MA.912.GR.6.5

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Apply transformations to prove that all circles are similar.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Geometric Reasoning
Standard: Use properties and theorems related to circles.
Date Adopted or Revised: 08/20
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Circle
  • Dilation
  • Similarity
  • Translation

 

Vertical Alignment

Previous Benchmarks

Next Benchmarks

 

Purpose and Instructional Strategies

In grade 8, students learned about similarity and similarity transformations. In Geometry, students apply transformations to prove that all circles are similar. 
  • Instruction includes presenting students with a pair of circles of different size and asking them to identify a sequence of transformations that would map one onto the other. Students should realize that a single translation and a single dilation is all that is needed in a sequence to map one onto the other. 
  • Instruction includes the connection to the coordinate plane by showing that two circles are similar using coordinates. 
  • Students should connect the definition of similarity in terms of corresponding parts applied to polygons and explore what parts of the circles will be in proportion between the preimage and image of a dilations.
    • For example, given two circles, their radii (r1 and r2) and their diameters (d1 and d2) would satisfy the proportional relationship Expression.

 

Common Misconceptions or Errors

  • Students may think that always need a formal proof to prove that all circles are similar.

 

Instructional Tasks

Instructional Task 1 (MTR.4.1)
  • Two concentric circles with point A as the center and circle B are given on the coordinate plane.
    circle
    • Part A. Describe the transformation(s) needed to map the smaller circle A onto the larger circle A.
    • Part B. List the transformation(s) that could be used to show that each circle A is similar to circle D. Compare your transformations with a partner.
    • Part C. What is the difference in the transformation(s) depending on the circle A chosen?

 

 

Instructional Items

Instructional Item 1
  • Circle A and circle D are given below.
    circle
    • Part A. Describe a set of transformations that could be used on circle A to show it is similar to circle D.
    • Part B. Describe a set of transformations that could be used on circle D to show it is similar to circle A.

 

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

Related Courses

This benchmark is part of these courses.
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 - 2024, 2024 and beyond (current))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

Formative Assessments

Similar Circles:

Students are given two circles with different radii and are asked to prove that the circles are similar.

Type: Formative Assessment

All Circles Are Similar:

Students are given two circles with different radius lengths and are asked to prove that the circles are similar.

Type: Formative Assessment

Lesson Plans

Why are Circles Similar?:

Students learn how to prove that all circles are similar using the three transformations (reflection, dilation, translation) on a unique circle. The lesson takes students through multiple-step transformations to show a proof of circle similarity. A Socratic discussion is used as closure to aid in the comprehension of circle similarity.

Type: Lesson Plan

Are All Circles Similar?:

This lesson allows students to prove that all circles are similar using transformations. Students will need prior knowledge of similarity, transformations, and the definition of a circle. The lesson begins with a warm up regarding dilations, then poses the question: Are all circles similar? The students are guided through the proof using a translation and dilation. The teacher emphasizes the details in the proof. The lesson closes with an exit ticket.

Type: Lesson Plan

Congruence vs. Similarity:

Students will learn the difference between congruence and similarity of classes of figures (such as circles, parallelograms) in terms of the number of variable lengths in the class. A third category will allow not only rigid motions and dilations, but also a single one-dimensional stretch, allowing more classes of figures to share sufficient common features to belong.

Type: Lesson Plan

Circle to Circle:

Students use coordinate-based translations and dilations to prove circles are similar.

Type: Lesson Plan

Perspectives Video: Professional/Enthusiast

All Circles Are Similar- Especially Circular Pizza!:

What better way to demonstrate that all circles are similar then to use pizzas! Gaines Street Pies explains how all pizza pies are similar through transformations.

Download the CPALMS Perspectives video student note taking guide.

Type: Perspectives Video: Professional/Enthusiast

MFAS Formative Assessments

All Circles Are Similar:

Students are given two circles with different radius lengths and are asked to prove that the circles are similar.

Similar Circles:

Students are given two circles with different radii and are asked to prove that the circles are similar.

Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.