More Dilations

5 min

Narrative

In this Warm-up, students think about how the size of the scale factor impacts the dilation of a figure. While students may notice and wonder many things about the image, the fact that it shows several dilations of one triangle, all with the same center but different scale factors is the important discussion point.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

In the digital version of the activity, students use an applet to adjust the scale factor of a dilation. The applet allows students to instantly see how increasing or decreasing a scale factor impacts the dilation of a triangle. 

Launch

Arrange students in groups of 2. Display the image for all to see. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder. Give students another minute to discuss their observations and questions.

Student Task

All of the triangles are dilations of Triangle D. What do you notice? What do you wonder?

A triangle D, six images after dilation, point P and three dashed projection rays.
A triangle D, six images after dilation, point P and three dashed projection rays. The dashed rays start at point P on the left, then to the right are increasing in size triangles A, B, C, D, E, F and G. The distance between triangles A, B and C is smaller than the distance between triangles E, F and G.

 

Sample Response

Things students may notice:

  • Triangles A, B, and C are smaller and closer to point PP than Triangle D is.

  • Triangles E, F, and G are larger and farther away from point PP than Triangle D is.

  • The center of dilation is point PP.

  • Corresponding sides of the triangles are parallel to each other.

  • There is a dashed line through each set of corresponding vertices.

  • The corresponding angles appear to be congruent.

Things students may wonder:

  • What scale factors were used? 

  • Are the triangles scaled copies of each other?

Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

Math Community
After the Warm-up, display the Math Community Chart and a list of 2–5 revisions suggested by the class in the previous exercise for all to see. Remind students that norms are agreements that everyone in the class shares responsibility for, so everyone needs to understand and agree to work on upholding the norms. Briefly discuss any revisions and make changes to the “Norms” sections of the chart as the class agrees. Depending on the level of agreement or disagreement, it may not be possible to discuss all suggested revisions at this time. If that happens, plan to discuss the remaining suggestions over the next few lessons.

Tell students that the class now has an initial list of norms or “hopes” for how the classroom math community will work together throughout the school year. This list is just a start, and over the year it will be revised and improved as students in the class learn more about each other and about themselves and math learners.

Standards
Addressing
  • 8.G.A·Understand congruence and similarity using physical models, transparencies, or geometry software.
  • 8.G.A·Understand congruence and similarity using physical models, transparencies, or geometry software.

20 min