Proofs about Perpendicular Bisectors
Lesson 9 of 10
Objective: SWBAT write proofs about points on a perpendicular bisector of a line segment.
For this Do Now, we link back to the previous unit on constructions. I ask the students to draw three line segments of different lengths, any lengths. I don’t specify the lengths so we have many different examples to work with for the Mini-Lesson. They then construct the perpendicular bisector of each segment using a compass and straightedge. Students will use their examples to investigate a theorem about points on the perpendicular bisector of a line segment.
Not specifying the length of the segments gives students some choice in what they do. It also allows students to compare their own results with those of their peers.
Using their examples from the Do Now, students plot three points on each perpendicular bisector. They then draw a line segment connecting the point to each endpoint on the bisected segment. Next, students measure the length of the line segments connecting the points on the bisector to the endpoint of the bisected segment. I then ask students to write a conclusion about what their example illustrates, such as: the lengths of the line segments connecting a point on the perpendicular bisector to the endpoints of the bisected segment are the same. We then reword the conclusion to the theorem, “Points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints” (G.CO.9).
My students should already be familiar with this theorem from investigating it using Geometer’s Sketchpad in a previous lesson. To remind them, I show them a Geometer's Sketchpad Demonstration that animates a point on the perpendicular bisector of a line segment and shows the lengths of the segments connecting the point to the endpoints of the bisected segment are the same no matter where the point is located on the Perpendicular Bisector.
By using Geometer's Sketchpad (see Proofs about Perpendicular Bisectors Mini-Lesson.gsp if you have this tool) and by having students perform their own constructions, students are able to experience this theorem. Constructing the theorem helps students see the connection between previous lessons and this one. The sketchpad presentation helps them to validate their own findings.
In the Proofs Involving Perpendicular Bisectors Activity, students will write paragraph proofs involving perpendicular lines. Students use given statements and diagrams and are asked to prove statements about the figures. They write the proofs independently in their notebooks. After about 10 minutes, students share their responses with a partner and make any necessary corrections. I then call on students to present their proofs to the class.
In the previous lessons, we used more formal methods of writing proofs. Paragraph proofs allow me to see if students understand the concept of proofs without the formality of writing a two column proof. Students can express themselves in a less formal way to demonstrate their understanding.
As a summary activity for this lesson, I have students write about the similarities and differences between paragraph proofs and 2-column proofs and decide which type they prefer. After three minutes, we discuss their answers. This helps students take ownership of their learning by choosing a method they prefer.