One-Day Module
Materials :The materials listed below are for all of the experiments. You may
choose to do only certain experiments, and therefore would not necessarily need all of
these materials.
TI-83 Calculator (or another version that will do graphing and exponential
regression) and overhead calculator display for the teacher
Golf or Tennis Balls
Meter or Yard Sticks
Paper Plates (1 per group)
Bags of Skittles Candy
Cup for each group
Population Growth/Decay Worksheet and Homework Assignment
Activities:
1. Background Information – The first thing that should be mentioned to
students, is that this module will involve percents. The teacher should quickly review
writing 2%, .5%, and 85% as a decimal. Next, most students are familiar with linear
equations and graphs, but some background might be needed to introduce/remind
students about exponential functions. One way to do this might be to present the
student with the graph of y = 60(.8)x (on the overhead calculator display.) This might
be the graph of the temperature of a hot soda that has been placed in the freezer. It
would show that it is cooling at a non-constant rate and then leveling off. The teacher
should discuss this exponential decay with the students, including the standard form
for an exponential function where y = a(b)x. The teacher should point out that a is the
starting value and b is the rate or base of an exponential. The rate is identified as a
growth rate if b is greater than one (ie 1+r) and a decay rate if b is less than one (ie 1r.) The teacher also needs to point out that the graph never actually touches zero (ie
the x-axis) and why ( there is no x value that will make the equation zero.) Also, they
should mention that this could also model an animal or human population. It should
be pointed out to the students an example of a population that would be decreasing
exponentially. A quick discussion should then follow about things that have been
taken into account for an equation such as this. It could model the growth of a
population by taking the birthrate and subtracting the death rate. It could not however,
model immigration or migration out of the population. The teacher should point out
that this would involve adding a constant on to the end of the original equation (y =
a(b)x +c.) This is not studied in this module, but may be considered as an extension.
Next, the teacher should present a graph of exponential growth such as y = 5(1.10)x.
They should note that this shows exponential growth and ask the students to come up
with possible scenarios that this could model. It is important that the students
understand that for each time period that passes, the population is 110% of the size it
was the period before. It should also be mentioned, that the growth cannot continue
indefinitely or overpopulation will occur, a discussion could then follow about ways
of controlling populations. It is very important for the teacher to point out that for
exponential decay, the rate or base of an exponential is less than one and for
exponential growth, the rate or base of an exponential is greater than one. (This
should take 15-20 minutes.)
2.
Investigation 1: Exponential Decay – I have included two possible
experiments. The first one is a great investigation that allows students to use
golf/tennis balls to model decay. It lets them get a feel for exponential equations.
The only drawback to this investigation is that it does not address the decay of a
population. Therefore if the focus of the lesson is to be on population decay,
I’ve included a second investigation that could be used to model a population.
Also, it would not include having to leave the room, so this may be a better
option for some people.
Experiment 1: In order to model exponential decay, there is a neat experiment
that involves students dropping a golf or tennis ball from a certain height and
measuring each rebound height. Students should be broken into groups of 3 or 4 for
this activity. (If the time limit becomes a problem, this could just be demonstrated
by one group of students and the whole class could use the same data.) This works
best for students to stand on a chair either in an uncarpeted area or outside. They
should measure the height in a unit designated by the teacher so that the whole class
is using the same measurements (inches or cm.) They should then drop the ball. The
second group member should catch the rebound and hold it still at the highest point
while the third member measures the height. The fourth member records the data in
a table. They should do this for 5 or 6 rebounds if possible. If there is time, then the
ultimate situation would be for each group to do a table for tennis balls and another
for golf balls and compare the results. Once all the groups have recorded their data,
the teacher should demonstrate entering the data into the lists of the calculator and
making a scatterplot. Students should copy the graph of their own data onto the
activity sheet. Next, the teacher should then go through how to calculate the
exponential regression on the calculator. Students should copy their own function
onto the worksheet and graph the function with the scatterplot to see if it is a good
fit. (This activity sounds like it takes a great deal of time, but it is actually not that
long. I would allow 30-40 minutes)
Experiment 2: Students should be in groups of 3 or 4 students. Give each group a
paper plate and 25 Skittles. Please inform them that they will be using the candy for
the next investigation as well, so do not allow them to eat it yet. Before the
experiment begins, the teacher will need to have folded the plates and colored in a
certain section of the area. I would suggest ¼, ½, and ¾, or any other portion that
would be easy for you to do. Some groups may have the same section colored in and
that’s OK. The students need to place all 25 Skittles in a cup. This is the beginning
population for any kind of species that the teacher wishes to use (bald eagles might
be a good example.) Next, the students should shake the Skittles onto the plate. Any
Skittles that land in the shaded region must be removed from the plate as they have
“died.” (It should be determined as a class beforehand what to do if the skittle lands
on the edge so that half of it is in the colored region and half is not.) The students
should then count the number of skittles that are left and record the number in the
table on their worksheet. They should continue this at least 5 times, or until the
population is zero. (If the population is zero, please remind students that they cannot
enter this data into the calculator. It will not perform exponential regression if zero
is a y-value.) Once all the groups have recorded their data, the teacher should
demonstrate entering the data into the lists of the calculator and making a scatterplot.
Students should copy the graph of their own data onto the activity sheet. Next, the
teacher should then go through how to calculate the exponential regression on the
calculator. Students should copy their own function onto the worksheet and graph
the function with the scatterplot to see if it is a good fit. (Note: This could be varied
by changing the beginning population depending on how much time there is. Also,
the students could do this until all the Skittles are gone as opposed to only doing it 5
times. Once again, if time limit is a problem have a few students come to the front
and demonstrate and let the whole class use the same data.) ( This entire experiment
should take about 30-40 minutes.)
Investigation 2: Exponential Growth – I have included two experiments for
exponential growth of a population. Both involve the Skittles used in Experiment
2 above. It is the teacher’s discretion as to which one they would rather use.
Experiment 1: Student should be in groups. Give each group several
skittles and a cup along with their worksheet. Have them put two skittles in the
cup and record this in the table on the worksheet under 0. They should consider
this as two frogs happily living in the local pond. They should then shake the cup
and pour the candy onto a desk. For every one with the s showing, the students
should add in another skittle and record the new population. They should continue
this process and fill in the table. They should then enter the data into the lists and
do exponential regression as before while they answer the questions on the
worksheet. Each group should share their equation with the class. ( Note: This
experiment could be varied by changing the beginning population from 2 to
another number. Keep in mind however that it’s about exponential growth, so
don’t start out too big!)(This should take about 30 minutes depending on how
well the students stay on task.)
Experiment 2: This experiment is much like Experiment 2 above, only
instead of decay, there will be growth. Students should be in groups of 3 or 4
students. Give each group a paper plate and 10 Skittles. Before the experiment
begins, the teacher will need to have folded the plates and colored in a certain
section of the area. I would suggest ¼, ½, and ¾, or any other portion that would
be easy for you to do. Some groups may have the same section colored in and
that’s OK. The students need to place all 10 Skittles in the cup, and record 10
under the zero in the table on the worksheet. This is the beginning population for
any kind of species that the teacher wishes to use (frogs or fish might be a good
example.) Next, the students should shake the Skittles onto the plate. Any Skittles
that land in the shaded region must be counted. These will be the ones to
reproduce. Once they’ve been counted they can all be returned to the cup and that
same number of Skittles that landed in the shaded region is added to the
population. (It should be determined as a class beforehand what to do if the
skittle lands on the edge so that half of it is in the colored region and half is not.)
The students should add in the number of Skittles added and record the new total
number in the table on their worksheet. They should continue this process until
they’ve filled in the table. They should then enter the data into the lists on the
calculator and do exponential regression as before while they answer the
questions on the worksheet. Each group should share their equation with the
class. ( Note: This experiment could be varied by changing the beginning
population from 10 to another number. Keep in mind however that it’s about
exponential growth, so don’t start out too big!)(This should take about 30 minutes
depending on how well the students stay on task.)
Conclusion
The teacher needs to discuss several things with the students at this point.
The general form of an exponential equation where a is the starting population, b
is the growth or decay rate, and x is the time needs to be reviewed. The fact that
the exponential function represents growth if the rate is greater than one and
decay if the rate is less than one needs to be reiterated. The teacher should make
up a few examples that are similar to the homework (see the link from the main
webpage) and go over them with the students. In particular, students have a hard
time with the idea of adding the rate to one for growth or subtracting it from one
for decay. Here is an example:
1. The birth rate for a colony of ants is 60%. The death rate is 46%. Assume that
at the beginning of the observations the population was 650. What is the
difference between the birth and death rates? (14%) Is the population increasing
or decreasing? (Increasing, because the birth rate is higher than the death rate.)
Write an equation that could model this situation. [y = 650(1+.14)x or y =
650(1.14)x] How long will it take for the population to reach 1000? Round to the
nearest year. (4 years – This could be found by using the tables on the calculator
or graph and find the intersection.)
2. The birth rate for a school of guppies is 56%. The death rate is 64%. Assume
that at the beginning of the observations the population was 800. What is the
difference between the birth and death rates? (8%) Is the population increasing or
decreasing? (Decreasing, because the death rate is higher than the birth rate.)
Write an equation that could model this situation. [y = 800(1-.08)x or y =
800(.92)x] How long will it take for the population to reach 300? Round to the
nearest year. (12 years)
Assessment:
The students will be provided with a worksheet that has modeling
questions on it that they will be required to complete for homework. I would
expect that most teachers would grade this as a homework assignment. I would
also give a quiz the next day if there were time. I have included a sample quiz and
the worksheets along with this module (see the links on the main webpage.)