Over the weekend, I was having a conversation with my extended family about high school mathematics (because this is obviously what you do when you are a math teacher and your brother is an engineer). While my brother and I were both very good at math in high school, we both admitted that we didn’t really understand math until college.
My brother said, “I wish more classes started with why we use the idea practically before they taught how to use it abstractly.” I had to agree with him. When I think back to my experiences in high school, nothing demonstrates this point more than trigonometry. I would bet that if you asked most adults to finish the phrase, “SOH. CAH. ____” they could. But would they have any idea at all about what it was used for or why it worked? When I was in high school geometry, I certainly didn’t! I know there are plenty of high school students today who are having the same experience that I did. We must teach students more than just “how”. We have to teach students “why”. How do we do that? Try these ideas.
Tie the new learning to previous learning.
For any new concept to have meaning, we should tie it back to what students already know. When first introducing trigonometry in geometry, focus on similarity in triangles. Students already know that similar triangles have proportional side lengths. This property explains why we can find the ratio of any two sides of a right triangle, aka sine, cosine, and tangent, just by knowing one of the angles, and can then find missing side lengths.
When creating the unit circle, connect values back to special right triangles. When introducing the graphs of trig functions, tie each ordered pair back to angles and coordinates on the unit circle.
Trig is always about RATIOS.
Whether you are teaching Geometry, Algebra 2, or Precalculus, it should all come back to this idea: a trigonometric ratio is a comparison of two sides of a right triangle. It’s important to understand the conceptual progression of trig ideas across the courses, and to use consistent language so that this progression is evident to students. Every trig unit should start with a refresher on what a trigonometric ratio represents. We must get students thinking proportionally.
Simple changes make a BIG difference.
Don’t just ask for x. Instead of asking to find x, ask for a specific side length or ratio of side lengths (Ex: “Find AB” or “Find DE/EF”). The more students are reminded that we are working with measurements on a triangle, and not just random values, the better.
Use fractions in your example problems. Start by writing trig ratios as fractions instead of as decimals. Then transition to decimals that are easily represented as a fraction, such as 0.5, 0.75, or 0.8. Have students practice interpreting statements like sin(A)=0.75 so they can confidently say that the side opposite Angle A is three-fourths as long as the hypotenuse of the triangle. Students are much more likely to think proportionally about the side lengths if the ratio is written as a fraction than if it is a decimal. Which brings me to my next point:
Wait to use the calculator! Because we so quickly jump to the sin, cos and tan buttons on our calculators, students lose the idea that we are dealing with ratios. A large reason for this is that the calculators give the ratios as a decimal and not a fraction.
We have four units on trigonometry available on Math Medic. Try them out with your students and let us know how it goes!
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