MATH 113: Practice With Symmetries


This worksheet guides you through creating objects with reflection symmetry and exploring the relationship between reflective and rotational transformations.

For best results, the figures you draw should be simple and should not have any mirror or rotational symmetry. The letter F is a good example of such a figure.


Work with a partner or two on the following activity.

Reflection

Draw a line down the middle of a sheet of paper.

Draw half of a figure touching that line, then give the paper to your partner.

Complete the half figure you receive from your partner to create a figure with mirror symmetry. In other words, reflect the figure your partner drew across the line.

Rotation

Draw a line down the middle of a sheet of paper. Put a dot on the line near the middle of the sheet of paper.

Draw a figure to one side of the line near the dot, then give the paper to your partner.

Copy the figure you receive from your partner to create an image with 180 degree rotational symmetry. In other words, rotate the figure 180 degrees about the dot on the line. Check your work by putting your pencil point at the dot on the line and rotating the paper 180 degrees around that point.


More Practice

Draw a small, complete figure on one side of a piece of paper, then give the paper to your partner.

On the sheet of paper you receive from your partner, choose a line of mirror symmetry and then draw the mirror image of the figure they drew.

Draw a small, complete figure on a piece of paper, then give the paper to your partner.

On the sheet of paper you receive from your partner, choose a point a short distance from the figure they drew. Draw the result of rotating their figure by 90 degrees about the point you chose. Check your work by putting your pencil point at the center of rotational symmetry you chose and rotating the paper 90 degrees around that point.


Combining Reflections

Draw two lines on a sheet of paper, close to the middle of the page.

Draw a small figure between the two lines. If the two lines intersect, draw your figure near the point of intersection.

Give your paper to a partner.

On the piece of paper you receive, mark one of the lines with the number 1.

Draw the reflection of the figure across line 1 and mark the reflection with the number 1.

Give the paper to a partner.

On the piece of paper you receive, draw the reflection of the figure marked 1 across the unmarked line. Mark the new figure with the number 2.

What relationship do you see between the original, unmarked figure and figure number 2?


Bonus:

We've studied rotation, reflection and the identity transformation. There are two other fundamental types of transformation. Draw a small figure on a sheet of paper and then draw another copy of the figure on the same sheet so that it is not related to the first by rotation, reflection or the identity transformation.