Experience First

The purpose of today’s activity is to get students meaningful practice with the Law of Sines and Law of Cosines in an applied context. Students must determine when each law should be used and must explore the validity of a model. In particular, how do actual driving distances compare to straight line distances on a map? Why might it be useful to find the area of a triangular region? (We look at population density as one possible reason).

Students will need access to some kind of device to check Google Maps. At first, students thought that the calculated distance between UNC and NCSU should be greater than the Google Maps answer since driving doesn’t happen on perfectly direct routes. This is a great line of reasoning but students were challenged to think about the data they were given, which already represented driving distances, not necessarily the sides of that triangle. It’s important that students learn to model with mathematics (MP4) and be prepared to explain differences between a model’s prediction and the true reality.

When finding the area of the triangle, students had no trouble drawing in the altitude that represented the height. Many used the Law of Sines instead of right triangle trig to find this value. 

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Formalize Later

Students look at one possible way of finding area during the activity by using right triangle trig to find the height of the triangle. This is then generalized into the area formula A=½(a)(b)sin C and its various forms. Discuss with students how each triangle has three heights and thus we get the three different forms. We tell students that there is no need to memorize this formula. They are welcome to first find the height of the triangle in any method they choose and then apply their traditional formula from Geometry for the area of a triangle.

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Heron’s formula is a way of finding the area when given three sides of a triangle, which does not allow for the use of right triangle trig to find the height. We discuss our knowledge of the prefix “semi” to conclude that semiperimeter means half the perimeter.

Students may struggle to draw an accurate picture in question 3 of the Check Your Understanding. Encourage students to check each other’s drawings.

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