Lesson 1

In middle school, students explored scale drawings including how measurements in a scaled copy of a figure relate to measurements in the original figure. In this activity students remind themselves of these relationships by studying an example and a non-example.

The important concepts in this discussion are that in a scale drawing: 

• the ratio of the distance between two points in the original figure to the distance between two corresponding points in the scaled figure is constant
• the corresponding angles are congruent

Remind students of the term scale drawing from middle school. Ask students what they could measure to figure out which hippo was a scale drawing of the original hippo. (If the distance between the original hippo’s pupils is twice the distance between the scaled hippo’s pupils, then the length of the original hippo’s tail should also be twice the length of the scaled hippo’s tail.) If no student mentions angles, ask students if the protractor could also help them confirm which hippo is the scale drawing. (The angles defined by points in the scaled figure should have the same measure as the angles defined by points in the original figure.) Students need not do any measuring.

Ask students to add this definition to their reference charts as you add it to the class reference chart:

Scale factor is the factor by which every length in an original figure is multiplied when you make a scaled copy.

Scale factor is 2 or \(\frac12\)

Students drew dilations in middle school. This activity is the first time students are asked to dilate a figure in this course. Monitor for students with misconceptions, such as making the distance from \(H\) to \(H’\) 120 mm, rather than making the distance from \(C\) to \(H’\) 120 mm so that the distance from \(H\) to \(H’\) is 80 mm

If students aren't sure how to get started, support them in understanding the definition of dilation given in the task. Draw an example using the labels in the definition and invite them to match the labels in the task to the example.

Invite students to share ideas they had for how to get started with the problem. (Draw a ray from \(C\) through \(H\).)

Display the images for all to see and ask students why they are incorrect. (The total distance \(CH'\) should be 120 mm. They used the wrong center of dilation, \(K'O\) should be \(\frac34\) as long as \(KO\).)

Make sure that all students understand how to draw rays from the center point through the point(s) to be dilated, and find the image of the points by measuring along the rays. Students will dilate more complex figures in subsequent activities.

Students dilated a single point in a previous activity. In this activity, they dilate a figure with multiple points and multiple shapes. The goal is for them to practice dilation and begin to confirm some properties of dilation. Students will have additional opportunities to formalize properties of dilation in subsequent lessons (A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter according to the same ratio given by the scale factor.)

Monitor for students who use the following (incomplete) strategies:

• use the two given lines to place \(D’\) and \(B’\), and then estimate all of the other shapes and points
• measure distances within the figure and try to replicate them, for example calculating the radii of the two circles and trying to place them without scaling \(A\) and \(C\) along rays through \(P\)

Monitor for students who use the precise strategy of drawing rays from \(P\) through each point in the original figure and placing points in the image halfway between \(P\) and the corresponding point.

It's okay if students do not have time to dilate every point, encourage them to record things they notice and wonder as they draw the dilation.

The goal of this discussion is to surface some observations. If most students were comfortable drawing a dilation move directly to that. If not, select multiple students who used different strategies (both precise and incomplete) to share their work. Ask students to explain what was precise in each student’s method. The key point is that the most precise way to draw a dilation is to draw rays from the center of the dilation through each point of the original figure, and then measure to figure out where the point goes in the image.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the relative length of segments, or measure of angles do not come up during the conversation, ask students to discuss these ideas.