Early Christian Fathers

MASSACHUSETTS
URRICULUM FRAMEWORK C
FOR
MATHEMATICS

Grades Pre-Kindergarten to 12

Incorporating the
Common Core State Standards for Mathematics



March 2011










This document was prepared by the
Massachusetts Department of Elementary and Secondary Education
Mitchell D. Chester, Ed. D., Commissioner


Board of Elementary and Secondary Education Members
Ms. Maura Banta, Chair, Melrose
Ms. Harneen Chernow, Vice Chair, Jamaica Plan
Dr. Vanessa Calderon-Rosado, Milton
Mr. Gerald Chertavian, Cambridge
Mr. Michael D’Ortenzio, Jr., Chair, Students Advisory Council, Wellesley
Ms. Beverly Holmes, Springfield
Dr. Jeffrey Howard, Reading
Ms. Ruth Kaplan, Brookline
Dr. James McDermott, Eastham
Dr. Dana Mohler-Faria, Bridgewater
Mr. Paul Reville, Secretary of Education, Worcester

Mitchell D. Chester, Ed.D., Commissioner and Secretary to the Board


This document was adopted by the Massachusetts Board of Elementary and Secondary Education
on December 21, 2010.


The Massachusetts Department of Elementary and Secondary Education, an affirmative action employer, is
committed to ensuring that all of its programs and facilities are accessible to all members of the public.
We do not discriminate on the basis of age, color, disability, national origin, race, religion, sex, or sexual orientation.
Inquiries regarding the Department’s compliance with Title IX and other civil rights laws may be directed to the
Human Resources Director, 75 Pleasant St., Malden, MA, 02148, 781-338-6105.



© 2011 Massachusetts Department of Elementary and Secondary Education
Permission is hereby granted to copy any or all parts of this document for non-commercial educational purposes.
Please credit the “Massachusetts Department of Elementary and Secondary Education.”

This document printed on recycled paper



Massachusetts Department of Elementary and Secondary Education
75 Pleasant Street, Malden, MA 02148-4906
Phone 781-338-3000 TTY: N.E.T. Relay 800-439-2370
www.doe.mass.edu


TABLE OF CONTENTS
Commissioner’s Letter ............................................................................................................. ii
Acknowledgements .......................................................................................... iii

Introduction ................................................................ 1

Guiding Principles for Mathematics Programs in Massachusetts .......................... 7

The Standards for Mathematical Practice ............................................................... 13

The Standards for Mathematical Content
Pre-Kindergarten–Grade 8
Introduction ................................................................................................................ 21
Pre-Kindergarten ........................ 23
Kindergarten ............................................................................... 26
Grade 1 ....................................................... 30
Grade 2 ............................................. 34
Grade 3 ............. 38
Grade 4 ...................................................................................... 43
Grade 5 ............. 48
Grade 6 ............................................. 53
Grade 7 ....................................................... 59
Grade 8 ............................................ 65
High School
Conceptual Categories
Introduction ........................................................................................................... 73
Number and Quantity ............................................................ 75
Algebra ................................. 79
Functions .............................................................................. 85
Modeling ........................................................................................ 90
Geometry ........................ 92
Statistics and Probability ....................................................................................... 98
High School Model Pathways and Model Courses
Introduction .................................................................................. 105
Model Traditional Pathway
Model Algebra I ............................................................. 108
Model Geometry .................................... 116
Model Algebra II ..... 123
Model Integrated Pathway
Model Mathematics I ..................................................... 129
Model Mathematics II ................................................................ 137
Model Mathematics III .................... 147
Model Advanced Courses
Model Precalculus ......................................................................................... 155
Model Advanced Quantitative Reasoning ..................................................... 161

Application of Common Core State Standards for
English Language Learners and Students with Disabilities .................................................. 167
Glossary: Mathematical Terms, Tables, and Illustrations ..................... 173
Tables and Illustrations of Key Mathematical Properties, Rules, and Number Sets .. 183
Sample of Works Consulted ................................................................................................. 187
Massachusetts Curriculum Framework for Mathematics, March 2011 i


Massachusetts Department of
Elementary and Secondary Education

75 Pleasant Street, Malden, Massachusetts 02148-4906 Telephone: (781) 338-3000
TTY: N.E.T. Relay 1-800-439-2370
Mitchell D. Chester, Ed. D., Commissioner


March 2011

Dear Colleagues,

I am pleased to present to you the Massachusetts Curriculum Framework for Mathematics, adopted by
the Board of Elementary and Secondary Education in December 2010. This framework merges the
Common Core State Standards for Mathematics with additional Massachusetts standards and other
features. These pre-kindergarten to grade 12 standards are based on research and effective practice,
and will enable teachers and administrators to strengthen curriculum, instruction, and assessment.

In partnership with the Department of Early Education and Care (EEC), we supplemented the Common
Core State Standards with pre-kindergarten standards that were collaboratively developed by early
childhood educators from the Department of Elementary and Secondary Education, EEC mathematics
staff, and early childhood specialists across the state. These pre-kindergarten standards lay a strong,
logical foundation for the kindergarten standards. The pre-kindergarten standards were approved by the
Board of Early Education and Care in December 2010.

The comments and suggestions received during revision of the 2000 Massachusetts Mathematics
Framework, as well as comments on the Common Core State Standards, have strengthened this
framework. I want to thank everyone who worked with us to create challenging learning standards for
Massachusetts students. I am proud of the work that has been accomplished.

We will continue to work with schools and districts to implement the 2011 Massachusetts Curriculum
Framework for Mathematics over the next several years, and we encourage your comments as you use it.
All Massachusetts frameworks are subject to continuous review and improvement, for the benefit of the
students of the Commonwealth.

Thank you again for your ongoing support and for your commitment to achieving the goals of improved
student achievement for all students.

Sincerely,


Mitchell D. Chester, Ed.D.
Commissioner of Elementary and Secondary Education
ii Massachusetts Curriculum Framework for Mathematics, March 2011 ACKNOWLEDGEMENTS
The 2011 Massachusetts Curriculum Framework for Mathematics is the result of the contributions of
many educators across the state. The Department of Elementary and Secondary Education wishes to
thank all of the Massachusetts groups that contributed to the development of these mathematics
standards and all of the individual teachers, administrators, mathematicians, mathematics education
faculty, and parents who took the time to provide thoughtful comments during the public comment
periods.

Lead Writers, Common Core State Standards for Mathematics
Phil Daro, Senior Fellow, America's Choice
William McCallum, Ph.D., University Distinguished Professor and Head, Department of Mathematics,
University of Arizona; Mathematics Consultant, Achieve
Jason Zimba, Ph.D., Professor of Physics and Mathematics, and the Center for the Advancement of
Public Action, Bennington College; Co-founder, Student Achievement Partners

Lead Writers, Massachusetts Department of Elementary and Secondary Education,
2011 Massachusetts Curriculum Framework for Mathematics
Barbara Libby, Director, Office for Mathematics, Science and Technolog