## Modeling a Position Finder - Key - Section 3: Practice and Application

# Angle Management

Lesson 4 of 15

## Objective: SWBAT use a straight-edge and compass to construct a copy of an angle and to construct adjacent angles. Students will understand the meaning of an angle in terms of its structure.

## Big Idea: Using constructions to understand the structure of an angle and to model the operation of a navigator's position-finder.

*59 minutes*

#### Lesson Open

*5 min*

The Warm-up prompt for the lesson asks students to define an angle. This lesson opener follows our Team Warm-up routine, with students writing individual answers in their Learning Journals. I display the prompt using the slideshow for the lesson.

**Goal-Setting**

After the team scribes have written definitions of an angle on the front board, I display the Agenda and Learning Targets. We compare some of the answers. It will probably turn out that an angle is a rather difficult thing to describe precisely, unless we describe it in terms of its parts (**MP6**, **MP7**). Students often confound the definitions of an 'angle' and an 'intersection', but at least they are on the right track (an angle is a composition of simpler geometric objects). Other students will not recognize the difference between an angle (object) and angle measure (a quantity). I tell the class that today we will be learning to construct angles using a straight-edge and compass. I expect that the definition of an angle will make more sense after we have constructed a few.

An overview of this lesson can be found in the accompanying video.

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#### Constructing Angles

*15 min*

As in previous lessons on constructions, I use direct instruction to teach students to construct angles. I distribute the Guided Notes and demonstrate the procedures for constructing the copy of an angle and for copying adjacent angles. As I demonstrate the steps of each construction, students follow along and construct the illustrations in their notes. The illustrations are ordered in a sequence, like a cartoon. As students complete the illustrations, they practice the steps of the construction "pyramid-fashion".

As I demonstrate the constructions, I remind student to use proper technique in order to make accurate constructions (**MP6**). In particular, they should "lay down lead" with their pencil whenever using their compass to reproduce a distance.

During this part of the lesson, we learn the following constructions:

- Construct a Copy of an Angle

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#### Practice and Application

*25 min*

**Constructions Practice**

Students practice the constructions they have just learned using the Rally Coach format. I display the instructions on the front board. I hand out the Construction Practice Problems and students take turns with their partner.

The first two pairs of problems are straight-forward. The last pair asks students to verify that two angles are identical by constructing a copy of one on top of the other. This exercise is based on the definition of congruence, which we will study in depth later in the semester. From the beginning, however, I want students to be able to compare angles by constructing a copy of one angle superimposed onto the other. Of course, students are already able to superimpose a copy of a line segment onto another: that is what a compass is for.

**Modeling a Position-Finder**

The next activity requires tracing paper.

In this Activity, students simulate using a navigational instrument called a position finder (**MP4**). Students construct a pair of adjacent angles such as could be measured directly by using a position finder (which is a three-legged compass that also acts as a sighting instrument). They copy the adjacent angles onto tracing paper and use it to find the position of a boat on a nautical chart (map). For a demonstration, see the accompanying video.

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#### Summarizing Angles

*10 min*

I distribute the Guided Notes on Structure and lead the class in completing the Big Ideas and the terms related to angles: *angle*, the *interior* of an angle, *adjacent *angles. We will return to the notes following lessons on polygons and polyhedra to reinforce or extend the meaning of these terms. Some of the features of the notes are highlighted in this video. More on how I use Guided Notes can be found in my strategies folder.

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**Team Size-Up**

The lesson close follows our Team Size-Up routine. The Lesson Close prompt asks students to define an angle.

**Homework**

I assign problems #15-17 from Homework Set 1. Problems #15 and #16 ask students to practice the constructions that were introduced in this lesson. Problem #17 offers practice in vocabulary, including terms which were introduced in this lesson.

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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: Previewing Dimension and Structure
- LESSON 2: Unto the 4th Dimension
- LESSON 3: New Directions
- LESSON 4: Angle Management
- LESSON 5: Triangle Construction Site
- LESSON 6: Diagonal Daze
- LESSON 7: Platonic Relationships
- LESSON 8: Reviewing Structure
- LESSON 9: Flatland Encounters
- LESSON 10: Intersection Derby
- LESSON 11: Mission Impossible
- LESSON 12: Intersection Logic
- LESSON 13: Cutting Planes
- LESSON 14: Reviewing Intersections
- LESSON 15: Dimension & Structure Unit Quiz