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# New Directions

Lesson 3 of 15

## Objective: SWBAT describe the figure that is generated by rotating or translating another figure and classify it according to number of dimensions. Students will understand how a shape can be created by the path of a figure moving through space.

#### Lesson Open

*15 min*

Using the slideshow, I display the Warm-Up prompt for the lesson as the bell rings. The prompt asks students to recall the explanation of 'dimension' given by Arthur Square in the movie Flatland, which students watched in the previous lesson.

After reviewing team answers, I use a--somewhat whimsical--set of animated slides to illustrate the concept of dimension (see Moving Parallel to Oneself and New Directions Slideshow).

I use the language of the film: a new dimension is created when an object moves "parallel to itself". While this language is confusing, I like it because it gives me an opportunity to explore the meaning of 'parallel' with the class. My goal is to lead students to a more precise definition of the term, but first I must develop the concept of an intersection (**MP6**). I will do this over the course of the next several lessons.

I ask, "What does it mean for objects to be parallel?" Someone will surely say that parallel lines do not cross. So, in this context, moving parallel means that the object does not cross any part of its own path. The object does not re-occupy any point in space that it has occupied before. The slides illustrate what this can mean. The last two slides in the set are non-examples.

The warm-up follows our Team Warm-up routine, with students writing their answers in their Learning Journals.

**Goal-Setting**

Displaying the agenda and learning targets, I tell the class that we will begin by summarizing some of the ideas that were explored in the movie 'Flatland'. Words like "dimension", "space", and "plane" show up everywhere in geometry, but they are not easy to define (**MP6**). To understand them better, we will look at examples. Following the lead of Arthur Square, we will explore how the path of an object as it moves through space can generate a new shape with a greater number of dimensions.

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#### Summarizing Dimension

*20 min*

In this section, the whole class completes Notes to summarize concepts and vocabulary related to the concept of 'dimension'. 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. Classifying objects according to the number of their dimensions is our starting point for analyzing the structure of a geometric figure(**MP7**).

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#### Generating Shapes

*15 min*

In this activity, students practice describing the result of moving (rotating and translating) figures in different ways.

I display the images of 1- and 2-dimensional figures using a set of slides in the slideshow. Students respond by writing on individual white boards.

The first slide includes some simple animations, which can be used to show that translating the figure within its own plane will cause it to intersect its own path. It must be moved in a *new* direction. The third slide includes a link to a video demonstration of a spade drill bit, which I use to help students visualize the solid of roation formed by rotating a rectangle.

While the standards only address rotations of 2-dimensional figures, I include translations, as well. Translating a 2-dimensional figure in space to form a solid helps to prepare some students to visualize 2-dimensional cross-sections of solids, which they will do later in this unit.

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**Team Size-Up**

Displaying the Lesson Close prompt, I ask students to describe the shape formed by rotating two circles around the bisector of the segment between their centers (a doughnut or torus). This activity follows our Team Size-Up routine.

**Homework**

I assign problems #12-14 from Homework Set 1. Problem #12 reviews the construction of a triangle using three given segments for sides, which we studied in the last unit. This problem is important preparation for the lesson after next, Triangle Construction Site, in which students must use a circle to model the possible locations of the endpoint of a side of the triangle. Problem #13 asks students to describe the 2-dimensional shape which can be rotated to generate a 3-dimensional solid. Problem #14 practices vocabulary which was introduced in this lesson.

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- UNIT 1: Models and Constructions
- UNIT 2: Dimension and Structure
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- UNIT 4: Triangles and Congruence
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- 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