10. Prove the Laws of Sines and Cosines and use them to solve problems.

The world is full of signs. Traffic signs, billboards, official John Hancocks, sign language, and plenty more. By the way, what's your sign?

Whether they're Capricorns or Cancers, students should be familiar with a very different kind of sign. We're talking about none other than sine and cosine and the laws that govern them.

The Laws of Sines and Cosines allow students to easily solve triangles. Depending on what information they are given, students can use these laws to find the missing angles or side lengths of a triangle. As you might expect, that means students have to know these laws and understand what they mean. Deriving and proving these laws might be an excellent start.

Law of Sines:

Law of Cosines:

Students will also need to be fluent in the language of proofs, whether writing a two-column or paragraph proof, and being able to justify their statements with legitimate reasons. They might enjoy a break from the rigor of regular class work to play a matching game in which they have to pick out which theorem, property, or definition justifies a given statement.

Of course, students will also need a working knowledge of the whole collective of other theorems and definitions related to triangles and trigonometry, including but most definitely not limited to the Angle Sum Theorem for triangles, the definitions of sine and cosine, and their relationship among complementary angles.

Satisfying this standard is especially important for students planning to take more advanced mathematics courses. This is one of the few standards that have been earmarked as critical for students intending to enroll in higher math classes. If anything, that's a sign of how critical these laws are.

All these signs might just open up your eyes.