Mapping an Image onto Itself

12 Sep


Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

We are incorporating this learning objective into our unit on Rigid Motions. And we are still learning how best to ask questions to get students thinking about how to carry an image onto itself.

We started with a rectangle and asked students to perform a reflection that would map the image onto itself.

A few students tried to “eyeball” where they could draw a line of reflection. But I was excited at how many used the Construction tools to construct a line of reflection.

Some chose to construct the perpendicular bisector of a side of the rectangle. Some constructed the midpoints of more than one side of the rectangle and then used the segment tool to connect. Some constructed the midpoint of one side of the rectangle and then constructed a perpendicular to the side through its midpoint.

One nice feature of the TI-Nspire CX is color. I made sure that the rectangle had a colored border so that when a student reflects the image onto itself, the result will default back to a black border, instead of the blue border of pre-image. For those without CX models, students can hover over the border of the rectangle and see that the reflection worked when they see the object selection “polygon” on top of the rectangle.

After reflecting the rectangle onto itself, we grabbed movable vertices of the rectangle to make sure that the reflection “stuck”. Those students who drew instead of constructed a line of symmetry for the rectangle soon saw two rectangles instead of only the one. So they undid their work and tried again constructing a line of symmetry.

I gave students a hexagon next…This time most didn’t have to actually construct any lines of symmetry to come up with a solution.

So how could we take this to the next level? Students know that reflecting an image about a line of symmetry will carry the image onto itself.

In the next lesson, we asked students to create an image that would map onto itself when reflected about a given line. We started with the line y=x.

I was surprised at how many students thought to create a circle with its center on the line. I would not have thought of a circle myself since we have mostly been reflecting, translating, and rotating polygons.

Others created triangles, regular pentagons, regular hexagons, squares. We got to see everyone’s work using the Class Capture feature of TI-Nspire Navigator. And we got to talk about the properties of the shapes (will any triangle work?) and the coordinates of the vertices.

Rotations are next…We will see what happens when we begin rotating an image onto itself. Or creating an image that can be rotated onto itself.

And so the journey continues….

1 Comment

Posted by on September 12, 2012 in Geometry, Rigid Motions


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One response to “Mapping an Image onto Itself

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