Reasoning About Rigid Motions

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SWBAT describe or sketch the result of translating a line or rotating a line by 180 degrees. Students will understand properties of rigid motions in the context of parallel lines.

Big Idea

Students learn how to use reason as well as experience to understand the result of transforming a figure. Deductive and inductive logic work hand-in-hand!

Lesson Open

8 minutes

The warm-up prompt for this lesson asks students to sketch the result of rotating a line by an angle of 180 degrees around a point that is not on the line.  The warm-up follows our Team Warm-up routine.  I choose students at random to write the team's answer on the board.  

I ask the class to look at the drawings each team has made on the board.  Who thinks that the image of the line under rotation would be parallel to the original line?  Can we be sure?  

I display the agenda and learning targets for the lesson.  I tell the class that today we will be looking closely at how lines behave when they are rotated or translated.  We will prove a pair of theorems which will help us to visualize and describe the results of transformations more accurately. More importantly, they will allow us to use transformations rigorously in proofs.

Examining the Transformations of a Line

20 minutes

Using guided notes we summarize the properties of rigid motions.  This includes completing conjectures, which I have named the Rigid Motions Postulate, the Line Rotation Theorem, and the Line Translation Theorem, with examples to illustrate the meaning of each one. I prepare carefully for this section of the lesson (see Video Narrative). 

Proving the Line Rotation Theorem

I use an animated Slide Show as I demonstrate an indirect proof of this theorem.  I make sure that I review the animations and practice the proof before class.  Indirect proofs can be hard enough to follow, so I want my demonstration to flow.  I try to make my exposition stand alone (with the help of the drawings).  I do not read from the slides, but bring up a written summary of each main points of the proof after I have presented it orally.

After proving that any point P' on the image of the rotated line cannot also be on the original line (pre-image), I point out that the same argument can be applied to any point on the image of the line.  Therefore, there is no point which exists on both the image of a line and the original line, which means that the line and its image are parallel.

I ask students if their heads hurt after paying attention to this proof.  If they are, it is a good sign that they understand the proof.

Proving the Line Translation Theorem

I introduce this property by asking students how they go about sketching the translation of a figure. (They completed such an exercise in the homework that was assigned following the last lesson.)  Do students know that they can sketch the translated image of a polygon one side at a time, by sketching the image of each line parallel to the original?  Some students will have noticed this property of the translation of a line.  Now we will prove it.

I again use animated slides as I demonstrate an indirect proof of this theorem (see Summarizing Properties of Rigid Motions Slide Show).  

I ask students to look for examples of these properties at work as we review the homework. They will have a chance to apply them in tonight's homework.

Homework Review

25 minutes

We use the Team Homework Review variation of our homework review procedure.

Most student questions will probably be about how to visualize the result of transforming a figure.  I use student questions as opportunities to point out how students can use their knowledge of the properties of transformations to help out their 'inner eye'.

If there is time remaining, students can get started on their new homework assignment.

Lesson Close and Homework

5 minutes

The lesson close follows our individual size-up routine.  The prompt asks students to describe the result of rotating a line by an angle of 180 degrees, which we have just summarized. 

Recognizing Good Work

While the class is completing the lesson close activity, I invite a student from each team to assign his or her team a score for the lesson.  Student scorekeepers write the score in a spot on the front board, and I write the scores I assign to each team next to them. 

You can read more about how I assign Team Points for cooperative learning activities in my Strategies folder.  


For  homework, I assign problems #1-2 of Homework Set 2 for this unit.  Problem #1 asks students to review the Line Translation Theorem and Line Rotation Theorem, while problem #2 previews the transformations which will be used to prove conjectures about angles in the next lesson.
I also hand out the Portfolio Problems for the unit: students should begin on portfolio problem #1.