# Lesson 1

Angles and Steepness

## 1.1: Ratios Galore (5 minutes)

### Warm-up

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 (\(\frac{AB}{DE} = \frac{AC}{DF}\)).

### Student Facing

Triangle \(ABC\) is similar to triangle \(DEF\). Write as many equations as you can to describe the relationships between the sides and angles of the 2 triangles.

### Student Response

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### Activity Synthesis

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.

## 1.2: Can You Calculate? (10 minutes)

### Activity

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 \(48^{\circ}\))

### Student Facing

Find the values of \(x, y, \text{ and } z\). If there is not enough information, what else do you need to know?

### Student Response

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### Anticipated Misconceptions

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.)

### Activity Synthesis

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 \(z\).)

*Speaking: MLR8 Discussion Supports.*Use this routine to support whole-class discussion. For each student that shares, ask peers to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.

*Design Principle(s): Support sense-making*

## 1.3: Is it Accessible? (20 minutes)

### Activity

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).

### Launch

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.

*Engagement: Develop Effort and Persistence.*Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “How did you get. . .?”

*Supports accessibility for: Language; Social-emotional skills*

### Student Facing

- Some buildings offer ramps in addition to stairs so people in wheelchairs have access to the building. What characteristics make a ramp safe?
- A school has 4 steps to the front door. Each step is 7 inches tall. Design a ramp for the school.
- Your teacher will give you the Americans with Disabilities Act (ADA) guidelines. Does your design follow the rules of this law? If not, draw a new design that does.

### Student Response

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### Student Facing

#### Are you ready for more?

- What is the angle measurement if the base is only 6 units long and the height is 1 unit tall?
- When the length is half as long does that make the angle half as big?
- What is the angle measurement if the base is 6 units long and the height is increased to 2 units tall?
- When the height is twice as tall does that make the angle twice as big?

### Student Response

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### Anticipated Misconceptions

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.)

### Activity Synthesis

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.

*Representing, Conversing: MLR7 Compare and Connect.*Use this routine to prepare students for the whole-class discussion. At the appropriate time, invite students to create a visual display that includes their ramp design and a justification for whether it meets ADA guidelines. Invite students to quietly circulate and read at least 2 other displays in the room. Next, ask students to find a new partner to discuss what is the same and what is different about the different methods groups chose to validate their designs. Listen for and amplify observations that include mathematical language and reasoning about angles and ratios of side lengths.

*Design Principle(s): Cultivate conversation; Optimize output (for justification)*

## Lesson Synthesis

### Lesson Synthesis

Ask students, “Will any ramp with one angle of 4.8 degrees have a slope ratio of \(1:12\)? Will any ramp with a slope ratio of \(1:12\) have an angle of 4.8 degrees?” (The ramp makes a right triangle. If we know the measure of one angle is 4.8 degrees, then we know any ramps with those measurements will be similar right triangles by the Angle-Angle Triangle Similarity Theorem. So they will all have the same ratio of the vertical side to the horizontal side. If we know the ratio of the vertical side to the horizontal sides in the ramps are \(1:12\), then we know that two pairs of sides are scaled copies of each other and the corresponding angles between them are congruent, so the triangles are similar by the Side-Angle-Side Triangle Similarity Theorem. So they will all have the same angles, because similar triangles have congruent corresponding angles.)

Ask students, “How do you think the people who wrote the Americans with Disabilities Act found the ratio of vertical length to horizontal length that is true for every 4.8 degree angle?” (They built ramps and measured them. They used math that we’re about to learn. They looked it up.)

“In the cool-down, you’ll see an example of another guideline for a different type of ramp with a different angle and ratio. Does every angle in a right triangle give you a fixed ratio of vertical length to horizontal length?” (Yes, because when you’re dealing with right triangles, all the triangles with the same angle measure for one of the other angles are similar, so they have the same ratios.)

## 1.4: Cool-down - Sidewalk Ramp (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

Because of the Pythagorean Theorem, if we know any 2 sides of a right triangle, we can calculate the length of the third side. But what if we know a side and an angle rather than 2 sides?

All right triangles with one pair of congruent acute angles are similar by the Angle-Angle Triangle Similarity Theorem. Knowing just one side length in addition to those angle measures is enough to uniquely define the triangle.

The Americans with Disabilities Act includes guidelines for safe and accessible wheelchair ramps. Ramps must form a maximum 4.8 degree angle with the ground, which creates a maximum \(1:12\) ratio for the legs of the right triangle.

Let's assume we are building a ramp for a 3 inch threshold.

To find length \(x\) we can use similarity. By corresponding sides, \(\frac{1}{12}=\frac{3}{x}\) so \(x\) is 36 units. To find length \(y\) we can use the Pythagorean Theorem, \(3^2+36^2=y^2\). So \(y=\sqrt{1,\!305}\) or about 36.1 units. To build a ramp that goes up 3 inches we need to start 36 inches, or 3 feet, out from the edge of the threshold and use a board that's about 36.1 inches long.