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.

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

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.

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.

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.

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.

**Homework**

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.