CCSS.Math.Content.HSG-C.A.1
The standard
Prove that all circles are similar.
Common Core State Standards for Mathematics
What this standard means
Students need to show, not just claim, that any circle can be mapped onto any other circle by transformations. They should use the center and radius, then describe a translation to match centers and a dilation to match radii.
Mastery looks like a clear proof with correct transformation language and a scale factor such as r2/r1. Students often get stuck thinking similarity means same size, or they say “all circles are round” without proving it. Watch for vague proofs that do not name the center, radius, and dilation factor.
Ways to teach it
- Hands-on: Give pairs two paper circles, ask them to mark centers, translate one center onto the other, then resize using a scale factor.
- Prompt: Explain why matching centers alone is not enough to prove two circles are similar.
- Quick assessment: Show two circles with radii 3 and 8, and ask students to write the needed dilation scale factor.
- Real-world connection: Compare coins or lids of different sizes, and ask which transformations would make one outline match the other.
Plan a lesson for CCSS.Math.Content.HSG-C.A.1
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Related standards
- CCSS.Math.Content.HSG-C.A
Understand and apply theorems about circles
- CCSS.Math.Content.HSG-SRT.A.3
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
- CCSS.Math.Content.HSG-SRT.B.5
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
- CCSS.Math.Content.HSG-SRT.B
Prove theorems involving similarity