If students are struggling to find , ask them what they notice about the triangles. (They are both right triangles.) If necessary, prompt them to check their reference chart for information about right triangles. (The Pythagorean Theorem applies.)

The purpose of this discussion is to consider whether it is possible to calculate .

Ask, “If we don’t have enough information to find the exact length of , does that mean that  could be any length?” (No.)

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response by correcting errors, clarifying meaning, and adding details. 

• Display this first draft:
  It has a 50-degree angle and a side with length 3. So there’s only one length for . 
  Ask, “What parts of the response need more details?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft. 
• Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

Students may approach this task using similarity or the Angle-Side-Angle Triangle Congruence Theorem. If similarity is not mentioned by students, ask, “If we had a triangle with the same angles as triangle but different side lengths, would that be helpful for finding ?” (Yes, then we could use the scale factor to find .)

In this activity, students are building skills that will help them in mathematical modeling (MP4). Students formulate a model by designing a ramp that they think will be accessible. Then they validate their results by comparing them to the Americans with Disabilities Act (ADA) guidelines. Students then use measurement and computation to determine if their ramp is acceptable and redesign it if necessary.

Monitor for students who validate their design using these strategies:

• Measure the angle of their ramp with a protractor.
• Use the slope information by measuring to make sure that for every inch of vertical length, there are 12 inches of horizontal length.
• Construct one or more slope triangles along their ramp to demonstrate that it meets the guidelines.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Some students may be struggling to design a ramp. Ask them what shape is a good model for the side view of a ramp. (a right triangle) 

The goal of this discussion is to see that a triangle with a 4.8-degree angle and a ratio of for the vertical to horizontal length are equivalent.

Display 2–3 ramp designs from previously selected students for all to see. Tell students, “Simplified diagrams of right triangles are mathematical models of the cross sections of ramps.” This previews the work with cross sections that students will do in a subsequent unit, but there’s no need to define “cross section” now.

Invite students who used the ratio of to check their ramps to share their process. Make sure that all students understand how to use the ratio to generate the horizontal length of the ramp given the vertical length, because students will need to do this in the Cool-down.

Next, invite students who used the angle of 4.8 degrees to share.

Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the approaches have in common? How are they different?”
• “Did anyone solve the problem the same way but would explain it differently?”
• “Why do the different approaches lead to the same outcome?”

Ask students to confirm that ramps that have a ratio of for the side lengths also have an angle of 4.8 degrees, and vice versa. Students should be convinced that a 4.8-degree angle and a ratio of for the vertical to horizontal length are equivalent. In the Lesson Synthesis, they will connect this concept to triangle similarity.