Introducing similarity - dilations and scale factors

Similarity was introduced today. I used the Photocopy Faux Pas investigation package. Here is a run-down of how the lesson unfolded.

First, I told students we were going to look at similarity. Since this was the first look, we would work through an investigation to better understand the concepts.

I held up a copy of the first two pages of the packet (copied 2-sided). I told students not to be concerned with the questions on the back side for now. I asked students to read through the text one time to help establish context. Basically, I wanted to have students read through the text like they would a story. I then told them that they needed to read through the story a second time, highlighting or annotating key words, math ideas and pertinent information.

I passed out a copy to each student and let them read through and annotate. I had to remind students that this was not a group activity at this point and to read through the text twice: one time for context and one time for detail. I walked around to check on student progress and to gauge their level of engagement with annotating the text. Overall, this piece progressed smoothly.

Next I told students to imagine they were the boss. I asked the class to think about why things went wrong and how they could be corrected. With this perspective in mind, I asked the class to complete four of the questions on the back-side of the page (skipping question 2 for now). I told students to answer to the best of their ability given their current understanding.

I had tables share their responses to the fifth question and then we did a class share out. The list included things like initial position, starting size, desired end size, size of paper, etc. Students were clearly thinking about relevant issues.

We then went back to question two. This step required use of a rubber band stretcher. A colleague found instructions for a Two Band Stretcher Activity. I handed out the first page instructions to each table group and then either page 3 or page 4 to every student. I passed out two rubber bands to each student and demonstrated how to tie the rubber bands together. After the second pass through the tying instructions there were only 3 students who didn't have their rubber bands tied. Table mates helped these individuals so everyone was set.

I explained the process of having the knot of the rubber bands trace along the figure. They were not to look at what their pencils were doing. At this point the class started tracing their figures. Some students complained that their rectangle drawing went off the paper. My response was, "Well, what happened in the photocopy situation?" Students realized the same issue arose.

I asked the class to consider what they were seeing and to go back through their responses to the other four questions. I wanted the class to consider whether any of their responses changed based upon the rubber band drawing results.

I then passed out pages 3 and 4 from the Photocopy Faux Pas investigation packet. I briefly connected dilations and scale factors to image manipulation in software packages. I discussed the meaning of scale factor and then had students work on determining scale factors and side lengths.

As the class drew to a close I asked students to think about where the center of dilation was for the photocopier. At this point, a student asked what impact having a longer rubber band would have on the image drawing. It was a good question that was asked right when class ended. I asked students to think about how they would respond to that question as well.

Tomorrow we'll address those questions, discussion their ideas on the location of centers of dilations described in the problems and then move into the next investigation piece.

A colleague said that her students were still confused about the idea of doubling an image size. Did this mean doubling side lengths of did it mean doubling area. It will be interesting to see how my class thinks about the idea of doubling as we move forward.