This example is adapted from the module The Equiangular Spiral at the Connected Curriculum Project.
In this variation the student uses an interactive Excel spreadsheet to fit an exponential function to measurements made on the shell below. Then the student types the results into the form below to see the curve superimposed on the shell. Begin by moving this window to the right leaving one or two inches at the left of your screen for the Excel spreadsheet. Then click here to open an Excel spreadsheet. Arrange the spreadsheet window and this browser window as shown below to make it easy to mave back-and-forth between the two windows.
The goal of this example is to describe the green curve in the image below. This curve was colored by hand to highlight a natural feature of the shell. Begin by clicking on the spreadsheet window. Notice the blue arrow at the top indicating where the coordinate information from your measurements should be placed. Now click on this browser window to make it active. You will make 19 measurements. The cross hairs are already positioned at the first point to be measured. Press the Mark point button to mark and record this point. Next move counterclockwise along the green curve. Mark, in turn, each point where the curve crosses the white lines indicating the lines theta = 0, pi/4, pi/2, 3 * pi/4, pi, 5 * pi/4, ... . There will be a total of 19 points to be marked. After you have marked all these points press the List points button. A new window will appear with the coordinates of the 19 points that you marked.
The next step is to copy and paste this data into the spreadsheet. The steps involved depend a bit on your operating system. First highlight the data by clicking and dragging or by clicking and shift-clicking. Then either drag the highlighted data to the point on the spreadsheet indicated by the blue arrow or copy it by pressing command-c (MacOS) or control-c (Windows) and then paste it at the point indicated by the blue arrow by pressing command-v (MacOS) or control-v (Windows). When you are done you should see 19 rows of data with four entries in each row.
Now following the instructions in the Excel spreadsheet, fit a curve of the form R(t) = A ekt to your data. After you have determined the values of the constants A and k enter them in the form below and press the Try it!!! button to see the results.
Notice the power this combination of a Lite applet and Excel gives to students. Because we are using a Lite applet and relying on Excel (or a computer algebra system) to do the analysis, students can investigate questions of their own. For example, a student might wonder about the relationship between the volume of successive chambers in the shell. He or she could investigate this by making various measurements on each chamber. If it appeared that the linear measurements seemed to follow a pattern, the student might make an assumption about the missing third dimension and based on that assumption make a conjecture about the relationship between the volumes of successive chambers.
In closing we repeat the three principles underlying the Lite Applet Collection.
The Lite Applet Collection is open source. All the files including the Java source code files used in this article are freely available and may be downloaded from this site. We invite you to submit your own Lite applets and modules for consideration for inclusion in this collection. We expect to add new Lite applets and new modules using those applets to the collection regularly.
There are a number of sources for maps that can be used in curriculum modules. Ideas for finding maps.
Lite applets, like Image_and_Cursor can be used in curriculum modules at many different levels. Some examples using this applet at the middle and high school levels.