The Shortest Segment

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SWBAT construct the perpendicular to a line through a point not on the line. Students will understand how the properties of a perpendicular are related to the properties of bisectors.

Big Idea

Students use the symmetry of a geometric model to reason that the shortest distance between a point and a line is a... bisector.

Lesson Open

9 minutes

I begin the lesson by displaying the warm-up prompt using the slide show for the lesson.

The prompt asks students to describe the symmetry in a set of non-concentric circles. The purpose of the warm-up is to help ensure that students see the symmetry in the concentric circles they construct as part of the tire swing problem.  It is also good practice in a skill that they learned several lessons ago.  

The warm-up follows our Team Warm-up routine.  I choose students at random to write the team's answer on the board. 

Reviewing the teams' answers on the board, I ask where the centers of the circles all have to be.  On the line of reflection.

I display the Agenda and Learning Targets for the lesson.  I tell the class that today we will be using the symmetry of a figure to solve an interesting problem.  At least, I hope it is interesting.  Has anyone ever built a tire swing?

Without further ado, I introduce the next activity: a pair of problems.


The Tire Swing and the Tent Fire

30 minutes

The purpose of this activity is to dramatize the construction of a perpendicular bisector with a real-world problem (MP4).  As much as possible, I want students to reason for themselves that the shortest segment between a point and a line--which is what a perpendicular is, of course--can be found by constructing a circle that intersects the line at two points first, and then constructing the bisector of those points (MP1).

To lead students to this idea the first problem (The Tire Swing) is designed to entice students to construct a pair of circles that have too great a radius, and one that is too small.  From the symmetry of the figure, I hope that they will think of constructing a bisector to find the point midway between the points of intersection (MP7)

As students go to work, display the instructions and circulate to answer questions or give out hints where necessary.  If students find a different method of solution, I give praise while asking them to see the reasoning behind the solution I have in mind.  

Students each complete their own solution, but I encourage them to work together and share their thinking.

The second problem (A Campfire Story) can be used as a check for understanding.  It also gives students a second opportunity to solve the problem and see the symmetry in the construction of the perpendicular to a line.

Note that the handout for the activity is meant to be reproduced on 11" x 17" legal-size paper.

If students finish early, I give them a problem from the Bisector Challenge activity.

Summarizing Perpendiculars

15 minutes

We use the Guided Notes to summarize the steps of the construction of a perpendicular.  Which, of course, the name of the bisector that students have just learned to construct for themselves.  I demonstrate the steps of the construction, and students follow along to create the diagrams in their notes.

For more on how I use Guided Notes in my lessons, see the article in my Strategies folder.



Lesson Close and Homework

5 minutes

Individual Size-Up

The lesson close follows our Individual Size-Up routine.  The prompt asks students to name the shortest path between a point and a line.


For  homework, I assign problems #18-20 of Homework Set 1 for this unit.  Since there is no time in this lesson to practice the construction of a perpendicular or fully explore its properties, the homework problems review and extend concepts and skills taught in earlier lessons.