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The LearningPathwayMathematics Instruction and Assessment for Grades K–6

Winnipeg School Division

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© 2012 by Winnipeg School Division

Pages of this publication designated as reproducible with the following icon ( ) may be reproduced under licence from Access Copyright. All other pages may only be reproduced with the express written permission of Portage & Main Press, or as permitted by law.

All rights are otherwise reserved and no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanic, photocopying, scanning, recording, or otherwise, except as specifically authorized.

Portage & Main Press gratefully acknowledges the financial support of the Province of Manitoba through the Department of Culture, Heritage, Tourism & Sport and the Manitoba Book Publishing Tax Credit, and the Government of Canada through the Canada Book Fund (CBF) for our publishing activities.

Printed and bound in Canada by FriesensCover and interior design by Relish New Brand Experience Inc.

Library and Archives Canada Cataloguing in Publication The learning pathway : mathematics instruction and assessment for grades K-6 / Winnipeg School Division.

Includes bibliographical references.ISBN 978-1-55379-358-8

1. Mathematics--Study and teaching (Primary). 2. Mathematics--Study and teaching (Elementary). 3. Mathematical ability--Testing. I. Winnipeg School Division

QA135.5.L38 2012 372.7’044 C2012-907910-3

100-318 McDermot AvenueWinnipeg, MB, Canada R3A 0A2Tel.: 204-987-3500 • Toll free: 1-800-667-9673Toll-free fax: 1-866-734-8477Email: [email protected]

C016245

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This book is dedicated to the teachers, administrators, and consultants in Winnipeg School Division who helped create, pilot, and implement the Mathematics Learning Pathway. Our role was to listen, think, and create a format for math instruction that would support teachers and schools in making a difference. We believe that the experience, questions, and ideas from educators have helped make The Learning Pathway a document that can be used to assist learners in achieving mathematical success. For this we are truly grateful.

—Assessment Steering Committee Winnipeg School Division

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ContentS

Preface vii

Part 1: The Learning Pathway: An Overview 1

Part 2: The Learning Pathway: The Signposts in Detail 11

Appendix A: Some Background Information 61

Appendix B: Planning Sheets 75

Glossary 101

References 109

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PrefaCe

In the field of education, a great deal of research has been done about the best ways to “teach” mathematics in elementary classrooms. This research confirmed for us that what students need to learn has remained virtually unchanged over the past few decades. What has changed, however, is how we present the content to our students. This was reflected in the Western and Northern Canadian Protocol (WNCP) curriculum document in 2006, when outcomes/expectations of different grade levels were re-aligned.

Manitoba, which uses WNCP, adopted the changes and integrated them into the provincial mathematics curriculum in 2008. Since then, teachers have been asking for a tool to help them effectively teach mathematics in a way that ensures both they and their students understand the concepts.

In 2008, we formed a writing team of mathematics teachers in Winnipeg School Division. Informed by international research from the field of mathematics, and using the WNCP curriculum as the foundation, we created The Learning Pathway. The Learning Pathway is the intellectual work of many, and it has been field tested, piloted, and revised according to the voices of our teachers.

The Learning Pathway is not for teachers in Manitoba only. Teachers who live and teach elsewhere can use The Learning Pathway as a bridge between the curriculum and assessment practices. They will find that it guides them in knowing what skills and strategies need to be taught according to a grade-based developmental continuum, and it informs the instructional assessment cycle. The pathway helps teachers plan next steps for instruction by focusing on key growth points that students travel through as they consolidate their content knowledge and develop/learn new strategies. At the same time, teachers will find that The Learning Pathway gives them the opportunity to focus on curricular outcomes/expectations and breaks them down into small manageable “bits.”

You will also find that The Learning Pathway allows your students to take an active role in their own learning. The “I can” statements that are found in each “signpost” help them understand that they can use what they already know to learn new material. As such, students are able to track their growth, as well as set their own goals. The Learning Pathway can also be used to guide student-led conferencing conversations. Further, parents will find it useful for understanding the math their children are learning in Number and Algebra.

As educators, we try to honour learning every day. It is our hope that in The Learning Pathway you will find materials you can use to support your students’ learning in meaningful ways.


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Part 1

The Learning Pathway : An Overview

Introduction

Welcome to The Learning Pathway. This mathematics resource was developed to assist K–6 teachers with the ongoing cycle of planning for assessment and instruction. It was also developed to be a resource for teachers in breaking down the curricular outcomes into a developmental model in response to the implementation of the new Western and Northern Canadian Protocol (WNCP). The Learning Pathway provides information to help you plan for whole-class, small-group, and individual instruction. You will find that using this resource will guide decisions you make about how to differentiate instruction in a meaningful way.

The Learning Pathway is intended as a companion document to your current mathematics instruction and as an assessment tool you can use throughout the school year. The Learning Pathway is based on extensive research (see References), Winnipeg School Division data, curriculum outcomes/expectations, and global developmental stages (see page 2). It is designed to help you do the following:• Plan ongoing assessment to determine your students’ points of strength, as well as plan

next steps for instruction.• Focus on key growth points that students travel through as they consolidate their

content knowledge and apply their learning in real-world contexts.• Plan focused instruction to meet your students’ needs, as well as guide formative

assessment and reporting to parents.• Differentiate your instruction for student learning, according to the math that is just

beyond each student’s current ability.• Highlight the important roles that Knowledge, Application of Knowledge, and

Modelling and Communicating Mathematical Thinking play as students construct their understanding of concepts in mathematics.

