Getting Started 
Misconception/Error The student does not demonstrate an understanding of the relationship between and . 
Examples of Student Work at this Level The student writes that because:
 The triangle with side lengths and is bigger.
 The triangles are similar not congruent.
The student writes that because:
 Both triangles have a angle.
 Both triangles are special right triangles.
 The sides “match up.”
 The angles are congruent so the sides must be congruent.

Questions Eliciting Thinking What do you know about these two triangles? Are they congruent? Similar?
If two triangles are similar, what do you know about the ratio of their corresponding sides?
What does mean? Does it mean that the corresponding sides must be congruent?
Are corresponding sides of two right triangles always proportional? Why or why not?
Can you be certain that both triangles are , , triangles? 
Instructional Implications Review ways to prove two triangles are similar (AA, SAS, SSS) and what must be established in order to conclude two triangles are similar using each method. Remind the student that once two triangles are proven similar, all corresponding angles are congruent and all corresponding sides are proportional.
Explain that the two given triangles are similar by the AA Similarity Theorem. Ask the student to write ratios of corresponding sides in the form of an extended proportion. Guide the student to use the extended proportion to deduce that . Model explaining that because corresponding sides of similar triangles are proportional.
Provide the student with pairs of triangles that can be shown to be similar. Ask the student to identify the theorem that justifies the similarity and then write an extended proportion relating the lengths of the sides and congruence statements that show how angles are related.
Review terminology related to right triangles (e.g., right angle, acute angles, legs, hypotenuse, side opposite an angle, and side adjacent to an angle) and the definitions of the sine, cosine and tangent ratios. Provide the student with a right triangle with the lengths of the sides given. Have the student identify the sine, cosine, and tangent ratios for both acute angles. Include a variety of right triangles in different orientations.
Have the student compare the cosine ratio for the same acute angle of several right triangles with sides of different lengths. Guide the student to understand that the trigonometric ratios for a given angle measure are the same in all right triangles with an acute angle of that measure. Remind the student that right triangles with the same acute angle measure are similar by the AA Similarity Theorem.
Consider implementing the MFAS task The Sine of 57 (GSRT.3.6) to further assess the student’s understanding of trigonometric ratios. 
Moving Forward 
Misconception/Error The student does not recognize the ratios as the cosine of a and the cosine of ß. 
Examples of Student Work at this Level The student recognizes that the triangles are similar and the ratio of corresponding sides is equal. However, the student does not recognize the given ratios as the cosine ratios. The student:
 May describe the lengths in the ratio as the lengths of the hypotenuse, opposite side, or adjacent side but does not identify the ratio as the cosine ratio.

Questions Eliciting Thinking How is the cosine ratio defined?
What is the cosine of ? What is the cosine of ? 
Instructional Implications Review terminology related to right triangles (e.g., right angle, acute angles, legs, hypotenuse, side opposite an angle, and side adjacent to an angle) and the definitions of the sine, cosine and tangent ratios. Provide the student with a right triangle with the lengths of the sides given. Have the student identify the sine, cosine, and tangent ratios for both acute angles. Guide the student to understand that the two given ratios are cosine ratios and since , then cos = cos .
Have the student compare the cosine ratio for the same acute angle of several right triangles with sides of different lengths. Guide the student to understand that the trigonometric ratios for a given angle measure are the same in all right triangles with an acute angle of that measure. Remind the student that right triangles with the same acute angle measure are similar by the AA Similarity Theorem.
Consider implementing the MFAS task The Sine of 57 (GSRT.3.6) to further assess the student’s understanding of trigonometric ratios. 
Almost There 
Misconception/Error The student is unable to deduce that if the cosines of two acute angles of two right triangles are congruent, then the triangles are similar. 
Examples of Student Work at this Level The student recognizes that the triangles are similar so that the ratio of corresponding sides are equal and identifies the given ratios as cosine ratios. However, the student is unable to deduce that if the cosines of two acute angles of two right triangles are congruent, then the triangles are similar. The student:
 Indicates that he or she does not understand what must be true of the two triangles.
 Writes that the triangles must be congruent.
 Writes that the triangles are similar but is unable to explain why.

Questions Eliciting Thinking What must be true of and if cos(E) = cos(Q)?
What is the difference between similar and congruent triangles? 
Instructional Implications Remind the student of the definition of the cosine ratio. Explain that the cosine ratio is a ratio of specific sides in a class of right triangles that are related by similarity. The class can be identified by an acute angle measure that is common to all right triangles in the class. So if cos(E) = cos(Q), then and are angles of similar right triangles. Consequently, since and are both right triangles such that cos(E) = cos(Q), then by definition of the cosine ratio.
Consider implementing the MFAS task The Sine of 57 (GSRT.3.6). 
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 explains that since both triangles are right triangles and = , the triangles are similar by the AA Similarity Theorem. The student writes that because corresponding sides of similar triangles are proportional. The student recognizes that as the cosine of and as the cosine of . The student is able to deduce from cos(E) = cos(Q) that right is similar to right .

Questions Eliciting Thinking Could you deduce that is similar to if they were not both right triangles?
Why are trigonometric ratios the same in right triangles with the same acute angle measure? 
Instructional Implications Ask the student to explore and explain the relationship between the sine and cosine of complementary angles.
Consider implementing the MFAS task The Sine of 57 (GSRT.3.6). 