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Massachusetts Comprehensive Assessment System

 Question 21: Open-Response Reporting Category: Number and QuantityStandard: 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.,√(32 - 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.,π2). 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. Vivian and William are approximating the values of square roots and cube roots.Vivian approximated the value of . What is the value of  to the nearest whole number? Show or explain how you got your answer.William approximated the value of . What is the value of  to the nearest whole number? Show or explain how you got your answer.Vivian approximated the value of  to be 5.8. William found a closer approximation to the value of  than Vivian did.What could be the approximation that William found? Show or explain how you got your answer.Vivian approximated the value of to be 10.0. William claimed that 10.1 is a closer approximation to the value of  than Vivian's approximation.Whose approximation, Vivian's or William's, is closer to the value of ? Explain your reasoning.

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

ScoreDescription
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. The student approximates an expression to a greater degree of accuracy than a given approximation of the expression.
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. 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. 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.
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 to merit any points.

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

Question 17:
Question 20:
Question 21:
Question 36:
Question 41:
Question 42: