Experience First

Teaching students about the ambiguous case of the Law of Sines has always been a challenge for me! We want students to understand that SSA is not a congruence shortcut because more than one triangle or no triangle at all could be made with a given set of conditions, that the relative sizes of the sides plays a role in determining which “case” we’re in, and that the math of the Law of Sines supports our initial conjectures about how many triangles can be made. That’s a tall order! After experimenting for many years, the approach described in this lesson has been the most successful with students.

In the first page of the activity, students practice applying the Law of Sines to three different triangles. In question 2, students will immediately recognize that such a triangle is not possible, but it is helpful for them to see that actually doing the calculation leads to an undefined answer since arcsin is only defined for values between -1 and 1. This is a nice review of inverse trig functions and the unit circle.

We recommend debriefing the front page before having students move on to the back page. Our students really enjoyed using the Geogebra applet to see the different triangles that could be made. Thank you to Jerome White who programmed this tool. I have been using it every year with my class and it does a great job of illustrating the various cases. Be careful that when students set angle A to be 150˚, they are actually forming a triangle that includes that angle, not its supplement on the left side of the angle. Students should see that side b needs to be greater than 9 to even form a triangle.

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Formalize Later

When debriefing question 2 on the front page, make sure to emphasize that not all triangles are drawn to scale and they won’t always be able to immediately tell if a side will “reach” or not. Fortunately, the Law of Sines will produce an undefined value for the missing angle, thus confirming that the triangle was not possible to begin with. For question 3, most students did not think about an alternate location for segment CB. It is okay to save this for the debrief and have students draw in the second triangle with their red pen.

On page 2, we use the informal language of “fixed side” and “swinging side” to identify the various pieces of the triangle. When using “side a” and “side b” we find that students have a hard time generalizing the patterns to other triangles. Side b is not ALWAYS the swinging side, and side a is not ALWAYS the fixed side. The swinging side and fixed side are determined by the side’s location relative to the given angle. We say that the fixed side is attached to the angle and it is “locked in” by the angle, and thus can’t move.