Massachusetts Mathematics Curriculum Framework — 2017

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MATHEMATICSGrades Pre-Kindergarten to 12iMassachusettsCurriculumFramework –2017

This document was prepared by the Massachusetts Department of Elementary and Secondary EducationBoard of Elementary and Secondary Education MembersMr. Paul Sagan, Chair, CambridgeMr. James Morton, Vice Chair, BostonMs. Katherine Craven, BrooklineDr. Edward Doherty, Hyde ParkDr. Roland Fryer, CambridgeMs. Margaret McKenna, BostonMr. Michael Moriarty, HolyokeDr. Pendred Noyce, BostonMr. James Peyser, Secretary of Education, MiltonMs. Mary Ann Stewart, LexingtonMr. Nathan Moore, Chair, Student Advisory Council,ScituateMitchell D. Chester, Ed.D., Commissioner and Secretary to the BoardThe Massachusetts Department of Elementary and Secondary Education, an affirmative action employer, iscommitted to ensuring that all of its programs and facilities are accessible to all members of the public. We donot 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 tothe Human Resources Director, 75 Pleasant St., Malden, MA, 02148, 781-338-6105. 2017 Massachusetts Department of Elementary and Secondary Education.Permission is hereby granted to copy any or all parts of this document for non-commercial educationalpurposes. Please credit the “Massachusetts Department of Elementary and Secondary Education.”Massachusetts Department of Elementary and Secondary Education75 Pleasant Street, Malden, MA 02148-4906Phone 781-338-3000 TTY: N.E.T. Relay 800-439-2370www.doe.mass.edu

Massachusetts Department ofElementary and Secondary Education75 Pleasant Street, Malden, Massachusetts 02148-4906Dear Colleagues,I am pleased to present to you the Massachusetts Curriculum Framework for Mathematics adopted by the Board ofElementary and Secondary Education in March 2017. This Framework builds upon the foundation of the 2010Massachusetts Curriculum Framework for Mathematics as well as versions of the Massachusetts Mathematics Frameworkpublished since 1995.The current Framework incorporates improvements suggested by Massachusetts educators after six years of experience inimplementing the 2010 standards in their classrooms. These revised pre-kindergarten to grade 12 standards are based onresearch and effective practice, and will enable teachers and administrators to strengthen curriculum, instruction, andassessment.The 2017 standards draw from the best of prior Massachusetts standards, and represent the input of hundreds of theCommonwealth’s K–12 and higher education faculty. The 2017 standards present the Commonwealth’s commitment toproviding all students with a world-class education.This revision of the Framework retains the strengths of previous frameworks and includes these improved features: Increased coherence across the grades and improved clarity of mathematical terms and language to describeexpectations for students.Clear expectations for student mastery of basic addition, subtraction, multiplication, and division facts.An enhanced high school section that includes: 1) clearer model course standards; and 2) guidance on makingdecisions for course sequences and the model Algebra I course, along with options for various course-takingpathways.Guidance for moving students into an Algebra I course by grade 8 and through Calculus in high school; andMore detailed descriptions about rigor and aspirations for students with a stronger emphasis on how the contentstandards, the standards for mathematical practice, and the guiding principles prepare students for college,careers, and civic participation.In the course of revising these standards, the Department received many valuable comments and suggestions. I want tothank everyone who contributed their suggestions and ideas to make these revised standards useful for educators,students, families, and the community. In particular, I am grateful to the members of the Mathematics Standards ReviewPanel and our Content Advisors for giving their time generously to the project to improve the learning standards forMassachusetts students. I am proud of the work that has been accomplished.We will continue to collaborate with educators to implement the 2017 Massachusetts Curriculum Framework forMathematics.Thank you again for your ongoing support and for your commitment to achieving the goal of improved studentachievement for all our students.Sincerely,Mitchell D. Chester, Ed. D.Commissioner of Elementary and Secondary Education

