Lesson 4

Ratios in Right Triangles

Lesson Narrative

This lesson is students’ first encounter with trigonometry although they won’t encounter the word trigonometry yet. They start with the essential concept of connecting angle measurements with the ratios of side lengths in a right triangle. In this lesson, students are focused on looking for patterns, estimating, and moving back and forth between predicting angle measures from ratios and ratios from angle measures. Taking time to build students’ intuition helps them make sense of trigonometry so that they will be able to ask themselves, “is that a reasonable answer?” in subsequent lessons (MP1).

Students start by measuring side lengths and calculating ratios in several different-sized triangles with the same angle measures. This reinforces students’ understanding of similarity. Do not name these ratios yet; the long descriptions are important for students to build understanding. The decision to put the columns in the order that will eventually be named “cosine, sine, tangent” is purposeful. Because cosine represents the \(x\)-coordinate in the unit circle, while sine represents the \(y\)-coordinate, tables with cosine first correctly correspond to the \((x,y)\) coordinates that students will see later. 

As students measure side lengths and compute ratios there is an opportunity to discuss measurement error and the relationships between precision in measurement and precision in values calculated with those measurements. In this unit, we recommend rounding side lengths to the nearest tenth and angle measures to the nearest degree. Students are instructed to calculate the ratios of side lengths based on measured lengths to the hundredths place, and, when using digital tools, to use ratios calculated to the thousandths place.

When students examine the class table they might notice that:

  • angles with larger measures have larger ratios of the opposite side to the adjacent side or hypotenuse
  • the larger the ratio of the opposite side to the hypotenuse, the smaller the ratio of the adjacent side to the hypotenuse
  • the ratio of adjacent side to hypotenuse is equal to the ratio of opposite side to hypotenuse for complementary angles, or angles which sum to 90 degrees.

These observations will be topics in subsequent lessons, so students need not justify their conjectures at this point.

Learning Goals

Teacher Facing

  • Comprehend that knowing one acute angle of a right triangle determines all the ratios of side lengths in that triangle.
  • Generate ratios of side lengths of right triangles (using words and other representations).

Student Facing

  • Let’s investigate ratios in the side lengths of right triangles.

Learning Targets

Student Facing

  • I can build a table of ratios of side lengths of right triangles.

CCSS Standards

Building On

Addressing

Building Towards

Glossary Entries

  • complementary

    Two angles are complementary to each other if their measures add up to \(90^\circ\). The two acute angles in a right triangle are complementary to each other.