Curriculum  /  Math  /  10th Grade  /  Unit 3: Dilations and Similarity  /  Lesson 4

Dilations and Similarity

Lesson 4

Math

Unit 3

10th Grade

Lesson 4 of 18

Objective

Divide a line segment into equal sections using dilation.

Common Core Standards

Core Standards

  • G.CO.D.12 — Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
  • G.SRT.A.1.A — 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.
  • G.SRT.A.1.B — The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Foundational Standards

  • 8.G.A.1

Criteria for Success

  1. Describe that a dilation of a line not passing through the center of the dilation will map to a parallel line. 
  2. Use the property of dilations with respect to parallel lines to describe that the benchmark line segment must be parallel to the line segment that is to be partitioned. 
  3. Construct a line parallel to the original line. 
  4. Use vectors from the center of dilation and the benchmark points of segmentation to dilate onto the segment to be partitioned. 
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Anchor Problems

25-30 minutes

Problem 1

Dilate the line segment $${\overline{ST}}$$ by $$2$$ from center point $$R$$. Label the new line segment as $${\overline{KM}}$$. 

Guiding Questions

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Problem 2

Divide segment $${AB}$$ into three segments of equal length. 

Below is line segment $${{AB}}$$ and line $${DG}$$, which is parallel to $${{AB}}$$. Points $$D$$, $$E$$, $$F$$, and $$G$$ are all equidistant from one another using the same compass setting. 

If points $$D$$ and $$G$$ are dilations of points $$A$$ and $$B$$ respectively, how can you use the points $$E$$, $$F$$, and $$G$$ to partition line segment $${{AB}}$$?

Guiding Questions

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Target Task

5-10 minutes

Divide the following line segment into a ratio of 1:4 using dilations. Then, describe the parallel relationship you have created through this process. 

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include problems where students are practicing constructing parallel lines from Unit 1. 
  • Include problems where students need to divide a line segment into a ratio of ____. Ensure that students identify the total number of equal pieces that need to be identified to do this correctly. 
  • Include problems where the construction is done for the students and steps are provided. Ask students to identify and justify the relationships in the diagram. 
  • Include problems where the construction is given and students need to provide the steps. 
  • Include problems where a fictitious student has made an error or has gotten stuck in partitioning a segment. Ask students to identify and correct the error. Focus on errors where the number of equal segments is not adequate for the ratio given.

Next

Dilate a figure from a point on the figure. Use the properties of dilations with respect to parallel lines to verify dilations and find the center of dilation.

Lesson 5
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Dilations off the Coordinate Plane

Topic B: Dilations on the Coordinate Plane

Topic C: Defining Similarity

Topic D: Similarity Applications

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