Table of ContentsAcknowledgments .5Introduction .8Guiding Principles for Mathematics Programs in Massachusetts. 14The Standards for Mathematical Practice . 16Standards for Mathematical Content Pre-Kindergarten–Grade 8 . 19Organization of the Pre-Kindergarten to Grade 8 Content Standards . 20Pre-Kindergarten . 22Introduction . 22Pre-Kindergarten Overview . 23Pre-Kindergarten Content Standards . 24Kindergarten . 25Introduction . 25Kindergarten Overview . 26Kindergarten Content Standards . 27Grade 1 . 29Introduction . 29Grade 1 Overview . 30Grade 1 Content Standards . 31Grade 2 . 33Introduction . 33Grade 2 Overview . 34Grade 2 Content Standards . 35Grade 3 . 37Introduction . 37Grade 3 Overview . 38Grade 3 Content Standards . 39Grade 4 . 42Introduction . 42Grade 4 Overview . 43Grade 4 Content Standards . 44Grade 5. 48Massachusetts Curriculum Framework for Mathematics1

Introduction . 48Grade 5 Overview . 49Grade 5 Content Standards . 50Grade 6 . 54Introduction . 54Grade 6 Overview . 56Grade 6 Content Standards . 57Grade 7 . 61Introduction . 61Grade 7 Overview . 62Grade 7 Content Standards . 63Grade 8 . 67Introduction . 67Grade 8 Overview . 68Grade 8 Content Standards . 69The High School Standards for Mathematical Content . 72Introduction . 73High School Content Standards by Conceptual Categories . 76Conceptual Category: Number and Quantity [N] . 77Introduction . 77Conceptual Category: Number and Quantity Overview [N] . 78Conceptual Category: Number and Quantity Content Standards [N] . 79Conceptual Category: Algebra [A] . 81Introduction . 81Conceptual Category: Algebra Overview [A] . 83Conceptual Category: Algebra Content Standards [A] . 84Conceptual Category: Functions [F] . 87Introduction . 87Conceptual Category: Functions Overview [F] . 88Conceptual Category: Functions Content Standards [F]. 89Conceptual Category: Modeling [ ] . 92Introduction . 92Conceptual Category: Geometry [G] . 94Massachusetts Curriculum Framework for Mathematics2

Introduction . 94Conceptual Category: Geometry Overview [G] . 96Conceptual Category: Geometry Content Standards [G] . 97Conceptual Category: Statistics and Probability [S] . 100Introduction . 100Conceptual Category: Statistics and Probability Overview [S] . 101Conceptual Category: Statistics and Probability Content Standards [S]. 102The Standards for Mathematical Content High School:Model Pathways and Model Courses. 105Introduction . 106Model Traditional Pathway: Model Algebra I [AI] . 108Introduction . 108Model Traditional Pathway: Model Algebra I Overview [AI] . 109Model Traditional Pathway: Model Algebra I Content Standards [AI] . 110Model Traditional Pathway: Model Geometry [GEO] . 115Introduction . 115Model Traditional Pathway: Model Geometry Overview [GEO] . 117Model Traditional Pathway: Model Geometry Content Standards [GEO] . 118Model Traditional Pathway: Model Algebra II [AII] . 122Introduction . 122Model Traditional Pathway: Model Algebra II Overview [AII] . 123Model Traditional Pathway: Model Algebra II Content Standards [AII] . 125Model Integrated Pathway: Model Mathematics I [MI] . 129Introduction . 129Model Integrated Pathway: Model Mathematics I Overview [MI] . 131Model Integrated Pathway: Model Mathematics I Content Standards [MI]. 133Model Integrated Pathway: Model Mathematics II [MII] . 137Introduction . 137Model Integrated Pathway: Model Mathematics II Overview [MII] . 138Model Integrated Pathway: Model Mathematics II Content Standards [MII] . 140Model Integrated Pathway: Model Mathematics III [MIII] . 146Introduction . 146Model Integrated Pathway: Model Mathematics III Overview [MIII] . 147Massachusetts Curriculum Framework for Mathematics3

Model Integrated Pathway: Model Mathematics III Content Standards [MIII] . 149Model Advanced Course: Model Precalculus [PC] . 154Introduction . 154Model Advanced Course: Model Precalculus Overview [PC] . 155Model Advanced Course: Model Precalculus Content Standards [PC]. 156Model Advanced Course: Model Advanced Quantitative Reasoning

Nancy Johnson, 7–12 Mathematics Teacher and Department Head, Hopedale Jr.-Sr. High School, Hopedale Public Schools (retired); President, Association of Teachers of Mathematics in Massachusetts Patty Juranovits, Supervisor of Mathematics, K–12, Haverhill Public Schools Elizabeth Kadra, Grades 7 & 8 Mathematics Teacher,

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