# Exploring Translations

Lesson 1 of 4

8th grade

40-60 minutes

**Description**

Develop an understanding of **translations** in the coordinate plane and determine general rules for **translations** through exploration.

**Materials**

### Introduce

Prepare students to use the visual thinking strategy by providing the “I see, I think, I wonder” map on the board. Ask them to observe your actions and note their thinking. Demonstrate translation by picking up an object and physically moving it to a new location. Repeat if needed. Students should notice that each translation’s **orientation**, size, and shape are preserved. The only change with **translations** is location.

**Teacher Notes:**

Possible guiding questions include:

- What are some characteristics that you noticed about the movements?

*[Possible student responses: the shape stayed the same, the object is not in the same spot, the object changed direction]*

- What changes do you notice?

*[Possible student responses: nothing about the object is different, it is just in a new place]*

- Are there any patterns that you notice?

*[Possible student responses: the shape always moves location but does not change in size]*

- What happens to the
**coordinates**?

*[Possible student responses: they always change ]*

### Explore

Have students work with their partner on Part I of the Translations Activity Sheet (download from Materials section above) to generalize a rule for each translation using x and y notation.

After a few minutes, bring students back together to share student generalizations. Expand on student thinking to connect to the general translation rules using x and y notation. For example, (x +2, y-3) for a translation of a point 2 units to the right and 3 units down.

Have students continue to work in partners to complete Part II Exploring Properties of **Translations**. For this part of the activity, students should be able to understand and apply a translation rule to a given image.

As a whole class, have partners share their observations.

**Teacher Notes:**

Possible guiding questions:

- What do you notice about the pre-image and new image?

*[Possible student responses: The combination of these ***translations** moved the image diagonally.*]*

- How are all the points on the translated image related to the corresponding points on the pre-image?

*[Possible student responses: They are the same distance apart, just moved down.]*

*[Possible student responses: All of the points on the shape are still the same distance away from each other.]*

### Synthesize

After students have completed the second part of the worksheet, work as a class to summarize rules for performing a translation. Teachers may want to post the rules for translation on a classroom chart for future reference.

Students should recognize the **orientation**, size, and shape are preserved in **translations**; that is, the resulting figure will have the same side lengths and interior angle measures as the original figure.

**Teacher Notes:**

Examples of rules include:

- When translating up, the y-coordinate will increase by the number of units moved.
- For example, if a point (x, y) is translated up 8 units, then the image point will be (x, y + 8).

- When translating down, the y-coordinate will decrease by the number of units moved.
- For example, if a point (x, y) is translated down 3 units, then the image point will be (x, y - 3).

- When translating right, the
**x-coordinate** will increase by the number of units moved.
- For example, if a point (x, y) is translated right 4 units, then the image point will be (x+4, y).

- When translating left, the x-coordinate will decrease by the number of units moved.
- For example, if a point (x, y) is translated left 5 units, then the image point will be (x-5, y).

### Assessment

**Teacher Notes:**

**Extension Questions**

- If you translate a figure three units to the right on the coordinate graph, how would the coordinates change? [three is added to each x-coordinate.]
- If you to translate a figure 2 units down on the coordinate graph, how would the coordinates change? [2 is subtracted from each y-coordinate.]
- How can you determine the coordinates of the points of the new figure given the translation without drawing the figure? [If a figure is translated left or right, a number should be added or subtracted from the x-coordinate, and if a figure is translated up or down, a number should be added or subtracted from the y-coordinate.]

### Teacher Reflection

- How were you able to gather and elicit evidence of student thinking and understanding from “Part I Exploring Translations with Patty Paper”? How could this evidence help during instruction for developing a general rule for translations?
- How did you connect the student-generated strategies and methods to the more formal procedure?
- What scaffolding or differentiation was needed in response to student thinking? How did you extend learning for students?

Leave your thoughts in the comments below.