Getting Started 
Misconception/Error The student does not use the Pythagorean Theorem (or the Distance Formula) to determine the distance between two points in the coordinate plane. 
Examples of Student Work at this Level The student:
 Attempts to count the number of diagonal unit lengths between the two points.
 Calculates slope instead of distance.

Questions Eliciting Thinking What is one unit of distance in the coordinate plane? Is a diagonal of a square the same length as its side?
Is slope the same as distance?
Can you apply the Pythagorean Theorem to determine the distance between the two points? How would you determine the length of the legs of the right triangle?
How would you calculate the distance between two points that have the same ycoordinate? How far is point A from (3, 5)? 
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. Explain why the hypotenuse must be the longest side of a right triangle in terms of the relative size of the right angle.
Assist the student in developing a thorough understanding of the Pythagorean Theorem before trying to apply it. Consider implementing the CPALMS Lesson Plan A Hypotenuse Is a WHAT???? (ID 31918) which guides the student through the history and discovery of the Pythagorean Theorem or Pythagoras’ Theorem (ID 7726) which introduces the Pythagorean Theorem and provides both visual and algebraic proofs for the theorem. Consider using the MFAS tasks Pythagorean Squares (8.G.2.6), Explaining a Proof of the Pythagorean Theorem (8.G.2.6), and Converse of the Pythagorean Theorem (8.G.2.6) to assess the student’s level of understanding regarding the Pythagorean Theorem. Also, address any specific misconceptions such as the confusion of slope with distance.
Then, 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 to determine the distance between two points in the coordinate plane. 
Moving Forward 
Misconception/Error The student applies the Pythagorean Theorem incorrectly. 
Examples of Student Work at this Level The student attempts to use the Pythagorean Theorem to write an equation of the form but makes a significant error in the process of solving. The student may:
 Write the formula incorrectly (e.g., 2a + 2b = 2c).
 Perform the wrong operation during the solving process.
 Substitute the length of a leg for the hypotenuse.
 Square the final value instead of taking the square root.
 Confuse taking a square root with dividing by two.

Questions Eliciting Thinking What formula are you trying to apply? Can you show me how you used this formula?
What variable in the formula represents the hypotenuse? What do you know about the hypotenuse?
How can you determine the lengths of the legs of the right triangle?
What does squared mean? Does the order of operations apply?
How do you square a number? Is squaring a number the same as multiplying by two?
What is the inverse of squaring? How do you take the square root of a number? Is taking the square root the same as dividing by two? 
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. Explain why the hypotenuse must be the longest side of a right triangle in terms of the relative size of the right angle.
Focus instruction on the specific algebraic errors observed in student work. Consider implementing the CPALMS Lesson Plan Applying the Pythagorean Theorem (ID 48973) and then the CPALMS Lesson Plan Origami BoatsPythagorean Theorem in the Real World (ID 49055).
Provide instruction, as needed, on evaluating squares and square roots. Emphasize the inverse relationship between squares and square roots. Model the use of the square root symbol and be sure the student understands the distinction between evaluating square roots and dividing.
Provide additional opportunities for the student to apply the Pythagorean Theorem to find distance between points in the coordinate plane. Consider using the MFAS task Coordinate Plane Triangle (8.G.2.8) or Calculate Triangle Sides (8.G.2.8). 
Almost There 
Misconception/Error The student makes minor errors when applying the Pythagorean Theorem. 
Examples of Student Work at this Level The student may:
 Make a minor calculation error.
 Miscount the length of a leg.
 List a positive and negative answer (e.g., ).
 Write an incorrect statement (e.g., ).
 Not show all work required to justify the answer.
 Not assign units.
 Use = instead of when representing an irrational number with a rational estimate.
The student may apply the Distance Formula but does not know how it relates to the Pythagorean Theorem.

Questions Eliciting Thinking I think you made an error. Can you go back and review your work?
Can distance be negative?
Does c represent the same value as ? Explain.
How did you solve the equation ? If you take the square root of one side of an equation, should you take the square root of the other side as well? Why or why not?
You did not show every step needed to justify your answer. Can you go back and fill in the missing steps?
It looks like you used the Distance Formula to find the distance between points A and B. Could you have also used the Pythagorean Theorem? How are these two approaches related?
What is the difference between = and ? 
Instructional Implications Provide feedback and allow the student to revise his or her work. Specify 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. 
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 determines the distance between the two points is units or ˜ 10.6 units.
The student applies the Distance Formula and can explain how it relates to the Pythagorean Theorem. 
Questions Eliciting Thinking Why did you use the Pythagorean Theorem to solve this problem?
Is there another triangle you could draw to help you determine your answer?
Does the triangle have to be a right triangle in order to use the Pythagorean Theorem?
How can you be sure it is a right triangle?
It looks like you used the Distance Formula to find the distance between points A and B. Could you have also used the Pythagorean Theorem? How are these two approaches related? 
Instructional Implications 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.
Introduce the student to the Distance Formula. Demonstrate the use of the Distance Formula to determine the distance between two points in the coordinate plane, and then challenge the student to explain the relationship between the Distance Formula and the Pythagorean Theorem. 