Lesson 3

Students drew dilations in previous lessons but this example is slightly different since the center of dilation is in the interior of the figure. This warm-up gives students an opportunity to practice dilating and do some error analysis.

Display the image for all to see and ask students why it is incorrect. (The image of \(H\) should be on the same side of \(C\) as \(H\). You can tell it's wrong because the triangles don't look like they're the same shape.)

Make sure that all students understand how to draw rays from the center point through the points to be dilated, and find the image of the points by measuring along the rays. Students will have more opportunities to dilate figures in other activities in this lesson so there is no need to have them correct any errors they made.

In this activity, students get another opportunity to practice drawing dilations precisely. They also get to see the effects of different scale factors on an image. By assigning students scale factors less than and greater than one, students get a chance to see how the image is taken closer to or farther away from the center of dilation. By assigning equivalent scale factors, such as \(\frac32\) and 1.5, students get a chance to explore how images dilated with the same scale factor are congruent. They also get a chance to remind themselves of decimal representations of fractions, which will be useful when they calculate scale factors from ratios of side lengths in later lessons.

Arrange students in groups of 2. Assign each student a scale factor from the list: \(\frac14\), \(\frac12\), \(\frac34\), \(\frac76\), \(\frac32\), \(\frac52\), 0.25, 0.5, 1.5, 2.5

If students are unclear about relative lengths, tell them to make a prediction and write a description such as “a little smaller” or “a lot bigger” before they calculate.

The goal of this discussion is to identify the effects of various scale factors. Collect and align several tracing papers on the original image. Invite students to make generalizations about different scenarios:

• compare scale factors which are greater and less than one
• compare equivalent scale factors with different representations (such as 0.5 and \(\frac12\))

In this activity, students dilate quadrilateral \(ABCD\) using center \(P\) by different scale factors \(k\). They notice that not only are \(\frac{PA’}{PA}, \frac{PB’}{PB}, \frac{PC’}{PC}\), and \(\frac{PD’}{PD}\) equal to \(k\) because that is how dilations are defined, but for example, \(\frac{B’A’}{BA}\) and \(\frac{C’B’}{CB}\) are also equal to \(k\). This assertion is not proved in this course. \(\frac{PA’}{PA} = k\) is a consequence of the definition of dilation, but the fact that dilation results in figures which are scaled copies of one another by scale factor \(k\) is taken as an assertion after students make a conjecture based on their measurements. Students should be familiar with this assertion from middle school. In this activity students extend that understanding by making a distinction between the distances from points in the original figure and image to the center of dilation (determined by the definition of dilation), and lengths in the image and original figure (an unproven property of dilations).

Arrange students in groups of 2–4. Assign each group a scale factor: 2, 3, \(\frac13\), \(\frac12\). Encourage students to split up the measuring tasks to save time.

Explain to students that although they may be used to a ratio referring to an association between two or more quantities, people often also use the word to refer to a quotient of two quantities in a ratio relationship. For example, we might say the ratio of juice to sparkling water in a punch is 3 to 2. We could also say that the ratio of juice to sparkling water is \(\frac32\) to 1, since this is equivalent. As a shorthand, people sometimes say the ratio of juice to sparkling water is three halves. In this activity, ratio is used to refer to the quotient of two side lengths, and in this unit we frequently use ratio as a shorthand for quotient.

The purpose of this discussion is to identify characteristics of a dilation. Invite students to explain their observations about the first table. Students might conjecture that the values of all the ratios are equal to \(k\), and can prove this using the definition of dilation. If no student attempts to justify their conjecture using the definition of dilation, remind students of the definition and ask how this explains the conjecture they made.

Then invite students to explain their observations about the second table. Students might conjecture that the values of all the ratios are still equal to \(k\). Students need to attend to precision when reading the definition of dilation (MP6) to realize that the definition of dilation doesn’t guarantee that corresponding sides from the image and original figure must have a scale factor of \(k\).

Explain to students that while it seems obvious that corresponding sides from the image and original figure must have a scale factor of \(k\), it’s actually tricky to prove. Remind them that just measuring a bunch of examples does not constitute a proof. Instead, explain that when we’ve verified something with examples and believe it seems obvious but aren’t going to prove it in this class, we can call it an assertion.

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

The dilation of a line segment is longer or shorter according to the same ratio given by the scale factor. (Assertion)

\(PC:PC'=3:1\), \(BC:B'C'=2: \frac23\)