Exploring Rotations 1

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Objective

SWBAT show how to find the rotation image of a figure around a point.

Big Idea

Using trace paper and geometry tools, students rotate a figure by turning its points a certain number of degrees.

Launch

20 minutes

To begin the day's lesson I hand each student a copy of the resource Rotation Task 1 along with a sheet of tracing paper. I make sure students have pencils, a ruler, a compass, and a protractor. 

The instructions read:

1. Place the tracing paper over Rotation 1 and trace the x and y axis using a ruler. Mark the center of rotation with a dot and then remove the tracing paper.

2. On the 3 Reflection Task 1 handout, place the tip of your pencil or compass on the "center of rotation" (in this case the origin) and rotate the whole paper the indicated number of degrees  (90 deg counter-clockwise in case 1) and keep it still. 

3. Place the tracing paper back on the 3 Polygons resource paper and carefully line up the center of rotations. (In this case the axis will also line up because the center of rotation is the origin.)

4. Trace the shape of the Polygon 1. The image of the rotation is now on the tracing paper.  Compare the pre-image and the image. 

I will have my students work independently, yet allow them to speak to their elbow partners for any help. I've previously prepared my Elmo (document camera),  the polygon resource and tracing paper, so someone can come up and show there task paper once completed.

To make sure they all finish the launch task in time, I ask the faster students to help me out aiding any other student that is struggling as I walk around assessing the students' work.  

Once the class is done, I give a few minutes to answer the discussion questions. My students always like to come up to the Elmo to show their work. I ask a volunteer to share their answer to Question 1 with the class. Today, I will be strict that accepted answers must involve measuring with the appropriate tools. Although the preservation of congruence may seem obvious, the distance of points from the center is probably not so obvious. I make sure students measure and provide exact measurements in their answers.

New Info / Note taking

15 minutes

During today's Launch, students should have found that a point and its rotation image are the same distance from the center of rotation. Since rotations are the last of the basic transformations, I make sure students check that their notes are complete. Their notes should contain examples of translations, reflections, and now rotations. I ask students to copy the following notes for this lesson so far: 

  1. Any point and its rotation image are equidistant from their center of rotation. 
  2. A figure and its rotation image are congruent. Therefore, an angle and its rotation have the same measure and a segment and its rotation have the same length.
  3. Rotating an object a positive amount of degrees is a counter-clockwise motion.
  4. Rotating an object a negative amount of degrees is a clockwise motion. 
  5. Rotation symmetry means that the figure coincides with its rotation image. A good example is many car hubcaps. (Lesson image is an example)
  6. Rotations preserve orientation. 

Common Misconception: Students may think that rotations do not preserve orientation because they see that the figure spins and in our launch case, points in another direction. I ask students to label both preimage and image using letters in order and show that the letter order remains the same when naming them in one direction. I always show how this is not true of reflections by actually showing the labeled pre-image (arrow) and its reflection so that the student(s) can see how the letter order is different this time. 

 

 

Activity

25 minutes

The Rotation 1 Activity involves more detail than the Launch task. I never expect every student to go through it with ease. I find that rotations are more difficult for most students than the other transformations we've studied. As students work, I will walk around assessing students' drawings and making sure that they understand the directions. 

Students may struggle with drawing an angle using their protractor. If I observe that students are struggling with this, I sometimes have these students draw a circle with center O through a point A, Then I talk about the images of A through rotations like 10°, 20°,  -90, and so on. 

Another strategy that I use with this lesson is to have students use different color pencils for rotating each vertex point. I find that this helps a lot, especially for small angles where there may be many close or overlapping lines. 

Once students are finished with the Rotation activity, I ask them to complete 

  1. List all the angles that measure 70 degrees below the drawing.
  2. What conclusions can you make about rotations from this activity?

Before completing this section of the class, I add an idea for the students to consider. With a rotation, the angle formed is the absolute value of the amount of rotation. I have students add this observation to their notes. 

Closure

10 minutes

To end the lesson, we will discuss several closure questions. I project the resource Rotations Closure Questions on the SmartBoard and I ask students to answer the questions. As individual students contribute responses, I write their answers on the board.

Certain students know they will be called on because I told them during the Activity that I was going to ask about the very question or concept they were struggling with. (This is something I do often. This helps students focus on the idea, ask if they are still confused, and it is good for their self-confidence when they are able to respond positively at the end of the lesson.)