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Massachusetts Comprehensive Assessment System

2012, Grade 10 Mathematics

  Question 41: Open-Response calculator allowed
The table below shows figures composed of circles. The number of circles in each figure and the diameter of each circle in each figure follow a pattern, as shown.
Table with four rows. The first row has four cells titled: Figure 1, Figure 2, Figure 3, and Figure 4. The second row has four cells. The first cell has a picture of one circle. The second cell has a picture of four circles.  The third cell has a picture of sixteen circles. The fourth cell has a picture of sixty four circles. The third row has five cells and the first cell is titled Number of Circles.  The second, third, fourth and fifth cells have the numbers 1, 4, 16, and 64. The fourth row has five cells. The first cell is titled Diameter of Each Circle (in inches). The second, third, fourth, and fifth rows have the numbers 1, ½, ¼, and 1/8.
  1. What is the number of circles in figure 5? Show or explain how you got your answer.
  2. What is the diameter, in inches, of each circle in figure 5? Show or explain how you got your answer.
  3. What is the ratio of the number of circles in figure 6 to the number of circles in figure 7? Show or explain how you got your answer.
  4. Write an algebraic expression that could be used to determine the number of circles in figure n.
  5. Write an algebraic expression that could be used to determine the diameter, in inches, of each circle in figure n.

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 Patterns, Relations, and Algebra concepts involved in completing, extending, and generalizing a pattern. The student determines the number of circles in a figure and the diameter of each circle and creates algebraic expressions to represent each pattern.
4
3 The student response demonstrates a good understanding of the Patterns, Relations, and Algebra concepts involved in completing, extending, and generalizing a pattern. 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 fair understanding of the Patterns, Relations, and Algebra concepts involved in completing, extending, and generalizing a pattern. 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 only minimal understanding of the Patterns, Relations, and Algebra concepts involved in completing, extending, and generalizing a pattern.
0 The student response contains insufficient evidence of an understanding of the Patterns, Relations, and Algebra concepts involved in completing, extending, and generalizing a pattern to merit any points.

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

Grade 10 Mathematics

Question 17: Patterns, Relations, and Algebra
Question 20: Data Analysis, Statistics, and Probability
Question 21: Number Sense and Operations
Question 31: Data Analysis, Statistics, and Probability
Question 41: Patterns, Relations, and Algebra
Question 42: Measurement

Return to the MCAS 2012 Student Work Directory



Last Updated: January 22, 2016
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