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 finds the coordinates of point P, the midpoint of  , but:
- Does not attempt any other portion of the task.
- Attempts to use the distance formula to find AP but is unable to use the ratio to partition
 .
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| Questions Eliciting Thinking Into how many congruent pieces should the segment be divided if the ratio of the parts is to be 2:1?
Can you find the rise and run as you move from point A to P?
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 a three inch horizontal segment (not in the coordinate plane) into three one-inch parts using a ruler. Demonstrate finding points on the segment that partition it in the ratios such as 1:2, and 2:1. Assist the student in observing that the parts of each ratio sum to three. 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 2:1. 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 2:1. 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 understands how the ratio relates to the segment but is not able to correctly calculate coordinates of point P. |
| Examples of Student Work at this Level The student finds the coordinates of point P, the midpoint of , and draws a point on that appears to divide in the ratio 2:1. Next, the student:
- Uses the distance formula to find the length of and divides this length by three. The student is unable to find the coordinate of point D.
- Correctly finds the differences in the x-coordinates and y-coordinates of the endpoints of , divides each by three (
 and  ), but does not understand how to use these values to calculate the coordinates of point D.
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| Questions Eliciting Thinking Is finding the length of  the same as finding the coordinates of point D?
How can you use and to calculate the coordinates of point D?
Will point D be closer to point A or point P? How can you tell? |
| Instructional Implications Model how to calculate the differences in the x-coordinates and y-coordinates of the endpoints of and use these values along with the coordinates of point A to calculate the coordinates of point D. 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.
Remind the student that if AD:DP=2:1, then AD is two times DP so point D will be closer to point P than point A. Help the student to recognize which endpoint D will be nearer.
Consider implementing the MFAS task Partitioning a Segment (G-GPE.2.6). |
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 calculates the coordinates of point D but writes mathematically incorrect statements in the course of showing work (e.g.,
 =  ).
- Calculates the x-coordinate of D as
 instead of 2 + .
- Converts to a mixed number incorrectly, writing it as 2
 .
- Reverses the scale and calculates the coordinates of point D so that AD:DP=1:2.
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| Questions Eliciting Thinking Is this statement really true: = ?
How could you show your work clearly and correctly?
I think you may have made an error in your work. Will you see if you can find it?
How can you check to see if you found D 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 AD and DP and determine if AD:DP = 2:1).
Consider implementing the MFAS task Partitioning a Segment (G-GPE.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 correctly calculates the coordinates of point D as follows:
- x-coordinate work: 2 +
 8=7
- y-coordinate work: 2 + 4=4
and concludes that the coordinates of point D are(7,4). All work is shown clearly and in a mathematically correct way.
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| Questions Eliciting Thinking Could you locate a point on this same segment that partitions the segment so that AF:FP = 1:2? How is locating this point different from locating point D?
How could you verify that your answer is correct? |
| Instructional Implications 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 Partitioning a Segment (G-GPE.2.6). |
