In this activity, students perform dilations on a square grid, which is particularly helpful when the center of dilation and the points being dilated are grid points. Students will again see that scale factors greater than 1 produce larger copies while scale factors less than 1 produce smaller copies. 

Monitor for students who use these strategies to find the dilated points:  

• Use a ruler or index card to measure distances along the rays emanating from the center of dilation

• Count how many squares to the left or right and up or down each point is from the center of dilation

The goal of the discussion is to make connections between the 2 strategies described in the Activity Narrative for finding a dilation. Measuring with a ruler or index card and reasoning about the grid and counting spaces will both result in the dilated point ending up in the same location. Display 2–3 approaches or representations from previously selected students for all to see. If time allows, invite students to briefly describe their approach orrepresentation. Use Compare and Connect to help students compare, contrast, and connect the different approaches and representations. Here are some questions for discussion:

• “What do the 2 strategies have in common? How are they different?” (Both strategies are measuring distance, but one method measures distance in a straight line while the other measures distance by counting horizontal and vertical squares.)

• “How does the scale factor show up in each method?” (When using a ruler or index card, the scale factor shows up as how many times farther the new point will be. When counting grid squares, the scale factor tells us how many more squares we need to count from the center of dilation.)

Students sort different graphs and descriptions into matching pairs during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP7). 

Students begin by sorting the cards into categories of their choosing. This allows students to familiarize themselves with the content of the cards before finding matching pairs. Monitor for different ways groups choose to match the graphs and the descriptions, but especially for groups who identify similarly for triangles and quadrilaterals and that the dilation of a circle is a circle. Monitor for students who systematically perform the dilations to help identify a match versus those who reason by structure and elimination of possibilities. 

As students work, encourage them to refine their explanations of how they made their matches using more precise language and mathematical terms (MP6).

Once all groups have completed the card sort, discuss the following: 

• “Which matches were tricky? Explain why.”

• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”

Highlight for students:

• A dilation maps a circle to a circle, a quadrilateral to a quadrilateral, and a triangle to a triangle.

• If the center of dilation for a polygon is one of the vertices, then that vertex is on the dilated polygon.

• If the scale factor is less than 1 then the dilated image is smaller than the original figure.

• If the scale factor is larger than 1 then the dilated image is larger than the original figure.