In this activity, students make connections between transformations and the coordinate grid. When defining transformations, they will notice and make use of the structure created by the grid (MP7). This task also presents an opportunity to refresh students’ memories of transformation language.

Monitor for students who draw a right triangle and use the Pythagorean Theorem to find the distance between points and . Throughout this unit, distance will be viewed as an application of the Pythagorean Theorem. Distance calculations using the Pythagorean Theorem will lead to the development of equations for circles and parabolas. There will be no need to introduce a separate distance formula.

Invite a student who drew in a right triangle to share that method. If a student suggests the distance formula as an alternate method, ask the class how the formula connects to the Pythagorean Theorem. If no one uses the distance formula, there is no need to mention it.

Ask a few students to share their transformations. There are many possibilities. Transformations that take multiple steps are as valid as single-step transformations. If students use descriptions such as “Move 3 units down and 4 units right,” connect this back to the language of translating and directed line segments. Remind students of this language by asking them to read the sentence frames for transformations from their reference chart:

• “Translate (object) along the directed line segment from (point) to (point).”
• “Rotate (object) (clockwise or counterclockwise) using (point) as the center by angle (measure).”
• “Reflect (object) across line (name/equation).”

Note that, during this unit, points could be named with letters (for example, point ) or with coordinates (for example, . Similarly, lines could be named in various ways, such as “-axis” or “.”

In this activity, students practice transforming a figure on the coordinate plane. Students may choose to use tracing paper and perform these transformations as if there were no grid. Other students may notice the structure of gridlines and look for patterns in the coordinates. During the Activity Synthesis, students are reminded that rigid transformations produce congruent figures. This helps prepare students for the next activity, in which they reason that given two congruent figures, there must be a sequence of transformations carrying one figure to the other.

Making dynamic geometry software and tracing paper available gives students an opportunity to choose appropriate tools strategically (MP5).

In this activity, students calculate side lengths of triangles on the coordinate plane. In the process they demonstrate that two given triangles are congruent. They recall some of the conditions needed to show that triangles are congruent, and they identify a sequence of rigid motions that will take one triangle to another.