Getting Started 
Misconception/Error The student is unable to explain what the given sine ratio indicates about . 
Examples of Student Work at this Level The student does not respond to the question or indicates he or she is uncertain of the meaning of the question. The student writes:
 It turns degrees into numbers.
 It is a right triangle.
 The .
 0.8387 is the angle measure of one of the acute angles.
 0.8387 is the length of one of the sides of the triangle.

Questions Eliciting Thinking What is a sine ratio? What two values are compared in a sine ratio?
How is related to the sine of 57? 
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 provide instruction on the definitions of the trigonometric ratios. Give the student experience identifying the ratios in specific right triangles and using the ratios to find lengths of sides and angle measures in right triangles. Emphasize that the values obtained from a calculator or table describe the ratio of the lengths of pairs of sides of a right triangle. Explain that one acute angle measure defines an entire class of similar right triangles. When an acute angle measure is used in a trigonometric ratio, it is identifying this class of triangles. However, the ratio refers to a specific ratio of lengths of sides (depending on whether it is a sine, cosine, or tangent ratio) within this class of right triangles. Explain that all right triangles with the same acute angle measure are similar (by the AA Similarity Theorem) so that corresponding ratios of sides are equal. Emphasize that this fact underlies the definitions of the trigonometric ratios. Illustrate this with a 345 and 6810 right triangle (both have an acute angle of approximately ).
Consider implementing the MFAS task The Cosine Ratio (GSRT.3.6) to further assess the student’s understanding of trigonometric ratios. 
Moving Forward 
Misconception/Error The student demonstrates a partial understanding of the ratio of side lengths represented by sine. 
Examples of Student Work at this Level The student describes 0.8387 as representing a ratio of side lengths but is unable to identify the sides or identifies the wrong sides.

Questions Eliciting Thinking What lengths are compared in the sine ratio? Where are these lengths with respect to ?
Can you describe the cosine and tangent ratios? 
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. Include a variety of right triangles in different orientations. Also give the student a trigonometric ratio, such as sin B = , and ask the student to draw and label right triangle ABC that corresponds to the ratio. Have the student write the other two trigonometric ratios for the triangle.
Have the student compare the sine ratio for the same acute angle for 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. If needed, explain to the student that right triangles with the same acute angle measure are similar by the AA Similarity Theorem.
Consider implementing the MFAS task The Cosine Ratio (GSRT.3.6) to further assess the student’s understanding of trigonometric ratios. 
Almost There 
Misconception/Error The student is unable to explain why the sine ratio of a given angle is the same in every right triangle with an acute angle of that measure. 
Examples of Student Work at this Level The student explains that sin = 0.8387 indicates that the ratio of the length of the side opposite and the length of the hypotenuse is 0.8387. However, when asked if the sin is the same value for every right triangle with an acute angle of measure , the student says:
 The sin is not the same because the lengths of the sides or the location of the angle in different triangles vary.
 The sin is the same but does not provide an explanation based on similarity and the proportionality of corresponding sides.

Questions Eliciting Thinking If you are given three right triangles all with an acute angle measure of , what would be true about those triangles?
What do you know about the lengths of corresponding sides of similar triangles?
Why does your calculator always give you the same value for the sine regardless of the triangle with which you are working? 
Instructional Implications Guide the student to understand that the trigonometric ratios for a given acute angle measure are the same in all right triangles with an acute angle of that measure. Explain to the student that right triangles with the same acute angle measure are similar by the AA Similarity Theorem so that corresponding ratios of sides are equal. Emphasize that this fact underlies the definitions of the trigonometric ratios. Illustrate this with a 345 and 6810 right triangle (both have an acute angle of approximately ).
Consider implementing the MFAS task The Cosine Ratio (GSRT.3.6) to further assess the student’s understanding of trigonometric ratios. 
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 writes that sin = 0.8387 indicates that the ratio of the length of the side opposite and the length of the hypotenuse is 0.8387. The student continues to write that sin will be the same value for every right triangle with an acute angle of measure since all right triangles with an angle of measure are similar by the AA Similarity Theorem. Consequently, corresponding ratios of side lengths are equal.

Questions Eliciting Thinking How can a sine ratio be used to solve for a missing angle or side in a right triangle?
Why are trigonometric ratios the same in right triangles with the same acute angle measure?
Suppose you have a triangle that is not a right triangle but it contains an angle of measure . What does sin indicate about that triangle? 
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 Cosine Ratio (GSRT.3.6) to further assess the student’s understanding of trigonometric ratios. 