Objective
Define a dilation as a non-rigid transformation, and understand the impact of scale factor.
Common Core Standards
Core Standards
The core standards covered in this lesson
8.G.A.4— Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
Geometry
8.G.A.4— Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
Foundational Standards
The foundational standards covered in this lesson
7.G.A.1
Geometry
7.G.A.1— Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7.RP.A.2
Ratios and Proportional Relationships
7.RP.A.2— Recognize and represent proportional relationships between quantities.
Criteria for Success
The essential concepts students need to demonstrate or understand to achieve the lesson objective
- Understand a dilation as a non-rigid transformation that proportionally changes the size of a figure while maintaining the same shape.
- Understand that corresponding angle measures in dilated figures are congruent.
- Understand that a scale factor greater than 1 will enlarge a figure and a scale factor less than 1 will shrink a figure.
- Sketch and identify images of dilated figures with a described scale factor.
Tips for Teachers
Suggestions for teachers to help them teach this lesson
- Students should be familiar with the concept of a scale drawing from seventh grade; however, it will be new for them to apply this concept as a transformation.
- This video,Understanding Dilations, by PBS Learning Mediaintroduces the concept of dilation and includes animation that can be helpful for students to visualize the movements.
Lesson Materials
- Optional: 180° Protractor (1 per student)
- Optional: Patty paper (transparency paper) (1 sheet per student)
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Anchor Problems
Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
Problem 1
Several figures are shown below.
a.What do you notice? What do you wonder about the relationships between the shapes?
b.A dilation is a transformation that enlarges or shrinks a figure in a proportional way so that the shape remains the same. Where do you see evidence of a dilation in the figures?
Guiding Questions
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Problem 2
Triangle$${ABC}$$is dilated to create similar triangle$${DEF}$$.
a.Indicate the corresponding angles in the diagram. What is the relationship between corresponding angles?
b.Name the corresponding sides in the diagram. What is the relationship between corresponding side lengths?
Guiding Questions
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Problem 3
Sketch an image of each figure after the dilations described below. The figures do not need to be drawn exactly to size but should include the lengths of the sides.
a.Triangle $${LMN}$$is dilated by a scale factor of 3.
b.Rectangle $${ABCD}$$ is dilated by a scale factor of$${\frac{1}{2}}$$.
Guiding Questions
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Problem Set
A set of suggested resources or problem types that teachers can turn into a problem set
Fishtank Plus Content
Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.
Target Task
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
Trapezoid $${ABCD}$$, shown below, is dilated by a scale factor of $${\frac{1}{4}}$$. Angle $$D$$is a right angle.
a.Which statements are true? Select all that apply.
a.$${\overline{C'D'}}$$will be $${48}$$units long.
b.$${\overline{B'C'}}$$will be$${2\frac{1}{2}}$$units long.
c.$${{\overline{A'D'}}}$$will be $${15}$$units long.
d.The measure of$${{\angle B}'}$$will be$${\frac{1}{4}}$$the measurement of $${\angle B}$$.
e.$${\angle D'}$$will be a right angle.
f.$${\overline {B'C'}}$$will be parallel to $${{\overline{A'D'}}}$$.
g.Figure $${A'B'C'D'}$$will be a trapezoid.
b.Explain your response to answer choices (b) and (d). Why did you decide that those answer choices were true or false?
Student Response
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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.
- Examples where students are given an original figure and then multiple other dilated figures; students identify which figures are dilations (proportional) and which are not dilations; be sure to include images with scale factor equal to 1
- Examples where students identify corresponding line segments and angles between a figure and a dilation; include examples where the dilated image is in a different orientation from the original
- Illustrative Mathematics Scaling Angles and Polygons—This task does a nice job of connecting to prior seventh-grade skills
- EngageNY Mathematics Grade 8 Mathematics > Module 3 > Topic A > Lesson 1—Teacher Version: The examples in this lesson use notation that students will likely not be familiar with and will require some explanation
- Open Up Resources Grade 8 Unit 2 Practice Problems—Lesson 1, #1-3
Lesson 10
Lesson 12
