Đó là bởi vì mục tiêu của Toán học Eureka là để tạo ra những học sinh không chỉ biết đọc biết viết mà còn thông thạo toán học. Thông thạo có nghĩa là không chỉ biết nên sử dụng quy trình nào khi giải quyết vấn đề mà còn hiểu tại sao quá trình đó hoạt động.
Eureka Math Study Guide Other Books Wheatley Portfolio English, Grades K–5, Second Edition English, Grades 6–8, Second Edition English, Grades 9–12, Second Edition Alexandria Plan United States History, Grades K–2 World History, Grades K–2 United States History, Grades 3–5 World History, Grades 3–5 Eureka Math Study Guide Grade Cover design by Chris Clary 1″ Cover image: Vincent van Gogh (1853–1890), The Starry Night Saint Rémy, June 1889 Oil on canvas, 29 × 36 (73.7 × 92.1 cm) Acquired through the Lillie P Bliss Bequest Location: The Museum of Modern Art, New York, NY, U.S.A Digital Image © The Museum of Modern Art / Licensed by SCALA / Art Resource, NY Copyright © 2015 by Great Minds All rights reserved Published by Jossey-Bass A Wiley Brand One Montgomery Street, Suite 1000, San Francisco, CA 94104-4594—www.josseybass.com No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600, or on the Web at www.copyright.com Requests to the publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, 201-748-6011, fax 201-748-6008, or online at www.wiley.com/go/permissions Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages Readers should be aware that Internet Web sites offered as citations and/or sources for further information may have changed or disappeared between the time this was written and when it is read Jossey-Bass books and products are available through most bookstores To contact Jossey-Bass directly, call our Customer Care Department within the U.S at 800-956-7739, outside the U.S at 317-572-3986, or fax 317-572-4002 For more information about Eureka Math, visit www.eureka-math.org Wiley publishes in a variety of print and electronic formats and by print-on-demand Some material included with standard print versions of this book may not be included in e-books or in print-on-demand If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com For more information about Wiley products, visit www.wiley.com Library of Congress Cataloging-in-Publication Data Eureka math study guide A story of units, grade education edition / Great Minds—First edition pages cm Includes bibliographical references and index ISBN 978-1-118-81397-3 (paperback) 1. Mathematics—Study and teaching (Preschool)—Standards—United States. I. Great Minds QA135.6.E83 2015 372.7′2—dc23 2014029344 Printed in the United States of America first edition PB Printing 10 9 8 7 6 5 4 3 2 1 Contents Introduction by Lynne Munson From the Writers by Hae Jung Yang and Marianne Strayton Foreword by Scott Baldridge How to Use This Book Chapter Introduction to Eureka Math Vision and Storyline Advantages to a Coherent Curriculum vii ix xi xiii 1 Chapter Major Mathematical Themes in Each Grade Band Year-Long Curriculum Maps for Each Grade Band Math Content Development for PreK–5: A Story of Units5 How A Story of Units Aligns with the Instructional Shifts 10 How A Story of Units Aligns with the Standards for Mathematical Practice 14 Chapter Grade-Level Content Review Rationale for Module Sequence in Grade 1 17 18 Chapter Curriculum Design Approach to Module Structure Approach to Lesson Structure Approach to Assessment 23 23 24 34 Chapter Approach to Differentiated Instruction Scaffolds for English Language Learners Scaffolds for Students with Disabilities Scaffolds for Students Performing below Grade Level Scaffolds for Students Performing above Grade Level 37 38 39 41 42 Chapter Grade-Level Module Summary and Unpacking of Standards Module 1: Sums and Differences to 10 Module 2: Introduction to Place Value Through Addition and Subtraction Within 20 Module 3: Ordering and Comparing Length Measurements as Numbers Module 4: Place Value, Comparison, Addition, and Subtraction to 40 Module 5: Identifying, Composing, and Partitioning Shapes Module 6: Place Value, Comparison, Addition, and Subtraction to 100 43 43 57 68 74 84 90 Chapter Mathematical Models 103 Array and Area Models 104 Base-Ten Blocks 105 Bundles106 Money107 Number Bond 108 vi | Contents Number Disks 110 Number Line 111 Number Path 113 Number Towers 114 Place Value Chart 115 Rekenrek117 Tape Diagram 118 Ten-Frame120 Chapter Terminology Grade PreK Grade K Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 123 123 126 128 130 133 135 138 Notes 141 Board of Trustees 143 Eureka Math Study Guide: A Story of Units Contributors 145 Index 147 Introduction When you know you really understand something? One test is to see if you can explain it to someone else—well enough that they understand it Eureka Math routinely requires students to “turn and talk” and explain the math they learned to their peers That is because the goal of Eureka Math (which you may know as the EngageNY math modules) is to produce students who are not merely literate, but fluent, in mathematics By fluent, we mean not just knowing what process to use when solving a problem but understanding why that process works Here’s an example A student who is fluent in mathematics can far more than just name, recite, and apply the Pythagorean theorem to problems She can explain why a2 + b2 = c2 is true She not only knows the theorem can be used