Select Program Area --Select Program Area-- DESE HOME Accountability, Partnership, & Assistance Adult & Community Learning Amazing Educators BESE Advisory Councils Board of Elementary & Secondary Education Career/Vocational Technical Education Charter Schools College and Career Readiness Compliance/Monitoring (PQA) Conferences, Workshops and Trainings Instructional Support Digital Learning District & School Turnaround District Review, Analysis, & Assistance Tools Educator Evaluation Educator Licensure Tests (MTEL) Educator Licensure Educational Proficiency Plan (EPP) Edwin ELAR Log In Employment Opportunities: DESE English Language Learners Every Student Succeeds Act (ESSA) Family Literacy Federal Grant Programs High School Equivalency (HSE) Testing Program Grants/Funding Opportunities Information Services Laws & Regulations Literacy LEAP Project MCAS MCAS Appeals METCO Office for Food and Nutrition Programs Performance Assessment for Leaders (PAL) Planning and Research Professional Development RETELL Safe and Supportive Schools School and District Profiles/Directory School Finance School Redesign Science, Technology Engineering, and Mathematics (STEM) Security Portal | MassEdu Gateway Special Education Special Education Appeals Special Education in Institutional Settings Statewide System of Support Student and Family Support Systems for Student Success (SfSS) Title I Part A
 Students & Families Educators & Administrators Teaching, Learning & Testing Data & Accountability Finance & Funding About the Department Education Board

# Massachusetts Comprehensive Assessment System

 Question 17: 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. 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 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, 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.

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