Massachusetts Comprehensive Assessment System
2012, Grade 10 Mathematics
Question 41: OpenResponse 

Reporting Category: Patterns, Relations, and Algebra Standard: 10.P.1  Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative, recursive (e.g., Fibonacci Numbers), linear, quadratic, and exponential functional relationships. (AI.P.1)

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.  What is the number of circles in figure 5? Show or explain how you got your answer.
 What is the diameter, in inches, of each circle in figure 5? Show or explain how you got your answer.
 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.
 Write an algebraic expression that could be used to determine the number of circles in figure n.
 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.
Score  Description 
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: April 28, 2017
