




Massachusetts Comprehensive Assessment System
2015, Grade 10 Mathematics
Question 17: OpenResponse 

Reporting Category: Number and Quantity Standard: 10.N.3  Find the approximate value for solutions to problems involving square roots and cube roots without the use of a calculator, e.g.,√(3^{2}  1) ≈ 2.8. (AI.N.3)
Standard: Mathematics.8.NS.1.02  Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g.,π^{2}). For example, by truncating the decimal expansion of √2 show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

A student is approximating the locations of square roots and cube roots of integers on a number line.  Between which two consecutive integers on the number line is located? Show or explain how you got your answer.
 What is the value of to the nearest tenth? Show or explain how you got your answer.
The value of where m is an integer, is located between 11 and 12 on the number line.  What could be the value of m? Show or explain how you got your answer.
The value of where n is an integer, is also located between 11 and 12 on the number line.  What could be the value of n? 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 Number and Quantity concepts involved in using rational approximations of irrational numbers to compare the size of irrational numbers, locating them approximately on a number line diagram, and estimating the value of expressions. 
4 
3 
The student response demonstrates a good understanding of the Number and Quantity concepts involved in using rational approximations of irrational numbers to compare the size of irrational numbers, locating them approximately on a number line diagram, and estimating the value of 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 a fair understanding of the Number and Quantity concepts involved in using rational approximations of irrational numbers to compare the size of irrational numbers, locating them approximately on a number line diagram, and estimating the value of 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 a minimal understanding of the Number and Quantity concepts involved in using rational approximations of irrational numbers to compare the size of irrational numbers, locating them approximately on a number line diagram, and estimating the value of expressions. 
0 
The student response contains insufficient evidence of an understanding of the Number and Quantity concepts involved in using rational approximations of irrational numbers to compare the size of irrational numbers, locating them approximately on a number line diagram, and estimating the value of expressions 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: Algebra and Functions
 Question 36: Statistics and Probability
 Question 41: Algebra and Functions
 Question 42: Geometry
Return to the MCAS 2015 Student Work Directory
Last Updated: November 3, 2011


