ALEX Classroom Resource

  

Geometry Module 2, Topic B: Dilations

  Classroom Resource Information  

Title:

 Geometry Module 2, Topic B: Dilations

URL:

 https://www.engageny.org/resource/geometry-module-2-topic-b-overview

Content Source:

 EngageNY
Type: Lesson/Unit Plan

Overview:

Module 2, Topic B is an in-depth study of the properties of dilations. Though students applied dilations in Topic A, their use in the ratio and parallel methods was to establish relationships that were consequences of applying a dilation, not directly about the dilation itself. In Topic B, students explore observed properties of dilations (Grade 8 Module 3) and reason why these properties are true. This reasoning is possible because of what students have studied regarding scale drawings and the triangle side-splitter and dilation theorems. With these theorems, it is possible to establish why dilations map segments to segments, lines to lines, etc. Some of the arguments involve an examination of several sub-cases; it is in these instances of thorough examination that students must truly make sense of problems and persevere in solving them (MP.1).

In Lesson 6, students revisit the study of rigid motions and contrast the behavior of the rigid motions to that of a dilation. Students confirm why the properties of dilations are true in Lessons 7–9. Students repeatedly encounter G.SRT.A.1a and b in these lessons and build arguments with the help of the ratio and parallel methods (G.SRT.B.4). In Lesson 10, students study how dilations can be used to divide a segment into equal divisions. Finally, in Lesson 11, students observe how the images of dilations of a given figure by the same scale factor are related, as well as the effect of a composition of dilations on the scale factor of the composition.

Content Standard(s):
Mathematics
MA2015 (2016)
Grade: 9-12
Geometry
17 ) Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity. [G-SRT4]
Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
26. Verify experimentally the properties of dilations given by a center and a scale factor.

a. Verify that 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.

b. Verify that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a center of dilation, a scale factor, and a polygonal image,
  • Create a new image by extending a line segment from the center of dilation through each vertex of the original figure by the scale factor to find each new vertex.
  • Present a convincing argument that line segments created by the dilation are parallel to their pre-images unless they pass through the center of dilation, in which case they remain on the same line.
  • Find the ratio of the length of the line segment from the center of dilation to each vertex in the new image and the corresponding segment in the original image and compare that ratio to the scale factor.
  • Conjecture a generalization of these results for all dilations.
Teacher Vocabulary:
  • Dilations
  • Center
  • Scale factor
Knowledge:
Students know:
  • Methods for finding the length of line segments (both in a coordinate plane and through measurement).
  • Dilations may be performed on polygons by drawing lines through the center of dilation and each vertex of the polygon then marking off a line segment changed from the original by the scale factor.
Skills:
Students are able to:
  • Accurately create a new image from a center of dilation, a scale factor, and an image.
  • Accurately find the length of line segments and ratios of line segments.
  • Communicate with logical reasoning a conjecture of generalization from experimental results.
Understanding:
Students understand that:
  • A dilation uses a center and line segments through vertex points to create an image which is similar to the original image but in a ratio specified by the scale factor.
  • The ratio of the line segment formed from the center of dilation to a vertex in the new image and the corresponding vertex in the original image is equal to the scale factor.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.26.1: Define dilation and scale factor.
GEO.26.2: Apply a scale factor.
GEO.26.2: Illustrate when given an original figure with a line (e.g., m) through it, not through the center, a parallel line to m will be created when the dilation is performed.
Example: Given a line x=, dilate the graph and line by 2. What happened to the line?
GEO.26.3: Illustrate when given an original figure with a line (e.g., m) through its center the line will remain unchanged when the dilation is performed.
GEO.26.4: Illustrate dilation.
Example: Find the distance of line AB, given A (0,0) and B (2,3), after dilating AB by a scale factor of 1/2.
GEO.26.5: Determine the change in length of a line segment after dilation.
GEO.26.6: Discuss the properties of parallel lines.
GEO.26.7: Dilate a line segment.
GEO.26.8: Recognize whether a dilation is an enlargement or a reduction.

Prior Knowledge Skills:
  • Recall how to name points on a Cartesian plane using ordered pairs.
  • Recognize ordered pairs (x, y).
  • Define similar.
  • Recognize dilations.
  • Recognize translations.
  • Recognize rotations.
  • Recognize reflections.
  • Identify similar figures.
  • Analyze an image and its dilation to determine if the two figures are similar.
  • Define dilation.
  • Recall how to find scale factor.
  • Give examples of scale drawings.
  • Identify parts of the Cartesian plane.
  • Recognize ordered pairs.
  • Define function, ordered pairs, input, output.
  • Demonstrate how to plot points on a Cartesian plane using ordered pairs.                     
Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.24 When given two congruent triangles that have been transformed (limit to a translation), determine the congruent parts. (Ex: Determine which leg on Triangle A is congruent to which leg on Triangle B.)
Tags: dilations, theorems, triangles
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Comments

There are six lessons in this topic.

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  This resource provided by:  
Author: Hannah Bradley
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