Getting Started 
Misconception/Error The student does not demonstrate an understanding of the Pythagorean Theorem or the need to use it to solve the problem. 
Examples of Student Work at this Level The student’s work contains no evidence of the use of the Pythagorean Theorem. For example, the student:
 Estimates the unknown length based on the lengths of the known sides.
 Adds the two known lengths to determine the unknown length.
 Multiplies the two known side lengths to determine the unknown side.
Note: The student may or may not make calculation errors.

Questions Eliciting Thinking How did you determine your answer? Can you think of a way to find an exact answer rather than estimating?
How would you describe the triangle in the diagram? What theorems do you know that involve the lengths of the sides of right triangles? 
Instructional Implications Provide the student with basic instruction on the Pythagorean Theorem. Be sure to review the parts of a right triangle (e.g., vertices, right angle, acute angles, hypotenuse, and legs). When initially introducing the Pythagorean Theorem, emphasize that it only applies to right triangles. Be very explicit about what the theorem states, describing it verbally and with mathematical symbols. Caution the student to be careful not to confuse the legs and hypotenuse when applying the theorem. Give the student the opportunity to find missing lengths in right triangles in both realworld and mathematical problems. Include problems in which the length of the hypotenuse is unknown, the length of a leg is unknown, unknown lengths are integers, unknown lengths are rational or irrational numbers, and diagrams must be sketched and labeled. Guide the student to show work completely and in an organized manner.
If needed, review finding and approximating square roots. 
Moving Forward 
Misconception/Error The student makes errors in applying the Pythagorean Theorem. 
Examples of Student Work at this Level The student recognizes the need to use the Pythagorean Theorem but makes significant errors in its application. For example, the student:
 Finds the square of a length but neglects to take its square root or reports the square as the actual length.
 Squares the sum of the squares of the legs.
 Multiplies the squares of the legs.
 Divides by two instead of taking a square root.
 Does not substitute the correct values for the lengths of the legs or the hypotenuse.

Questions Eliciting Thinking Do you think your answer is reasonable? Why or why not?
How did you decide which value to substitute for each variable? Which dimension would represent the hypotenuse (or legs)?
Can you tell me what you know about the Pythagorean Theorem? How did you apply it? 
Instructional Implications Review the properties of a right triangle, as needed. Be sure the student is able to identify the legs and hypotenuse of any right triangle.
Provide instruction, as needed, on evaluating squares and square roots. Emphasize the inverse relationship between squares and square roots. Use the square root symbol and be sure the student understands the distinction between evaluating square roots and dividing.
Give the student the opportunity to find missing lengths in right triangles in both realworld and mathematical problems. Include problems in which the length of the hypotenuse is unknown, the length of a leg is unknown, unknown lengths are integers, unknown lengths are rational or irrational numbers, and diagrams must be sketched and labeled. Guide the student to show work completely and in an organized manner. 
Almost There 
Misconception/Error The student makes minor errors when applying the Pythagorean Theorem or does not completely justify each step. 
Examples of Student Work at this Level The student:
 Writes an incorrect statement (e.g., or ).
 Uses = instead of when using a rational number to estimate an irrational answer.
 Does not assign units to the measurement.
Note: The student may have more than one minor error.

Questions Eliciting Thinking Can you explain and justify each step of your work?
How does ? If you take the square root of one side of an equation, should you take the square root of the other side as well?
What is the difference between = and ? Is this difference important? 
Instructional Implications Provide feedback to the student regarding any errors made and allow the student to revise his or her work. Encourage the student to carefully label diagrams. Remind the student to show work neatly and completely to avoid careless errors.
Review the definition of an irrational number and discuss the need to approximate irrational numbers in models of realworld lengths. Review how to approximate an irrational number more precisely by using more of its place values.
Encourage the student to check for the reasonableness of the answer within the given context and to label the units of measure. 
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 applies the Pythagorean Theorem and provides work to show Lily walks miles or miles or 0.9 miles. 
Questions Eliciting Thinking Can you apply the Pythagorean Theorem to any triangle? Explain.
Can you use the Pythagorean Theorem to determine lengths of two sides if you are only given the length of the hypotenuse? Explain.
Did you expect Lily to walk a shorter or longer distance than Matthew? Why?
Why is used instead of = ? 
Instructional Implications Challenge the student to use the given information to solve problems. For example, ask the student:
 How much farther did Matthew walk than Lily?
 How far would Matthew have walked in total if he went directly home after school?
Consider using the MFAS task New Television (8.G.2.7) to assess if the student can apply the converse of the Pythagorean Theorem.