to find the length of a right triangle’s hypotenuse, but can apply it more broadly—such as to find the distance between any two points in the coordinate plane, for example She also can see the theorem as the glue joining seemingly disparate ideas including equations of circles, trigonometry, and vectors By contrast, the student who has merely memorized the Pythagorean theorem does not know why it works and can little more than just solve right triangle problems by rote The theorem is an abstraction—not a piece of knowledge, but just a process to use in the limited ways that she has been directed For her, studying mathematics is a chore, a mere memorizing of disconnected processes Eureka Math provides much more It offers students math knowledge that will serve them well beyond any test This fundamental knowledge not only makes wise citizens and competent consumers, but it gives birth to budding physicists and engineers Knowing math deeply opens vistas of opportunity A student becomes fluent in math—as they in any other subject—by following a course of study that builds their knowledge of the subject, logically and thoroughly In Eureka Math, concepts flow logically from PreKindergarten through high school The “chapters” in the story of mathematics are “A Story of Units” for the elementary grades, followed by “A Story of Ratios” in middle school and “A Story of Functions” in high school This sequencing is joined with a mix of new and old methods of instruction that are proven to work For example, we utilize an exercise called a “sprint” to develop students’ fluency with standard algorithms (routines for adding, subtracting, multiplying, and dividing whole numbers and fractions) We employ many familiar models and tools such as the number line and tape diagrams (aka bar models) A newer model highlighted in the curriculum is the number bond (illustrated below), which clearly shows how numbers are comprised of other numbers viii | Introduction Eureka Math is designed to help accommodate different types of classrooms and serve as a resource for educators, who make decisions based on the needs of students The “vignettes” of teacher-student interactions included in the curriculum are not scripts, but exemplars illustrating methods of instruction recommended by the teachers who have crafted our curricula Eureka Math has been adopted by districts from East Meadows, New York, to Lafayette, Los Angeles, to Chula Vista, California At Eureka Math we are excited to have created the most transparent math curriculum in history—every lesson, all classwork, and every problem is available online Many of us have less than joyful memories of learning mathematics: lots of memorization, lots of rules to follow without understanding, and problems that didn’t make any sense What if a curriculum came along that gave children a chance to avoid that math anxiety and replaced it with authentic understanding, excitement, and curiosity? Like a New York educator attending one of our trainings said: “Why didn’t I learn mathematics this way when I was a kid? It is so much easier than the way I learned it!” Eureka! Lynne Munson Washington, DC Notes Chapter 3 When a cluster is referred to in this chart without a footnote, the cluster is addressed in its entirety In this module, work is limited to within 10 1.OA.2 is addressed in Module From this point forward, fluency practice is part of the students’ ongoing experience The balance of this cluster is addressed in Module Focus in this module is on numbers to 20 The balance of this cluster is addressed in Modules and 6 The balance of this cluster is addressed in Module While the use of pennies and dimes will be used throughout the module, 1.MD.3 is not a focus grade-level standard in Module Instead, this standard will become a focal standard in Module 6, when all coins are introduced and used The balance of this cluster is addressed in Module Focus on numbers to 40 10 Focus on numbers to 40; 1.NBT.2b is addressed in Module 11 Focus on numbers to 40 12 Time is addressed in this module and money in Module Note that “and money” is not part of the cluster heading in the Common Core State Standards for Mathematics (CCSSM) but has been included here 13 “Recognize and identify coins, their names, and their value” is not part of Standard 1.MD.3 in the CCSSM but has been included here 14 The balance of this cluster is addressed in Module 15 1.NBT.2b is addressed in Module 16 Money is addressed in this module Time is addressed in Module Note that “and money” is not part of the cluster heading in the CCSSM but has been included here 17 “Recognize and identify coins, their names, and their value” is not part of standard 1.MD.3 in the CCSSM but has been included here 142 | Notes Chapter 6 For an analysis of addition and subtraction word problem types used in Grades K–2, refer to the Counting and Cardinality progressions document, pp and 9, and the Common Core State Standards, p 88 All progressions documents can be found at http://ime.math.arizona.edu/progressions/ 1.OA.2 is addressed in Module For an analysis of addition and subtraction word problem types used in Grades K–2, refer to the Counting and Cardinality progressions document, pp and 9, and the Common Core State Standards, p 88 Refer to the Counting and Cardinality progressions document, p Refer to the