Ministry of Education
                                                                                                                                                                                                     REVISED
                                                                                                                         TheOntario Curriculum
                                                                                                                         Grades1-8
                                                                                                                                            Mathematics
                                              Printed on recycled paper
                                                   04-163
                                               ISBN 0-7794-8122-4
                                           © Queen’s Printer for Ontario, 2005
                                                                                                                         2005                                                  ISBN 0-7794-8121-6 (Print)
                                                                                                                                                                               ISBN 0-7794-8122-4 (Internet)
                    Contents
                                             Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     3
                                             The Importance of Mathematics  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                 3
                                             Principles Underlying the Ontario Mathematics Curriculum  . . . . . . . . . . . . . . . . . . . .                                3
                                             Roles and Responsibilities in Mathematics Education . . . . . . . . . . . . . . . . . . . . . . . . . .                          4
                                             The Program in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                 7
                                             Curriculum Expectations  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .             7
                                             Strands in the Mathematics Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                    8
                                             The Mathematical Processes  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .               11
                                             Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       11
                                             Reasoning and Proving  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .          14
                                             Reflecting  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   14
                                             Selecting Tools and Computational Strategies  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                   14
                                             Connecting  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     16
                                             Representing  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     16
                                             Communicating  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        17
                                             Assessment and Evaluation of Student Achievement . . . . . . . . . . . . . . . . . . . . . . . . . .                            18
                                             Basic Considerations  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       18
                                             The Achievement Chart for Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                     19
                                             Some Considerations for Program Planning in Mathematics  . . . . . . . . . . . . . . . . . . .                                  24
                                             Teaching Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         24
                                             Cross-Curricular and Integrated Learning  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                   26
                                             Planning Mathematics Programs for Exceptional Students . . . . . . . . . . . . . . . . . . . . . . .                            26
                                             English As a Second Language and English Literacy Development (ESL/ELD) . . . . . . . .                                         28
                                             Antidiscrimination Education in Mathematics  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                      28
                   Une publication équivalente est disponible en français sous le 
                                                                          re      e
                   titre suivant : Le curriculum de l’Ontario de la 1       à la 8 année –
                   Mathématiques, 2005.
                   This publication is available on the Ministry of Education’s 
                   website, at http://www.edu.gov.on.ca.
                   Every effort has been made in this publication to identify mathematics resources and tools (e.g., manipulatives) in generic terms. In cases where a
                   particular product is used by teachers in schools across Ontario, that product is identified by its trade name, in the interests of clarity. Reference to
                   particular products in no way implies an endorsement of those products by the Ministry of Education. 
            2                                                                                           THE ONTARIO CURRICULUM, GRADES 1–8: MATHEMATICS
                                              Literacy and Inquiry/Research Skills  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                 29
                                              The Role of Technology in Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                       29
                                              Guidance and Mathematics  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .               30
                                              Health and Safety in Mathematics  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                 30
                                              Curriculum Expectations for Grades 1 to 8
                                              Grade 1  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    31
                                              Grade 2  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    41
                                              Grade 3  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    53
                                              Grade 4  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    64
                                              Grade 5  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    76
                                              Grade 6  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    86
                                              Grade 7  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    97
                                              Grade 8  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  109
                                              Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  120
                                                          3
       Introduction
               This document replaces The Ontario Curriculum,Grades 1–8: Mathematics,1997. Beginning in
               September 2005,all mathematics programs for Grades 1 to 8 will be based on the expectations
               outlined in this document.
               The Importance of Mathematics
               An information- and technology-based society requires individuals who are able to think 
               critically about complex issues,analyse and adapt to new situations,solve problems of various
               kinds,and communicate their thinking effectively.The study of mathematics equips students
               with knowledge,skills,and habits of mind that are essential for successful and rewarding partic-
               ipation in such a society.To learn mathematics in a way that will serve them well throughout
               their lives, students need classroom experiences that help them develop mathematical under-
               standing; learn important facts,skills,and procedures; develop the ability to apply the processes
               of mathematics; and acquire a positive attitude towards mathematics.The Ontario mathematics
               curriculum for Grades 1 to 8 provides the framework needed to meet these goals.
               Learning mathematics results in more than a mastery of basic skills. It equips students with a
               concise and powerful means of communication. Mathematical structures,operations,processes,
               and language provide students with a framework and tools for reasoning,justifying conclusions,
               and expressing ideas clearly.Through mathematical activities that are practical and relevant to
               their lives, students develop mathematical understanding,problem-solving skills,and related
               technological skills that they can apply in their daily lives and,eventually,in the workplace.
               Mathematics is a powerful learning tool. As students identify relationships between mathema-
               tical concepts and everyday situations and make connections between mathematics and other
               subjects,they develop the ability to use mathematics to extend and apply their knowledge in
               other curriculum areas,including science,music,and language.
               Principles Underlying the Ontario Mathematics Curriculum 
               This curriculum recognizes the diversity that exists among students who study mathematics.
               It is based on the belief that all students can learn mathematics and deserve the opportunity 
               to do so. It recognizes that all students do not necessarily learn mathematics in the same way,
               using the same resources,and within the same time frames. It supports equity by promoting
               the active participation of all students and by clearly identifying the knowledge and skills stu-
               dents are expected to demonstrate in every grade. It recognizes different learning styles and 
               sets expectations that call for the use of a variety of instructional and assessment tools and
               strategies. It aims to challenge all students by including expectations that require them to use
               higher-order thinking skills and to make connections between related mathematical concepts
               and between mathematics,other disciplines,and the real world.