In this activity, students continue studying dilations on a circular grid by focusing on what happens to points on a polygon. They discover that just as the image of a grid circle is another circle, the dilation of a polygon is another polygon, and the dilated polygon is a scaled copy of the original (MP7). These important properties of dilations are not apparent in the definition.

In the digital version of the activity, students use an applet to visualize the dilation of a polygon. The applet allows students to dilate each vertex by the provided scale factor on a circular grid.

The goal of this discussion is to make observations about a figure and its image under dilation. Begin by displaying this figure. 

Ask students to share what they notice about the new polygon. Encourage students to verify their responses using geometry tools like tracing paper, a ruler, or a protractor. Here are some questions for discussion:

• “What do you notice about corresponding sides, such as segment and segment ?” (They look parallel. The side of the new polygon is twice the length of the original.)

• “How can we verify this?” (We can use a ruler to measure each side and the sides of the new figure should be twice the length of the original figure.)

• “Is our new polygon a scaled copy of the original? (Yes, because all the sides of the new polygon are twice the length of the original polygon.)

• “What do you notice about corresponding angles?” (They look the same.)

• “How can we verify this?” (We can measure them with a protractor.) 

• “What happened when we dilated additional points that were on the sides of the original polygon?” (They landed on the side of the dilated polygon.)

In this activity, students investigate dilations with no grid. Students make sense of a new problem as they perform dilations of points, determine scale factors, and find centers of dilation (MP1). 

After doing a few of the problems, students may notice that the requested point is always one of the labeled points and then use this observation to expedite the work.

Some students may think that for a point to be a dilation of itself, the scale factor is 0. Prompt them to consider multiplying the distance by 0. If they want the distance to be the same, they need to multiply by 1 instead.

The goal of this discussion is to make sure students understand how the location of the center of dilation and the scale factor work together, and to prepare students to draw their own dilations without a grid. Begin by discussing any strategies used to solve the problems. Ask students who noticed that all of the answers were labeled points to share how their observation helped them answer the questions. Then discuss with students: 

• “What do you notice about the scale factors for dilating to and dilating back to in the second and third questions?” (They both used as the center of dilation and the scale factors are reciprocals of each other.)

• “What do you notice about the center of dilation, the point being dilated, and the image of the point after dilation?” (All 3 of those points lie on the same line.)

• “What scale factor results in an image that doesn’t move?” (A scale factor of 1 does not move any points.) 

Show students 2 points and and explain that we want to dilate point using as the center and a scale factor of 3. Ask students to explain how to perform the dilation and if time allows, have a student demonstrate the process.

Be sure students understand that the image of point will lie on the same line as points and and be located 3 times as far from as . 

In this activity, students find an appropriate way to take measurements in order to perform a dilation (MP5), most likely by using a ruler or the edge of an index card. Different students will work with different scale factors to produce perspective drawings of a box. 

Monitor for students who draw these different representations:

• An accurate diagram using a scale factor greater than 1

• An accurate diagram using a scale factor less than 1

The goal of this discussion is for students to connect that scale factors less than 1 result in an image that is smaller and closer to the center of dilation while scale factors greater than 1 result in an image that is larger and farther away from the center of dilation.

Display 2–3 perspective drawings that all have a scale factor less than 1 from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the representations. Here are some questions for discussion:

• “What do the dilations have in common?” (The images all got smaller. The images are all closer to the center of dilation.)

• “How are they different?” (The center of dilation is in a different location. They have different scale factors.)

• “How does the scale factor show up in each dilation?” (The scale factor determines how much smaller the dilation will be.)

Then display 2–3 perspective drawings that all have a scale factor greater than 1 from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the representations. Here are some questions for discussion:

• “What do these dilations have in common?” (The images all got bigger. The images are all farther away from the center of dilation.)

• “How are these dilations different from the first set?” (These dilations all got bigger while the other dilations all got smaller.)

• “How does the scale factor show up in each dilation?” (The scale factor determines how much larger or smaller the dilation will be.)