Getting Started 
Misconception/Error The student does not have an effective strategy for determining the distance between points in the coordinate plane (without graphing). 
Examples of Student Work at this Level The student:
 Graphs the points and counts interval units to determine the distances between the pairs of points.
 Says he or she does not know how to find the distances without graphing.
 Attempts a calculation unrelated to distance in the coordinate plane.

Questions Eliciting Thinking What is an ordered pair (coordinate pair)? What does (5, 3) represent?
How would you determine your answer if you were allowed to use graph paper?
What do you notice about the xcoordinates (ycoordinates)? 
Instructional Implications Provide opportunities for the student to represent, using absolute value symbols, distances between points on the number line and then calculate the distances. For example, to calculate the distance from 4 to 3, model representing this distance as either 4 – 3 = 7 = 7 or 3 – (4) = 7 = 7. Be sure the student understands that distance is represented by a nonnegative value. Guide the student to apply this approach to determining the distance between points with the same x or ycoordinate in the coordinate plane. Initially, have the student graph ordered pairs with the same x or ycoordinate. Then using the graph, explain the similarity to finding distances between points on a number line. Guide the student again to represent distances using absolute value symbols [e.g., to represent the distance from (5, 3) to (1, 3) as 5 – (1) = 4 = 4]. Provide the student with additional opportunities to represent distances between points with the same x or ycoordinates using absolute value symbols and to calculate the distances. 
Moving Forward 
Misconception/Error The student does not understand how to represent distance. 
Examples of Student Work at this Level The student subtracts coordinates to find the distances between the pairs of points but:
 Subtracts both the x and ycoordinates and represents the distance as an ordered pair.
 Represents one or both distances as a negative number.

Questions Eliciting Thinking How is distance typically represented? If I asked you how far you live from school, would you describe that distance as an ordered pair of numbers?
How is distance typically represented? If I asked you how far you live from school, would you describe that distance as a negative number? 
Instructional Implications Explain that distance is typically represented by a nonnegative value. Guide the student to represent distances on both the number line and in the coordinate plane using absolute value symbols [e.g., to represent the distance from (5, 3) to (1, 3) as 5 – (1) = 4 = 4]. Provide the student with additional opportunities to represent distances between points with the same x or ycoordinates using absolute value symbols and to calculate the distances.
Review operations with integers as needed [e.g., 3 – (2)]. Consider implementing CPALMS Unit/Lesson Sequence Integers: Quick, Fun, and Easy to Learn (ID 6459). 
Almost There 
Misconception/Error The student makes a minor computational error. 
Examples of Student Work at this Level The student subtracts the appropriate coordinates to determine the distances and represents the distances with a positive value but makes a computational error. Upon questioning, the student indicates that distance is represented by a positive value.

Questions Eliciting Thinking I think you may have made a subtraction error. Can you check your work?
Would you get the same answer if you subtracted in the other order?
Why did you write four as your answer when in your work you wrote: 5(1)= 4?
Can distance be represented by a negative integer? Why or why not? 
Instructional Implications Provide feedback to the student regarding any errors made and allow the student to revise his or her work. If the student did not use absolute value symbols to represent the distances, model how to do so. Encourage the student to interpret expressions of the form a – b as meaning the distance between two points whose coordinates are a and b or the distance from a to b on the number line. Provide the student with additional opportunities to represent distances between points with the same x or ycoordinates using absolute value symbols and to calculate the distances. 
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 the distance from André to Boris is four units while the distance from André to Carlos is five units. The student either:
 Explicitly uses absolute value [e.g., writes 5 – (1) = 4 = 4],
 Implicitly uses absolute value and writes 1 – (5) = 4, recognizing that the smaller coordinate should be subtracted from the larger coordinate in order to get a positive difference, or
 Subtracts in either order [e.g., 5 – (1) = 4] but adjusts the final answer so that it is positive, recognizing that distance cannot be represented by a negative integer.

Questions Eliciting Thinking Would you get the same answer if you subtracted in the other order?
Why did you write four as your answer when in your work you wrote: 5(1)= 4?
Can distance be represented by a negative integer? Why or why not?
What do you suppose absolute value has to do with determining distances? 
Instructional Implications Encourage the student to interpret expressions of the form a – b as meaning the distance between two points whose coordinates are a and b or the distance from a to b on the number line. Provide the student with additional opportunities to represent distances between points with the same x or ycoordinates using absolute value symbols and to calculate the distances. 