Getting Started 
Misconception/Error The student is unable to determine the lengths of the sides of a right triangle in the coordinate plane. 
Examples of Student Work at this Level The student:
 Graphs the triangle incorrectly and is unable to calculate any lengths.
 Counts unit lengths to determine the lengths of the legs but attempts a similar strategy to determine the length of the hypotenuse.
 Uses some contrived formula or combination of operations rather than the Pythagorean Theorem.
 Substitutes coordinates of vertices for either a or b in the equation .
 Finds the area of the triangle or the slopes of the sides rather than the lengths of the sides.
 Attempts to use a proportion to determine the lengths of the sides.

Questions Eliciting Thinking Can you explain how you calculated these lengths?
Did you estimate this length? Can you think of a way to calculate it instead?
What do the slopes you calculated tell you about the sides of the triangle? Are slope and length the same?
What kind of triangle is this? What theorem relates the lengths of the sides of a right triangle? 
Instructional Implications If needed, address graphing in the coordinate plane. Review the unit of measure for length and assist the student in counting unit lengths to find the lengths of vertical or horizontal segments.
Review terminology related to right triangles. Be sure the student is able to identify the legs and hypotenuse of right triangles in a variety of orientations. Explain that the longest side, the hypotenuse, is opposite the largest angle, the right angle.
Introduce the Pythagorean Theorem and provide opportunities for the student to apply the theorem to calculate unknown lengths in right triangles. Then introduce segments in the coordinate plane. Assist the student in applying the Pythagorean Theorem to find the length of a segment or the distance between two points in the coordinate plane.
Consider implementing the CPALMS Lesson Plan As the Crow Flies (ID 43471), a twopart lesson which guides the student to apply the Pythagorean Theorem to determine the length of an unknown side in a right triangle, and then to apply the Pythagorean Theorem to determine the distance between two points in the coordinate plane. 
Moving Forward 
Misconception/Error The student makes an error when applying the Pythagorean Theorem. 
Examples of Student Work at this Level The student uses the Pythagorean Theorem to write an equation of the form but applies the theorem incorrectly or makes an error when solving the resulting equation. For example, the student:
 Substitutes the length of a leg for the length of the hypotenuse writing .
 Divides by two or squares to find the length of c.
 Attempts to apply the distance formula but subtracts a yvalue from an xvalue, adds instead of subtracts coordinates, or neglects to square the differences.

Questions Eliciting Thinking Can you explain what each variable in the formula you are using represents? How did you decide which value to substitute for each variable?
Do you think your answer is reasonable? What do you know about the lengths of each side of a right triangle? Which side must be the longest and why?
Is taking a square root the same as dividing by two? What is the difference between squaring and taking the square root of a number? 
Instructional Implications Review terminology related to right triangles. Be sure the student is able to identify the legs and hypotenuse of right triangles in a variety of orientations. Explain that the longest side, the hypotenuse, is opposite the largest angle, the right angle. As needed, review the Pythagorean Theorem and how to correctly apply it. Be sure the student understands the meaning of the variables in the equation .
If needed, provide instruction on evaluating squares and square roots. Emphasize the inverse relationship between squaring and taking a square root. Model the use of the square root symbol and be sure the student understands the distinction between evaluating square roots and dividing by two.
Address any errors related to the application of the distance formula. If not done so already, derive the distance formula from the Pythagorean Theorem. Provide opportunities to use the distance formula to calculate lengths of segments or distances between two points in the coordinate plane. 
Almost There 
Misconception/Error The student makes a minor error when calculating a length. 
Examples of Student Work at this Level The student makes a minor mathematical error in some step of the solution. The student:
 Incorrectly finds the sum of 49 and 169.
 Calculates the length of a leg incorrectly but all other work is correct.
 Plots one vertex incorrectly.
 Describes a length as negative (e.g., PQ = 14.8).
 Writes a mathematically incorrect statement (e.g., 169 + 49 =14.8 or ).
 Rounds ZX incorrectly.

Questions Eliciting Thinking You made an error in your work. Can you find and correct it?
Is it possible for a length to be negative?
Is this statement true: 169 + 49 = (or )? What might be a better way to show your work? 
Instructional Implications Provide feedback concerning any errors made and allow the student to revise his or her work. Explain that equations of the form = p have two solutions although one or both may not make sense in the context of the problem. If needed, assist the student in showing work in a manner that justifies strategies and answers. Remind the student to label the units of measure.
Provide the student with additional practice using the Pythagorean Theorem to solve realworld and mathematical problems. Consider implementing the MFAS task New Television (8.G.2.7) or How Far to School (8.G.2.7).

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 determines that XY = 13 and YZ = 7. The student correctly applies the Pythagorean Theorem or the distance formula to determine that (or an appropriately rounded approximation).
Note: The student may have used the Distance Formula to correctly calculate all three lengths not recognizing that there is a more efficient way to find the lengths of the legs of the triangle.

Questions Eliciting Thinking Can you apply the Pythagorean Theorem to any triangle? Explain.
How can you prove this triangle is a right triangle?
Can you find the lengths of the legs without using the distance formula? 
Instructional Implications If needed, assist the student in recognizing that the distance formula is not needed to calculate the lengths of the legs of the triangle.
Introduce the concept of a Pythagorean Triple and challenge the student to find examples of triples.
Challenge the student to find the perimeter of a figure with no sides parallel to an axis. Give the student the coordinates of the vertices and guide the student to decompose the shape into right triangles whose hypotenuses correspond to the sides of the figure. 