Counting and Cardinality progressions document, p 14 The balance of this cluster is addressed in Module The focus in this module is on numbers to 20 The balance of this cluster is addressed in Modules and Just as the Common Core State Standards note, students not learn or use these formal terms The balance of this cluster is addressed in Module 10 See the Geometry progressions document 11 The balance of this cluster is addressed in Module 12 Focus on numbers to 40 13 Focus on numbers to 40; 1.NBT.2b is addressed in Module 14 Focus on numbers to 40 15 Integrates the 1.MD.3 standard for dime and penny This standard will become a focal standard in Module 6, when all four coins have been introduced 16 Time is addressed in this module and money in Module 17 This excellent image plus further clarification is found in the Geometry progressions document, p 18 This excellent image plus further clarification is found in the Geometry progressions document, p 19 Focus on money 20 Found in the Counting and Cardinality: Operations and Algebraic Thinking progressions document, p 21 Focus on money Board of Trustees Erik Berg, Second-Grade Teacher, John D Philbrick Elementary School in Boston, Massachusetts Barbara Byrd-Bennett, Chief Executive Officer, Chicago Public Schools Dan Cookson, Founder, BansheeBox, LLC Pascal Forgione, Jr., Executive Director of the Center on K–12 Assessment and Performance Management, Educational Testing Service Lorraine Griffith, Title Reading Specialist, West Buncombe Elementary School in Asheville, North Carolina Jason Griffiths, Director of Programs, The National Academy of Advanced Teacher Education Bill Honig, President, The Consortium on Reading Excellence Carol Jago, Associate Director, California Reading and Literature Project at University of California — Los Angeles William Kelly, Co-founder and CEO, ReelDx Richard Kessler, Executive Dean of Mannes College, The New School for Music Nell McAnelly, Co-director, Gordon A Cain Center for STEM Literacy at Louisiana State University Lynne Munson, President and Executive Director Maria Neira, Former Vice President, New York State United Teachers Eureka Math Study Guide: A Story of Units Contributors Katrina Abdussalaam, Curriculum Writer Tiah Alphonso, Program Manager — Curriculum Production Kelly Alsup, Lead Writer / Editor, Grade Catriona Anderson, Program Manager — Implementation Support Debbie Andorka-Aceves, Curriculum Writer Eric Angel, Curriculum Writer Leslie Arceneaux, Lead Writer / Editor, Grade Kate McGill Austin, Lead Writer / Editor, Grades PreK-K Adam Baker, Lead Writer / Editor, Grade Scott Baldridge, Lead Mathematician and Lead Curriculum Writer Beth Barnes, Curriculum Writer Bonnie Bergstresser, Math Auditor Bill Davidson, Fluency Specialist Jill Diniz, Program Director Nancy Diorio, Curriculum Writer Nancy Doorey, Assessment Advisor Lacy Endo-Peery, Lead Writer / Editor, Grades PreK-K Ana Estela, Curriculum Writer Lessa Faltermann, Math Auditor Janice Fan, Curriculum Writer Ellen Fort, Math Auditor Peggy Golden, Curriculum Writer Maria Gomes, PreKindergarten Practitioner Pam Goodner, Curriculum Writer Greg Gorman, Curriculum Writer Melanie Gutierrez, Curriculum Writer Kelley Isinger, Curriculum Writer Nuhad Jamal, Curriculum Writer 146 | Eureka Math Study Guide: A Story of Units Contributors Mary Jones, Lead Writer / Editor, Grade Halle Kananak, Curriculum Writer Tam Le, Document Production Manager Susan Lee, Lead Writer / Editor, Grade Jennifer Loftin, Program Manager — Professional Development Soo Jin Lu, Curriculum Writer Nell McAnelly, Project Director Ben McCarty, Lead Mathematician / Editor, PreK-5 Cristina Metcalf, Lead Writer / Editor, Grade Susan Midlarsky, Curriculum Writer Pat Mohr, Curriculum Writer Victoria Peacock, Curriculum Writer Jenny Petrosino, Curriculum Writer Terrie Poehl, Math Auditor Robin Ramos, Lead Curriculum Writer / Editor, PreK-5 Cecilia Rudzitis, Curriculum Writer Tricia Salerno, Curriculum Writer Chris Sarlo, Curriculum Writer Ann Rose Sentoro, Curriculum Writer Colleen Sheeron, Lead Writer / Editor, Grade Gail Smith, Curriculum Writer Shelley Snow, Curriculum Writer Kelly Spinks, Curriculum Writer Marianne Strayton, Lead Writer / Editor, Grade Theresa Streeter, Math Auditor Lily Talcott, Curriculum Writer Kevin Tougher, Curriculum Writer Saffron VanGalder, Lead Writer / Editor, Grade Lisa Watts-Lawton, Lead Writer / Editor, Grade Erin Wheeler, Curriculum Writer MaryJo Wieland, Curriculum Writer Jessa Woods, Curriculum Writer Hae Jung Yang, Lead Writer / Editor, Grade Index Page references followed by fig indicate an illustrated figure Accommodations: English language learners (ELLs) scaffolds and, 38–39; IEPs or Section 504, 39; students performing above grade level scaffolds and, 42; students performing below grade level scaffolds and, 41–42; students with disabilities scaffolds and, 39–41; Universal Design for Learning (UDL) use of, 37–42 Addition (Module 1): the equal sign and commutative property of, 45, 52; word problems, 44, 50–51 Addition (Module 2): aligned with the Standards, 17; Grade-Level Content Review on, 17 Addition (Module 4): aligned with the Standards, 17; Grade-Level Content Review on, 17; A Story of Units curriculum on, 9; ten-frame model used for, 103, 120–121 See also Composition Addition (Module 6): aligned with the