Visualizing Transformations

The warm-up prompt for this lesson asks students to draw the image of a rectangle under reflection over a line that passes through the figure.  The purpose of the warm-up is to help me make the point that knowing the properties of rigid motions is the key to being able to visualize the result of a transformation with accuracy.  Since this sort of reflection is normally outside our experience, most students will find that they are not able to visualize the reflected image correctly.  However, by applying the properties of a reflection to the figure, one vertex at a time, students will see that a little knowledge can help them be successful.

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

After reviewing the sketches each team scribe has written on the front board, I suggest that, for most of us, our 'inner eye' is not very good at predicting what the image of a transformation will look like.  I then use a think-aloud to model the way students can use their knowledge of transformations as an aid to visualization.  

If any of the team solutions are correct, I ask the class to choose between two solutions.  I then think aloud about how to apply the properties of reflections to check the candidates for accuracy. 

Displaying the Agenda and Learning Targets slide for the lesson, I tell the class that today we will practice sketching the results of reflecting, rotating, or translating a figure.  I ask students to get out their guiided notes on the properties of rigid motions before they begin, as they will find them helpful.

Displaying the slide, I ask students to work in pairs during the next activity.  Students will practice drawing the result of a given reflection, rotation, or translation.  This activity uses the Rally Coach format.

I advise students not to rely solely on spacial visualization ability.  I remind them  of some of the tools that are available to them: compass, straight-edge, tracing paper (MP5).

I am on the lookout for:

• Do students understand that when the image of a point A coincides with another point B, then point B is considered to be the image of point A?  (In other words, the two points are not thought of as laying on top of one another.)  For this reason, the image of the first point is labeled B, rather than A' (A prime).  However, I do not prevent students from writing both B and A' next to the vertex in question.
• Are students drawing the image under reflection correctly?  This often happens when the line of reflection intersects the figure, since reflections of this sort are usually outside our experience.  I advise students to remember the properties of reflections and focus on one vertex at a time (MP7).  Draw a perpendicular from one vertex of the pre-image to the line of reflection.  Extend the perpendicular on the opposite side of the line of reflection and plot the image of the vertex the same distance from the line as the pre-image is.  Repeat with the remaining vertices until the image can be visualized.  If students do this faithfully, they will draw the image of the figure accurately.  The ability to visualize the reflected image comes with practice.
• Are students drawing the image under rotation correctly?  Students will often draw the image with the correct orientation but in the wrong location.  Once again, it helps to remember the properties of rotations and focus on one vertex at a time (MP7).  Draw a radius from a vertex of the pre-image to the center of rotation.  Draw a second radius so that it makes the required angle with the first, and plot the image of the vertex the same distance from the center as the pre-image.  Repeat with the remaining vertices until the image can be visualized.

Both types of problems above are helpful in getting students to think analytically about the properties of rigid motions, but they are not transformations that will be used often to show that figures are congruent.  My goal is to highlight several of those "strategic" transformations in the next section.  

During this section, I use student work to highlight transformations that will be useful in proving figures congruent or in describing the symmetry of a figure.  They are useful, because the properties of rigid motions guarantee that key points, lines, rays, or segments will coincide under these transformations.  

I call these transformations "strategic" or "precise", and I highlight them in the course notes.  In the Guided Notes, they are presented in the form of conjectures, and all could be proven from the definitions of rotations, reflections, or translations.  Rather than subject my students to so many tedious proofs, however, I plan to call students' attention to them where they make their appearance in the exercises.  

My goal is to use student work whenever possible.  I expect that students will find these demonstrations convincing, especially when students have left arcs and other construction marks on their work to illustrate the properties of rigid motions that apply. 

Today, I present these "strategic" transformations as useful shortcuts or rules of thumb that students can use to assist them or to confirm their prediction of the result of certain transformations.  I tell students that they are worth remembering, as these situations will come up again.  

My goal is for students to recognize the usefulness of these "shortcuts" and apply them when they complete the homework problems.  In the next lesson, we will review these strategic transformations again when we summarize them in the guided notes.

The lesson close follows our individual size-up routine.  The prompt asks students to sketch the image of an isosceles triangle under rotation.

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.   

Homework

For homework, I assign problems #35-37 of Homework Set 2 for this unit.  Problems #35 and #36 review the properties of transformations which were presented in the lesson. Problem #36 also previews the concept that rotating an angle around its vertex by the measure of the angle carries one side to the other.  Problem #37 reviews the definition of a translation and applies it to a real-world context.