If you’ve ever taught high school math before, you’ll easily recognize the problem below as being the bread and butter of a right triangle trigonometry unit.

Despite the fact that we often offer a clear list of steps to help students get to the answer, the truth is that students still struggle with questions like this. So what’s the issue? Perhaps the problem is not a fluency issue that can be corrected with more practice problems. Perhaps the problem is that we need a more compelling story of trigonometry—one that builds upon the fundamental ideas of similarity students already know and will lay a sturdy foundation for the future trigonometric concepts students will encounter in Algebra 2, Precalculus, and beyond.

This was the topic of my 2023 NCTM session titled “Trigonometry from the Ground Up”. Trigonometry is one of the most over-proceduralized topics in high school mathematics and yet hinges on just a handful of interconnected concepts. If we leverage the power of these few ideas and give students opportunities to see concepts visually, make connections, and notice mathematical relationships, I argue that students would have a much stronger conceptual understanding of trigonometry. So let’s get started.

The study of trigonometry generally starts in Geometry but builds on ideas from earlier grades.

## Middle school

Proportional relationships & geometric dilations, the idea of a scale factor, ratios *between* figures are generally emphasized (e.g. the sides of this rectangle are 1/2 the length of the sides of another rectangle, or the actual dimensions of a room are 50 times larger than the drawing).

## Triangles (Unit 4)

Relative sizes of sides and angles in triangles (**Geo Lesson 4.1**); the largest sides are across from the largest angles, the smallest sides are across from the smallest angles.

## Similarity (Unit 6)

Similar figures (with a focus on triangles) have congruent angles and proportional sides. If two triangles have two sets of congruent angles, then the triangles are similar (AA Similarity Criterion).

## Trigonometry (Unit 7)

Having grasped these big ideas, students are ready for what I consider the 4 most important days of Geometry. These are the first 4 days of **Geo Unit 7**, which follow immediately after the unit on similarity.

## Days 1-2: Special right triangles

Using inductive and deductive reasoning, students discover the patterns in the sides of 45˚, 45˚, 90˚ and 30˚, 60˚, 90˚ triangles. Instead of just asking students to solve for missing sides of these triangles, we ask them for ratios of sides and how many times bigger one side is than another. The goal is to help students think about proportional relationships. We also extend students’ understanding of ratios from “nice” numbers like ½ to more complex ratios like sqrt(3) or ratios represented by crazy decimals like 0.707107…

## Day 3: Defining the trigonometric ratios for right triangles

In **this lesson**, each group of students is given a different triangle to work with and asked to record the ratio of specific sides on a class chart. All the triangles are really scaled versions of 2 types of triangles. Students get a chance to notice that all the ratios for similar triangles are the same, but this idea is actually not new to them, since it follows from the definition of similar triangles studied in the last unit. What is new is that the specific ratios of sides have *names* and are defined by their location with respect to one of the acute angles in the triangle. In this lesson, students do NOT learn about the calculator keys for sine, cosine, and tangent.

## Day 4: Unlocking the Triangle: A new way of solving for the missing lengths of a right triangle.

In the NCTM session, participants went through **this lesson** acting as students and I acted as the teacher facilitating a debrief of the activity and adding margin notes. This lesson emphasizes the transition from having a second, similar triangle to determine the ratio of sides and solve the triangle, to using a trig ratio to get this same information. Throughout the activity, students are prompted to consider whether certain triangles are similar based on their angle measures and ratios of sides, reviewing key concepts from **Unit 6**. First, students were given a “nice” sine ratio of ⅖ and later students were given the acute angle of a right triangle and realized that they needed a way to “look up” the ratios of sides associated with that type of triangle. It is at this point that the calculator buttons are introduced. The big take-away is that a trigonometric ratio imparts all the same information as having a second similar triangle in front of you, but in a shorter, more efficient way. Looking up the trig ratio unlocks the triangle!

## A Problem String

Participants then looked at a sequence of 4 problems that made this progression of ideas even more explicit (see page 2 of the **participant handout**). The big idea was that the special right triangles are really the key to helping students transition from using two similar triangles to solve for sides to using only a trig ratio. This is because in special right triangles, students see for the first time that *some ratios of sides can be known*, rather than deduced from a similar triangle.

There is nothing inherently “special” about these triangles (except that the ratios can be expressed as exact values, but is that so special?). The ratios of particular sides in any right triangle can be known! By using language like “In a 20˚, 70˚, 90˚ triangle…” students continue to internalize that the ratios of sides in a right triangle are a unique property of the angles in that triangle, are fixed, and can be known (or looked up).

## Moving Beyond Right Triangles

In Algebra 2 and Precalc students are asked to grapple with more big questions about trigonometry. Specifically,

What is the meaning of an expression like “sin(150˚)” when 150˚ could never be an angle in a right triangle? How do we define trigonometric ratios for angles greater than 90˚?

Why do we care about the unit circle???

In the workshop we looked at **this lesson** to respond to both of these questions. We extended the definition of trigonometric ratios to represent the ratios of horizontal and vertical displacements of points on the terminal ray of an angle in standard position, as well as the true distance from the origin (the radius). It is critical that we are still talking about lengths and ratios, not just coordinates when we make the transition to angles on a circle! For this reason, students are given a coordinate plane with grid lines so they are actually estimating distances. Amazingly, the x- and y-coordinates of a point *do* represent the horizontal and vertical displacements of a point from the y-axis and x-axis, respectively. By considering circles of radii 1, 2, and 3 units, students realize that any size circle can be used to determine the sine, cosine, or tangent of an angle. But mathematicians often look for efficient strategies, and what’s more efficient than a denominator that is always 1?

## Graphing Trig Functions

The next progression is for students to **graph the trigonometric functions**, realizing that the input is the angle and the output is the ratio. This represents a significant transition because the coordinates of a point have a new meaning. No longer is it true that x=cos and y=sin! Functions are unique in that they show the covariation between two changing quantities. Instead of looking at all the trig ratios of one type of triangle, the graph of a function like the sine function shows one ratio (the sine ratio) of all types of right triangles (a right triangle with an acute angle of 1˚, 2˚, 3˚, etc.)

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