Modeling a Spiral, and enjoying Desmos

At a recent meeting, I got to tell people about an old, non-finished book, EGADs (Enriching Geometry and Algebra through Data). The idea of the book is that there are geometrical constructions that have relationships under them—usually a relationship about length—that you can model using a symbolic formula.

Like that spiral. How does the length of the “spokes” of this spiral depend on the spoke number?

This post has two purposes:

1. To get you to try the spiral example.
2. To show how you can use the Desmos graphing calculator to do the graphing and calculation.

The draft of the book (link above) is free for now, but it occurred to me that you could do at least one activity (integrates trig, geometry, data, exponential functions) easily using Desmos’s cool new technology. Read on!

The activity: Spiral 20

Here’s what you do:

1. Download the PDF with the one-page handout. Print it out.
2. Measure the long legs of the triangles as suggested on the handout.
3. Go to the Desmos graphing calculator.
4. Enter the data.
   • To do that you need a table. Click the “<” button under the little panel, upper left, where you will eventually enter a function.
   • Click “table.” A table appears.
   • Enter the data. You can change the variable names if you want.
5. Enter a function in that top panel. Try to match the data!
   • If you leave parameters in the formula, it will ask if you want sliders. I love sliders. You will too.
   • You can type values in, even if you have sliders.
   • You can change the range of sliders by clicking the limits at the slider ends.
6. Play with the sliders and all the other features of this site.

Here is a graph of mine from Desmos, with only a little of the data: