Lesson 1

In the previous unit, students studied similar triangles and used the properties of rigid transformations and dilations to establish that in similar triangles:

• All pairs of corresponding angles are congruent.
• Lengths of all pairs of corresponding sides are proportional.

This warm-up helps students to recall that prior learning and to practice expressing ratios of side lengths in similar triangles and noticing they are equal. Monitor for students who use ratios comparing lengths within each triangle (\(\frac{AB}{AC} = \frac{DE}{DF}\)) and students who use ratios comparing lengths of corresponding sides (\(\frac{AB}{DE} = \frac{AC}{DF}\)).

Invite students to share their equations using angles. Then invite students to share their equations using side lengths. Continue collecting examples until there are some ratios comparing lengths within each triangle and some ratios comparing lengths of corresponding sides.

As students complete this activity, they ask themselves whether they have enough information to find unknown side lengths in right triangles, and if not, what information they might need. Students may notice that without two side lengths given, they can’t use the Pythagorean Theorem to find the length of the third side. In the synthesis, there is an opportunity to connect back to what students know about triangle similarity and congruence, and to preview the fact that while we don’t yet have enough information to find the length \(z\), the length \(z\) is fixed once we know the measures of angles \(G\) and \(H\) and the length of \(GH\).

Monitor for students who:

• attempt to measure or estimate the length of \(z\)
• assume that angles \(H\) and \(E\) are congruent and use similarity to find the length of \(z\) (the angles are not congruent, angle \(E\) actually measures \(48^{\circ}\))

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

The purpose of this synthesis is to discuss whether it is possible to calculate \(z\). Invite the previously selected students to share their attempts to calculate \(z\).

Ask, “If we don’t have enough information to find the exact length of \(z\), does that mean that \(z\) could be any length?” (No, by the Angle-Side-Angle Triangle Congruence Theorem, any right triangle with a 50 degree angle and a side with length 3 between the known angles has to be congruent to this one. You can’t draw any other triangle with those measurements.)

If similarity is not mentioned by students, ask, “If we had a triangle with the same angles as triangle \(GHJ\) but different side lengths, would that be helpful for finding \(z\)?” (Yes, then we could use the scale factor to find \(z\).)

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 the methods students choose to validate their design:

• Use the angle to check their ramps by measuring the angle 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).

You may want to do an image search for “bad wheelchair ramp” to show some examples that are not safe to help students come up with good characteristics.

Arrange students in groups. After students make their design, distribute one copy of the ADA guidelines, cut from the blackline master, per group.

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 having a 4.8 degree angle and a ratio of vertical to horizontal length of \(1:12\) are equivalent.

Display previously selected drawings of student ramp designs. Tell students, “Simplified diagrams of right triangles are a mathematical model of the cross section of a ramp.” 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 \(1:12\) 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, as students will need to do this in the cool-down.

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

Ask students to confirm that ramps in which students checked for a ratio of \(1:12\) also have an angle of 4.8, and vice versa. Students should end convinced that having a 4.8 degree angle and a ratio of vertical to horizontal length of \(1:12\) are equivalent. In the lesson synthesis, they will connect this concept to triangle similarity.