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Adventures With The Fish Pond: Population Modeling
(Grade Levels: 7 - 9)

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angelfish This activity builds on the population decay M&M activity, introduces NOW-NEXT or recursive equations, and uses calculators as an efficient tool for exploring population models. To do this activity you need a calculator that has an Answer Key, usually designated by ANS. Most graphing calculators have this feature.

This activity is best done by pairs of students (but can be done individually). Each pair of student of students needs:

  • an activity sheet, and
  • their own calculator (sharing is okay, but it is better for each to have their own).
Population modeling is one type of mathematical modeling in which you produce a pattern or rule that could be used to help describe the situation. The pattern usually doesn't fit exactly, but closely describes what is happening in the situation.

1. Using the data from the Population Decay simulation determine a general rule to describe what happens to the fish population from one year to the next. Finish the following sentence.

Next year's fish population will be __________________

NOW - NEXT equations are another name for recursive equations which are useful in modeling. You want to write an equation to describe NEXT year's fish population if you know the fish population NOW. In this equation, NOW represents the fish population now, and NEXT represents the fish population next year.

2. Your goal is to translate your sentence from problem #1 into a NOW - NEXT equation that describes the Population Decay simulation. Start with NEXT =, which means the same as "Next year's fish population will be". Fill in the right half of the mathematical equation, and use the word NOW where you would plug in the number of fish currently in the pond.

NEXT = ___________________

(Have students share their solutions. Some possible solutions include: NEXT = NOW - .5 NOW, NEXT = .5 NOW, NEXT = NOW/2)

Now you need an initial condition to test it out. Suppose the initial condition is that the pond has 80 fish. This represents the first NOW for this simulation.

3. Using the NOW-NEXT equation, how many fish will be in the pond next year?

4. And the following year?

5. And the following year?

Many calculators these days have an Answer key that plugs the last answer calculated. The Answer function is often designated by ANS.

To model this on the calculator we need to tell it two things:

  • the initial condition (which is the same as the initial NOW value).
  • how to calculate NEXT.

6. Our initial condition was that the pond had 80 fish. To store the value on the calculator, type in the value 80 and press the appropriate key (ENTER, RETURN, =,...). For us, 80 is now stored as the initial NOW value.

7. To tell the calculator how to calculate NEXT we need to enter in the right half of our NOW-NEXT equation. However we need to use ANS in place of "NOW" in our equation. Then press ENTER. For example if your equation was NEXT * .5 then you would put in ANS *.5 and hit ENTER.

8. Did the calculator return the correct value? If not you need to start over, you will need to reenter your initial condition (80) and then ANS expression.

9. If you hit ENTER again and again the calculator should give you the value for the next years population. This is because we taught it the "rule" for our model.

10. How many years does it take before our population is under 5?

11. If we started with 1000 fish, how many years would it be before the population was below 10?

12. If we started with 2000 fish, how many years would it be before the population was below 10?

13. Knowing what you know about 2000 fish, how many years would it be before a population of 8000 fish was below 10?

Activity adapted from lessons in:

Coxford, A., Fey, J., Hirsch, C., Schoen, H., Burrill, G., Hart, E., Watkins, A., Messenger, M., & Ritsema, B. (1997). Contemporary mathematics in context: A unified approach (Core-Plus Mathematics Project). Chicago: Everyday Learning.