Patterning and Algebra

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  • Kindergarten to Grade 3

    Patterning and Algebra

  • Every effort has been made in this publication to identify mathematics resources andtools (e.g., manipulatives) in generic terms. In cases where a particular product is usedby teachers in schools across Ontario, that product is identified by its trade name, in theinterests of clarity. Reference to particular products in no way implies an endorsementof those products by the Ministry of Education.

    Printed on recycled paper

    ISBN 978-1-4249-4589-4

    07-006

    Queens Printer for Ontario, 2007

    Ministry of Education

  • Contents

    Une publication quivalente est disponible en franais sous le titre suivant :Guide denseignement efficace des mathmatiques, de la maternelle la 3e anne Modlisation et algbre.

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

    Purpose and Features of the Document . . . . . . . . . . . . . . . . . . . . . . . 2

    Big Ideas in the Curriculum for Kindergarten to Grade 3 . . . . . . . . . 2

    The Big Ideas in Patterning and Algebra . . . . . . . . . . . . . . . . . . . . 4

    Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    General Principles of Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    Working Towards Equitable Outcomes for Diverse Students . . . . . . . . 6

    Patterns and Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    Experiences With a Variety of Patterns . . . . . . . . . . . . . . . . . . . . . . . . 9

    Growing and Shrinking Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    Number Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    Characteristics of Student Learning and Instructional Strategies by Grade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

    Kindergarten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Grade 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Grade 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Grade 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    Expressions and Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

    Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

    Characteristics of Student Learning and Instructional Strategies by Grade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    Kindergarten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Grade 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

  • iv A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 3 Patterning and Algebra

    Grade 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Grade 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

    References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    Learning Activities for Patterning and Algebra . . . . . . . . . . . . . . . . 39

    Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41The Mathematical Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Accommodations and Modifications . . . . . . . . . . . . . . . . . . . . . . . 44A Special Note About Kindergarten . . . . . . . . . . . . . . . . . . . . . . . . 47

    Appendix A: Kindergarten Learning Activities . . . . . . . . . . . . . . . . 49

    Patterns and Relationships: Colourful Caterpillars . . . . . . . . . . . . . . . . 51Blackline master: PRK.BLM1

    Expressions and Equality: What a Handful! . . . . . . . . . . . . . . . . . . . . . 57Blackline masters: EEK.BLM1 EEK.BLM6

    Appendix B: Grade 1 Learning Activities . . . . . . . . . . . . . . . . . . . . . 63

    Patterns and Relationships: Dragon Tail Patterns . . . . . . . . . . . . . . . . . 65Blackline masters: PR1.BLM1 PR1.BLM3

    Expressions and Equality: Balance It! . . . . . . . . . . . . . . . . . . . . . . . . . 73Blackline masters: EE1.BLM1 EE1.BLM5

    Appendix C: Grade 2 Learning Activities . . . . . . . . . . . . . . . . . . . . . 79

    Patterns and Relationships: Growing Ts . . . . . . . . . . . . . . . . . . . . . . . 81Blackline masters: PR2.BLM1 PR2.BLM3

    Expressions and Equality: Combo Quest . . . . . . . . . . . . . . . . . . . . . . . 91Blackline masters: EE2.BLM1 EE2.BLM3

    Appendix D: Grade 3 Learning Activities . . . . . . . . . . . . . . . . . . . . 99

    Patterns and Relationships: Pattys Pattern Machines . . . . . . . . . . . . . 101Blackline masters: PR3.BLM1 PR3.BLM3

    Expressions and Equality: Zeros and Ones Game . . . . . . . . . . . . . . . . 111Blackline masters: EE3.BLM1 EE3.BLM6

    Appendix E: Correspondence of the Big Ideas and the CurriculumExpectations in Patterning and Algebra . . . . . . . . . . . . . . . . . . . . . 119

    Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

  • 1

    This document is a practical guide that teachers will find useful in helping

    students achieve the curriculum expectations for mathematics outlined in

    The Kindergarten Program, 2006 (on page 47, under the subheading Patterning)

    and the expectations outlined in the Patterning and Algebra strand for Grades 1

    to 3 in The Ontario Curriculum, Grades 18: Mathematics, 2005. It is a companion

    document to A Guide to Effective Instruction in Mathematics, Kindergarten to

    Grade 6, 2006.

    The expectations outlined in the curriculum documents describe the knowledge

    and skills that students are expected to acquire by the end of each grade. In

    Early Math Strategy: The Report of the Expert Panel on Early Math in Ontario

    (Expert Panel on Early Math in Ontario, 2003), effective instruction is identified

    as critical to the successful learning of mathematical knowledge and skills, and

    the components of an effective program are described. As part of the process of

    implementing the panels vision of effective mathematics instruction for Ontario,

    A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6, 2006

    provides a framework for teaching mathematics. This framework includes specific

    strategies for developing an effective program and for creating a community of

    learners in which students mathematical thinking is nurtured. The strategies

    described in the guide focus on the big ideas inherent in the expectations;

    on problem solving as the main context for mathematical activity; and on

    communication, especially student talk, as the conduit for sharing and developing

    mathematical thinking. The guide also provides strategies for assessment, the use

    of manipulatives, and home connections.

    Introduction

  • 2 A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 3 Patterning and Algebra

    Purpose and Features of the DocumentThe present document was developed to provide practical applications of the

    principles and theories behind good instruction that are elaborated in A Guide

    to Effective Instruction in Mathematics, Kindergarten to Grade 6, 2006.

    The present document provides:

    an overview of each of the big ideas in the Patterning and Algebra strand;

    four appendices (Appendices AD), one for each grade from Kindergarten

    to Grade 3, which provide learning activities that introduce, develop, or help

    to consolidate some aspect of each big idea. These learning activities reflect

    the instructional practices recommended in A Guide to Effective Instruction in

    Mathematics, Kindergarten to Grade 6, 2006;

    an appendix (Appendix E) that lists the curriculum expectations in the Patterning

    and Algebra strand under the big idea to which they correspond. This clustering

    of expectations around each of the two big ideas allows teachers to concentrate

    their programming on the big ideas of the strand while remaining confident

    that the full range of curriculum expectations is being addressed;

    a glossary that provides definitions of mathematical terms used in this document.

    Big Ideas in the Curriculum for Kindergarten to Grade 3In developing a mathematics program, it is vital to concentrate on important

    mathematical concepts, or big ideas, and the knowledge and skills that go

    with those concepts. Programs that are organized around big ideas and focus

    on problem solving provide cohesive learning opportunities that allow students

    to explore concepts in depth.

    All learning, especially new learning, should be embedded in well-chosen

    contexts for learning that is, contexts that are broad enough to allow

    students to investigate initial understandings, identify and develop relevant

    supporting skills, and gain experience with varied and interesting applica-

    tions of the ne