The purpose of this activity is to activate students’ prior knowledge of rigid transformations. Students also see that the image of a polygon is determined by the images of each of its vertices. Students build toward the concept that transformations are functions that take points as inputs and produce points as outputs so that distances and angles are preserved.

The important idea for discussion is that rigid transformations preserve distances and angles. Display a student’s work for all to see and ask:

• “For the reflection of across line , how do the side lengths of compare to the corresponding side lengths of its image?” (The lengths are equal.)
• “What is the measures of the angle in the upper left corner of ? How does this compare to the corresponding angle measure in any of the images of ?” (This angle and all of its images measure 120 degrees.)

In this activity, students use what they know about constructing perpendicular lines to determine where a reflection should take a point to become the image. This activity strengthens students’ understanding of reflections without reference to a coordinate grid. Recognizing that a reflection takes a point to another point the same distance from the line, and that the distance between a point and a line is measured on the perpendicular line, leads to a rigorous definition of reflections that students will use to prove theorems in the next several lessons.

Monitor for strategies students use to determine the image of point , including:

• Constructing circles centered around 2 or more points on that go through point . The point will be the common intersection of all these circles because the points on are each the same distance from as they are from .
• Constructing a line perpendicular to going through , then marking a point that is the same distance away from as is.

The important idea for discussion is that line is the perpendicular bisector of the segments connecting each point in the original figure to its image. Remind students that the distance between a point and a line is the perpendicular distance, so for a reflection to give points the same distances away, they need to use a perpendicular.

Ask students to share their strategies for locating . If not mentioned by students, discuss the strategy of constructing a line perpendicular to going through and marking the point that is the same distance away from as .

Some students may have trouble reflecting on the isometric grid. Ask these students to use tracing paper to fold across the line of reflection to find the image.

If students are stuck as to how to translate when isn't connected to a vertex of the figure, remind students that is telling them the direction and the distance, but the location doesn't matter.

If a student has trouble getting started, suggest connecting to and asking themself what they notice about the distances to the line. If they are still stuck, ask them to mark some point on the line of reflection and think about the distance from that point to and .