Standards, 17; Grade-Level Content Review on, 17; to 100 using place value understanding, 91, 96–97 Algebraic thinking: Grade-Level Content Review on operations and, 17–22; Standard emphasis cluster on, 18 Alignment Chart: Module 1: Sums and Differences to 10, 17, 19–20; Module 2: Introduction to Place Value Through Addition and Subtraction Within 20, 17, 20; Module 3: Ordering and Comparing Length Measurements as Numbers, 17, 20–21; Module 4: Place Value, Comparison, Addition, and Subtraction to 40, 17, 21; Module 5: Identifying, Composing, and Partitioning Shapes, 17, 21; Module 6: Place Value, Comparison, Addition, and Subtraction to 100, 17, 22; sequence of Kindergarten modules aligned with the Standards, 17 Application Problem (lesson component): Lesson 15, 29fig–30; lesson order of the, 25, 26 Application problems: description and strategies for, 27; Distribution of Instructional Minutes chart on, 25fig; Lesson 15, 29fig–30; read-drawwrite (RDW) steps for, 27, 29; structured lesson structure on, 25fig Application rigor: description of, 13; new types of fluency through, 13–14; practicing and understanding intensity of, 13; Problem Sets used for, 13 Array and area models (Grade Level 1–5): instructional strategies for using, 104–105; multiplication application of, 103; overview of the, 104 Assessment Summary, 24 Assessments: curriculum design approach to, 34–35; Daily Formative, 34, 38; End-of-Module, 24, 35, 38; Mid-Module, 24, 35, 38; module structure component of, 23, 24; purpose and goals of, 34; rigor in the, 35 Base 10 number and operations, 17–22 Base-ten blocks (Grade Level K–2): GradeLevel Content Review on Base 10 number and operations, 17–22; instructional strategies for using, 106; overview of the, 105–106; place value and standard algorithms applications of, 103 Bundles model (Grade Level K–2): instructional strategies for using, 107; overview of the, 106–107; place value and standard algorithms applications of, 103 Circles (halves and quarters), 85, 89 Coherence Links: description and function of, 24; Module 1: sums and Differences to 10, 48, 49, 51, 52, 53, 54, 55, 56, 57; Module 2: Place Value Through Addition and Subtraction Within 20, 62, 64, 66, 67; Module 3: Ordering and Comparing Length Measurements as Numbers, 71, 72, 73, 74; Module 4: Place Value, Comparison, Addition, and Subtraction to 40, 78, 79, 80, 81, 83, 84; Module 5: Identifying, Composing, and Partitioning Shapes, 87, 88, 89, 90; Module 6: Analyzing, Comparing, and Composing Shapes, 96, 97, 98, 99, 100, 101 Coherence shift, 11–12 148 | Index Coherent curriculum: advantages of a, 2–3; coherence shift description and strategies, 11–12; supported by use of concrete and pictorial models, 12 Coin values, 91, 98–99 Comparing numbers (Module 3), 17 Concept Development: description and strategies for, 26–27; Distribution of Instructional Minutes chart on, 25fig; Lesson 15, 29fig, 31–33; structured lesson structure on, 25fig Conceptual understanding rigor, 12–13 Counting: on from embedded numbers, 44, 48–50; on or making ten to solve result unknown and total unknown problems to 20, 58–59, 61–63; strategies for to 10, 44–45, 51–52 Culminating experiences (Grade 1), 91, 100–101 Curriculum: A Story of Functions (9–12 grades), 1, 2, 5, 9fig; A Story of Ratios (6–8 grades), 1, 2, 5, 8fig See also Eureka Math; A Story of Units (PreK–5 grades) Curriculum design: approach to assessment, 34–35; approach to lesson structure, 24–32; approach to module structure, 23–24; description and purpose of, 23 Curriculum maps: A Story of Functions (9–12 grades), 5, 9fig; A Story of Ratios (6–8 grades), 5, 8fig; A Story of Units (PreK–5 grades), 5, 6fig–7fig Daily Formative Assessments, 34, 38 Data See Measurement and data Debriefing See Student Debrief Decomposition: subtraction and strategies for, 45, 55–56; varied problems of ten numbers as 1 ten and some ones with, 60, 66–67 See also Subtraction (Module 4) Dienes blocks, 105 Differentiated instruction: scaffolds for English language learners (ELLs), 38–39; scaffolds for students performing above grade level, 42; scaffolds for students performing below grade level, 41–42; scaffolds for students with disabilities, 39–41; Universal Design for Learning (UDL) used for, 37–42 Distribution of Instructional Minutes: chart of, 25ig; description of, 23 Embedded numbers: counting on from, 44, 48–50; decompositions and, 44, 47–48 End-of-Module Assessment, 24, 35, 38 English language learners (ELLs), 38–39 Eureka Math: coherent curriculum approach of, 2–3; online version of, 2; overview of, 1–2 See also Curriculum; Great Minds Exit Tickets: Daily Formative Assessments using, 34, 38; description and purpose of, 28; Lesson 15, 33fig Fluency Practice: description and strategies for, 26; Distribution of Instructional Minutes chart on, 25fig; Lesson 15, 29fig, 30; structured lesson structure on, 25fig Fluency rigor: conceptual understanding paving the way for new types of, 13–14; daily Fluency Practice for, 15; description and strategies used for, 12; development of addition fluency within 10, 45, 52–53; development of subtraction fluency with 10, 45–46, 56–57; Grade-Level Content Review on, 17 Focus: Focus Grade Level Standards on, 11; GradeLevel Content Review on, 17; PARCC emphasis on, 11 Focus Grade-Level Standards: description of, 23; Foundational Standards to support