• Follow the principles of student global developmental stages of growth in mathematics.• Establish a bridge between actual classroom practice and the curriculum.

The Learning Pathway is divided into two sections. Part 1 provides an overview of how children learn mathematics. In Part 2, you will find detailed information about the developmental stages and 10 signposts your students will be progressing through.

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2 The Learning Pathway Portage & Main Press, 2012, The Learning Pathway, ISBN: 978-1-55379-358-8

Following Part 2 are the appendices and a glossary. Appendix A contains important background information about empty number lines, fractions, mental math, multiplicative thinking, and place value. We suggest you become familiar with the contents of this appendix before using the signposts, and refer back to appendix A whenever necessary. In appendix B, you will find reproducible planning sheets. You can use the planning sheets as is or adapt them to fit your assessment needs and the needs of your students. The glossary includes easy-to-understand definitions, descriptions, explanations, and examples of many of the mathematical terms used in The Learning Pathway.

The poster will provide you and your students with a visual to the 10 signposts. You can also refer to the poster when you want to quickly check the WNCP end-of-grade outcomes/expectations – they are reflected in the statements on the poster.

Keys to Learning Mathematics

The Learning Pathway focuses on two curricular areas: number reasoning (number sense) and algebraic reasoning. Number sense and algebra are the gateway understandings toward learning mathematics (Bobis, Mulligan, and Lowrie 1999; Wright, Martland, and Stafford 2000; Winnipeg School Division). Success in the other strands often depends on the understanding of number and patterns and algebra. For example, measurements (millimetres, centimetres, kilometres, and so on) make far more sense to young students when they understand the patterns in the number system, relative size, base ten and place value, and counting. Students who understand “number” have an easier time transferring that knowledge into helping them understand measurement than do students who do not understand “number.”

Global Developmental Stages

When children are learning mathematics, they move through three developmental stages: (1) counting, (2) part-whole thinking, and (3) proportional reasoning. At each stage, students develop skills, learn strategies, and achieve significant competencies that enable them to make sense of their learning. It is important to understand the difference between skill and strategy. Today’s math curricula are strategy focused – that is, they focus on the “thinking” behind how students solve a problem. This book is explicitly set up with a Knowledge component (which is skill focused) and an Application of Knowledge component (which is strategy focused).

Learning strategies allows students to acquire more sophisticated knowledge; likewise, knowledge leads to the development of more sophisticated strategies. This cycle continues, enabling students to become proficient with numbers in meaningful ways that will last a lifetime. Using drill, rather than learning strategies, may promote speed, but not necessarily accuracy. Drills result in temporary acquisition of facts. Practising math facts is appropriate only after students have acquired known facts through strategic thinking (Manitoba Education, Citizenship and Youth 2009).

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Part 1 The Learning Pathway: An Overview 3Portage & Main Press, 2012, The Learning Pathway, ISBN: 978-1-55379-358-8

Students who learn successful mental addition and subtraction strategies use known facts and compatibles to assist them in mental calculations. These students have many strategies in their repertoire and use them efficiently. They can easily determine which strategy is the most sensible to use based on the context of the problem and the numbers given. This knowledge, along with the students’ ability to apply the knowledge, in turn, allows them to continue onto the next developmental stage. This developmental continuum is similar to continuums found in other disciplines such as English and is instrumental in helping students become proficient in mathematics. When students can model and communicate their mathematical thinking, they are starting to take control of their own learning.

Mathematical proficiency…has five components, or strands (the five strands are interwoven and interdependent in the development of proficiency in mathematics):• conceptual understanding – comprehension of mathematical concepts, operations,

and relations• procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently,

and appropriately• strategic competence – ability to formulate, represent, and solve mathematical

problems• adaptive reasoning – capacity for logical thought, reflection, explanation, and

justification• productive disposition – habitual inclination to see mathematics as sensible, useful,

and worthwhile, coupled with a belief in diligence and one’s own efficacy.

As they go from pre-kindergarten to eighth grade, all students should become increasingly proficient in mathematics. That proficiency should enable them to cope with the mathematical challenges of daily life and enable them to continue their study of mathematics in high school and beyond (Kilpatrick, Swafford, and Findell 2001).

The Learning Pathway is designed for all students in K–6, regardless of their aptitude in mathematics. Most students progress through the three developmental stages at approximately the same age and at the same grade level. There will always be some students who struggle with mathematics and some who are more advanced.

The Learning Pathway is also for all K–6 teachers. Teachers who do not have a mathematics background or who are teaching mathematics for the first time will find Part 2: The Signposts in Detail especially helpful. Teachers with a strong mathematical background may choose to use Part 2 as a reference and as a refresher.

The Significance of Using the Metaphor “Signposts”

As someone travels along a road or highway, he or she passes “signposts” that mark the direction of travel, or the name of or distance to the next town. In The Learning Pathway, the signposts mark each student’s developmental journey on his or her way to mathematical proficiency.

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Global Developmental Stage 1: Counting

Counting is the first major developmental stage young students progress across. Students from kindergarten to the end of grade 2 are usually within this stage. The Learning Pathway identifies five signposts that learners pass through in the Counting stage (see Part 2 for more detailed descriptions of these signposts):

1. Pre-Counter

2. Uses 1:1 Correspondence

3. Uses Manipulatives

4. Uses Visualization

5. Capable Counter

Many students start school with some knowledge of counting. Some students, however, remain counters far into their middle-school years and struggle with grade-level based instruction, because they know how to solve problems by counting only. As practitioners in early-years classrooms, it is vital that teachers provide rich counting experiences for their students. Counting maintains a sharp focus in the curricular outcomes/expectations of the early grades. Students need to learn both “how to count” (oral rote-counting sequences) and how to recognize “what to count.”

