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# Tangent Ratio Investigation

Lesson 2 of 10

## Objective: Students will understand the relationship between the measure of the acute angle of a right triangle and the lengths of the sides opposite and adjacent to the angle. Students will be able to use the tangent ratio to solve for the missing sides of a right triangle.

Present the lesson opener (10 minutes). The opener for this lesson (found in the slide show) asks students to predict how doubling the height of a right triangle while holding the base constant affects the measure of the (acute) base angle of the triangle (**MP2**). This activity follows the **Team Warmup** class routine, which is described in my **Strategies Folder**.

The lesson opener refers to the skateboard ramp problem that was the center of the previous lesson. The goal is for the class to see that the relationship between the sides and angle of a right triangle is positive but non-linear. If students have no idea how to make a prediction about the relationship between the sides and angle of the triangle, or if they come up with a conjecture too quickly without showing any real thought, ask them to refer to the scale drawings they made to answer questions 1, 2, and 3 of the Building a Kicker Ramp activity.

While students complete the lesson opener, I take attendance and circulate around the classroom noting who does not have their homework or their materials.

Review learning targets and the agenda for the lesson. Display the agenda and learning goals for the lesson while distributing the handout for the Investigating Tangent Ratio activity.

I tell students that they will be using a virtual simulation to test the predictions they made about the relationship between the angle of a right triangle and the lengths of its legs. This is a very important ratio, one that is at the heart of trigonometry, and they will be using it soon to solve problems.

Have the class move to the computer lab and log in to the computers (5-10 minutes).

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#### Conducting the Investigation

*30 min*

Have students complete the tangent ratio investigation (25-30 minutes). Have students work in pairs or individually. The Internet tool is an applet available on the College Preparatory Mathematics (CPM) website as a Teaching Resource (**MP5**). I demonstrate its use in the accompanying video.

In the CPM Geometry Connection textbook, the concept of the tangent function is developed as a relationship between the slope of a line on the coordinate plane and the angle formed by the line and the x-axis. While I recognize the merit of this approach, I find that many of my students are not entirely comfortable with the concept of slope. Therefore, I choose to develop the tangent ratio as property of a right triangle, drawing on students’ familiarity with the properties of similar polygons. I refer to the skate board ramp problem whenever I want to students use a concrete example.

To help students learn how to use the Internet tool in the least amount of time, I lead the class through parts 1 and 2 a-h of the activity. Then I circulate while they complete part 3. I emphasize to students that their goal is to answer the question: How is the acute angle of a right triangle related to the lengths of its legs? (**MP2**)

Things to look out for: Are students plotting points accurately enough to recognize the curve (**MP6**).

##### Resources (5)

#### Resources

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Have students reflect on the lesson in pairs (3 minutes). Students share the conclusions they reached from the tangent ratio investigation in pairs. Students write their own reflections in their learning journals.

Homework. The homework assignment is in the syllabus for this unit.

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- UNIT 1: Models and Constructions
- UNIT 2: Dimension and Structure
- UNIT 3: Congruence and Rigid Motions
- UNIT 4: Triangles and Congruence
- UNIT 5: Area Relationships
- UNIT 6: Scaling Up- Dilations, Similarity and Proportional Relationships
- UNIT 7: Introduction to Trigonometry
- UNIT 8: Volume of Cones, Pyramids, and Spheres

- LESSON 1: Building a Kicker Ramp
- LESSON 2: Tangent Ratio Investigation
- LESSON 3: Applying the Tangent Ratio
- LESSON 4: Understanding Tangent as a Function
- LESSON 5: Progress Check and Homework Review 1
- LESSON 6: Properties of Sine and Cosine
- LESSON 7: Solving Triangles with Trigonometry
- LESSON 8: Modeling with Trigonometry
- LESSON 9: Measuring the Flag Pole
- LESSON 10: Introduction to Trigonometry Unit Quiz