Getting Started 
Misconception/Error The student does not understand the angle relationships in the diagram nor how to use trigonometric ratios to find the indicated angle. 
Examples of Student Work at this Level The student cannot express the relationships of the angles in the diagram. The student:
 Assumes the angle is a particular measure such as 60°.
 Finds the length of the hypotenuse of the triangle using the Pythagorean Theorem but does not find the indicated angle.

Questions Eliciting Thinking What are you given? What are you asked to find in this problem?
Will the Pythagorean Theorem help you find an angle?
Could you use a trig ratio to help you find the missing angle? 
Instructional Implications Guide the student to understand how the angles in the diagram are related. Explain the angle relationships in terms of concepts the student knows and understands. For example, complements of the same angle or in terms of parallel lines intersected by a transversal.
Review the definitions of the trigonometric ratios. Provide opportunities to apply the definitions to right triangles (presented in various orientations) by asking the student to identify the sine, cosine, and tangent ratios associated with each of the two acute angles. Then model finding an unknown length or angle measure in a right triangle by using an appropriate trigonometric ratio. Caution the student to carefully select an appropriate ratio and substitute measures correctly to write an equation. If needed, review solving equations of the form a = with the unknown in all positions.
Provide additional opportunities to find an unknown length or angle measure in right triangles by using the Pythagorean Theorem or an appropriate trigonometric ratio.
Consider implementing other MFAS tasks for GSRT.3.8. 
Moving Forward 
Misconception/Error The student makes an error in interpreting the given information or in using a trigonometric ratio to solve the problem. 
Examples of Student Work at this Level The student observes that the angle the stringer makes with the wall is the complement of but:
 Does not use the correct trigonometric ratio in the equation.
 Assumes the triangle is 454590 or 306090.
 Writes the inverse of the ratio in the equation.
 Does not know how to solve for the angle measure.

Questions Eliciting Thinking Could you label the angles in the diagram 1, 2 and 3? Would these labels help you explain the relationships among the angles?
How did you find these angle measures? Why did you label the angles this way?
If the two angles of a right triangle are 45 and 45, what will be true about the side lengths opposite those angles?
What is the tangent ratio? What value should go in the numerator? What value should go in the denominator?
How do you solve a trig equation when you do not know the measure of the angle? 
Instructional Implications Assist the student in labeling the angles in the diagram with numbers or variables to make the written explanation of the relationships among the angles easier to compose. Guide the student to clearly explain the relationship between the angle the stringer makes with the wall and each of and . Assist the student in composing a justification for each described relationship.
Review the definitions of the trigonometric ratios. Then provide feedback to the student regarding any error made and allow the student to revise his or her work. Guide the student to carefully label given measures in diagrams and to consider if final answers make sense.
If needed, review how to solve for the angle measurement when both values in the trig ratio are known, both with a calculator and with a table of trigonometric values.
Provide additional opportunities to find an unknown length or angle measure in right triangles by using the Pythagorean Theorem or an appropriate trigonometric ratio.
Encourage the student to explore trigonometry using an interactive online site such as http://www.mathopenref.com/trigsummary.html. 
Almost There 
Misconception/Error The student makes a minor computational or rounding error or is not able to fully explain his or her reasoning. 
Examples of Student Work at this Level The student labels the diagram correctly, writes a correct equation involving a trigonometric ratio but: Does not round according to the directions of the problem.
 Calculates in radians instead of degrees when using a calculator.
 Makes an error using the table of trigonometric values.
The student correctly finds the angle, but is not able to clearly explain or justify the relationships among the angles in the diagram.

Questions Eliciting Thinking What mode should your calculator be in?
There is a small error in your work. Can you find and correct it?
Can you show me how you used the trig table to find the angle measure?
According to the problem, how should you express your final answer?
Could you label the angles with numbers to help you with your explanation of their relationships? 
Instructional Implications Provide feedback to the student regarding any error made and allow the student to revise his or her work. Provide additional opportunities to find unknown lengths or angle measures in right triangles.
Assist the student in labeling the angles in the diagram with numbers or variables to make the written explanation of the relationships among the angles easier to compose. Guide the student to clearly explain the relationship between the angle the stringer makes with the wall and each of and . Assist the student in composing a justification for each described relationship.
Consider implementing other MFAS tasks for GSRT.3.8. 
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 understands the angle relationships in the diagram and explains that the angle the stringer makes with the wall is the complement of (since together they form a right angle). But is also the complement of (since they are the two acute angles of a right triangle). So, the angle the stringer makes with the wall must be congruent to (since complements of the same angle must be congruent). The student may also reason using angle relationships involving parallel lines and a transversal.
The student writes an equation involving a trigonometric ratio to find the indicated angle, solves the equation, and rounds the answer to the nearest whole degree:

Questions Eliciting Thinking Can you think of another correct way to write the equation? How?
Could you find all of the other angles in the diagram? 
Instructional Implications Introduce the student to the concept of solving a triangle (i.e., finding all angle measures and side lengths). Ask the student to solve right triangles given minimal information (e.g., the measure of an acute angle and one side or the measures of two sides).
Consider implementing other MFAS tasks for GSRT.3.8. 