In today's Do Now, I ask my students to use coordinates to work out transformations. First, they are given the coordinates of a pre-image, and, a scale factor. They are asked to find the image after a dilation centered at the origin. Next, they are given the coordinates of a center that is not the origin and the coordinates of an image after a dilation about that center. I ask my students to figure out the translation between the two images.
For this warmup the information is organized in a chart. We will utilize this chart during the Mini-Lesson as well. In the Mini-Lesson I will show my students how to work out a rule for dilating an object about a center that is not the origin (G.SRT.1).
At the beginning of the Mini-Lesson, I have my students analyze their Do_Now_Table to look for patterns (MP8). After two minutes of think time, I have the students share their what they notice with the person next to them. I then call on students to share their conversations with the rest of the class (MP3).
Many of my students can see there is a relationship between the translations and the center. The translation vector is a multiple of the center. I ask students to identify the multiple. This prompt usually enables them to make observations about how the multiple relates to the scale factor. The multiple is one less than the scale factor.
Using our discussion, I call on a student to generalize the rule. The rule we generate is usually something like:
When we dilate a figure about a center that is not the origin, we multiply the coordinates of the pre-image by the scale factor and then subtract the product of the x-coordinate of the center and one less than the scale factor. Then, we repeat the process for the y-coordinate.
Next, students will use our rule to figure out the coordinates of an image after a dilation about a point that is not the origin.
The goal of today's Activity is for students to perform a dilation about a center that is not the origin, without graphing the pre-image. Using what we have learned so far today, I want my students to try to apply their knowledge of transformations to work out the correct coordinates of the image.
For the first task, I ask students to dilate a triangle about the origin. Students can multiply the coordinates of the pre-image by the scale factor. Next, they figure out the coordinates of the image after a dilation about the point (-1, 6) by calculating the vector that maps the first image to the second image. Then they verify their work by graphing the coordinates of the pre-image and the image. Students compare their dilations with a partner to ensure they are correct.
At the end of this lesson, I will have my students take a Quiz. I use this quiz to assess my students' understanding of dilations. Dilations is one of those topics that leads to unpredictable results on assessments. I will review the work carefully and consider next steps.