Two jogging paths go around the perimeters of two similar triangles in a park. The dimensions of the path around the smaller triangle are shown in the figure below. One complete lap around the outer edge of the longer path is 3,600 yards. Use the answer box to complete the following. (You will need to use additional paper to complete the answer satisfactorily.) What are the dimensions of the path around the larger triangle?   Use mathematics to justify your answer.

The following 15 Sample Student Responses represent a range of score points.

Score for Sample Student Response #1: Rubric Score 2

Annotation: This response demonstrates conceptual understanding and analysis of the problem. The correct dimensions, 1500, 1200, and 900, are evident. Work reveals that the student found the scale factor of 3 and applied the scale factor to each of the sides, maintaining the ratio in an appropriate strategy. No justification is provided. Compare to Anchor Paper #3.

Score for Sample Student Response #2: Rubric Score 1

Annotation: This response demonstrates minimal understanding and analysis of the problem. The student arrives at the incorrect dimensions of "1100, 1200, 1300." The explanation reveals that the student recognizes that "the perimeter of the large one is 3,600," but the inappropriate strategy ("the pattern in the small one goes up by 100 hundred") does not maintain proportional sides. The attempt to account for the 3600-yard perimeter demonstrates minimal understanding. No justification is present. Compare to Anchor Paper #1.

Score for Sample Student Response #3: Rubric Score 3

Annotation: This response demonstrates complete understanding and analysis of the problem. The correct dimensions of "900, 1200, and 1500" are found using an appropriate strategy. The label that denotes the unit measure of the dimensions is missing, but this is not a significant mathematical error. Justification is present in the statements, "the two triangles were similar" and "I found the ratio between the two triangles and found the dimensions of the larger triangle's path." Compare to Anchor Paper #6.

Score for Sample Student Response #4: Rubric Score 2

Annotation: This response demonstrates conceptual understanding and analysis of the problem. The student has arrived at the correct dimensions of "(1500 x 900 x 1200) yds." Work provides a complete explanation of an appropriate strategy with the student applying proportion to each of the sides. Justification is absent. Compare to Anchor Paper #5.

Score for Sample Student Response #5: Rubric Score 3

Annotation: This response demonstrates a complete understanding and analysis of the problem. The student states the correct dimensions: "the side in front of you is 1200 yds long, the angled path is 1500 yds long, and the shorter side is 900 yds long." The student describes an appropriate strategy that maintains the ratio: "Because the smaller triangle has a perimeter of 1200 and the larger triangle a perimeter of 3600 the larger path must be 3 times larger than the smaller path. So, by multiplying the values of the sides of the small triangle by 3 and translating which sides go where you find the value of the sides." Justification for the strategy is provided in the statement, "The smaller path and larger path are both perimeters of like triangles." The student uses the term "like" rather than "similar" and then clarifies with the statement, "The triangles share a shape and angles, but not size." Compare to Anchor Paper #7.

Score for Sample Student Response #6: Rubric Score 1

Annotation: This response demonstrates a minimal understanding and analysis of the problem. The student has provided the incorrect dimensions and an explanation of an inappropriate strategy, "...1300 yds, 1200 yds, and 1100. I found thes because the two triangles are similar which would mean that the differences in the sides are the same. So I figured out a way to make the sides of the large triangle add up to 3,600 and keep the same difference in the sides that the small triangle had." The student has accounted for the 3600-yard perimeter without maintaining the ratio, a serious flaw in the reasoning to solve this problem. No justification is present. Compare to Anchor Paper #1.

Score for Sample Student Response #7: Rubric Score 2

Annotation: This response demonstrates conceptual understanding and analysis of the problem. The correct dimensions, "a=900yds, b=1200yds, and c=1500 yds," were found by applying an appropriate strategy. A complete explanation of the strategy is provided: "One complete lap around the smaller Δ was 1,200. One complete lap around the bigger lap was 3,600. 1,200 (the smaller Δ) x 3 = 3,600 (the larger Δ). Now all I did was multiplied the 3 dimensions on the smaller Δ by 3 and got the larger dimensions." No justification is provided. Compare to Anchor Paper #3.

