Experience First

Today’s experience again ties back to important ideas from Geometry class. First, students recall that a triangle is a right triangle if it satisfies the Pythagorean theorem. Students may or may not remember how to tell if a triangle is acute or obtuse based on the measures of a^2, b^2, and c^2, but we expect students to be able to visualize how increasing the angle from 90˚, as in an obtuse triangle, would stretch out the third side, whereas decreasing the angle form 90˚, as in an acute triangle, would shorten the third side.

In question 2, we build curiosity by having students first make a claim about a triangle given very limited information. Only in question 3, will you provide them the side lengths of the triangle (AB=50, BC=44, and AC=29) so they can prove or disprove their conjecture.

As you are monitoring groups and while debriefing the lesson, continually ask students what the importance of the “2ab cos C” value is. Students should be able to articulate that if you know how much you are “off” by (the correction factor) you can find the true value. We use the analogy of a student asking their friend about some numerical value (temperature, grade in class, etc.) and them knowing from experience that their friend always lies by a certain amount. If you know how much a student lies by, can you figure out the true temperature/grade/etc.?

In the final question, we ask groups to find a more general relationship between a, b, and c, using the fact that they now know how much each triangle is “off” from the Pythagorean theorem. Many students wrote their equation as c^2+2ab cos C=a^2+b^2. It makes a lot of sense to write the equation in this way, as it clearly illustrates that when c^2 is too small, we need to add to it, and when c^2 is too big we need to subtract from it (since 2ab cos C will be negative when C is obtuse). Essentially students are writing the Law of Cosines before it is given to them.

One extension for groups that are ready for a challenge is to ask them to find the measure of angle C in the triangle from question 2, by knowing the size of the correction factor.

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Formalize Later

When debriefing the lesson, make sure you have identified some students that were able to write an equation relating sides a, b, and c, that would work for every triangle using the idea of the correction factor. Note that the correction factor could really be put on either side of the equation, but conventionally it is subtracted from a^2 +b^2. Students may not immediately realize why the correction factor in question 6 is negative, so give students time to think about when the correction factor will be positive or negative and what that has to do with whether the given triangle is acute or obtuse. The unit circle really never goes away!