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Massachusetts Comprehensive Assessment System

2014, Grade 10 Mathematics

  Question 41: Open-Response calculator allowed
A ladder is leaning against the side of an office building, as shown in the diagram below.
A diagram of a ladder leaning against the side of an office building. The building is in the shape of a vertical rectangle. The ladder is placed against the side of the rectangle and the ladder forms a triangle with the ground. The ladder is labeled x. The base of the triangle formed is labeled 4 feet and the height of the triangle is 16 feet.
The top of the ladder reaches a point on the building that is 16 feet above the ground. The bottom of the ladder is 4 feet from the base of the building.
  1. Write an equation that could be used to find x, the length in feet of the ladder.
  2. Use the equation you wrote in part (a) to find x, the length, to the nearest tenth of a foot, of the ladder. Show or explain how you got your answer.
A second ladder that is 32 feet in length will be leaned against the same building. The bottom of the second ladder will be placed 7 feet from the base of the building.
  1. What is the height, to the nearest tenth of a foot, of the point the top of the second ladder will reach on the building? Show or explain how you got your answer.

Scoring Guide and Sample Student Work
Select a score point in the table below to view the sample student response.

ScoreDescription
4 The student response demonstrates an exemplary understanding of the Geometry concepts involved in using the Pythagorean Theorem to solve right triangles in applied problems. The student uses the Pythagorean Theorem to determine the length of a ladder and the height to which a second ladder reaches on a building.
4
3 The student response demonstrates a good understanding of the Geometry concepts involved in using the Pythagorean Theorem to solve right triangles in applied problems. Although there is significant evidence that the student was able to recognize and apply the concepts involved, some aspect of the response is flawed. As a result the response merits 3 points.
2 The student response demonstrates a fair understanding of the Geometry concepts involved in using the Pythagorean Theorem to solve right triangles in applied problems. While some aspects of the task are completed correctly, others are not. The mixed evidence provided by the student merits 2 points.
1 The student response demonstrates a minimal understanding of the Geometry concepts involved in using the Pythagorean Theorem to solve right triangles in applied problems.
0 The student response contains insufficient evidence of an understanding of the Geometry concepts involved in using the Pythagorean Theorem to solve right triangles in applied problems to merit any points.

Note: There are 2 sample student responses for Score Point 4.

Grade 10 Mathematics

Question 17: Number and Quantity
Question 20: Geometry
Question 21: Statistics and Probability
Question 36: Algebra and Functions
Question 41: Geometry
Question 42: Algebra and Functions

Return to the MCAS 2014 Student Work Directory



Last Updated: July 24, 2017
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