Overall summary of the lesson -
This lesson would be appropriate for trigonometry students who were studying the Law of Sines. Students are comfortable using the Law of Sines when the given information is two angles and a side (ASA, or AAS). They can recall from geometry that they could prove triangles congruent with this information. The student should also recall that being given two sides and a non-included angle (SSA) was not enough information to prove congruent triangles. The confusing list of possible cases in the SSA situation would be better understood by the student if they have actually constructed the triangles using the possible cases and have tried to reach a generalization on their own. Before the students begin the worksheet, they would work through a guided practice of constructing triangles given two sides and a non-included angle using Geometer's Sketchpad. The guided practice would include examples that would produce one, two, or no triangles. After the students finish the worksheet, there should be a summary of the conclusion. Here, Geometer's Sketchpad would be used again to look at the special case were side a is less than side b and angle A is acute. This allows the student to see that the size of A together with the relative difference of a and b determines how many triangles are possible.
It might avoid confusion to have the students always construct their triangles in the form seen below.
Connection to NCTM Curriculum Standards and Illinois State Goals - This lesson plan would specifically target Goal #9 which would have the student be able to: "Analyze, categorize and draw conclusions about objects and spatial relationships using geometric methods and drawings, sketches, graphs, models, symbols, calculators and computers."