The term "dilation" is formally defined in this activity. Students measure the distance between the center of a circular grid and corresponding points on two circles. They observe that the distances to the smaller circle are all multiplied by the same scale factor to get the distance to the larger circle. Students need to explain their reasoning when finding the scale factor (MP3). 

In the digital version of the activity, students use an applet to find the scale factor between a smaller circle and a larger circle. The applet allows students to practice using the digital tools to visualize both circles and the scale factor. The digital version may be helpful as a tutorial for the digital tools. If students don't have individual access, displaying the applet for all to see would be helpful in the Launch.

The purpose of this discussion is to define dilation. Begin by asking students to share their answers to the last question and what they noticed about the values in the table. Explain to students that this is an example of a dilation and share the following definition:

A dilation is a transformation in which each point on a figure moves along a line and changes its distance from a fixed point, called the center of dilation. The scale factor determines how far each point moves.

In this activity, the larger circle (K) is a dilation of the smaller circle (J). Point is the center of dilation, and the scale factor is 3 because every point on Circle K is 3 times as far from point than every point on Circle J.

Here are some questions for discussion:

• “What would happen if Circle J were dilated with point as the center of dilation and a scale factor of 2?” (The result would be a circle with twice the radius and the same center.)

• "What would happen if Circle J were dilated with point as the center of dilation and a scale factor of ?" (The result would be a circle with the radius and the same center.) 

• “What would happen if Circle J were dilated about point with a scale factor of 1?” (The result would be a circle with the same radius and the same center.)

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.)

The purpose of this optional activity is to work on dilations of polygons on a circular grid but without the radial lines from the center. Students will need to use a ruler or other straightedge to connect each point to the center of the circular grid. If time allows, they can experiment with dilating points other than the vertices and check that the dilation of a side of the polygon is still a line segment (though there may be small deviations due to measurement error).

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

The purpose of this discussion is to highlight the need to draw in the rays going through point and each of the vertices of the polygon, and the fact that a scale factor of resulted in an image that was smaller than the original figure. Here are some questions for discussion:

• “What effect did a scale factor of have on the dilation of polygon ?” (The resulting figure is smaller than the original and located closer to the center of dilation.)

• “What scale factor would result in an image that was the same size and the same distance from the center of dilation?” (a scale factor of 1)

• “How would you dilate a polygon if you knew the center of dilation but did not have the circles or radial lines printed for you?” (Draw rays starting at the center of dilation and going through each vertex of the polygon, and then measure the appropriate distances along each ray.)