In this activity, students find distances between points in the coordinate plane. The points are shown on a coordinate plane without a grid to encourage reasoning about the coordinates rather than counting grid squares. If connected, the three points are the vertices of a right triangle, which will help students recognize this situation as one where the Pythagorean Theorem can be used.

The goal of this discussion is for students to see how they can find the distance between any two points in the coordinate plane by adding a third coordinate that allows them to draw a right triangle.

Begin by displaying the image from the Task Statement. Before students share their solutions, display an incorrect solution based on a common error you observe for finding the distance between points with negative coordinates. For example, “The distance between and is 6, because .”  Ask students to identify the error, critique the reasoning, and revise the original statement. As students discuss in partners, listen for students who reference the points in the coordinate plane to show that the distances are incorrect. Then invite groups to share their solutions.

Next, draw two points on the coordinate plane, for example and . Ask students how we can use the problem we just solved to find the distance between these two points. (Draw the point or and draw a right triangle between the points, then use the Pythagorean Theorem.) 

If time allows, ask students to calculate the distance between these two new points.

In this optional activity, students calculate the perimeters of two triangles on the coordinate plane using the Pythagorean Theorem. Use this activity if time allows and students need practice calculating the distance between two points in the coordinate plane.

Ask students which figure they think has the longer perimeter to see how the results have changed. For each triangle, select 1–2 students to share their calculations. 

The purpose of this task is for students to think about a general method for finding the distance between two points in the coordinate plane. Students do not need to formalize this into a more traditional representation of the distance formula, but they should be able to communicate precisely about their methods (MP6).

In groups of 4, each student will find the distance between two coordinate pairs and then share how they completed their calculations. The coordinates are carefully chosen so that the distances are all equal. Each pair represents a possible diameter for a circle centered at with radius 5, though students do not need to know this in order to complete their calculations.

Identify students who clearly explain their thinking as they work with their group. Monitor for any groups that discover the points are all on the perimeter of a circle.

The purpose of this discussion is for students to compare their methods for finding the distance between two points. Select 2–3 previously identified students to share how they found the distance between their points.

If any groups figured out that the points lie on a circle, ask them to share how they did so. Then ask students to find the distance between one (or both) of their points and the point using the method they described in the second problem. If students' calculations are correct, they should get a distance of 5 units, which is the radius of the circle.