The goal of this activity is for students to experience a data-driven, inductive investigation leading to the cosine law. This could be used in addition to or instead of the traditional proof to deepen the understanding of the behavior of triangles and make the concepts more accessible to more students.
Before the Activity
Launch the activity by asking the students what they know about the relationship between the sides of a right triangle. Then ask them what happens as the right angle becomes acute or obtuse. Have a brief discussion about their conjectures. Then tell them they will be investigating the relationship between the sides of a triangle as it changes from a right triangle to an oblique triangle.
During the Activity
Students will be investigating how the Pythagorean relationship changes as the triangle changes from a right triangle to an oblique triangle. Directions are on the student activity sheet and in the TI-Nspire TNS file. If students are new to TI-Nspire they may also want to use the beginner directions.
There are 4 problems on the TI-Nspire TNS file. Ask the students to make further conjectures and do a mini-summary at the end of problem #1 and problem #2 on the TI-Nspire TNS file.
After the Activity
After problem #3 on the TI-Nspire tns file, summarize the activity and make sure all students understand the graphical verification of the cosine law. We can look at the cosine law as kind of a correction factor to the Pythagorean Theorem as the triangle changes from a right triangle to an oblique triangle. The goal is to deepen students' understanding of the behavior of triangles by looking at them numerically, graphically, and geometrically as well as analytically.
Problem #4 on the TI-Nspire TNS file is an extension where students allow different parts of the triangle to vary, and the graph is linear rather than a cosine curve.