Counting strategies can be extremely complex. To be a successful counter, a student has to know more than how to count the number words forward and backward. He or she also has to understand quantity, one-to-one correspondence, cardinality, and order irrelevance. Much research has been devoted to how students develop the ability to be confident counters, some of which is embedded in The Learning Pathway.

While in the Counting stage, students progress from counting objects (including fingers) to pictorial representations of objects, to visualizing the sets of objects, to being able to hold a set of objects in their head and count on. Therefore, once students learn the rote-counting sequence, it is a very significant step for them to be able to identify one item to one verbal count.

Global Developmental Stage 2: Part-Whole Thinking

By the time students enter the second developmental stage, Part-Whole Thinking, they have gone beyond counting to solve problems. This is important, because part-whole thinking is foundational in solidifying a student’s number sense. Students in grades 3 to 5 are usually within this stage. The Learning Pathway identifies four signposts that students move across at the Part-Whole Thinking stage (see Part 2 for more detailed descriptions of these signposts):

1. Uses Additive Reasoning: Beginning

2. Uses Additive Reasoning: Capable

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3. Uses Multiplicative Reasoning: Beginning

4. Uses Multiplicative Reasoning: Capable

According to Young-Loveridge and Wright (2002), “Research evidence supports the idea that students have difficulty reasoning proportionally unless they can use multiplicative part-whole strategies, and find it difficult to reason multiplicatively unless they have a good grasp of additive part-whole strategies.”

Students at this stage of the learning pathway have emerging arithmetical strategies that involve use of part-whole thinking; they no longer rely on counting alone. As part-whole thinking becomes more sophisticated, so too does students’ understanding of place value: they are able to use the structure of the base-ten system more effectively to solve problems.

Successful use of strategies – such as skip-counting by groups, doubling and halving, estimating part-whole thinking – is the gateway to a student’s ability to acquire more difficult mathematics knowledge with confidence and understanding. When students begin to count in groups and see that numbers are “made up of” smaller groups, they then can begin to invent mental-math strategies that involve “breaking apart” and “rearranging” numbers to make “friendlier numbers.”

Note: There is a growing body of research that suggests teaching pencil-and-paper algorithms before fundamental part-whole thinking is established damages students’ development of number sense. That is not to say that having a functional written algorithm at some point is undesirable. Research evidence simply supports delay in teaching algorithms until appropriate part-whole understanding is fluently established with lesser numbers before written methods are applied to larger numbers.

Global Developmental Stage 3: Proportional Reasoning

Proportional reasoning is the ability to understand and work with relationships in situations involving proportions. This ability results after a student has acquired confidence in many aspects of math, and it needs to develop over many years. Proportional reasoning is the final developmental stage that students enter in elementary-school mathematics. By the time students enter grade 6, they have reached this stage. Students have more success with formal mathematics if they have an understanding of proportional reasoning. Students progress through only one signpost at this stage (see Part 2 for a more detailed description of this signpost):

1. Uses Proportional Reasoning: Beginning

Signposts

As mentioned above, each developmental stage is made up of one or more signposts. Signposts include key growth points that students reach as they progress across the developmental stages of counting, part-whole thinking, and proportional reasoning. Each signpost contains significant competencies, skills, and strategies that students need to

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Global Developmental Stages and Signposts

Global DevelopMenTal STaGeS

SiGnpoSTSStage 1: Counting

Stage 2: Part-Whole Thinking

Stage 3: Proportional Reasoning

Pre-Counter

1:1 Correspondence

Uses Manipulatives

Uses Visualization

Capable Counter

Additive Reasoning: Beginning

Additive Reasoning: Capable

Multiplicative Reasoning: Beginning

Multiplicative Reasoning: Capable

Proportional Reasoning: Beginning

As students move through the developmental stages, they become more proficient with numbers in meaningful ways.

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become proficient in as they make sense of their learning. For example, once a student learns the rote-counting sequence, it is a very significant step for him or her to be able to identify one item to one verbal count. This is classified as one-to-one correspondence and is the key indicator that a student may be progressing from the Pre-Counter signpost to the Uses 1:1 Correspondence signpost. The Learning Pathway identifies 10 signposts in total.

Signpost Components

Each signpost is made up of three components: (1) Knowledge, (2) Application of Knowledge, and (3) Modelling and Communicating Mathematical Thinking. The three components are nested together conceptually in the sense that they are not exclusive of one another – as one aspect of a learner’s thinking is developed it gives way to or is supported by the other aspects of knowledge. Each nested component helps deepen mathematical understanding. The skill-focused Knowledge component supports the strategy-focused Application of Knowledge component, which can be observed in the third component – how students model and communicate mathematical thinking. It is important for both teachers and students to reflect on how learning is evident in each of these different components and to understand how the components connect.

Knowledge Component

Content knowledge leads to the development of more sophisticated skills. The Knowledge component focuses on four key topics:

1. Rote Counting: Rote counting is counting the number sequence forward and backward, as well as counting by equal groups. With rote counting, students eventually gain confidence in counting with rational numbers.

