This lesson is geared toward an Algebra II type class at the high school level. To use this lesson, students should already be familiar with exponential and logarithmic functions to the extent that they can relate to model problems regarding those functions. This lesson uses spreadsheet technology to illustrate exponential functions both algebraically and graphically. As stated in the NCTM Standards, the mathematics curriculum in grades 9-12 should include the study of functions so that students can model real-world phenomena with functions. The Standards also state that students should be able to translate among tabular, symbolic, and graphical representations of functions. This lesson also bridges mathematics and science so that students can value the connections between mathematics and other disciplines as written in the Standards. Coordinating this lesson with the science department at your school would really help your students see that math is not an isolated field. Thus, this lesson incorporates those recommendations in the investigation of exponential functions.

As stated above, students should already be familiar with exponential functions prior to this lesson. Using the TI-83 graphing calculator to explore exponential functions would provide students with a basis of representing functions symbolically and graphically. The TI-83 has a split screen option that shows the graph of the function along with the table of values for the function. Hence, students get an opportunity to see how the table of values corresponds to the graph.

One could also use the TI-83 to illustrate the concept of asymptotes. Students can trace along the exponential graph by using the trace command to show an asymptote. The table of values would show the y-coordinates getting smaller and smaller without reaching zero. Knowing the characterisitcs of exponential graphs will lead to interesting discussions about the limitations of mathematical modeling in the lesson that follows.

PROBLEM STATEMENT: Through this investigation, we will see how many atoms remain after a given number of half-lives during the radioactive decay of an isotope. The equation y = A(1/2)^x represents this function, where x = the number of half-lives, A = the starting amount of atoms, and y = the amount of atoms that survived after a half-life.

Materials Needed for a class of 30 students:

--Spreadsheet program such as Microsoft Excel
--15 small boxes with lids like shoeboxes
--1500 or more pennies (can be substituted with 2 different sided tokens) to represent the number of atoms of an isotope. Coins are a good representation of an atom since the atom either survives during the half-life or it decays.

Format: Divide the students into pairs and give each pair a shoebox along with 100 coins. Each pair should have pencil and paper to record data.

Procedure: After discussing the problem statement, the teacher should present the model to the students. When the discussion of the model is complete, the students should place their 100 coins in the shoebox and close the lid. They should then shake the shoebox, open the lid, and take out all the coins that display a head. Students should then count the coins taken out, subtract that number from 100, and record the amount of pennies left in the box. Each shake represents one half-life so that first piece of data should be recorded as the ordered pair (1, the amount of coins left in the box). Next, students should shake the box again (without replacing the coins they took out) and take out any coins that display a head. Once again the coins taken out should be counted, subtract the cumulative amount of coins taken out in both shakes from 100, and record as the ordered pair (2, the amount of coins left in the box). This process should continue until the box is empty.

When all students are done with the experiment, the class should then total their results to generate one set of data for 1500 atoms. So the amounts left in the box after the first half-life should be added together, then the second amounts left in the box and so forth. Once the data is compiled, the students can then enter the data into a spreadsheet using Microsoft Excel or a similar program. Students can also use the spreadsheet to compare the experimental data with the mathematical data one would gather from simply using the equation of the function.

USING SPREADSHEETS

Entering Data:

(1) To enter the data gathered by the students, one needs to establish an X and Y column in the spreadsheet. So, in cell A1, place an X. Then fill in the column under X with the number of half-lives from the data (remember a half-life was represented by a shake of the box).

(2) In cell B1, place a Y. Then fill in the Y column with the data regarding the number of coins left in the box after a given shake.

Creating a graph of the data:

(1) In order to have the computer generate a graph of the data, the data entered in columns 1 and 2 needs to be highlighted. To highlight the data, move the cursor to the top left corner of cell A2 and push on the mouse. While holding the mouse down, drag the mouse so that all the data entered is highlighted in black. You should end up at the bottom right corner of the last piece of data entered in the second column.

(2) Go to the INSERT menu and pull down the menu. Highlight the CHART option and move over to highlight "On this sheet" and release the mouse. Move the mouse onto the spreadsheet and you should see that the cursor is now a plus sign with a little bar chart. With the mouse draw the region in which you would like your chart to be and release.

(3) A series of chart wizard boxes will now appear on the screen. First, hit NEXT, then choose a SCATTERPLOT and hit next, then choose OPTION 1 and hit next. You should now have a sample chart in front of you and hit FINISH. A graph of your data should now appear on your scatterplot.

(4) To have the computer form a trendline for the data, click twice on the graph and then click once on a data point so that all the points become highlighted. Go to the INSERT menu and highlight the TRENDLINE option. Select EXPONENTIAL. The trendline should now appear on the graph. ***Important Note: If a point on the graph has a y-coordinate of zero, the computer will not allow an exponential trendline. This is a limitation of the model that you can discuss with your students regarding the asymptotic nature of exponential functions.

Extensions: Creating a third column of values generated from the equation y = 1500(1/2)^x will allow the students to compare the data they gathered to the values given from using the equation. Then can then repeat the procedures above to see the similarities between the two graphs. This would help initiate a discussion about the limitations of the model compared to what actually occurs in a real-life situation of radioactive decay.

Here is a sample spreadsheet including the extension section:

x	y from model	y from equation
0	1500	1500
1	832	750
2	518	375
3	278	187.5
4	142	93.75
5	68	46.875
6	45	23.4375
7	30	11.71875
8	30	5.859375
9	22	2.9296875
10	8	1.46484375
11	7	0.73242188
12	7	0.36621094
13	4	0.18310547
14	1	0.09155273