Reconsidering Rotations

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Objective

SWBAT accurately describe or draw the result of rotating a figure. Students will understand a precise definition of rotation in terms of arcs and angles.

Big Idea

Students learn how rotations work by constructing one--then use their insights to deduce the location of a missing Mars rover.

Lesson Open

5 minutes

The warm-up prompt for this lesson asks students to recall the meaning of a rotation. Following our Team Warm-up routine, I ask students to discuss the question in their teams and agree upon the best answer they can come up with.  Then, 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.  I offer praise for any that show an attempt to be precise or complete in defining the term.  

Following the warm-up, I display the Learning Goals and Agenda for the lesson.  I tell the class that today we will begin a unit on congruence and rigid motions.  Students are already familiar with both concepts.   The lessons in this unit will both deepen and expand their understanding.  This is the beginning of a unit-long conversation in which students are asked to open their minds to new, more rigorous understandings about rigid motions and congruence.  At the end of today's lesson, students should not only have a more precise understanding of the meaning of a rotation, but they will learn how they can use the properties of rotations to solve a real-world problem.

 

Constructing a Rotation

25 minutes

During this part of the lesson, students use construction tools to create the image of a triangle under rotation.  The goal of this hands-on activity is to help students to understand the properties of rotations better by getting them to focus on the details.  In the next activity, they will see how the properties of rotations can be applied to plot the location of the missing Mars rover on the map.  

I display this slide while giving instructions.  Students will be working in pairs using the Rally Coach format so that they can help each other with the constructions.  They will need to put their desks in pinwheels, and they will need a straight-edge and compass.  I distribute the handout for this activity, 2 copies 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 high-performing student in each pair, because they will need to follow directions carefully. 

As students get started, I circulate around the classroom offering help where needed. If many students are struggling to understand the directions for this activity, I call the class's attention to the front board, where I can demonstrate the first steps of the construction using the document camera. As students work I am on the lookout for:

  • Are students copying the angle correctly?  Students will often forget to use the compass to measure the width of the angle between intersections.  Instead, they will use the same compass setting they used to draw the arcs from each vertex and end up with a 60 degree angle.
  • Are students labeling key intersections as they go?  There is enough going on in this construction that they will often confuse which rays go with which angles. 
  • Do students see that each point and its image are the same distance from the center of rotation?  Do they see that each point and its image are on the two sides of an angle, and that the measure of each such angle is the same?  In later lessons, the students will visualize an angle whose sides pass through corresponding points of two figures to determine the angle by which the first figure must be rotated so that it is carried onto the second. 

Once students have completed the construction, I ask them how they expect the sides and angles of the rotated image to compare to the sides and angles of the original.  I ask them to use their compass or tracing paper to verify that corresponding sides and angles of the image and pre-image are congruent.  They will probably find that corresponding angles are only approximately congruent, because most students find it hard to construct an accurate copy of an angle (MP6).  

It is okay if not every pair of students completes this part of the activity.  (Most will take it on faith that the dimensions of the image and pre-image are the same, anyway.) Understanding the structure of the rotation is the focus. At the end of the allotted time, I display student work so that all students can see the finished construction.  I point out the properties of the rotation and that the lengths and angles of the original (pre-image) are preserved in the rotated image.

If students need more than 25 minutes  to complete the construction, I weigh the benefit of continuing the activity.  Ideally, every student will have enough time to construct at least two of the three vertices of the triangle, at least.  At that point, they can get what they need by seeing the completed work of other students, which I display using a document camera.

 

Summarizing Properties of Rotations

10 minutes

In this section, we use Guided Notes to summarize the properties of rotations.  

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

Rover Found

5 minutes

Showing the slide for this section on the front whiteboard, I ask the class how the properties of rotations could be used to discover the location of the lost Mars rover, which students encountered in Portfolio Problem 2 for the unit.

If they need another prompt, I remind them that, in a rotation, every point of the pre-image is rotated in the same manner. How could this be applied to the route of the rover?  I am looking for a student to suggest that we don't have to rotate the entire route of the robot, just the point where the robot would be, had it followed the planned route.  I demonstrate this with the help of a large protractor (the kind used to demonstrate on a chalkboard--to measure the angle) and a compass or piece of string (to keep the distance from the center constant).  I mark the robot's actual location.

Then, I rotate the entire route with the help of an animation in the slide. 

Next I ask about the second part of the problem.  How should the programmer modify the rover's navigation program to bring it safely home.  I ask students what happened to the starting and finishing point of the rover's route when the route was rotated.  Students see that the point does not change, since it is the center of rotation.  

Now all students must do to complete the problem is to write an explanation in their own words.

Lesson Close and Homework

3 minutes

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 #29-31 of Homework Set 2.  Problem #29 gives students an opportunity to review the properties of rotations.  I encourage them to look for the answers in thier notes.  Problem #30 asks students to visualize and draw the image of a polygon under rotation. Problem #31 reviews the constructions of perpendiculars and parallels, which students will need in the next lesson when they construct the image of a polygon under translation.