Rotations and Reflections and Translations, Oh My!

I must confess, I had not considered rotations, reflections, and translations of congruent figures until I settled in with a big cup of tea and looked into the CCRS.  I realized that we need to move away from defining congruency as “same shape, same size”, to ‘two figures in the plane are congruent if there exists a finite composition of basic rigid motions that maps one figure onto the other figure.’  I, and my fellow adult education math instructors, need students to understand that Geometry exists as an axiomatic system.  It’s a cool math word that means ‘the establishment of a new fact comes strictly from basic assumptions of existing facts.’

So, how do I teach this in my ABE classroom?  I stumbled across a very thorough and professional website, Engage^NY, that really helped me wrap my head around math and the common core.  I'll share a few ideas from the website that you can use in your ABE math classroom.

Activity

Draw the following on a piece of paper:

• A line
• A ray
• A segment
• A point
• An angle
• A curved figure
• A simple drawing of your choice

Then draw a vector AB, length and direction is your choice.  Now trace everything you have just drawn onto a transparency.  Demonstrate on the document camera how to translate along the vector to show the exact direction of the vector and the exact length of the segment represented by the vector.  Ask:  Did the line change in shape or size? The ray? The segment? The point? The angle? The curved figure? Your simple drawing? In each case, no.  Translation is movement without real change; it's only a location change.  If you have access to computers, you can go to The National Library of Virtual Manipulatives to virtually translate, reflect, and rotate shapes (see images below).

You have just verified some basic properties of translation.

A translation:

• Maps lines to lines, rays to rays, segments to segments, and angles to angles.
• Preserves lengths of segments.
• Preserves angles, measures of angles.

 Take out your paper and transparency again.  This time, reflect each of the images you drew by flipping your transparency across the line you drew.  Notice that the figures did not move randomly, they moved to the opposite side of the line of reflection.  They are the exact distance from the line of reflection as they were before, just on the other side.  Ask:  Did the line change in shape or size? The Ray? The segment? The point? The angle? The curved figure? Your simple drawing? In each case, no.  Reflection is like translation in that they are both a rigid movement without real change, just location change.

Now you have just verified some basic properties of reflection.

A reflection:

• Maps lines to lines, rays to rays, segments to segments, and angles to angles.
• Preserves lengths of segments.
• Preserves angles, measures of angles.
• When you connect a point and its reflected image, the segment is perpendicular to the line of reflection (see the white line in the image above).

There is one more activity.  Take out your paper and transparency.  This time, rotate each of the images you drew by placing your finger on top of the point you drew and carefully rotate your transparency in one direction and then the other.  Again, these figures did not move randomly, they moved in a circle around the center of rotation. Did the line change in shape or size? The ray? The segment, etc.? Nope.

We have just verified that a rotation:

• Maps lines to lines, rays to rays, segments to segments, and angles to angles.
• Preserves lengths of segments.
• Preserves angles, measures of angles.

To increase the difficulty, you can begin sequencing the motion.  You can begin with two translations, then, once reflection is learned, students sequence two reflections followed by a translation and a rotation, or a rotation and a reflection, etc.

Students have just learned that congruence is defined in terms of a sequence of rigid motions, performed using a transparency that shows the mapping of one figure onto another.

Now, as an adult education math instructor, I look at this and wonder how I can extend this activity by applying it to my adult students’ real-life world.  Album covers came to mind.  I did a Google image search of album covers and came up with the images below. 

The students’ task is to:  analyze the geometric transformations that were used to create the designs on the covers. In other words, describe the images or designs that you see and then describe what transformations were used to move those images or designs to other locations.

Let us know if you come up with another clever idea by commenting on our blog.