Repeated Reflections of an "R"

Dr. Susan Addington
California Math Show

http://www.math.csusb.edu/faculty/susan/susan.html
formatted and edited by Suzanne Alejandre

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Objective: [NCTM Standards Grades 6-8: Geometry]

To draw a design with reflectional symmetry and rotational symmetry.


Materials:

Version 1
prepared "R" sheets with two lines and an "R" already drawn on them
pencils
optional: Mira (manipulative)

OR

Version 2

blank paper
protractors
pencils
optional: Mira (manipulative)


Procedure:

Version 1: - If using blank paper, direct students to follow the examples shown below
  1. Draw a horizontal line on the paper.
  2. Using a protractor, construct a line that intersects the first line at a 60-degree angle.
  3. Draw an "R" in the first quadrant.

    Continue directing the students using steps 1 through 5 shown below.

Version 2:

    If the teacher has already prepared "R" sheets, direct students to follow steps 1 through 5 as shown below.

Optional:

    If using the Mira manipulatives, before the students draw their R's, have them view the reflections using the Mira. Rather than darkening, folding and rubbing, using the Mira the students can easily trace the reflected "Rs".


Variations:

  1. This procedure can also be done with lines that intersect to form other angles. You can use lines that form angles of 45 degrees or 30 degrees or 90 degrees or parallel lines or ....

  2. An "R" was chosen for this exercise because it is not itself symmetric. What other letters could you use?


Evaluation:

View the drawing.
Use the questions for evaluation or discussion.
Ask the students to vary the procedure.
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Directions for students:

1. To make a reflection of a shape across a line, draw the shape in pencil and go over it several times so it is very dark. Fold on Line 1 with the shape on the inside of the fold.

Rub the outside of the paper with the pencil. The shape you drew should rub off onto the paper on the other side of the fold. Open the paper and darken the print.



2. Fold on Line 2 with the shape on the inside of the fold. Again rub the outside of the paper with the pencil. The shape you drew should rub off onto the paper on the other side of the fold.

Open the paper and darken the print. How are the R's related - translation, rotation, reflection, or glide reflection?

Answer: Rotation by 120 degrees.




3. Fold on Line 1 with the shape on the inside of the fold. Again rub the outside of the paper with the pencil. The shape you drew should rub off onto the paper on the other side of the fold.

Open the paper and darken the print.





4. Fold on Line 2 with the shape on the inside of the fold. Again rub the outside of the paper with the pencil. The shape you drew should rub off onto the paper on the other side of the fold.

Open the paper and darken the print.





5. Fold on Line 1 with the shape on the inside of the fold. Again rub the outside of the paper with the pencil. The shape you drew should rub off onto the paper on the other side of the fold.

Open the paper and darken the print.





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Questions to pose to students:

  1. What kinds of symmetry does your final design have?
  2. Where are the lines of reflection?
  3. Rotations are by how many degrees?

Answers:

  1. Reflections and rotations.
  2. Three lines
  3. Rotations by 0 degrees, 120 degrees and 240 degrees.


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