2. Place Value: Place value focuses on knowing the place-value system and teaches the concept of thinking in groups.

3. Basic Arithmetical Learning: This topic focuses on the student’s development of facts and related facts, learning doubles with benchmarks, and compatibles.

4. Algebraic Reasoning: Algebraic reasoning focuses on (a) understanding what equality means in mathematics, and (b) the ability to recognize, create, and extend patterns and relationships in different contexts.

The three components of each signpost are nested together. Each component helps deepen student’s mathematical understanding.

MoDellinG anD CoMMUniCaTinG MaTheMaTiCal ThinkinG

appliCaTion of knowleDGe

knowleDGe

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Application of Knowledge Component

Mathematical content knowledge is one aspect of a well-rounded mathematician. However, acquisition of knowledge and skill alone are not enough. When students learn the strategies with which they can apply their knowledge, they gain access to new content knowledge and skills. This, in turn, helps them develop more sophisticated strategies to apply knowledge. This pattern continues as learners zigzag their way through the signposts (see page 9). The Application of Knowledge component focuses on two key topics:

1. Flexible Thinking: This focuses on the strategies that students learn as they are gaining understanding of the operations. Students learn the importance of a variety of mental-math strategies and can understand when it is appropriate to apply specific strategies in different situations/contexts. As well, students learn the importance of benchmark numbers, the commutative property, related facts, and how to rename numbers in order to make them easier to “think” about.

2. Pre-Proportional Reasoning: This focuses on the learner’s ability to reason about the numbers he or she is using in proportional settings rather than just on applying rote rules and procedures. Pre-proportional reasoning competencies help students progress toward the last signpost in The Learning Pathway.

Modelling and Communicating Mathematical Thinking Component

The Modelling and Communicating Mathematical Thinking component is connected to WNCP, and, specifically, the seven processes: visualization, communication, reasoning, connections, mental math and estimation, problem solving, and technology. These processes can be used to monitor students’ ability to engage in meaningful “math talk” – for example, communicate their understanding of a process, as well as have the ability to explain another student’s reasoning.

Students who are successful mathematicians use mathematical language, symbols, and visual representations to communicate their thinking, strategies, and knowledge to others. In order to support students’ ongoing mathematical discussions, you need to encourage your students to engage in this kind of discussion on a regular basis. It is through daily conversations that you can use a variety of strategies to engage students in discussing mathematical content, as well as assess student strengths for planning next steps.

Moving Through the Signposts

It is important to remember that movement through the 10 signposts is not linear. Students may be farther along in the Knowledge component than they are in the Application of Knowledge component. This can occur when students acquire knowledge and perform skills/rote procedures before they have a depth of understanding. While students may be in several signposts at one time, instruction should assist students in narrowing the gap between key competencies, as well as focus on students’ ability to take skills learned, apply them in meaningful ways, and communicate their thinking to others.

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Three Components Within a Signpost

In Part 2, you will explore the 10 signposts in greater detail. As you are reading, think about how you can apply the learning pathway to your instructional and assessment practices.

Pre-Counter

Modelling and CoMMuniCating MatheMatiCal thinkingAt centres, students can use mathematical language while “playing” with materials that represent a real-world context.

During snack time, students can model mathematical situations.

Uses 1:1 Correspondence

Modelling and CoMMuniCating MatheMatiCal thinking (Using tools such as beaded number lines (5 and 10 structure), rekenrek, five frames, manipulatives, musical instruments, etc.)K.N.3/K.N.4/K.PR.1*

I can use manipulatives to represent and describe a numeral in two parts [to 5] and to 10.

I can construct a set of objects corresponding to a given numeral.

I can create a repeating pattern.

Uses Manipulatives

Modelling and CoMMuniCating MatheMatiCal thinking (Using tools such of beaded number lines to 20, ten frames, rekenrek, manipulatives, musical instruments, etc.) 1.N.4*/1.N.9*/1.PR.1/1.PR.3*

I can represent the results of counting and operations to 20 by: • drawing a picture/diagram. • matching numerals to quantity.

I can represent and describe numbers to 20 by: • using a variety of manipulatives, including ten frames and base-10 materials. • modelling a number, using two different objects.

I can create or act out a story problem orally or through a shared reading.

I can create and add to a repeating pattern.

I can construct two equal sets.

Uses Visualization

Modelling and CoMMuniCating MatheMatiCal thinking (Using tools such as beaded number lines, ten frames, double ten frames, ENL, rekenrek, manipulatives, etc.) 1.N.4*/1.N.9*/1.PR.3*/1.PR.4

I can represent and describe the results of counting and operations to 20 by: • drawing and labelling diagrams.• writing ± number sentences.

I can represent and describe numbers to 20 by: • using a variety of manipulatives, including ten frames and base-10 materials. • modelling a number, using two different objects.

I can solve addition and subtraction problems to 20 from screened/partially screened collections.

I can describe equality and inequality as a balance.

Capable Counter

Modelling and CoMMuniCating MatheMatiCal thinking (Using tools such as ENL, rekenrek, manipulatives, double ten frames, dotty arrays, coins, tallies, technology, etc.) 2.N.4/2.N.7/2.N.9*/2.PR.2/2.PR.3*

I can represent a number, using a variety of manipulatives.

