in your calculator (again use 1950 as a "benchmark")
b) Do an exponential regression on them.
c) Paste the regression equation into the Y= menu of your calculator. Y =
d) Set the window & sketch graph.
e) Use your graph to estimate (nearest 1000) the number of EI's in the year 2000.
E(50) =
3. Assuming the population of the U.S. to be about 313,000,000 in 2012 (62 years
after 1950) and an annual growth rate of about 1.2%, ...
a) Write an exponential equation of the form P(t) = a * bt
P(t) =
b) Graph the equation on the same grid and sketch in the grid above.
c) Answer the question: Will we ever all become Elvis Impersonators?
d) If your answer to "c" is yes, in what year will it happen?
Appendix C: Exponential Modeling Worksheet
Exponential Applications
NAME___________________________
1. When rabbits were first brought to Australia last century, they had no natural
enemies so their numbers increased rapidly. Assume that there were 60,000 rabbits in
1865, and that by 1867 the number had increased to 2,400,000. Assume that the
number of rabbits increased exponentially with the number of years that elapsed since
1865.
a) Write the particular equation for this function by writing data points and
doing an expreg
y1= ___________________
b) How many rabbits would you predict in 1870? Try this using the table
feature. 2nd window accesses the table setup. Table start at 0 and Table at 1. Indpt
and dpnt at auto. 2nd graph gives you access to the table.
c) According to your model, when was the first pair of
introduced into Australia?
y2=2 ( 2 bunnies)
rabbits
window: [-10, 5] [-1, 3 ]
Try the 2 trace (calc button) feature. Try the intersect option. Select it and press
enter 3 times.
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2. Oliver Sudden is driving along a straight, level highway at 64 km/h when his car runs
out of gas. As he slows down, his speed decreases exponentially with the number of
seconds since he ran out of gas, dropping to 48 km/h after 10 seconds.
a) Write the particular equation expressing speed in terms of time. Use the
above method.
b) Predict Oliver's speed after 25 seconds.
Try 2 trace (calc) value: x= 25
nd
c) At what time will Oliver's speed be 10 km/h?
For #3 and #4, use the continuous growth model: 𝑃 = 𝑃0 𝑒 𝑟𝑡
3. In a research experiment, a population of fruit flies is increasing as follows: After 2
days there are 100 flies, after 4 days there are 300 flies. What is the daily rate of
growth and how many flies will be present after 5 days?
4. The population P of Charlotte, North Carolina (in thousands) was 548 thousand in the
year 2000. In 1970, the population was 271 thousand. Use the continuous growth model
to find the annual rate of growth, r, and use this to predict the population in 2015.
(Use 1970 as year 0)
5. Complete the table for each radioactive isotope.
Isotope
226
Ra
C
239
Pu
14
Half-life
(years)
1599
5715
24,100
Initial Quantity
Amount after
1000 years
1.5 g
3g
2.1 g
Annual rate of
decay