Re-Examining Reflections

The warm-up prompt for this lesson asks students to review how to construct the perpendicular to a line through a point 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 invite the class to compare the answers on the front board.  This is an opportunity to clear up confusion about this construction, which students will have to perform correctly during the lesson.

Following the warm-up, I display the Agenda and Learning Targets for the lesson.  I tell class that today we will focus on the properties of reflections.

Visiting the Gateway Arch

Displaying the slide, I ask who has visited the Jeffersonian Museum of Westward Expansion in St. Louis, Missouri (the famous Gateway Arch).  We will be using the properties of reflections to answer a few questions about the Arch.

During this part of the lesson, students use construction tools to create the image of a triangle under reflection.  The goal of this hands-on activity is to help students to focus on the properties of reflections.  These details are reinforced in the next activity, in which students apply those properties in a real-world context.

I give instructions with the help of the slide for the activity.  Students will need a straight-edge, compass, and protractor.  I ask students use their notes on constructing perpendiculars from the previous day's lesson and arrange their desks in pinwheels.  They will be working in pairs, using the Rally Coach routine so that they can help each other with the constructions. I use  2 copies of the handout for every team of 3-4 students.

I often let students choose their partners within their cooperative learning teams for paired activities, but for this activity I direct them to work with their shoulder partners.  I want a more-capable student in each pair,

because they will need to follow directions carefully. As students get started on the activity, I circulate around the classroom offering help where needed. I am on  the lookout for: 

• Are students constructing perpendiculars to the line of reflection correctly?  Students will often use the wrong points as the centers when drawing arcs to construct a perpendicular bisector.
• Are students locating the image of each vertex on the perpendiculars correctly?  Instead of measuring the distance from a vertex to the line of reflection along the perpendicular and copying this distance to find the location of the image of the vertex, students will often use construction marks left over from constructing the perpendicular bisector as the location of the image of the vertex.  
• Do students see that each point and its image are the same distance from the line of reflection?  Do they see that this distance is measured along a perpendicular to the line of reflection?  (MP7)  In later lessons, they will use the fact that the line of reflection is the perpendicular bisector of any segment drawn between two corresponding points of the image and pre-image. 

If many students are struggling to understand the directions, I call the class's attention to the front board, where I can demonstrate the first steps of the construction using the document camera.

Once students have completed the construction, I ask them how they expect the sides and angles of the reflected image to compare to the sides and angles of the original.  I ask them to use their compass and a protractor to verify that corresponding sides and angles of the image and pre-image are have equal measures.

In this section, we use guided notes to summarize the properties of reflections.  

I begin by displaying a model sample of student work from the Constructing a Reflection activity on the front board using a document camera and asking the class to help me summarize its properties. 

As students complete the previous activity, they begin the Gateway Arch revisited activity and with the directions I display.  The goal is for students to apply the properties of reflections to answer a few questions about the Gateway Arch in St. Louis.  

Students work with a partner and share their answers within their cooperative learning team of 3-4 students.  As students work, I circulate around the classroom. I am on the lookout for: 

• Are students making drawings/ sketches of the Arch that can be used effectively to answer the questions?  This is not an art project!  Effective drawings should be simple.  Important information should be included: line of reflection, dimensions of the Arch (MP1).
• Do students recognize that each problem involves corresponding objects or points in one leg of the Arch (pre-image) and in its reflected image?  Are they able to use the properties of a reflection to answer the questions (MP4)?

At the end of the activity, I display student work on the front board using a document camera and ask the class to review their answers. 

Here are some additional useful links for this section of the lesson:

•  http://www.jug.net/wt/arch.htm 
•  http://en.wikipedia.org/wiki/Gateway_Arch

Individual Size-Up

The lesson close follows our Individual Size-Up Routine.  The Lesson Close prompt asks students to describe the properties of a reflection.

Homework

For homework, I assign problems #26-28 of Homework Set 2.  Problem #26 gives students an opportunity to review the properties of reflections.  I encourage them to look for the answers in thier notes.  Problem #27 asks students to visualize and draw the image of a polygon under reflection over a line.  Problem #28 gives students the opportunity to apply the properties of reflections to solve a real world problem.