I can represent the results of problem solving by: • drawing and labelling a picture/diagram. • using and describing my personal strategy. • writing ± number sentences/equations. • modelling relationships between addition and subtraction. • representing a number in a variety of ways. • modelling addition and subtraction, using concrete materials or visual representations

(technology), and recording the process symbolically.

I can create and extend increasing patterns.

I can construct two unequal sets.

Uses Additive Reasoning: Beginning

Modelling and CoMMuniCating MatheMatiCal thinking (Using tools such as ENL, arrays, manipulatives, technology, etc.) 2.N.9*/2.N.10*/3.N.2*/3.N.4/3.N.5/3.N.6/3.N.7/3.N.11/3.N.12/3.N.13/3.PR.2

I can use diagrams, words, and equations to represent and describe my personal mental strategies, and my results, for problems involving: • addition and subtraction (to 100). • multiplication facts with answers to 5 x 5. • division facts with answers (to 25 ÷ 5).

I can demonstrate an understanding of fractions by: • representing a portion of a whole divided into equal parts.

I can represent a number in different ways to 1000 (concrete, pictorial, symbolic) by: • using proportional and non-proportional, and explaining how they are equivalent. • estimating a quantity and justifying the reasoning (supply students with a referent).

I can represent and describe numbers concretely, pictorially, and symbolically (as an expression).

I can create and extend decreasing patterns.

Uses Additive Reasoning: Capable

Modelling and CoMMuniCating MatheMatiCal thinking (Using tools such as ENL, arrays, equations, expressions, pictures, manipulatives, technology, etc.) 4.N.1/4.N.2/4.N.5/4.N.6/4.N.7/4.N.8/4.N.9/4.N.10/4.PR.1/4.PR.2*/4.PR.3/4.PR.4/4.PR.5

I can use diagrams, words, and equations to represent my personal mental strategies, and my results, for whole-number problems involving: • addition, subtraction (to 10 000). • multiplication, division (2- or 3-digit by 1-digit). • mental-math strategies (to 9 x 9). • modelling multiplication, using the distributive property and arrays.

I can use numbers to 10 000 in a variety of ways to: • represent and describe math relationships using charts and diagrams to solve

problems. • identify, describe, reproduce, represent, and explain patterns and relationships

in a variety of ways.

I can represent (concretely, pictorially, symbolically): • decimals (10ths, 100ths) and fractions in a variety of ways. • decimals to relate to fractions.

Uses Multiplicative Reasoning: Beginning

Modelling and CoMMuniCating MatheMatiCal thinking (Illustrate, represent, explain, model, describe, record, discuss in context, and use technology, etc.) 4.PR.2*/4.PR.6/5.N.1*/5.N.2/5.N.3/5.N.4/5.N.5*/5.N.6*/5.N.9/5.N.11*

I can use diagrams, equations, and words to represent my personal mental strategies/results for problems involving: • whole-number addition/subtraction to 4 digits. • whole-number multiplication (up to 2-digits by 2-digits)/division (up to 3-digits

by 1-digit). • decimal number addition/subtraction to hundredths (with $).

I can identify and explain math relations using charts and diagrams to solve problems.

I can relate decimals to fractions concretely, pictorially, and symbolically.

I can represent and describe whole numbers to 1 000 000.

I can: • create a concrete representation of a pattern from a table or chart. • solve one-step equations using manipulatives.

Uses Multiplicative Reasoning: Capable

Modelling and CoMMuniCating MatheMatiCal thinking (Illustrate, represent, explain, model, describe, record, discuss in context, and use technology, etc.) 5.N.1*/5.N.5*/5.N.6*/5.N.8/5.N.11*/5.PR.1/5.PR.2

I can use diagrams, equations, and words to represent my personal mental strategies, and my results, for problems involving: • operations with whole numbers and decimal numbers (to 1000ths).

I can describe and represent decimals (10ths, 100ths, 1000ths) concretely, pictorially, and symbolically by: • expressing a given numeral in expanded notation (standard and non-standard).

I can:• describe, orally or in writing, a pattern using mathematical language. • create a problem in context for an equation.

Uses Proportional Reasoning: Beginning

Modelling and CoMMuniCating MatheMatiCal thinking (Ratio table, double number line, bar models, ENL, illustrate, represent, explain, model, describe, record, discuss in context, and use technology, etc.) 5.N.11*/6.N.1/6.N.3/6.N.4/6.N.5/6.N.6 /6.N.8 /6.PR.1/6.PR.2

I can: • represent, describe, and write numerals for numbers of any magnitude (greater than

one million; less than one-thousandth). • explain how that pattern of the place-value system works. • provide and explain a concrete or pictorial representation for a ratio. • describe, using everyday language, orally or in writing, the relationship shown on a graph. • state, using math language, the relationship in a table of values. • translate and graph a pattern to a table of values. • identify the factors for a number and demonstrate/explain the strategy used

(concretely, pictorially, symbolically). • represent ratio and proportion concretely, pictorially, symbolically.

I can use diagrams, equations, and words to represent my personal mental strategies, and my results, for problems involving:• improper fractions that represent a number greater than 1. • operations with whole and decimal numbers (to 1000ths).

The LeARning PAThwAy: Planning for instruction and Assessment in number Sense and Algebraic Reasoning (K–6)

appliCation of knowledgeflexible thinking 1.N.3*/1.N.5*/1.N.7*/1.N.9*/1.N.10*/l.PR.3*

I can use strategies to determine addition and subtraction problems to 20 by: • counting on/back. • making 10 as a strategy. • using 1 more/1 less.

