In this activity, students connect opposite signs in coordinates to reflections across one or both axes. Students investigate relationships between several pairs of points in order to more generally make this connection (MP8).

The use of the word “reflection” is used informally to describe the effect of opposite signs in coordinates. In grade 8, students learn a more precise, technical definition of the word “reflection” as it pertains to rigid transformations of the plane.

The goal of this discussion is for students to see that coordinates with opposite signs correspond to reflections across the axes. Begin by asking students what patterns they noticed for pairs of points whose -coordinates had opposite signs. Ask students to give specific examples of pairs of points and their coordinates when describing the pattern they saw. Record students’ explanations for all to see. Students may use phrasing like “the point flips across the -axis.” Introduce the word “reflection,” and discuss similarities between reflections across the -axis and reflections in a mirror. Note that rigid transformations, including reflections, will be studied further in a later course, so it is not necessary for students to use this term fluently.

Repeat this discussion for pairs of points where the -coordinates had opposite signs to see that they are reflections across the -axis.

Close by discussing the relationship between points and , where both the - and -coordinates have opposite signs. Ask students how they might describe the relationship between and visually on the coordinate plane. While students may describe the relationship in terms of two reflections (once across the -axis and again across the -axis, or vice versa), it is not expected that students see this relationship in terms of a rotation.

In this activity students develop strategies for finding the distance between two points in the coordinate plane when the coordinates might not be integers. The distances used restricted to horizontal and vertical distances—use of the general two-dimensional distance formula is not expected, nor are students expected to add or subtract negative numbers fluently. More general strategies for finding distance in the coordinate plane and rational number arithmetic is developed more completely in later courses.

If some students claim that a diagonal distance across a certain number of squares is equal to a vertical or horizontal distance across the same number of squares, consider asking:

• “How did you determine the distances between both sets of points?”
• “Does a ruler (or tracing paper) confirm that those distances are the same?”

The purpose of this discussion is for students to share strategies for determining the distance between two points on the same horizontal or vertical line. Begin by inviting students to share the distances they found between pairs of points and their reasoning. Record and display their responses for all to see. Some common strategies include:

• Counting grid spaces in a horizontal or vertical direction.
• Determining the distance between each point and the - or -axis and adding the distances.
• Reasoning using each point’s coordinates.
• Using tracing paper.

Here are some questions for discussion:

• “How are these strategies the same or different?”
• “Did anyone use the same strategy but would explain it differently?”
• “Would these strategies still work if both points were located in the same quadrant?” (Yes.)