Massachusetts Comprehensive Assessment System
2011, Grade 8 Mathematics
Question 9: OpenResponse 

Reporting Category: Patterns, Relations, and Algebra Standard: 8.P.1  Extend, represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic expressions. Include arithmetic and geometric progressions, e.g., compounding.

Chris used triangles to make the first four steps of a pattern, as shown below.
The pattern continues to grow. What will be the total number of triangles in Step 5 of the pattern? Show or explain how you got your answer.
 Write an expression in terms of n that represents the total number of triangles in Step n of the pattern.
 Use your expression from part (b) to find the total number of triangles in Step 10 of the pattern. Show or explain how you got your answer.
 Chris made one step of the pattern that had 15 more triangles than the step before it. What step of the pattern had 15 more triangles than the step before it? 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.
Score  Description 
4 
The student response demonstrates an exemplary understanding of the Patterns, Relations, and Algebra concepts involved in extending, representing, analyzing, and generalizing a variety of patterns with tables, graphs, words, and symbolic expressions. The student continues a pattern, writes a rule for the pattern, and solves problems involving the pattern. 
4 
3 
The student response demonstrates a good understanding of the Patterns, Relations, and Algebra concepts involved in extending, representing, analyzing, and generalizing a variety of patterns with tables, graphs, words, and symbolic expressions. 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 extending, representing, analyzing, and generalizing a variety of patterns with tables, graphs, words, and symbolic expressions. 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 extending, representing, analyzing, and generalizing a variety of patterns with tables, graphs, words, and symbolic expressions. 
0 
The student response contains insufficient evidence of an understanding of the Patterns, Relations, and Algebra concepts involved in extending, representing, analyzing, and generalizing a variety of patterns with tables, graphs, words, and symbolic expressions to merit any points. 
Note: There are 2 sample student responses for Score Point 4.
Grade 8 Mathematics
 Question 9: Patterns, Relations, and Algebra
 Question 21: Number Sense and Operations
Return to the MCAS 2011 Student Work Directory
Last Updated: December 23, 2015