I can count all concrete or pictorial objects to: • solve addition [to 10] and to 20. • solve subtraction [to 10] and to 20. • compare two sets [to 10] and to 20.

confirm benchmark to 10 before going to 20.

I can count all objects (up to 20) to solve simple problems involving: • FINDING the number of objects in

the whole set using groups of 2s, 5s, or 10s.

pre-proportional Reasoning 1.N.3*/1.N.7*

I can: • take any given numbered quantity

(2s, 4s, 6s) and share into two equal groups.

I can count all objects (up to 20) to solve simple problems involving: • GROUPING a set of counters into equal

groups and saying the quantity in each share.

knowledgeRote Counting K.N.1*/1.N.1*/1.N.8

I can: • rote count forward to 30 to 100**. • rote count backward from 20 from 100**. • say number before/after any given number to 20. • rote count by 2s to 10 to 30** and by 5s to 20 to 100**.

place Value 1.N.2*

I can subitize familiar patterns/arrangements: • to 6 on dice. • to 10 on ten frames, using finger patterns, on beaded number line/rekenrek.

Basic arithmetical learning 1.N.8*/1.N.10*

I can: • name 2 more/2 less to 10 to 20**.

I know: • doubles to 3 x 3.

algebraic Reasoning K.PR.1*/1.PR.1/1.PR.3*

I can: • add to a repeating pattern, using manipulatives, sounds, or actions. • describe equality as a balance to 20, concretely and pictorially.

appliCation of knowledgeflexible thinking 1.N.3*/1.N.5*/1.N.7*/1.N.9*/1.N.10*/1.PR.3*

I can use strategies to determine addition and subtraction problems to 20 by: • starting from known doubles. • using addition to subtract (think

addition).

I can visualize and count all objects when: • using pairs to 5 and 5 + strategies. • solving addition [to 10) and to 20. • solving subtraction [to 10] and to 20. • comparing two sets [to 10) and to 20.

confirming benchmark to 10 before going to 20.

I can visualize and count all objects (up to 20) to solve simple problems by: • counting the total number of objects in

a set by counting on, using groups of 2s, 5s, and 10s.


I can visualize and count all objects (up to 20) to solve simple problems using: • a variety of groups with and without

singles.

I can use doubles to solve simple problems involving: • naming half of an even-numbered set

to 10.

knowledgeRote Counting 1.N.1*

I can rote count: • from any starting point forward to 100 and backward from 100. • starting from 0 by 2s to 30 and by 5s to 100 and by 10s to 100.

place Value 1.N.2*/1.N.4*

I can identify familiar patterns/arrangements: • to 20, and describe the number’s relationship to 5 and to 10.

I can determine: • compatible number pairs for 5, 10, and 20.

Basic arithmetical learning1.N.l0*

I know: • doubles to 5 + 5. • doubles ± 1 to 5 + 5.

algebraic Reasoning 1.PR.2/1.PR.3*/1.PR.4/2.PR.1

I can:• predict the next element in a pattern, and translate a repeating pattern from one

representation to another. • identify the core of a repeating pattern. • record equalities using the = symbol.

appliCation of knowledgeflexible thinking 1.N.3*/1.N.7*/1.N.9*/1.N.10*/2.N.9*

I can use skip-counting, often in conjunction with 1:1 counting, to solve problems that involve: • finding the number in the whole set. • finding the number in each equal

share. • finding the total number of equal

shares.

I can count on or count back to solve addition and subtraction problems by: • 1s with answers to 100

(single-digit quantity). • 10s and 1s with answers to

100 (2-digit numbers).

I can solve addition and subtraction problems by: • using commutative property

for addition. • relating to addition facts to solve

subtraction (think addition).


I can: • use doubles to solve simple problems

involving naming half of an even-numbered set to 20.

• give examples of fractions in a real-world context.

knowledge Rote Counting 2.N.1*/2.N.5*

I can: • rote count to 100 forward and backward from any respective multiple by 2s and by 5s

and by 10s. • rote count to 100 forward by 2s from 1 and by 10s from 1 to 9. • order numbers in the range 0–100.

place Value 2.N.4/2.N.7

I can identify: • the number of groups of 10s in decade numbers. • the number of 10s and 1s in 2-digit numbers (standard and non-standard).

I can determine: • compatible number pairs to 50 involving benchmarks involving any number.

Basic arithmetical learning 1.N.10*/2.N.8/2.N.9*/2.N.10*

I know: • doubles to 10 + 10.• 10 + any single digit. • addition and subtraction facts with answers to 10 to 18**.

I can demonstrate: • an understanding of addition with answers to 100 and corresponding subtraction facts.

I can determine: • addition and subtraction facts with 0.

algebraic Reasoning 2.PR.2/2.PR.3/2.PR.4

I can: • create, extend increasing patterns with numbers up to 100. • record equalities/inequalities using the equal symbol or non-equal symbol.

appliCation of knowledgeflexible thinking 2.N.9*/2.N.10*/3.N.6/3.N.7/3.N.9/3.N.10/ 3.N.11/3.N.12

I can mentally solve (+ and –) problems using a limited range of part-whole strategies involving: • 2-digit numbers (place-value

partitioning, bridge over a decade, think addition to subtract) 100 to 1000**.

• hidden doubles.

