Getting Started 
Misconception/Error The student does not understand how the diagram relates to a statement of the Pythagorean Theorem. 
Examples of Student Work at this Level The student may state the Pythagorean Theorem and explain aspects of it or attempt to apply it. However, the student is unable to relate the theorem to a geometric demonstration of it. The student:
 Substitutes the lengths of the triangle’s sides into the equation .
 States that the figure demonstrates the Pythagorean Theorem because the given lengths work in the formula.
 Provides a general description of the Pythagorean Theorem or instructions for how it can be used.
 States that the Pythagorean Theorem is for finding a missing side length.
 States that the grids in the figure show the length of the triangle’s sides.
 States that the Pythagorean Theorem can be used because the triangle is a right triangle.

Questions Eliciting Thinking What might be the purpose of the square attached to the side of the triangle with a length of three?
What do each of the squares represent in relation to the sides of the triangle?
How do the areas of the squares compare? 
Instructional Implications If needed, review the Pythagorean Theorem. Focus on the distinction between the assumptions (a triangle is a right triangle with legs of lengths a and b and hypotenuse of length c) and the conclusion (). Have the student confirm the theorem holds for the right triangle on the worksheet by writing . Then, use the demonstration included in this task to relate the Pythagorean Theorem to areas of squares. Guide the student to observe that:
 The lengths of the sides of each square correspond to the lengths of the sides of the triangle.
 The sum of the areas of the squares with sides of length 3 units and 4 units (the legs of the right triangle) is equal to the area of a square with sides of length 5 units (the hypotenuse of the right triangle).
 The above is reflected in the statement of the Pythagorean Theorem applied to this triangle, .
Ask the student to construct a right triangle on grid paper with sides of lengths 5, 12, and 13. Then have the student draw squares with sides of lengths 5, 12, and 13, and verify that the sum of the areas of the two smaller squares equals the area of the larger square.
Introduce the student to a proof of the Pythagorean Theorem. Consider implementing MFAS task Explaining a Proof of the Pythagorean Theorem (8.G.2.6). 
Making Progress 
Misconception/Error The student does not demonstrate an understanding of the contrapositive of the Pythagorean Theorem. 
Examples of Student Work at this Level The student can explain how the demonstration relates to the Pythagorean Theorem. However, the student does not recognize that the triangle in the second figure is not a right triangle. The student states that it is a right triangle because:
 It has an angle that measures 90 degrees.
 The two sides are perpendicular.
 You can use the Pythagorean Theorem.
 You can put a little box in the corner.
The student states that it is not a right angle because there is no little box in the corner.

Questions Eliciting Thinking How did you determine that it is a right triangle?
Did you examine the relationship among the lengths of the sides?
Can you explain what the Pythagorean Theorem says? What would it mean if the sum of the squares of the legs did not equal the square of the hypotenuse?
How do you know that the angle measures 90° (the two sides are perpendicular, you can use the Pythagorean Theorem, or you can put a little box in the corner)?
Does the absence of the box mean that the angle cannot be a right angle? 
Instructional Implications Review the Pythagorean Theorem and explain that it is logically equivalent to its contrapositive: If the lengths of the sides of a triangle are not related by the equation , then the triangle cannot be a right triangle. Guide the student to observe that and to conclude that the second triangle cannot be a right triangle.
If needed, explain to the student that the presence of the “right angle box” in a diagram indicates that an angle is a right angle but the absence does not indicate anything about the angle’s measure. When the right angle box is missing, the Pythagorean Theorem and its converse can be used to determine if an angle is right or not. Provide the student with the lengths of the three sides of a triangle and ask the student to determine if the triangle is right or not.
Introduce the student to the converse of the Pythagorean Theorem. Consider implementing MFAS task Converse of the Pythagorean Theorem (8.G.2.6). 
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 the sum of the areas of squares with sides of length 3 units and 4 units (the legs of the right triangle) is equal to the area of a square with sides of length 5 units (the hypotenuse of the right triangle) demonstrating that as the Pythagorean Theorem would predict.
In the second figure, the student calculates the areas of the three squares and notes that the sum of the areas of the two smaller squares is not equal to the area of the larger square or determines that . Consequently, the second triangle is not a right triangle.

Questions Eliciting Thinking How is the Pythagorean Theorem different from its converse?
What do you think might be true of the angle opposite the side of length eight in Figure 2? Do you think its measure is greater or less than 90°? 
Instructional Implications Ask the student to completely state the Pythagorean Theorem. Then, ask the student to distinguish between the assumptions and the conclusion. Next, introduce the student to a proof of the Pythagorean Theorem. Consider implementing MFAS task Explaining a Proof of the Pythagorean Theorem (8.G.2.6).
Ask the student to completely state the converse of the Pythagorean Theorem. Then, ask the student to distinguish between the assumptions and the conclusion. Next, introduce the student to a proof of the converse of the Pythagorean Theorem. Consider implementing MFAS task Converse of the Pythagorean Theorem (8.G.2.6). 