the, 24; Module 1: Numbers to 10, 46; Module 2: Place Value Through Addition and Subtraction Within 20, 61; Module 3: Ordering and Comparing Length Measurements as Numbers, 69; Module 4: Place Value, Comparison, Addition, and Subtraction to 40, 76; Module 5: Identifying, Composing, and Partitioning Shapes, 85; Module 6: Analyzing, Comparing, and Composing Shapes, 92–93 Focus shift: description and strategies for, 11; ensuring fluency rigor, 12 Focus Standards: Module 1: Numbers to 10, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57; Module 2: Place Value Through Addition and Subtraction Within 20, 62, 64, 66, 67; Module 3: Ordering and Comparing Length Measurements as Numbers, 71, 72, 73, 74; Module 4: Place Value, Comparison, Addition, and Subtraction to 40, 78, 79, 80, 81, 83, 84; Module 5: Identifying, Composing, and Partitioning Shapes, 87, 88, 89, 90; Module 6: Analyzing, Comparing, and Composing Shapes, 95, 97, 98, 99, 100, 101 Focus Standards for Mathematical Practice: description and function of the, 24; Module 1: Numbers to 10, 47; Module 2: Place Value Through Addition and Subtraction Within 20, 61; Module 3: Ordering and Comparing Length Measurements as Numbers, 70; Module 4: Index Number Pairs, Addition, and Subtraction to 10, 7; Module 4: Place Value, Comparison, Addition, and Subtraction to 40, 76–77; Module 5: Identifying, Composing, and Partitioning Shapes, 86; Module 6: Analyzing, Comparing, and Composing Shapes, 92–93; Topic Overview list of the, 24 Formative Daily Assessment, 38 Foundational Standards: description and function of, 24; Module 1: Numbers to 10, 46–47; Module 2: Introduction to Place Value Through Addition and Subtraction Within 20, 61; Module 3: Ordering and Comparing Length Measurements as Numbers, 70; Module 4: Place Value, Comparison, Addition and Subtraction to 40, 76; Module 5: Identifying, Composing, and Partitioning Shapes, 85–86; Module 6: Analyzing, Comparing, and Composing Shapes, 92 See also Standards or Mathematical Practice Fractions: area models used for, 103; number line model used for, 103; A Story of Units curriculum on, 10 Grade 1: culminated experiences of, 91, 100–101; terminology for, 128–130 See also specific module Grade-Level Content Review: Alignment Chart with grade-level standards, 17, 19–22; overview and components, 17; rationale for module sequence in Grade 1, 18–19; Summary of Year, 17 Grade-level terminology: Grade PreK, 123–126; Grade K, 126–128; Grade 1, 128–130; Grade 2, 130–133; Grade 3, 133–135; Grade 4, 135–138; Grade 5, 138–139 Great Minds: coherent curriculum approach by, 2–3; teaching philosophy and support by, 1–2 See also Eureka Math Halves: application to telling time, 85, 89–90; of rectangles and circles, 85, 89 Homework (Daily Formative Assessment), 34–35 Individualized Education Programs (IEPs), 39 Instructional Days: Module 2: Place Value Through Addition and Subtraction Within 20, 62, 64, 66, 67; Module 3: Ordering and Comparing Length Measurements as Numbers, 71, 72, 73, 74; Module 4: Place Value, Comparison, Addition, and Subtraction to 40, 78, 79, 80, 81, 83, 84; Module 5: Identifying, Composing, and Partitioning Shapes, 87, 88, 89, 90; Module 6: | 149 Analyzing, Comparing, and Composing Shapes, 96, 97, 98, 99, 100, 101; A Story of Functions (9–12 grades), 9fig; A Story of Ratios (6–8 grades), 8fig; A Story of Units (PreK–5 grades), 6fig–7fig See also Time frames Instructional Shifts: 1: focus of the standards, 11; 2: coherence to connect learning within and across grades, 11–12; 3: rigor aspects of fluency, conceptual understanding, and application, 12–14; description of the, 10–11; Standards for Mathematical Practice integrated into the, 14–15 Length measurement: indirect comparison in, 68, 70–71; nonstandard and standard length units, 69, 72–73; standard length units, 68, 71–72 Lesson Objectives: description of, 24; Lesson 15, 33 Lesson structure: curriculum design balanced approach to, 24–32fig; distribution of instructional minutes, 25; dual intensity balancing practice and understanding in, 13–14; Standards for Mathematical Practice woven into the, 28–29; suggested time spent on each component of, 25; time frames used as part of the, 25fig, 38 Lesson structure components: Application Problem, 25fig, 26, 29fig–30; Concept Development, 25fig, 26–27, 29fig, 31–33; Fluency Practice, 25ig, 26, 29fig, 30; Student Debrief and Exit Ticket, 25fig, 28, 29ffig, 33 Lessons: balanced approach to designing, 24–32fig; debriefing built into, 14–15, 25fig–26, 28; module structure component of, 23, 24; sample Lesson 15, 29fig–33fig; Standards for Mathematical Practice woven into each, 28–29 See also Curriculum design Major Emphasis Clusters, 17 Mathematical models: array and area (Grade Level 1–5), 103, 104–105; base-ten blocks (Grade Level K–2), 103, 105–106; bundles (Grade Level K–2), 103, 106–107; categories indicating primary applications for specific, 103; money (Grade Level 2), 103, 107–108; number bond (Grade Level K–5), 103, 108–110; number disks (Grade Level 2–5), 103, 110–111; number line (Grade Level K–5), 103, 111–112; number path (Grade Level PreK–1), 103, 113; number towers (Grade Level PreK–3), 103, 114–115; place value chart (Grade Level 2–5), 103, 115–116; Rekenrek (Grade Level PreK–5), 103, 117–118; Standards or 150 | Index Mathematical Practice use of, 15; A