I can derive answers using known multiplication facts in combination with repeated addition for problems involving: • multiplication with answers to 5 x 5. • division with answers to 25 ÷ 5. • commutative property. • subtraction. • skip-counting. • partitioning.

pre-proportional Reasoning 3.N.12/3.N.13

I can use: • repeated halving in context with even

numbers less than 20.

I can determine: • a half of a REGION. • one-fourth or one-quarter of a REGION.

knowledgeRote Counting 3.N.1/3.N.2*

I can rote count forward and backward to 1000 from any starting point: • by 1s, by 10s, by 100s. • from respective multiples by 5s, by 25s. • to 25 from 0 by 3s, by 4s.

place Value 3.N.2*/3.N.5

I can identify: • the number of 10s in 3-digit numbers (standard and non-standard).

I can determine: • compatible number pairs for 100 involving benchmarks involving any number.

Basic arithmetical learning 3.N.10/3.N.11

I can determine: • addition facts with answers to 18. • subtraction facts with answers to 18. • doubles ± 1 to 20. • multiplication facts [for 2, 5, and 10] with answers to 5 x 5.

algebraic Reasoning 3.PR.2/3.PR.3

I can create and extend decreasing patterns: • with manipulatives, diagrams, and numbers.

I can solve: • one-step addition or subtraction equations involving unknown numbers.

appliCation of knowledgeflexible thinking 4.N.1/4.N.3/4.N.5/4.N.6/4.N.7/4.PR.4

I can use mathematical relationships to solve addition and subtraction problems: • with place-value understanding. • using personal strategies. • using compensation.

I can develop recall of multiplication facts to 81 using strategies such as: • doubling. • halving. • repeated doubling. • doubling and adding one more group. • skip-counting from a known fact. • using part-whole place-value reasoning. • using ten facts and five facts.

I can solve multiplication and division problems by: • using arrays to represent multiplication. • relating to division. • using the commutative property. • renaming. • partitioning.

pre-proportional Reasoning 4.N.5/4.N.8

I can use repeated halving in context involving: • identifying halves, fourths, or quarters

of a set.

I can use halving: • to determine fractions of a set to 20.

knowledgeRote Counting 4.N.8/4.N.9

I can rote count forward:• 1–2 s, 1–4 s, 10ths (0–10).

I can read and order: • decimal numbers involving tenths. • fractions with like denominators using fourths.

place Value 3.N.2*/4.N.8

I can identify: • the number of l0s and 100s in 4-digit numbers (standard and non-standard). • compatible fractions with 1–2 ,

1–4 , 1–3 (concrete and pictorial).

Basic arithmetical learning 4.N.5

I can: • recall multiplication facts for 2, 5, and 10 to 80. • develop multiplication facts to 81. • use strategies such as skip-counting, doubling, halving, doubling and adding

1 more group, repeated doubling, using ten facts and five facts.

algebraic Reasoning 4.PR.1/4.PR.3/4.PR.5

I can: • determine patterns in tables and charts. • extend patterns in tables and charts.• create an equation based on a context with a symbol to represent an unknown

(addition/subtraction).

appliCation of knowledgeflexible thinking 4.PR.6/5.N.2/5.N.3*/5.N.4

I can use mathematical relationships: • when using constant difference.

I can apply estimation strategies, such as: • front-end rounding. • compatibles. • compensation.

I can determine multiplication facts to 81 by: • halving and doubling. • relating facts. • doubling and adding one more group. • annexing zero.

I can solve: • problems using arrays and part-whole

thinking (distributive property).

pre-proportional Reasoning 4.N.8*/5.N.2/5.N.7*

I can use proportional reasoning with the AREA model for fractions to: • reason from part to whole. • reason from whole to part.

I can use: • repeated addition to determine

fractions of a set to 20.

knowledgeRote Counting 5.N.7*/5.N.8*/5.N.10

I can: • read, order, and compare fractions with unlike denominators using benchmarks

0, 1–2 , 1. • order decimals to thousandths. • say 1–10 more than or less than any decimal number.

place Value 5.N.1/(4.N.9–5.N.8)

I can identify: • the number of 10s and 100s in 5-digit numbers (standard and non-standard). • the number of 10ths and 100ths in decimals to two places. • compatible fractions for 3–4 , 2–3 , and 7–10.


I can recall: • multiples of 0, 1, 2, 3, and 5 to 81. • multiplication facts that are squares with answers to 9 x 9 [10 x 10].

I can derive: • division facts using their related multiplication facts with answers to 81 ÷ 9. • multiplication facts with 10s.


I can: • reproduce a pattern with concrete materials from information in tables and charts. • solve one-step equations with unknowns (multiplication).

appliCation of knowledgeflexible thinking 5.N.3*/5.N.4*/5.N.5*/5.N.8*/5.N.11/5.PR.2

I can use part-whole mental-math strategies in whole-number and decimal contexts to: • bridge to one with decimals. • partition using place value. • use front-end estimation. • predict sums and differences using

other estimation strategies.

I can determine multiplication facts and related division facts to 81 by: • halving and doubling with decimals.

I can use strategies to solve multiplication (2- by 2-digit) and division (3- by 1-digit) problems in context using: • the distributive property. • open arrays (partitioning). • renaming.


I can use proportional reasoning with the SET model for fractions involving: • reasoning from part to whole. • reasoning from whole to part. • fractions as division. • halving in a fraction context. Tenths

can be made by halving fifths. • linking proportional reasoning to

multiplication.

