Drawing dilations using a compass

Today we looked at the ideas connected with drawing dilations and finding centers of dilations. Class started by checking on responses to the Photocopy Faux Pas questions. Students seemed comfortable with identifying scale factors and finding corresponding side lengths.

One issue that needed clarification was the idea that a scale factor is actually a numeric value. This came about on the first problem in which the scale factor used was 2. When asking the class the scale factor, students replied that the scale factor doubled. While this was the result of using a scale factor of 2, double was not the scale factor.

Another issue that presented itself was that students did not understand the idea of center of a dilation. The scenarios presented in the last set of questions baffled students. The common thinking was the center of dilation was between the pre-image and image figures. I didn't press the issue but knew that we needed more work to build the understanding of center of a dilation.

I also checked their understanding about the issue of lengthening the rubber band chain that was used in the last class. Some thought the image would get larger but there was little confidence in the responses.

The next investigation I was planning on using was Triangle Dilations. Since this investigation required students to construct dilations and know centers of dilation, I didn't want to tackle this investigation just yet. First, I wanted students to learn how to construct dilations using a compass and practice a bit more with scale factors and work with centers of dilation.

A colleague had used the Drawing Dilations exploration from the On Core Mathematics curriculum. This exploration explains how to use a compass to draw dilations and provides work with using dilations in a coordinate plane. The dilation constructions require the construction of segment bisectors, which is a nice tie-in to previous geometric construction work. By the end of the exploration and work, students have a much better understanding of the center of a dilation.

This work took the entire class. Students had some basic questions and some needed reminding about segment bisector construction. The most difficulty occurred with dilations in the coordinate plane. Students repeatedly wanted to add the scale factor rather than multiply the scale factor. I tried to relate the idea of a scale factor to a continuous expansion or contraction. This seemed to help.

By the end of class students were feeling very comfortable with the compass constructions and understood how to apply scale factors in the coordinate plane. More importantly, they also had a better understanding about the center of dilations and what would happen with using longer rubber band chains to draw dilations.

When I asked the class what would happen if we used a longer rubber band chain, the class confidently stated the image would become larger. When I asked the class where the center of dilation is for the photocopy scenario, students recognized that the upper left corner was the center of the dilation. With these ideas solidified, I will proceed to the Triangle Dilations investigation next class.