Getting Started 
Misconception/Error The student does not understand how to use the ratio to partition the segment. 
Examples of Student Work at this Level The student:
 Indicates he or she does not understand.
 Attempts to use the distance formula to find the length of the segment but is unable to use the ratio to partition the segment.

Questions Eliciting Thinking Into how many congruent pieces should the segment be divided if the ratio of the parts is to be 1:7?
Can you find the rise and run as you move from point M to N?
I see that you were trying to use the distance formula. Can you explain how that would help you partition the segment? 
Instructional Implications Assist the student in understanding what it means to partition a segment in a given ratio by partitioning an eight inch horizontal segment (not in the coordinate plane) into eight oneinch parts using a ruler. Demonstrate finding points on the segment that partition it in various ratios such as 1:7, 2:6, 3:5, and 4:4. Assist the student in observing that the parts of each ratio sum to eight. Allow the student to partition another segment in a given ratio in this way. Ask the student to use the ratio to first reason which endpoint the point that partitions the segment is nearer.
Next, place a segment in the coordinate plane horizontally or vertically and model how to find the coordinates of the point that partitions the segment in the ratio 1:7. Then reposition the segment so that it is no longer horizontal or vertical, and model finding the coordinates of the point that partitions the segment in the ratio 1:7. Help the student develop a general method for finding the coordinates of a point that partitions a segment in a given ratio. Provide additional opportunities for the student to partition segments in given ratios. 
Moving Forward 
Misconception/Error The student is unable to use the differences in the xcoordinates and ycoordinates of the endpoints to find the coordinates of point P. 
Examples of Student Work at this Level The student correctly finds the differences in the xcoordinates and ycoordinates of the endpoints, divides each by eight ( and ), but does not understand how to use the resulting values (2 and 1) to calculate the coordinates of point P.

Questions Eliciting Thinking What do these values that you have found tell you?
How can you use these values to calculate the coordinates of point P?
How are these values related to the slope of the segment?
Will point P be closer to point M or point N? How can you tell? 
Instructional Implications Model using the values 2 and 1 to calculate the coordinates of point P. Relate partitioning a segment to using the rise and run of the slope to move from one point to another on a line. Help the student develop a general method for finding the coordinates of a point that partitions a segment in a given ratio. Provide additional opportunities for the student to partition segments in given ratios. 
Almost There 
Misconception/Error The student’s work is insufficiently shown or the student makes a minor error. 
Examples of Student Work at this Level The student:
 Correctly finds the coordinates of point P but does not adequately show supporting work. For example, the calculation of the coordinates (e.g., (4 + 2, 7  1) = (2,6)) is not shown.
 Graphs (12, 1) as (12, 1) but correctly calculates the coordinates of point P given this error.
 Calculates the coordinates of point P as (4 + 2, 7 + 1) = (2, 8).
 Finds the coordinates of point P so that MP:PN = 7:1.

Questions Eliciting Thinking Can you explain to me, step by step, how you found the coordinates of point P?
How could you show your work clearly and concisely to communicate your method on paper?
I think you may have made an error in your work. Will you see if you can find it?
What is the difference between MP:PN = 1:7 and MP:PN = 7:1?
How can you check to see if you found point P correctly? 
Instructional Implications Assist the student in determining how to appropriately show work when writing up solutions to problems. Show the student examples of written work from “Got It” classmates and describe the features of their work that make it exemplary.
Provide feedback to the student concerning any errors made and allow the student to revise his or her work. Model a strategy for checking an answer to a problem of this type (e.g., use the distance formula to calculate MP and PN and determine if MP:PN = 1:7).
Consider implementing the MFAS task Centroid Coordinates (GGPE.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 finds the differences in the xcoordinates and ycoordinates of the endpoints, divides each by eight ( and ), and uses these values and the coordinates of M to find the coordinates of point P as (4 + 2, 7  1) = (2, 6).

Questions Eliciting Thinking How could you verify that your answer is correct?
Can you find a point on that divides the segment in the ratio 7:1? 
Instructional Implications Challenge the student to find a point that divides a directed segment into ratios of the form a:b where a and b are both different from one, such as 3:5.
Challenge the student to find and describe a general method for finding the coordinates of a point that divides the segment into the ratio 1:n. Have the student use this general method to derive the midpoint formula.
Consider implementing the MFAS task Centroid Coordinates (GGPE.2.6). 