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 plot points in a coordinate plane to create a rectangle. They find the lengths of each side by finding the horizontal or vertical distance between points. Students attend to precision in language as they distinguish between distance (which is unsigned) and difference (which is signed) (MP6). 

In the digital version of the activity, students use an applet to plot points in a coordinate plane. The digital version may reduce barriers for students who need support with fine-motor skills.

The goal of this discussion is for students to compare the distance between two numbers with the difference between two numbers. Here are some questions for discussion:

• “How can you find the difference between two numbers?” (Subtract one number from the other.)
• “When finding the difference between two numbers, does the order matter?” (Yes, we need to subtract the final number from the beginning number.)
• “How can we find the distance between two numbers?” (We can use a number line and count the distance between the two numbers. We can subtract the two numbers.)
• “When finding the distance between two numbers, does the order matter?” (No, we are just looking for the magnitude of the change, not the direction of the change.)
• “How can we see this distinction between difference and distance in the points we drew in the coordinate plane?” (When finding the difference between the -coordinates of and , the order we subtracted mattered, but when finding the distance between the two points, the order did not matter.)

If not mentioned in students' explanations, emphasize that differences can be positive or negative (or 0) depending on the order of the numbers subtracted. Distances cannot be negative.

In this activity, students practice plotting points in the coordinate plane to make polygons.

In the digital version of the activity, students use an applet to plot points in the coordinate plane. The applet allows students to drag points to their location in the coordinate plane and quickly check their accuracy. The digital version may be helpful for students to quickly plot and adjust points of polygons without needing to erase.

The purpose of the discussion is to emphasize the connection between numbers, the coordinate plane, and geometry. To highlight these connections, ask:

• “How is the coordinate plane related to the number line?” (The coordinate plane has two axes that are both number lines.)
• “How are we able to make polygons in the coordinate plane?” (The vertices of a polygon are plotted as points in the coordinate plane.)

Complete the connection by explaining to students that the coordinate plane allows us to describe shapes and geometry in terms of numbers. This is how computers are able to create two- and three-dimensional images even though they can only interpret numbers.