I recently introduced my Geometry class to the basic concepts of Trigonometry. When I taught my lesson on finding the measurement of the missing angle using ratios, I noticed that many of my students were still struggling with the concepts of adjacent and opposite. I thought maybe they were confusing adjacent with hypotenuse since the hypotenuse is also adjacent to two of the angles, but they all were able to identify the hypotenuse. Even when I shared the adage SohCahToa (or SOHCAHTOA), they still struggled at identifying the adjacent leg and the opposite leg to the angle of reference. So, I created a different way to approach this. Instead of focusing purely on the values assigned to each leg, I had them represent the legs used in determining each ratio.
First, have the students draw a dot (preferably in a bright color) indicating the angle of reference.
Second, using SohCahToa, have the students determine which leg is used in the numerator and which leg is used in the denominator of the trigonometric ratio.
Third, have the students draw whichever leg is used in denominator in blue.
Fourth, have the students draw whichever leg is used in the numerator in black.
I had my students complete this process every time they worked on trigonometric ratios and it greatly helped their ability to visualize and identify the adjacent leg and the opposite leg to the angle of reference. Visualization is such a critical skill to understanding mathematics. We rely so much on visualization when we solve problems. In the early years it is one of the primary ways that we teach students to approach mathematics. What I found, though, is that visualization has a much more profound role in mathematics than in problem solving. Visualization allows us to create new pathways in our understanding of mathematics.
After having my students work through these visualization strategies, I found them identifying patterns (similar to trigonometric identities) without any knowledge of the identities themselves.