Story of Units support of coherence through, 12, 103; Tape Diagram (Grade Level 1–5), 103, 118–120; ten-frame (Grade Level PreK–3), 103, 120–121 Measurement and data: data interpretation (Module 3), 69, 73–74; Grade-Level Content Review on, 17–22; indirect comparison in length measure, 68, 70–71; nonstandard and standard length units, 69, 72–73; standard length units, 68, 71–72 Mental math, 16 Mid-Module Assessment: approach to, 35; description of, 24; differentiated instruction use of, 38 Module (Numbers to 10): aligned with the Standard, 17, 19–20; Focus Standards for Mathematical Practice, 47; Focus Grade-Level Standards, 46; Focus Standards, Instructional Days, and Coherence Links, 48, 49, 50–51, 52, 53, 54, 55, 56, 57; Foundational Standards, 46–47; Grade-Level Content Review of, 17–20; number bond model used for, 103, 108–110; number path model used for, 103, 113; number towers model used for, 103, 114–115; overview of A Story of Units curriculum on, 9; overview of the, 43–46; terminology for, 128; topic summaries of the, 47–57 Module topic summaries: Topic A: embedded numbers and decompositions, 44, 47–48; Topic B: counting on from embedded numbers, 44, 48–50; Topic C: addition work problems, 44, 50–51; Topic D: strategies for counting on, 44–45, 51–52; Topic E: the commutative property of addition and the equal sign, 45, 52; Topic F: development of addition fluency within 10, 45, 52–53; Topic G: subtraction as an unknown addend problem, 45, 53–54; Topic H: subtraction word problems, 45, 54–55; Topic I: decomposition strategies for subtraction, 45, 55–56; Topic J: development of subtraction fluency within 10, 45–46, 56–57 Module (Place Value Through Addition and Subtraction Within 20): aligned with the Standards, 17, 20; Focus Grade-Level Standards, 60; Focus Standards for Mathematical Practice, 61; Focus Standards, Instructional Days, and Coherence Links, 62, 64, 66, 67; Grade-Level Content Review on, 17; overview of, 57–60; rationale for module sequence of, 18–19; terminology for, 128; topic summaries of the, 58–60, 61–67 Module topic summaries: Topic A: counting on or making ten to solve result unknown and total unknown problems, 58–59, 61–63; Topic B: counting on or taking from ten to solve result unknown and total unknown problems, 59, 63–65; Topic C: strategies for solving change or addend unknown problems, 59, 65–66; Topic D: varied problems with decompositions of ten numbers as ten and some ones, 60, 66–67 Module (Ordering and Comparing Length Measurements as Numbers): aligned with the Standards, 17, 20–21; Focus Grade-Level Standards, 69–70; focus Standards for Mathematical Practice, 70; Focus Standards, Instructional Days, and Coherence Links, 71, 72, 73, 74; Foundational Standards, 70; Grade-Level Content Review on, 17; number bond model used for, 103, 108–110; number path model used for, 103, 113; number towers model used for, 103, 114–115; overview of, 68–69; rationale for module sequence of, 19; terminology for, 129; topic summaries of the, 68–69, 70–74 Module topic summaries: Topic A: indirect comparison in length measurement, 68, 70–71; Topic B: standard length units, 68, 71–72; Topic C: nonstandard and standard length units, 69, 72–73; Topic D: data interpretation, 69, 73–74 Module (Place Value, Comparison, Addition, and Subtraction to 40): aligned with the Standards, 17, 21; Focus Grade-Level Standards, 75–76; Focus Standards for Mathematical Practice, 7, 76–77; Focus Standards, Instructional Days, and Coherence Links, 78, 79, 80, 81, 83, 84; Foundational Standards, 76; Grade-Level Content Review on, 17; overview of, 74–75; rationale for module sequence of, 19; ten-frame model used for, 103, 120–121; terminology for, 129; topic summaries of the, 75, 77–84 Module topic summaries: Topic A: tens and ones, 75, 77–78; Topic B: comparison of pairs of two-digit numbers, 75, 79; Topic C: addition and subtraction of tens, 75, 79–80; Topic D: addition of tens or ones to a two-digit number, 75, 80–82; Topic E: varied problem types within 20, 75, 82–83; Topic F: addition of tens and ones to a two-digit number, 75, 83–84 Module (Identifying, Composing, and Partitioning Shapes): aligned with the Standards, 17, 21; Focus Grade-Level Standards, 85; Focus Standards for Mathematical Practice, 86; Focus Standards, Instructional Days, and Index Coherence Links, 87, 88, 89, 90; Foundational Standards, 85–86; number bond model used for, 103, 108–110; number path model used for, 103, 113; number towers model used for, 103, 114–115; overview of, 84–85; rationale for module sequence of, 19; terminology for, 129–130; topic summaries of the, 84–85, 86–90 Module topic summaries: Topic A: attributes of shapes, 84–85, 86–87; Topic B: Part-Whole Relationships Within Composite Shapes, 85, 87–88; Topic C: Halves and Quarters of Rectangles and Circles, 85, 89; Topic D: application of halves to tell time, 85, 89–90 Module (Analyzing, Comparing, and Composing Shapes): aligned with the Standards, 17, 22; Focus Grade-Level Standards, 91–92; Focus Standards for Mathematical Practice, 92–93; Focus Standards, Instructional Days, and Coherence Links, 95–96, 97, 98, 99, 100, 101; Foundational Standards, 92; Grade-Level Content Review on, 17; overview of, 90–91; rationale for module sequence of, 19; terminology for, 130; topic summaries, 93–101 Module topic summaries: Topic