Getting Started |
| Misconception/Error The student does not understand the theorem or the statements of its proof. |
| Examples of Student Work at this Level When asked, the student cannot state the AA Similarity Theorem or the definition of similarity in terms of similarity transformations. Consequently, the student provides no or incorrect justifications for most or all of the statements.
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| Questions Eliciting Thinking Can you state the AA Similarity Theorem in your own words?
What is the definition of similarity in terms of similarity transformations? How can two triangles be shown to be similar? |
| Instructional Implications Review the definition of similarity in terms of similarity transformations and explain how the definition can be used to show two triangles are similar. Provide opportunities to the student to show two triangles are similar using the definition. Then clearly state the AA Similarity Theorem and ask the student to identify the assumption and the conclusion. Be sure the student understands that a theorem cannot be used as a justification in its own proof.
Review each of the following:
- The Fundamental Theorem of Similarity,
- The Corresponding Angles Theorem, and
- The ASA Congruence Theorem.
Next review the overall strategy of the proof and guide the student through its steps prompting the student for justifications of key statements.
If needed, implement the MFAS task Describe the AA Similarity Theorem (G-SRT.1.3) to assess the student’s understanding of the theorem. Eventually, ask the student to repeat this task. |
Moving Forward |
| Misconception/Error The student understands the theorem and the general flow of the proof but is unable to correctly justify all of the statements. |
| Examples of Student Work at this Level When asked, the student can state the AA Similarity Theorem and the definition of similarity in terms of similarity transformations. However, the student is unable to correctly justify each statement of the proof. The student:
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| Questions Eliciting Thinking Can you state the AA Similarity Theorem in your own words?
What is the definition of similarity in terms of similarity transformations? How can two triangles be shown to be similar?
What is the relationship between point B and point B'? What is the relationship between point C and point C'? Do you remember the Fundamental Theorem of Similarity? Do you recall what is says?
What kind of angles are  and  ? What do you know about corresponding angles formed by parallel lines and a transversal? |
| Instructional Implications Provide feedback to the student concerning any incorrectly justified statements. Review the definition of similarity and any theorems related to incorrect justifications (e.g., the Fundamental Theorem of Similarity, the Corresponding Angles Theorem, or the ASA Congruence Theorem). Ask the student to explain why the definition of similarity or the relevant theorems justify each statement of the proof.
Eventually ask the student to repeat this task. |
Almost There |
| Misconception/Error The student provides an incomplete or unclear justification of a statement. |
| Examples of Student Work at this Level The student correctly justifies all statements in the proof but provides an incomplete or unclear justification of:
- Statement four (
 ) or five ( ).
- Statement three (
 ).
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| Questions Eliciting Thinking What is the definition of similarity in terms of similarity transformations? If one figure is the result of a dilation of another figure, are the figures similar? If one figure is the result of a dilation and a congruence of another figure, are the figures similar?
Are these angles corresponding because the lines are parallel? What makes these angles corresponding? What makes them congruent? |
| Instructional Implications Provide feedback to the student concerning the incomplete or unclear justification and allow the student to revise his or her work.
Provide a pair of triangles named differently from those on the worksheet and ask the student to write a complete proof of the AA Similarity Theorem. Then, consider implementing the MFAS task Prove the AA Similarity Theorem (G-SRT.1.3). |
Got It |
| Misconception/Error The student provides complete and correct responses to all components of the task. |
| Examples of Student Work at this Level The student is able to justify each step of the proof by providing the following reasons:
 is parallel to  by the Fundamental Theorem of Similarity. by the Corresponding Angles Theorem. by the ASA Congruence Theorem.- by the definition of similarity (
 is the result of a dilation of  ).
- by the definition of similarity (since dilation D maps to and congruence G maps
 ) .
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| Questions Eliciting Thinking What is the Fundamental Theorem of Similarity? What does it say?
Why can’t you use the AA Similarity Theorem as your justification for statement four? |
| Instructional Implications Provide a pair of triangles named differently from those on the worksheet and ask the student to write a complete proof of the AA Similarity Theorem.
Consider implementing the MFAS task Prove the AA Similarity Theorem (G-SRT.1.3). |
