# Defining Translations

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## Objective

SWBAT show that when a figure is translated, all segments connecting points on the preimage to their corresponding points on the image are congruent and have the same slope.

#### Big Idea

Putting the 'if and only if' in translation. In this lesson, students discover the defining characteristics of translations.

## Activating Prior Knowledge

10 minutes

Where We've Been: Students have been introduced to rigid transformations. We have described translations as slides, rotations as turns, and reflections as flips. We've also introduced important terminology like vector, center and angle of rotation, and line of reflection.

Where We're Going: Now we're taking a second pass at rigid transformations, this time defining them more rigorously.

In this particular lesson, students will need to use the distance, midpoint, and slope formulas correctly. They will also need to recall the meaning of translation and the slope criteria for parallel lines.

To get students primed for the lesson, I use the Activating Prior Knowledge: Defining Translation resource. I give students the first five minutes to work on it independently. Students who are already on point should finish within the five minutes. During the second five minutes, I provide a very direct refresher aimed particularly at students who are struggling. At the same time, I also aim to provide something for students who got all the answers correct already. I do this by discussing the concepts behind the correct answers, modeling precise academic language usage, and showing my work in precise and organized fashion (MP 6).

## Self-Guided Activity

45 minutes

For this section of the lesson, I create test-like conditions...at least at the beginning. I let the students know that this is an independent activity. They are to "get into the zone". They are to focus entirely on their own work. I let them know that I am not enforcing this strict order to be mean, but because I want them to concentrate fully on the activity and get the most out of it.

I provide each student with a ruler, a sheet of grid paper, a calculator, and the Define Translations student resource. Once I've done that, my job is to maintain order in the room and to make sure students are reading and following the directions.  For example, many students need intervention when it comes to "showing their work in organized fashion". So I do intervene in those cases.

When students are confused, I try my best not to think for them, but rather to ask them open-ended questions that will catalyze their own thought process. See  Helping Students Make Sense of Problems, a video that explains how I attempt to do this.

About half-way, or maybe two-thirds of the way through this section, students seem to naturally start turning to each other for help, confirmation, etc...even though I've asked them to work independently.  I allow this to happen. If it doesn't happen, I explicitly encourage it. Still, though, I remind students, "Read and try for yourself before turning to someone else for help."

## Closure

15 minutes

The purpose of this section is to make sure that everyone gets the important take-aways from the lesson. I choose to collect the papers before this section of the lesson so that I can scan the papers quickly assess how students did on their own. My goals in this quick scan are to (1) identify common strengths and weaknesses and (2) identify some candidates for student exemplars. While I'm doing this, I have the students write a quick reflection on what they did, what they learned, and what questions they still have.

Once I've scanned the papers, we're ready to begin the closure section. Projecting the resource on the whiteboard, I start with item #1: the image vs. preimage distinction. We've been over this before, so I call on a student to explain which is the image, which is the preimage, and why.

Next we discuss the type of transformation, again calling on a student. If necessary, I'd use a sentence starter like..."This transformation is a __________ because ________________."

For items #4 and #5, I select one to a few exemplary student work samples to model correct, precise, well-organized work. It's usually the case that even the exemplars are not perfect. They usually have their strong points and aspects that need refinement. I call attention to the strong points of each exemplar so that students get a composite view of what makes a strong response. The main things I look for in an exemplar are:

1. Precise labeling (e.g., "Slope of segment AA'=...)

2. Neatness

3. Thoughtful layout/arrangement

4. Going above and beyond to show understanding

After this, we move on to an important discussion of items #6 and #7, which ask students to make conjectures based on their observations (inductive reasoning). I provide a model response that goes something like this:

"Whenever a figure undergoes translation, all of the segments that connect a vertex on the preimage to its corresponding vertex on the image are congruent and have the same slope (usually parallel)"

Moving on, a good number of students do not fully understand the thrust of item #7 so it's important to clarify it at this point.  I tell students:

"In item #6, we were able to generalize only about the vertices of the figure because we had only dealt with vertices at that point. So in this step we are analyzing a segment that connects two corresponding points on the figures that are not vertices. For convenience, we chose the midpoint of segment DE because it has nice integer coordinates. It is important to understand though, that we could pick any arbitrary point on the figure. After analyzing this segment, we found that it obeys the same rules as the segments connecting vertices. Therefore, we can conclude (although very informally):

If a figure is translated, all line segments connecting corresponding points on the preimage and image are parallel and congruent.

Or stated another way, when a figure is translated, all points on the preimage are taken to their corresponding points on the image by congruent, parallel (or overlapping) line segments."

This is the take-home lesson, so I make sure that students record it in their notes.

As a take-home activity, I ask students to develop and rehearse a Translations Elevator Speech. See this video for an example of what they should look like: Sample Translation Elevator Speech