Exponential Modeling

Expone ntial Modelin g

This activity has 3 parts. In Part 1, you will be investigating the area of a region of paper after folding the paper several times. Part 2 is an investigation about radioactive decay using M&Ms. Part 3 will give you the opportunity to learn about a certain endangered species.

Part 1: Paper Foldin g

1. Fold a sheet of paper in half and determine the area of the smallest section. Assume that the full sheet of paper has area 1 square unit. 2. Fold the paper in half again then determine the area of the smallest section. Continue in this manner until it becomes too difficult to fold the paper.

3. Plot the data from your table on the grid below. Be sure to label your axes and the scale that you use.

4. Using the table and the graph, determine an equation that represents the data.

Part 2: Radioactive Nuclei

In this activity, we will be using M&M’s to simulate nuclei. You will notice that the M&M’s are marked on one side with the letter M. They will be your “radioactive nuclei.” You also need a paper towel on which to place your “nuclei.”

1. Dump out your bag of M&M’s and place them all so that the letter M is facing up. Count how many radioactive nuclei you have. Write that number in the data table under the heading “Number of Radioactive Nuclei” for toss number 0. 2. Place your nuclei in a paper cup, cover and shake the cup. Pour the nuclei onto your paper towel. Separate the nuclei into two piles, one with the marked side up and the other with the marked side down. Count the number of nuclei in each pile. On your data table, record the number of radioactive nuclei. 3. Return only the radioactive nuclei to your paper cup. (You decide what to do with the decayed nuclei, or those with the marked side down.) 4. Continue this process until there are no radioactive nuclei left. Add additional rows to the table if needed.

0 1 2 3 4 5 6

5. After you have recorded your data, work with your group to fill in the third column of the table. 6. Use a graphing calculator to make a scatter plot of your group data. Make a sketch of your scatter plot below. Be sure to label your axes and the scale that you use. 7. Use your calculator to determine a model which will fit your group data.

8. Discuss the relevance of the values of the constants in your equation above.

9. Discuss the similarities and differences between this activity and the paper-folding activity.

Ext ension Q uest ions 10. Each toss is considered a “half-life” because about half of the radioactive nuclei remain after each toss. If you started with a sample of 600 radioactive nuclei, how many would remain after 3 half- lives?

11. If 175 radioactive nuclei remain from a sample of 2800 nuclei, how many half-lives have passed?

12. Strontium-90 has a half-life of 28.8 years. If you start with a 10-gram sample of strontium-90, how much will be left after 115.2 years? Part 3: Extinction of Species

The following table gives population data estimates for the endangered African Black Rhinos. Year

1960 1970 1980 1991 1993 1. In your group, discuss the significance of the population and what kind of mathematical model could be used to describe it.

2. Use a graphing calculator to make a scatter plot of your group data. Make a sketch of your scatter plot below. Be sure to label your axes and the scale that you use.

3. Use your calculator to determine a model which will fit your data.

4. Discuss the relevance of the values of the constants in your equation above. 5. According to your model, what is the current African Black Rhino population?

6. The actual current African Black Rhino population is about 5055. Explain why this number does not fit your model. Also discuss possible reasons why there was such a drastic change in the population.

7. Discuss the similarities and differences between this activity and the paper-folding and radioactive decay activities.