Unit 2, Lesson 3
Dilations with No Grid
Grade 8
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To Gather
Geometry toolkits
In this Warm-up, students apply a dilation to points on a ray without a circular grid. Without the grid, students need to come up with a way to measure distances. Strategies they might try include using a ruler, marking off distances on an index card, or folding paper in half. Providing access to geometry toolkits gives students an opportunity to use appropriate tools strategically (MP5).
Provide access to geometry toolkits. Give students 2 minutes of quiet work time followed by a whole class discussion.
The goal of this discussion is for students to understand that points
Invite students to present their methods for finding the points
“How is this work similar to previous work with dilations on a circular grid?” (The points lie on the same ray at different distances.)
“How is this work different?” (There are no marked distances.)
“How can we describe
“How can we describe
MLR8: Discussion Supports
To Gather
Geometry toolkits
In this activity, students investigate dilations with no grid. Students make sense of a new problem as they perform dilations of points, determine scale factors, and find centers of dilation (MP1).
After doing a few of the problems, students may notice that the requested point is always one of the labeled points and then use this observation to expedite the work.
Arrange students in groups of 2. Give students 1–2 minutes to work on the first question and then pause for a whole-class discussion.
Remind students that the image of a point under dilation must lie on the same line as the point being dilated and the center of dilation. Demonstrate (or have a student demonstrate) how drawing a ray from point
Give students 3–4 minutes to work on the remaining questions, followed by a whole-class discussion.
The goal of this discussion is to make sure students understand how the location of the center of dilation and the scale factor work together, and to prepare students to draw their own dilations without a grid. Begin by discussing any strategies used to solve the problems. Ask students who noticed that all of the answers were labeled points to share how their observation helped them answer the questions. Then discuss with students:
“What do you notice about the scale factors for dilating
“What do you notice about the center of dilation, the point being dilated, and the image of the point after dilation?” (All 3 of those points lie on the same line.)
“What scale factor results in an image that doesn’t move?” (A scale factor of 1 does not move any points.)
Show students 2 points
Be sure students understand that the image of point
MLR7: Compare and Connect
To Gather
Geometry toolkits
In this activity, students find an appropriate way to take measurements in order to perform a dilation (MP5), most likely by using a ruler or the edge of an index card. Different students will work with different scale factors to produce perspective drawings of a box.
Monitor for students who draw these different representations:
An accurate diagram using a scale factor greater than 1
An accurate diagram using a scale factor less than 1
Provide access to geometry toolkits.
Give students 3–4 minutes to complete the first 2 questions. Pause to check student work and then display this perspective drawing for all to see.
Tell students they will create their own perspective drawing and assign each student a scale factor to use. Appropriate scale factors include
Select work from students with different scale factors, such as those described in the Activity Narrative, to share later.
The goal of this discussion is for students to connect that scale factors less than 1 result in an image that is smaller and closer to the center of dilation while scale factors greater than 1 result in an image that is larger and farther away from the center of dilation.
Display 2–3 perspective drawings that all have a scale factor less than 1 from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the representations. Here are some questions for discussion:
“What do the dilations have in common?” (The images all got smaller. The images are all closer to the center of dilation.)
“How are they different?” (The center of dilation is in a different location. They have different scale factors.)
“How does the scale factor show up in each dilation?” (The scale factor determines how much smaller the dilation will be.)
Then display 2–3 perspective drawings that all have a scale factor greater than 1 from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the representations. Here are some questions for discussion:
“What do these dilations have in common?” (The images all got bigger. The images are all farther away from the center of dilation.)
“How are these dilations different from the first set?” (These dilations all got bigger while the other dilations all got smaller.)
“How does the scale factor show up in each dilation?” (The scale factor determines how much larger or smaller the dilation will be.)
In the figure, point
Since point
Since point
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Here is a diagram that shows 9 points.
Nine points. Points E, F, G and H are mostly horizontal, spread out evenly and form a line. Point A above F, and point B above G, point H and Point I below and to the right also form a line. Point F, point D, point B and point C far above between point G and H also form a line.
Some students may think that for a point to be a dilation of itself, the scale factor is 0. Prompt them to consider multiplying the distance by 0. If they want the distance to be the same, they need to multiply by 1 instead.
Draw the images of points
Pause here so your teacher can review your diagram. Your teacher will then give you a scale factor to use in the next part.
Let's make a perspective drawing. Here is a rectangle.
Choose a point inside the shaded circular region but outside the rectangle to use as the center of dilation. Label it
Use your center
Draw segments that connects each of the original vertices with its image. This will make your diagram look like a cool three-dimensional drawing of a box! If time allows, you can shade the sides of the box to make it look more realistic.
Compare your drawing to other people’s drawings. What is the same and what is different? How do the choices you made affect the final drawing? Was your dilated rectangle closer to
Some students may try to make their drawing match the example drawing shown in the launch. For example, if the center of dilation shown is above and to the right, some students may place their center of dilation above and to the right of the rectangle. Emphasize that the center of dilation can be located anywhere, but its location will affect the resulting image.