A: comparison word problems, 90, 93–95; Topic B: numbers to 120, 90, 95–96; Topic C: addition to 100 using place value understanding, 91, 96–97; Topic D: varied place value strategies for addition to 100, 91, 98; Topic E: coins and their values, 91, 98–99; Topic F: varied problem types within 20, 91, 99–100; Topic G: culminating experiences, 91, 100–101 Module Overview: components of the, 23–24; description of the, 23 Module structure: Assessments component of, 23, 34–35; curriculum design of, 23–24; Lessons component of, 23, 24–32fig; Module Overview component of, 23–24; Topic Overview component of, 23, 24 Money mode (Grade Level 2): instructional strategies for using, 108; overview of, 107–108; place value and standard algorithms applications of, 103 Money mode/coin values (Grade 1), 91, 98–99 Multiplication: array and area models used for, 103, 104–105; Rekenrek model used for, 103, 117–118; A Story of Units curriculum on, 10 New York State Education Department, Number bond model (Grade Level K–5): instructional strategies for using, 109–110; | 151 numbers through 10 application of, 103; overview of the, 108–109 Number disks model (Grade Level 2–5): instructional strategies for using, 111; numbers through 10 application of, 103; overview of the, 110–111 Number line model (Grade Level K–5): fractions application of, 103; instructional strategies for using, 112; overview of, 111 Number path model (Grade Level PreK–1): instructional strategies for using, 113; numbers through 10 application of, 103; overview of, 113 Number towers model (Grade Level PreK–3): instructional strategies for using, 115; numbers through 10 application of, 103; overview of, 114–115 Numbers to 10: Grade-Level Content Review on, 17–20; number bond model used for, 103, 108–110; number path model used for, 103, 113; number towers model for, 103, 114–115; overview of A Story of Units curriculum on, 9; tens and ones, 75, 77–78 Numbers to 20: counting on or making ten to solve result unknown and total unknown problems, 58–59, 61–63; counting on or taking from ten to solve result unknown and total unknown problems, 59, 63–65; strategies for solving change or addend unknown problems, 59, 65–66; varied problem types with 20, 91, 99–100; varied problems with decompositions of ten numbers a ten and some ones, 60, 66–67 Numbers to 40: addition and subtraction of tens, 75, 79–80; addition of tens and ones to a two-digit number, 75, 83–84; comparison of pairs of two-digit numbers, 75, 79; tens and ones, 75, 77–78; varied problem types within 20, 75, 82–83 Operations: Base 10 number and, 17–22; GradeLevel Content Review on, 17–22; Standard emphasis cluster on algebraic thinking and, 18 Overview of Module Topic: description of, 24; Module (Numbers to 10), 43–46; Module (Place Value Through Addition and Subtraction Within 20), 57–60; Module (Ordering and Comparing Length Measurements as Numbers), 68–69; Module (Identifying, Composing, and Partitioning Shapes), 84–85; Module (Analyzing, Comparing, and Composing Shapes), 90–91 152 | Index PARCC (Partnership for Assessment of Readiness for College and Careers Content Emphases): evidence of focus in integral use of, 11; Problem sets using tasks of, 13 PARCC tasks: Type I, 13; Type II, 13; Type III, 13 Part-whole model, 119 Place value: Module 2: Introduction to Place Value Through Addition and Subtraction Within 20, 17; Module 4: Place Value, Comparison, Addition, and Subtraction to 40, 17; Module 6: Place Value, Comparison, Addition, and Subtraction to 100, 17, 91 Place value chart model (Grade Level 2–5): instructional strategies for using, 116; overview of, 115–116; place value and standard algorithms applications of, 103 Place value units: addition to 100 using place value understanding, 91, 96–97; base-ten blocks model used for, 103, 105–106; bundles model used for, 103, 106–107; money model use for, 103, 107–108; number disks model (with place value chart) used for, 103, 110–111; place value chart model used for, 103, 115–116; A Story of Units curriculum on, Practicing daily fluency, 15 PreK–5 grades See A Story of Units (PreK–5 grades) Problem Sets: application rigor using, 13; Daily Formative Assessment using, 34; Lesson 15, 33 Publishers’ Criteria: description of the, 2; evidence of focus in, 11; three-pronged nature of rigor as supporting them of the, 12 Rational for Module Sequence, 17, 18–19 RDW (read-draw-write) problem solving, 27, 29 Rectangles, halves and quarters of, 85, 89 Rekenrek model (Grade Level PreK–5): instructional strategies for using, 117–118; multiplication application of, 103; overview of the, 117 Rigor aspects: application, 13–14; assessment, 35; conceptual understanding, 12–13; fluency, 12 Rigor shift: description of the, 12; strategies used for aspects of, 12–14 Scaffolding: for English language learners (ELLs), 38–39; Eureka Math use of, 2; for students performing above grade level, 42; for students performing below grade level, 41–42; for students with disabilities, 39–41; Universal Design for Learning (UDL) use of, 37–42 Section 504 Accommodation Plans, 39 Shapes: attributes of, 84–85, 86–87; halves and quarters of rectangles and circles, 85, 89; Module 5: Identifying, Composing, and Partitioning Shapes, 17; Module 6: Place Value, Comparison, Addition, and Subtraction to 100, 17; part-whole relationships within composite, 