Nancy J. Schmitt
This lesson uses frog populations to demonstrate exponential growth and decay. The lesson uses M&Ms or skittles (good to use if you have a student with a peanut allergy) as a tasty manipulative. Put the students into workable group sizes of 3 to 4 students. Both activities can be done in a block schedule day. This way you only need to get enough M&Ms for each class. You will need small cups, and plastic gloves, and plates for this activity. Since the students will be eating the candies afterwards, many do not want candies that have been handled by others. The M&Ms will represent the frogs in the pond. If you are not allowed to use candy in your classroom, pennies would also work.
Healthy pond
The students from New Haven have been monitoring a pond for the last several years. They have been testing the water and counting the frogs. The water appears to be clean, has been testing to be relatively free of chemicals, and the frogs appear to be healthy. They have discovered that each year the population of the frogs has increased by 25%. They decide to do a simulation of the growth of the frog population using m& ms. The students record their data in tables and graph the data.
This is what they did. (No M&Ms are to be eaten yet)
1.
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First they started with 10 M&Ms They put these in the cup. This is trial number zero.
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2.
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Then they spilled the M&Ms onto a clean surface.
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3.
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They counted the number of M&Ms that are facing up with the M&M showing. They add half as many M&Ms as they counted in step 3 to their current m& ms and put them all back into their cup. This is the next trial.
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4.
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They record the current number of M&Ms in their table along with trial number.
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5.
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They continue to repeat steps 2 through 4.for 5 more times
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6.
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The information in the table is then graphed with trial number on the horizontal axis and total number of M&Ms on the vertical axis.
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7.
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The students are asked to come up with the equation that models the simulation, predict how many M&M's after 10 trials.
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8.
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Use the equation to determine if in year one there are 100 frogs, how many frogs would there be on year 10.
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9.
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The students are to explain why the simulation depicts an increase of 25% each time.
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10.
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The students then compare and contrast exponential growth to a linear function y=10+.25x
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Unhealthy pond
The students from Waterbury have been monitoring a pond that is located near an old fertilizer factory. They have been noticing that the levels of chemicals in the water they are testing are noticeable. They have been catching frogs and have noticed that the frogs do not seem to be healthy. In fact one of the frogs was missing a leg. They have been keeping track of the number of frogs in the pond for several years and have noticed a decline in the number of frogs. They believe that the number of frogs seems to be decreasing by about 25% each year.
They decide to a simulation of what is happening in the pond and start by using the cup of M&Ms they have from the last activity.
1.
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They count the number of M&M's they currently have. This is trial zero.
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2.
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Then they spilled the M&Ms onto a clean surface.
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3.
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They counted the number of M&Ms that are facing up with the M&M showing.
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4.
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They take half as many M&Ms they just counted and set aside. They they count their current m& ms and put them all back into their cup.
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5.
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They record the number of the trial and the current number of M&Ms in their table.
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6.
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They continue to repeat steps 2 through 5.for 5 more times
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7.
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The information in the table is then graphed with trial number on the horizontal axis and total number of M&Ms on the vertical axis.
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8.
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The students are asked to come up with the equation that models the simulation, predict how many M&M's after 10 trials.
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9.
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Use the equation to determine if on year one there are 100 frogs, how many frogs would there be on year 10.
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10.
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The students are to explain why the simulation depicts an decrease of 25% each time.
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11.
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The students then compare and contrast exponential decay to a linear function y=100-.25x
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The students then compare the two graphs and write a short description of how they are similar and different. Students are encouraged to share their findings and how their data may be somewhat different from other groups. The students may now share the M&Ms within their groups.
Teachers can modify the percentages to others that seem reasonable and fit in with the number of M&Ms or pennies they have. Graphing calculators could be used to produce the graphs for special needs students.