2.1: Midpoints in the Coordinate Plane
This activity is intended to supplement Geometry, Chapter 1, Lesson 4.
ID: 8614
Time required: 40 minutes
Topic: Points, Lines & Planes
- Given the coordinates of the ends of a line segment, write the coordinates of its midpoint.
Activity Overview
In this activity, students will explore midpoints in the coordinate plane. Beginning with horizontal or vertical segments, students will show the coordinates of the endpoints and make a conjecture about the coordinates of the midpoint. This conclusion is extended to other segments in the coordinate plane.
Teacher Preparation
- This activity is designed to be used in a high school or middle school geometry classroom.
- The screenshots on pages 1–3 demonstrate expected student results.
- The Coordinate Midpoint formula for the midpoint of \begin{align*}(x_1, y_1)\end{align*} and \begin{align*}(x_2, y_2)\end{align*} is \begin{align*}\left (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2} \right )\end{align*}. This can also be expressed as “The coordinates of the midpoint of a line segment are the averages of the coordinates of the endpoints.”
Classroom Management
- This activity is designed to be student-centered with the teacher acting as a facilitator while students work cooperatively. The student worksheet provides a place for students to record their observations.
- Depending on student skill level, you may wish to use points with integer coordinates, or only positive values.
- Note: The coordinates can display 0, 1, or 2 decimal digits. If 0 digits are displayed, the value shown will round from the actual value. To ensure that a point is actually at an integer value rather than a rounded decimal value, do the following:
- Move the cursor over the coordinate value so it is highlighted.
- Press + to display additional decimal digits or - to hide digits.
- The ideas contained in the following pages are intended to provide a framework as to how the activity will progress.
- To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
Associated Materials
- Student Worksheet: Midpoints in the Coordinate Plane http://www.ck12.org/flexr/chapter/9686
- Cabri Jr. Application
Problem 1 – Midpoints of Horizontal or Vertical Segments
Step 1:
- Students should open a new Cabri Jr. file. If the axes are not currently showing, they should select Hide/Show > Axes.
They will construct a horizontal segment in the first quadrant using the Segment tool.
Step 2:
- Students will select Coord. & Eq. and show the coordinates for the endpoints of the segment.
If the coordinates of the endpoints are not integers, they need to use the Hand tool to drag the endpoints until the coordinates are integers.
Step 3:
- Students should now make a prediction about the coordinates for the midpoint of the segment.
- To check their predictions, students will select Midpoint, construct the midpoint of the segment, and then show its coordinates.
Step 4:
- Before moving on, students need to hide the coordinates of the midpoint with the Hide/Show > Object tool.
- They should use the Hand tool to drag the segment to another location. If you drag the entire segment, it will remain horizontal.
Students can make a prediction about the new coordinates of the midpoint and check their prediction by showing the coordinates of the midpoint.
Step 5:
- Repeat this exploration with a new segment. Use a vertical segment.
If desired, have students explore midpoints of segments whose endpoints do not have integer coordinates, or are not in Quadrant 1.
Problem 2 – Midpoints of Diagonal Segments
Step 1:
- Instruct students to open a new Cabri Jr. file. If needed, select Hide/Show > Axes to show the coordinate axes.
- They should begin by using the Segment tool to construct a diagonal segment in the first quadrant.
Step 2:
- Students need to select Coord. & Eq. and show the coordinates for the endpoints of the segment.
If the coordinates of the endpoints are not integers, they should use the Hand tool to drag the endpoints to make the coordinates integers.
Step 3:
- Students should now make a prediction about the coordinates for the midpoint of the segment.
To check their predictions, students will construct the midpoint of the segment and show the coordinates of the midpoint.
Step 4:
- Students should now hide the coordinates of the midpoint with the Hide/Show > Object tool.
- Using the Hand Tool, students should drag the segment to another location. If the entire segment is selected, it will keep the same diagonal slant.
Students will make a prediction about the new coordinates of the midpoint and check their prediction by showing the coordinates of the midpoint.
Step 5:
- Repeat this exploration with a new segment.
If desired, direct students to again consider segments whose endpoints do not have integer coordinates or are not in Quadrant 1.
Step 6:
- In pairs, students should discuss the following topic:
Describe in words how to find the coordinates of the midpoint of a segment if you know the coordinates of the endpoints. Try to write a formula or a rule for midpoints.