I can use: • the 2s, 5s, 10s facts to determine 1–2 , 1–4 ,

1–10 of a set.

knowledgeRote Counting 5.N.7*/5.N.8*/5.N.9/5.N.10*

I can: • express a decimal 10th as an equivalent 100th or 1000th. • order common fractions using benchmarks. • read and relate fractions to decimals. • order decimals on a number line (between 0–2). • name equivalent fraction/decimal/percent for 1–2 , 1–4 , 1–10 of a set.

place Value 5.N.1*/5.N.7*/5.N.8*

I can identify: • the number of 10s, 100s, 1000s, and so on with any whole number up to 7 places

(standard and non-standard). • compatible decimals (10ths) to make one.


I can recall: • related division facts for multiples for 2s, 5s, and 10s.

I can determine: • related division facts. • factors of numbers [to 50] and to 100.


I can: • write an expression based on patterns in a table/chart. • solve single-variable equations with one unknown.

appliCation of knowledgeflexible thinking 5.N.5*/6.N.3*/6.N.8

I can: • use previously learned strategies to

develop more sophisticated strategies (e.g., combining strategies).

• extend strategy use to higher numeral ranges.

• apply strategies with problems involving decimal numbers.

developing proportional Reasoning 6.N.4/6.N.5/(7.N.5)*/6.PR.1/6.PR.3

I can use proportional reasoning with the AREA and SET model for fractions involving: • halving in a fraction context: sixths

by halving thirds, eighths by halving fourths.

• renaming improper fractions to a mixed number.

• explaining the part-whole and part-part ratio of a set.

• using a ratio table for solving problems.

NotE: outcomes/expectations involving rational number in grade 6 are sparse. In order for students to be successful in grade 7, they need more experience in becoming flexible with fraction. this outcome/expectation is not being assessed but to honour developmental instruction students need exposure.

knowledgeRote Counting 6.N.4

I can: • read, order, and relate fractions, including mixed and improper.

place Value 6.N.1/6.N.6/5.N.7*

I can identify: • the number of 10ths (standard and non-standard) and 100ths, 1000ths with decimal

numbers to 1000ths (standard). • solve a problem using percents (compatibles).

Basic arithmetical learning 6.N.3*/6.N.8

I can determine: • factors of numbers to 100.

I can determine products or quotients: • involving decimals (x or ÷ by 10).

algebraic Reasoning 6.PR.1/6.PR.2/6.PR.3/6.PR.4

I can: • demonstrate an understanding of the relationships within table of values. • represent and describe patterns and relationships using tables. • represent generalizations arising from number relationships using equations. • demonstrate and explain the meaning of preservation of equality.

appliCation of knowledgeI can make sets by matching:• 0 to 5.• 0 to 10.

knowledgeRote Counting

I can rote count: • to 5 to 10.

place Value

I can subitize:• familiar dot patterns to 3.

appliCation of knowledgeflexible thinking K.N.5/K.N.6

I can count objects 1:1 to: • find “how many” in a set up [to 5] and up to 10. • compare two sets up [to 5] and up to 10.

I can use manipulatives to compare quantities that are: • same as • more than • fewer than/less than

knowledge Rote Counting K.N.1*

I can rote count: • forward to 20 to 30**. • backward from 10. • saying number before/after any given number to 10.

place Value K.N.2

I can subitize familiar patterns to 5 to 6**. • on dice, on five frames, using finger patterns.

Basic arithmetical learning K.N.2

I can: • name 1 more / 1 less to 5.

algebraic Reasoning K.PR.1*

I can: • create a repeating pattern with manipulatives, sounds, or actions.

Outcomes listed are based on END-of-grade achievement expectations. Teachers should refer to the Manitoba Curriculum Framework of Outcomes or their provincial/territorial curriculum framework of expectations/outcomes to assess and plan for breadth and depth of instruction. * WNCP outcome appears in more than one signpost ** When learner reaches the competency before the arrow, he or she is ready to progress to the next signpost

LP_poster_v4bw.indd 1 12/19/12 11:32 AMSome mathematical learning can be linear across a specific component (as indicated by the broken arrows). At the same time, students zigzag their way through the different components of one signpost to the next (as indicated by the solid arrows).

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The Learning Pathway Mathematics Instruction and Assessment for Grades K–6

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The Learning Pathway is both a companion document to current mathematics instruction and an assessment tool that teachers can use throughout the school year. It is based on extensive research that shows all students progress through three global developmental stages – counting, part-whole thinking, and proportional reasoning – when they are learning the key topics in mathematics. With The Learning Pathway, teachers can do the following:

• Plan ongoing assessment to determine students’ points of strength, as well as plan next steps for instruction.

• Focus on key growth points that students travel through as they consolidate their content knowledge and apply their learning in real-world contexts.

• Plan focused instruction to meet students’ needs, as well as guide formative assessment and reporting to parents.

• Differentiate instruction for student learning, according to the math that is just beyond each student’s current ability.

• Highlight the important roles that Knowledge, Application of Knowledge, and Modelling and Communicating Mathematical Thinking play as students construct their understanding of concepts in mathematics.

• Use “signposts” that follow the principles of student global developmental stages of growth in mathematics.

• Establish a bridge between actual classroom practice and the curriculum.

Teachers will find The Learning Pathway is suitable for whole-class, small-group, and individual instruction. As a resource, it will help guide decisions about how to differentiate instruction in meaningful ways.

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6dpsoh3djhv the learning pathway - portage & … · teachers who live and ... they will find...

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