Score for Sample Student Response #8: Rubric Score 3

Annotation: This response demonstrates complete understanding and analysis of the problem. The student has arrived at the correct dimensions, "...one side of the triangle is 1500 yds, one is 1200 yds, and one is 900 yds," using an appropriate strategy. The explanation fully supports the strategy to find the variable by which the sides of the smaller triangle should be multiplied to arrive at the desired 3600-yard perimeter. Full justification is present in the statement, "I knew that these two triangles were similar. This means that they have the same proportions." Compare to Anchor Paper #6.

Score for Sample Student Response #9: Rubric Score 2

Annotation: This response demonstrates conceptual understanding and analysis of the problem. The correct dimensions, "900 yds, 1200 yds, & 1500 yds," were found "if you triple the smaller triangle dimensions they all add up to 3600 the same as the larger triangle" — an appropriate strategy using the correct scale factor. No justification is provided. Compare to Anchor Paper #3.

Score for Sample Student Response #10: Rubric Score 1

Annotation: This response demonstrates a minimal understanding and analysis of the problem. The student arrives at incorrect dimensions and provides an explanation of an inappropriate strategy: "The dimensions of the second path is 1200. I found this by first adding up all the dimensions on the other path just to see what they were since their similar paths, then I divided 3600 by three and got 1200." The student does not apply the scale factor of three to the individual sides. Accounting for the 3600-yard perimeter without maintaining the ratio reveals a serious flaw in reasoning. No justification is present. Compare to Anchor Paper #1.

Score for Sample Student Response #11: Rubric Score 3

Annotation: This response demonstrates a complete understanding and analysis of the problem. The student has arrived at the correct dimensions of the triangle, "900 x 1200 yds on the legs of the triangle and 1500 yds as the hypotenuse." The student has utilized an appropriate strategy of applying the perimeter ratio to each side of the small triangle. Full justification is provided in the statement, "...the ratio between the perimeter of two similar triangles has the same ratio as one side, to its corresponding side." Compare to Anchor Paper #7.

Score for Sample Student Response #12: Rubric Score 1

Annotation: This response demonstrates a minimal understanding and analysis of the problem. The student has provided the incorrect dimensions, "...1200 yds, 1100 yds, and 1300 yds." Explanation reveals an inappropriate strategy: "...It is first necessary to find the distance of a complete lap around the smaller triangle. When this is found by subtracting the larger triangle distance, and the smaller triangle distance this gives the difference between the 2 triangles. Then, if you multiply (here, the student actually divided) this number by 3, this gives you the number needed to add each dimension by." The student has accounted for the 3600-yard perimeter without maintaining the ratio, a serious flaw in reasoning. No justification is present. Compare to Anchor Paper #1.

Score for Sample Student Response #13: Rubric Score 2

Annotation: This response demonstrates conceptual understanding and analysis of the problem. The correct dimensions, "1200 yds, 1500 yds, and 900 yds," were found by applying an appropriate strategy using the correct scale factor: "The perimeter of the larger triangle divided by the perimeter of the smaller one (3600 ÷ 1200) equals 3. So I multiplied each side of the smaller triangle by three to find my answer." No justification is provided. Compare to Anchor Paper #3.

Score for Sample Student Response #14: Rubric Score 3

Annotation: This response demonstrates a complete understanding and analysis of the problem. The student's verbal explanation supports a reasonable strategy to arrive at the correct dimensions. The student's explanation reveals that the student set up a proportion for each of the sides, using the ratio of the two perimeters. Justification is provided in the statement, "Because the triangles were similar, I could find all the lengths by using ratios of similitude." Compare to Anchor Paper #5.

Score for Sample Student Response #15: Rubric Score 1

Annotation: This response demonstrates minimal understanding and analysis of the problem. The student has arrived at incorrect dimensions: "The height is 600yds, the base is 800 yds, and the hypotinas is 1,000 yds." The explanation reveals that the student applied an inappropriate strategy, "...the bigger triangle is two times the size of the smaller triangle...so I multiplied the dimensions of the smaller triangle to get the bigger triangles dimension," does not account for the 3600-yard perimeter. The attempt to maintain the ratio demonstrates minimal understanding. The student has not provided any justification. Compare to Anchor Paper #2.

Anchor Papers used in scoring