The Department of Public Works wants to put a water treatment plant at a point that is an equal distance from each of the three towns it will service. The location of each of the towns is shown below. Use the answer box to complete the following. (You will also need to use additional paper to complete the answer satisfactorily.) Locate the point that is an equal distance from each of the towns.   Explain how you determined this location. Use words, symbols, or both in your explanation.   Use mathematics to justify your answer.

The following 9 Anchor Papers represent a range of score points and are used in conjunction with the rubrics to assess student responses.

Score for Anchor Paper #1: Rubric Score 1

Annotation: This response demonstrates a minimal understanding and analysis of the problem. The representation is incorrect. The diagram and explanation reveal that the student applied an inappropriate strategy of finding the medians to locate the position of the water treatment plant. Arc marks are present for the perpendicular bisectors, but the perpendicular bisectors were used to locate the midpoints between the towns. There is no justification.

Score for Anchor Paper #2: Rubric Score 1

Annotation: This response demonstrates a minimal understanding and analysis of the problem. The representation is fundamentally correct; the point is close, but not correct. There is little evidence of a correct strategy. The student has stated the point would be located where "the measure of the length of each line would be the same," but this strategy was not employed as the lines do not all have the same measure. There is no justification.

Score for Anchor Paper #3: Rubric Score 2

Annotation: This response demonstrates a conceptual understanding and analysis of the problem. The representation is correct; the student has correctly located the point where the water treatment plant should be. The student has "basically used trial and error." While not the best method to locate the point, trial and error is an acceptable strategy if the concept of equidistance is applied. There is no justification.

Score for Anchor Paper #4: Rubric Score 2

Annotation: This response demonstrates a conceptual understanding and analysis of the problem. The representation is fundamentally correct. The point is close, but not correct. The diagram and explanation reveal that the student has applied an incomplete correct strategy of using the concept of a circle's radius to find a point equidistant from all three towns. There is no justification.

Score for Anchor Paper #5: Rubric Score 2

Annotation: This response demonstrates a conceptual understanding and analysis of the problem. The representation is fundamentally correct; the point is close, but not correct. The arc marks reveal incomplete application of a reasonable strategy to construct perpendicular bisectors. The explanation of the strategy provides no further indication of a complete reasonable strategy. There is no justification.

Score for Anchor Paper #6: Rubric Score 3

Annotation: This response demonstrates a clear understanding and analysis of the problem. The representation is essentially correct; the point is close without being exact. The diagram and explanation reveal that the student has applied a reasonable strategy of "using perpendicular bisectors" to find the location of the water treatment plant. However, the explanation for finding the perpendicular bisector is missing. Justification is provided in the statement, "Each side has a perpendicular bisector. I got all of them and found out where they intersected. This point is equidistant from all three towns."

Score for Anchor Paper #7: Rubric Score 3

Annotation: This response demonstrates a clear understanding and analysis of the problem. The representation is correct; the student has successfully located the point designating the location of the water treatment plant. At least two complete sets of arc marks provide a full explanation of a correct strategy employing perpendicular bisectors. Justification is incomplete. There is neither support of the equidistance with measurement, nor has it been conveyed that the perpendicular bisectors, not just the bisectors, provide the point that is equidistant from the three towns.

Score for Anchor Paper #8: Rubric Score 4

Annotation: This response demonstrates complete understanding and analysis of the problem. The representation is correct. A fully developed explanation of a reasonable strategy to locate the position of the water treatment plant is provided in the arc marks as well as a full description. Full justification is provided in the statement, "where all of the bisectors intersected is the point equidistant from each town."

Score for Anchor Paper #9: Rubric Score 4

Annotation: This response demonstrates a complete understanding and analysis of the problem. The representation is correct, indicating the correct location of the water treatment plant. Full explanation of a complete correct strategy is present, with at least two complete sets of arc marks. Full justification of the correct location is supported by measure, "It is 4½ cm from each town."

Additional Scored Student Responses