To the Teacher
The essential tool for these activities is the 10 cm-radius circle, which includes:
- a built-in protractor
- accurately marked x- and y-axes, as well as tangent and cotangent lines
The ten-centimeter circle is available as the CircleTrig Geoboard, from Nasco, or in a paper version which you can duplicate for your students. Download one or more of these:
- Geometer's Sketchpad First Quadrant (contributed by Roger Gemberling)
- Geometer's Sketchpad Four Quadrants (contributed by Roger Gemberling)
- SmartBoard Notebook file
Or find a hard copy in Geometry Labs. Depending on which original you use, and the properties of your printer, you may need to use your copy machine to adjust the size so that the radius is really 10 centimeters.
Acknowledgments: I was inspired to develop this approach by Sherman Stein, Cal Crabill, and Don Chakerian's Trigonometry.
The activities start with an introduction to the tangent ratio, which is more accessible than sine and cosine and is related to the already familiar concept of slope. The sine and cosine ratios follow. (At the Urban School of San Francisco where I used to work, we introduced the tangent ratio in the first semester of our Math 2 course, and the sine and cosine in the second. Separating them that way increases student exposure to the concepts, offers an opportunity to review, and focuses on the concepts rather than on mnemonics.)
Students build tables of those functions and their inverses with the help of the circle geoboard, and then use the table and the 10-cm circle itself to solve "real world" problems. (Possibly more effective sequence: introduce "slope angles", use the 10-cm circle to solve word problems, and then make the tables.) The words "tangent", "sine", and "cosine" are not used right away, mostly as a way to keep the focus on the geometric and mathematical content. Of course, these names need to be introduced sooner or later, and you will have to use your judgment as to whether to do that earlier than is suggested here. If you are using these labs in conjunction with other trig lessons, you may not have the option of postponing the introduction of the names, given that students will recognize the ratios.
In any case, when you do introduce the names, you will need to address the students' question "why are we doing this when we can get the answer on the calculator?" For me, the answer "you can use your calculator if you want, but for these first few activities, show me that you understand the geometric basis of these ideas" has worked. The calculator is the method of choice if you're looking for efficiency, but the geometric representation in the circle is a powerful mental image, which will help students understand and remember the meaning of the basic trig functions.
The ten-centimeter circle is a flexible tool in the introduction of trigonometry. You will no doubt find your own uses for it. To get you started, I include three activities -- more are available in the book, in a ready-to-duplicate format.
1. Slope Angles
Prerequisites: The concept of slope. The concept of similarity.
Teacher's Notes: This is an introduction to the tangent ratio, and to the arctangent. You should probably demonstrate the correct procedure (as shown in the figures). It is possible to get a reading accurate to the nearest degree or millimeter, so the tables can be accurate to two significant digits.
2. Using Slope Angles
Prerequisite: "Slope Angles".
Teacher's Notes: This lab provides applications for the work in the previous lab. The last two questions open up the issue of interpolating and extrapolating, since the tables we made then were clearly insufficient to deal with the full range of problems of this type.
In doing this work, it is helpful to orient the figures so that the rise is vertical, the run is horizontal, and you are dealing with positive slopes only. Eventually, when the tangent ratio is officially introduced, the formulation "opposite over adjacent" will be more flexible, as it can be applied to a right triangle in any position.
3. Ratios Involving the Hypotenuse
Prerequisite: "Slope Angles".
Teacher's Notes: This lab parallels "Slope Angles", but it is about the sine and cosine. Note however that this time we limit ourselves to angles between 0° and 90°.
After doing these activities, you can pursue any or all of the following options if you want to do more work on this topic:
- Students can use the 10-cm circle to make a full table for slope angles, with 5° intervals. This is somewhat tedious, but team work and patterns can help speed up the process.
- You can use applications of the trig ratios from any trig book, using the 10-cm circle in lieu of trig tables or calculator.
- You can use the 10-cm circle to introduce unit circle trigonometry -- just define the unit to be one decimeter (10 centimeters). See Geometry Labs, Section 11, for more.
- You can reveal the official names of the trig ratios and their inverses, which will allow students to use calculators to solve these problems.