Day 1 – Introduction to Exponential Growth
We already know what linear growth is, but has any one ever heard of exponential growth?
Students will begin the lesson doing an experiment in groups to discover exponential growth, exponential decay, and spread of contagious disease.
· Students are given approximately 100 M&Ms and a small container.
· Two M&Ms are placed in the container and the container is shaken. This is the first generation
· For every M facing up, one more M&M is added to the container
· All data is recorded for each generation (generation, total M&Ms)
· This is repeated until all of the M&Ms are used (~ 10-15 generations)
· The data is used to create a graph, which should be an approximation of the exponential function
· Star ting with 100 M&Ms, students will remove any face up Ms
· All data is recorded and the game is finished when no M&Ms are left
· The data is graphed
· Each group begins with 50 M&Ms in the container, only one of which is red (the infected M&M)
· After each shake, any M&M that is touching a red M&M is now infected and replaced by another M&M
· The number of red M&Ms is recorded each time, and the experiment is finished when all of the M&Ms are red
The students will then be asked to compare their data with other students to see if they have similar data.
1. The students will use their graphing calculators to graph the functions of y = exp(x)and y = exp(-x) to look at the parent functions of growth and decay.
2. Students will then be asked to manipulate the parent functions
a. What happens if you change between neg/pos?
b. What happens if you change the constant in front of e?
c. Each group will add to a master list of generalities to be put up on the room
3. After becoming familiar with the parent function, students will be reminded of the equation they are already familiar with,
A = P(1 + r/m)^(mt)
P = principal amount invested
A = accumulated amount
r = interest rate
m = number of times interest is compounded per year
t = time in years
1. They will be shown that, when taken to the limit, this function becomes the exponential function A = Pe^rt, which is primarily used to calculate continuously compounded amounts.
2. Students will, in their groups, work on a few example problems together relating to compounding interest rates.
3. After practice, students will be shown how this equation comes from a more general exponential growth equation, Q(t) = Qoe^kt.
Q(t) = amount at time t
Qo = starting amount
k = constant
t = time
This equation is where we will relate to medieval times (tomorrow)
4. Homework given with practice problems relating to continuously compounded interest
Day 2 – Growth and Decay Comparison
1. Students will consider the spread of bacteria in food (relating to food they look at in chemistry). Students will use real data (provided) to calculate the constant k (example chicken with Salmonella)
2. Using this constant, students will be asked to consider about estimated growth given the constant k of some other diseases.
3. Students will now look at exponential decay, using carbon dating and half-life. They will be asked what might be different in the equation for growth and decay. They should come up with the following:
a. It is impossible to take the natural logarithm of a negative number, so Q(t) can’t be negative
b. The initial is larger than the current amount
c. k must be negative so as to not affect Q(t)
4. They will follow through an example relating to Medieval France.
ex. The half-life of Carbon 14 is 5770 years. Bones found from an archeological dig in a castle in France were found to have 92.4% of the amount of Carbon 14 that living bones have. Determine the year the person died. (Ans. 658 years old, which is the year 1351)
5. As practice students will create their own problems, manipulating the constants and percent remaining to create other things archeologists may have found from Medieval France, and they will give their problems to other groups to solve. These will be collected as a quick assessment to be sure students are not struggling.
Day 3 – Growth/Decay Rates
1. Students will now look explore how the rate of growth changes as time changes
2. Because students likely do not know about derivatives, the equation can be presented to them as Q’(t) = kQoe^kt for growth and
Q’(t) = -kQoe^-kt for decay.
3. Given an example problem, students will calculate the instantaneous growth rates every year for 10 years of bacteria. The students, in groups, will create graphs of the growth rate against the year.
4. Students will then be asked to write down (in a journal), why it is they believe the rates change the way the do, (both intuitively and mathematically)
5. Students will now apply decay rates to the Black Plague in Medieval Europe using their knowledge of history. They should know how many years the plague affected Europe, and how large the population of Europe was before and after the plague. Using this knowledge they will create a graph showing the year against the population of Europe. Using their newest knowledge of growth and decay rates, they will be able to determine an approximate number of people who died each year.
6. Students will present and explain their graphs using mathematical examples to the class.
The students will look back to the beginning of the unit and review (in their groups) the basic ways to manipulate the exponential function. From here they will lead the teacher through the progression they took to reach exponential growth and decay, and then growth and decay rates. They will also compare their M&Ms data to an actual exponential graph to see how close their experimentations were.