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Right Angle Trigonometry (Lesson 9.1)

Learning Targets
  • Find missing side lengths and angles in right triangles. 
  • Understand that an angle measure in a right triangle uniquely determines the ratio of the side lengths.
  • Use trigonometric ratios to find missing side lengths for right triangles.

Today we’re starting the scariest unit of Algebra 2…TRIGONOMETRY. Muahahahha! At least that’s how my students imagine me laughing when we start our trig unit. But just because this unit has a scary mathematical title, it doesn’t mean it has to be confusing. Now when I think back to my experience with trigonometry in high school, I will admit it was very confusing. I simply memorized my way to an A by using SOH CAH TOA and I had no idea what I was actually doing. It wasn’t until college that I developed a conceptual understanding of trigonometry. And much to my surprise, it wasn’t confusing!


So how do we make trig less confusing for our students? We’re going to focus on developing a strong conceptual understanding of trigonometry as ratios using proportional thinking. This unit is intentionally designed so that students won’t even be aware of the sine, cosine, and tangent buttons on their calculators until Day 2. We’ve chosen nice numbers to work with so that students can think their way through problems intuitively. As you teach this unit, do your best to rely on proportions. This will help students understand that a trigonometric ratio is just that, a ratio of side lengths.

Activity: Trig-o-no-what??

Experience First

The goal of this lesson is for students to understand that a trigonometric ratio is just the ratio of the side lengths of a right triangle and that this ratio changes based on the angles of the triangle.


Intro the lesson briefly. Not a lot of explanation should be given at the start

of the lesson. In fact, you really should be able to give students the printout and tell them to get going! That’s why we say Experience First.


  • Angles of a triangle add to 180 degrees.

  • How to use Pythagorean Theorem to find side lengths.


Students will need a red, blue and green colored pencil or marker to color the sides of the triangles. Students should work on questions #1-3 first and then #4-5 in their small groups. As they are working, walk around the room and check in on groups. If groups are struggling, try using guiding questions to help them keep moving. When you see groups with a question done, ask them to fill in their work on the board.


Guiding Questions:

  • What do you know about the angles of a triangle?

  • I notice this triangle has a right angle. What do you remember about right triangles from geometry?

  • If we know two sides of a right triangle, can we find the third side? I remember something about sides and right triangles from when I used to teach geometry. What was that again?

  • What side lengths are used for the tangent ratio? Do we know any of those sides? What does the ratio of those sides have to be?

  • What is 0.4 written as a fraction?

Formalize Later

By time all of the groups have finished #1-3, all of the work should already be filled in on the board. Go over the students’ work on the board. Ask groups to explain their solutions to the group. Point out important parts of the solutions and formalize the work with MARGIN NOTES by explaining the sine, cosine and tangent trig ratios. Have students write the margin notes using a different color pen. We highly recommend the Papermate Flair, the greatest of all writing utensils


After this first formalization, students can then complete the rest of the page. Since there is more than one possible answer for #5, ask multiple groups to give their solutions. Formalize the work with margin notes.


In addition to formalizing the experience with margin notes, we formalize the content in the QuickNotes box. Everything in this box should relate directly to the learning targets for the lesson. We’ll add things like vocabulary and formulas to this box. You’ll share this content with students through direct instruction. It should be as concise as possible! Once complete, have students work on the Check Your Understanding (CYU) in their groups. Have groups write their work on the board. When most groups have finished, go over the solutions with the class.

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