Dilations on the Coordinate Plane, Center (h, k)
Lesson 4 of 10
Objective: SWBAT describe and perform dilations where the center is any point
In the Do Now, students are instructed to draw a grid, plot points, and then dilate the triangle by a given scale factor about the origin. Students can multiply the scale factor by the coordinates in order to perform the dilation. They will build on their dilations from the Do Now in the Mini-Lesson. On the second page of the presentation is an example of how the dilation should look. I project the example and have students check their work. The task is challenging for students, but I expect that most will complete the task successfully and produce a graph like this Student Work Example.
Since we are moving from Sketchpad to pencil and paper today, to begin the Mini-Lesson I recap yesterday's lesson to help students make the necessary conenctions. I will say something like, "We used Geometer's Sketchpad to investigate dilations about centers other than the origin (G.SRT.1). Today we are going to apply what we learned with pencil and paper."
Although my students were first introduced to dilations in the eighth grade, they only performed dilations about the origin. In this lesson, students dilate triangles about centers other than the origin by hand. We use the pre-image from the Do Now and dilate it by the same scale factor about a different center. I plan to show students two different procedures.
- I show students the process for performing this dilation. After they complete the dilation, they identify the transformation that maps the first image to the second image.
- I demonstrate the procedure for finding the center of a dilation given a pre-image and an image. I show the students an example and then they work on a guided practice on their own. They also practice dilating an image given a scale factor and center.
At the end of the Mini-Lesson, I plan to have two students present their dilations. I answer any questions and address any misconceptions they may have.
My students often have more difficulty finding an image given a pre-image, center and scale factor than finding the scale factor and center given a pre-image and an image (MP1). The difficulty lies when students try to find the scale factor of diagonal line. I review the method of for finding the scale factor. They draw lines connecting the center to through the vertices of the pre-image. They can count the the vertical and horizontal distance between the center and each vertex and then multiply it by the scale factor in order to find the coordinate of the image.
For today's Learning Activity, students work independently to practice dilations about various centers on the coordinate plane. Rather than have each of the students complete every problem, I have the students choose two from part A and one from part B. They can work on the same problem as students at their table or different ones. As they work, I circulate around the room and check to see their dilations are accurate (MP6). Sometimes the image and the pre-image are not in proportion. I have students re-check each of their sides.
After completing their problems in parts A and B, they work on part C. If they do not have time to finish part C, I will ask them to finish this work for homework. At the end of the activity, we go over the dilations. I will call on different students to show their dilations on the document projector.
As an Exit Ticket, students sum up the lesson by describing the transformation that maps one image to a second image. I then have them explain how the procedure differs for finding the image of an object that is dilated about the origin and the image of an object that is dilated about a different center. When I read their responses, I look to see if they understand that they can multiply the coordinates of the pre-image by the scale factor when the center of the dilation is the origin, but a different method when dilating an object about a center which is not the origin.