Dilations with No Grid
5 min
Teacher Prep
Setup
Required Preparation
Narrative
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).
Launch
Provide access to geometry toolkits. Give students 2 minutes of quiet work time followed by a whole class discussion.
Student Task
- Find a point on the ray whose distance from A is twice the distance from B to A and label it C.
- Find a point on the ray whose distance from A is half the distance from B to A and label it D.
Sample Response
Activity Synthesis (Teacher Notes)
The goal of this discussion is for students to understand that points C and D are dilations of point B, and to describe those dilations using precise language.
Invite students to present their methods for finding the points C and D, which may include the strategies mentioned in the Activity Narrative. Here are some questions for discussion:
-
“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 C as a dilation of B?” (C is a dilation of B with a center of dilation at A and a scale factor of 2.)
-
“How can we describe D as a dilation of B?” (D is a dilation of B with a center of dilation at A and a scale factor of 21.)
Standards
- 8.G.A·Understand congruence and similarity using physical models, transparencies, or geometry software.
- 8.G.A·Understand congruence and similarity using physical models, transparencies, or geometry software.
10 min
20 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Dilations with No Grid
5 min
Teacher Prep
Setup
Required Preparation
Narrative
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).
Launch
Provide access to geometry toolkits. Give students 2 minutes of quiet work time followed by a whole class discussion.
Student Task
- Find a point on the ray whose distance from A is twice the distance from B to A and label it C.
- Find a point on the ray whose distance from A is half the distance from B to A and label it D.
Sample Response
Activity Synthesis (Teacher Notes)
The goal of this discussion is for students to understand that points C and D are dilations of point B, and to describe those dilations using precise language.
Invite students to present their methods for finding the points C and D, which may include the strategies mentioned in the Activity Narrative. Here are some questions for discussion:
-
“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 C as a dilation of B?” (C is a dilation of B with a center of dilation at A and a scale factor of 2.)
-
“How can we describe D as a dilation of B?” (D is a dilation of B with a center of dilation at A and a scale factor of 21.)
Standards
- 8.G.A·Understand congruence and similarity using physical models, transparencies, or geometry software.
- 8.G.A·Understand congruence and similarity using physical models, transparencies, or geometry software.