The purpose of this activity is for students to recall how to plot points a given distance away from a particular point. Students may also make the observation that a set of points a given distance from a single point makes a circle. Students will continue building toward this understanding in subsequent activities, so it is not necessary for students to make that observation at this time.

Making scientific calculators available for this activity gives students the opportunity to choose appropriate tools strategically (MP5).

The purpose of this discussion is for students to explain how they found points a given distance away from a particular point, and to make observations about their results. Display the coordinate plane with the point . Poll the class for the points they found and annotate the graph with those points.

Here are some questions for discussion:

• “What were the first points you found?” (I found the points directly in line vertically and horizontally with the given point without much calculation.)
• “How did you find other points that ended up on a grid line?” (Since the distance from the set point was 5, I used a right triangle with legs of length 3 and 4 to find more points. Some of these were , , and .)
• “What do you notice or wonder about the points that are a distance of 5 from the point ?” (I notice that you can use the Pythagorean Theorem to find points. It appears to make a circle centered at . I wonder whether the circle has a radius of 5.)

If no students make use of a right triangle with legs of length 3 and 4 to find additional points, invite students to do so now. Recognizing that will be helpful in later activities.

In this activity, students make observations about the distance of points from the origin. As students categorize points by their distance from the origin, they are making sense of distances in the coordinate plane and persevering in solving problems (MP1). Then students graph the set of points the same distance from the origin on the same plane and generalize that the set of all points the same distance from a center creates a circle (MP8).

In this activity, students use the Pythagorean Theorem to test whether points are on a circle with a given center and radius. 

Monitor for student strategies in describing the horizontal and vertical distances between points, including: 

• Drawing a right triangle and labeling the legs.
• Identifying the horizontal and vertical changes between the points.
• Subtracting to find the distance between the horizontal and vertical coordinates of each point.

Students will build on this process in a following activity to develop a general equation for a circle.