Lesson 1

Angles and Steepness

Lesson Narrative

The goal of this lesson is for students to recognize that the ratio of the legs of a right triangle with a given acute angle is fixed. They are building connections to similar right triangles in the previous unit. Students should leave the lesson wondering how the people who wrote the Americans with Disabilities Act guidelines knew what the ratio of two sides of a right triangle would be for a given angle, or what the angle would be for a given ratio. 

Note on language: In previous courses, students may have learned that a ratio is an association between two or more quantities. However in more advanced work, such as this course, ratio is commonly used as a synonym for quotient. This expanded use of the word ratio first came into play in the previous unit on similarity. This usage continues in this unit.

Students have a chance to do a portion of the modeling cycle (MP4) as they consider all of the possible factors that make a ramp safe (railings, steepness, width, surface) and then focus on how to account for a single aspect (steepness) as they begin to design a ramp to accompany a specific set of stairs.

In addition, students have a chance to construct viable arguments (MP3) as they argue that any ramp that rises at the angle given by the Americans with Disabilities Act guidelines must have the vertical length to horizontal length ratio given by the guidelines, based on their understanding of triangle similarity.

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

Learning Goals

Teacher Facing

  • Calculate side lengths in right triangles.
  • Explain why the acute angles in a right triangle determine the ratio of the side lengths (orally).

Student Facing

  • Let’s solve problems about right triangles.

Learning Targets

Student Facing

  • I can explain why knowing one acute angle in a right triangle determines the ratio of the side lengths.

CCSS Standards

Building On


Building Towards

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