Lesson 3

Students apply a dilation to points on a ray. The scaffold of the circular grid has been removed but the structure of dilations is the same. 

Without the grid, students will need to come up with a way to measure in order to find the point twice as far from \(A\) as \(B\) and half as far from \(A\) as \(B\). They can use a ruler or the edge of an index card.

Monitor for these methods:

• using a ruler to measure distances
• marking off distances on an index card (for problem 1)
• folding paper in half (for problem 2)

Select students who use these methods and invite them to present.

Provide access to geometry toolkits.

Invite selected students to present their methods for finding the points which may include:

• using a ruler to measure distances
• marking off distances on an index card (for problem 1)
• folding paper in half (for problem 2)

Point out how this is similar to work with dilations on a circular grid (the points lie on the same ray at different distances) and how it is different (there are no marked distances). For the next activity, it is important for students to understand that \(C\) is the dilation of \(B\) with center \(A\) and scale factor 2. And \(D\) is the dilation of \(B\) with center \(A\) and scale factor \(\frac{1}{2}\).

This activity investigates dilations with no grid. Students have seen these for the first time in the warm-up, which had a ray drawn between two points. That scaffold has been removed here so the teacher may need to provide guidance by suggesting that students draw appropriate rays. 

Encourage students to measure distances carefully at first, since the problem statement does not state that the image of each point, after the dilation indicated, is one of the labeled points. After doing a few of the problems, the students should notice that the dilated point is always one of the labeled points and then use this observation to expedite the work. Also monitor for students who see the relationship between the scale factors used to send \(G\) to \(E\) and \(E\) to \(G\) (both with center \(H\)).

Students might need to be reminded that the image of a point under dilation must lie on the same line as the point being dilated and the center of dilation. Students might think that for a point to be a dilation of itself, the scale factor is 0. If this happens, ask them to consider multiplying the distance of the point by 0. (If they want the distance to be the same, they actually need to multiply it by 1 instead.)

Discuss any strategies used to solve the problems. Ask selected students who noticed that the answers to all of the questions were labeled points to share their observation and how it helped them answer the questions. Next ask selected students to share their observation about the scale factors for dilating \(G\) to \(E\) and dilating \(E\) back to \(G\). One way to reverse or “undo” a dilation is to use the same center and reciprocal scale factor.

Other important ideas to bring out include:

• The center of dilation, the point being dilated, and the image of the point after dilation must all lie on the same line.
• A scale factor of 1 does not move any points. If the scale factor is not 1, only one point does not move (the center of dilation).

In this activity, students continue to apply dilations without a grid. Unlike in the previous activity, the dilated images of the points are not plotted. So rather than identifying the correct point, they will need to find an appropriate way to take measurements (MP5), most likely with the aid of a ruler or the edge of an index card. Different students will work with different scale factors and will produce perspective drawings of a box. 

Watch for students who pick a point close to one vertex of the given rectangle. If the point is too close, it will be more difficult to visualize the box. Suggest that they move the point further away. Monitor for students who produce accurate drawings with different scale factors and invite them to share during the discussion.

Students may all try to make their drawing match any example drawings shown in the launch. For example, if the center of dilation in an example is above and to the right, everyone might place their center of dilation above and to the right of the rectangle. Any point is fine as a dilation point, but the effect on what the picture looks like may vary.

Students may not recall that to dilate a polygon, they can first dilate the vertices and then connect them in the proper order. It may be necessary to show students how to dilate one of the vertices and allow them to perform the dilation on the other three vertices.

Display the work of several students selected based on the different scale factors. Then ask students:

• “What are the effects of using a scale factor greater than 1?” (The image is larger than the original and farther away from the center of dilation than the original.)
• “What are the effects of using a scale factor less than 1?” (The image is smaller than the original and closer to the center of dilation than the original.)
• “What effect does the location of \(C\), the center of dilation, have?” (It impacts the size and location of the dilated rectangle: if the scale factor is less than 1 then the dilated rectangle is closer to \(C\) than the original and if the scale factor is larger than 1 then the dilated rectangle is further away from \(C\) than the original.)

Time permitting, consider showing several student drawings with the same scale factor but a different location for the point \(C\). How are they the same? How are they different? Two faces of these boxes (the original rectangle and the scaled copy) are congruent but the point of view or perspective on them is different.