85, 87–88 Sprint patterns, 15 Standard algorithms: base-ten blocks model used for, 103, 105–106; bundles model used for, 103, 106–107; mathematical models used for, 103, 105–108, 115–116; money model use for, 103, 107–108; number disks model (with place value chart) used for, 103, 110–111; place value chart model used for, 103, 115–116; A Story of Units curriculum on, Standards or Mathematical Practice: 1: make sense of problems and persevere in solving them, 14; 2: reason abstractly and quantitatively, 14; 3: construct viable arguments and critique the reasoning the others, 14–15; 4: model with mathematics, 15; 5: use appropriate tools strategically, 15; 6: attend to precision, 15; 7: look for make use of structure, 15; 8: look for and express regularity in repeated reasoning, 16; Focus Standards for Mathematical Practice to highlight which standard to focus on, 24; woven into each lesson, 28–29 See also Foundational Standards A Story of Functions (9–12 grades): introduction to, 1, 2; year-long curriculum map for, 5, 9fig A Story of Ratios (6–8 grades): introduction to, 1, 2; year-long curriculum map for, 5, 8fig A Story of Units (PreK–5 grades): alignment with standards for mathematical practice, 14–16; balanced approach to lesson design as part of, 24–25; concepts introduced during the, 8–10; differentiated instruction of, 37–42; fluency as shaping instructional vision of, 12; Grade-Level Content Review, 17–22; how knowledge is sequenced in, 38; Instructional Shifts alignment with, 10–14; introduction to, 1, 2; math content development for, 5, 8; terminology related to, 123–139; time frames used in, 25fig, 38; yearlong curriculum map for, 5, 6fig–7fig See also Curriculum A Story of Units (PreK–5 grades) concepts: addition and subtraction, 9; fractions, 10; multiplication, 10; numbers through 10, 9; place value and the standard algorithms, 9; Index progressive approach to teaching the, 8; word problems, 10 A Story of Units terminology: Grade 1, 128–130; Grade 2, 130–133; Grade 3, 133–135; Grade 4, 135–138; Grade 5, 138–139; Grade PreK, 123–126; Grade K, 126–128 Student Debrief: description and purpose of, 14; Distribution of Instructional Minutes chart on, 25fig; Exit Ticket component of, 28, 33ig, 34; Lesson 15, 29fig, 33; structured lesson structure on, 25fig–26; teaching strategies for, 14–15 Students: culminated experiences of Grade 1, 91, 100–101; scaffolds for English language learners (ELLs), 38–39; scaffolds for students with disabilities, 39–41; scaffolds for those performing above grade level, 42; scaffolds for those performing below grade level, 41–42 Students with disabilities: Individualized Education Programs (IEPs) or Section 504 Plans for, 39; scaffolds for, 39–41 Subtraction (Module 1): decomposition strategies for subtraction, 45, 55–56; development of fluency within 10, 45–46, 56–57; subtraction as an unknown addend problem, 45, 53–54; word problems, 45, 54–55 Subtraction (Module 2): aligned with the Standards, 17; Grade-Level Content Review on, 17 Subtraction (Module 4): aligned with the Standards, 17; Grade-Level Content Review on, 17; A Story of Units curriculum on, 9; ten-frame model used for, 103, 120–121 See also Decomposition Subtraction (Module 6): aligned with the Standards, 17; Grade-Level Content Review on, 17 Suggested Tools and Representations, 24 Summary of Year, 17, 18 Summative End-of-Module Assessment, 24 Tape diagram model (Grade Level 1–5): instructional strategies or using, 120; overview of the, 118–119; word problem application of, 103 Telling time instruction, 85, 89–90 | 153 Ten-frame model (Grade Level PreK–3): addition and subtraction applications of, 103; instructional strategies for using, 121; overview of the, 120–121 Terminology: Grade PreK, 123–126; Grade K, 126–128; Grade 1, 128–130; Grade 2, 130–133; Grade 3, 133–135; Grade 4, 135–138; Grade 5, 138–139; module section on, 24 Time frames: Distribution of Instructional Minutes, 25fig; Lesson 15, 29fig–31; A Story of Units use of, 38 See also Instructional Days Topic Overview: Coherence Links included in, 24; Focus Standards listed in the, 24 Two-digit numbers: addition of tens and ones to a, 75, 83–84; addition of tens or ones to a, 75, 80–82; comparison of pairs of, 75, 79; numbers to 120, 90, 95–96 Universal Design for Learning (UDL): differentiated instruction using, 38–39; scaffolds and accommodations as part of, 37–38; scaffolds for English language learners (ELLs), 38–39; scaffolds for students performing above grade level, 42; scaffolds for students performing below grade level, 41–42; scaffolds for students with disabilities, 39–41 Unknown problems: counting on or making ten to solve result unknown and total, 58–59, 61–63; counting on or taking from ten to solve result unknown and total, 59, 63–65; solve compare with bigger or small types of, 95; solve compare with difference types of, 94; strategies for solving change or addend, 59, 65–66; varied problem types within 20, 91, 99–100 Word problems: addition, 44, 50–51; comparison, 90, 93–95; solve compare with bigger or small unknown problem types, 95; solve compare with difference unknown problem types, 94; A Story of Units curriculum on, 10; subtraction, 45, 54–55; tape diagram model used for, 103, 118–120 WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley’s ebook EULA ... Ten-Frame 12 0 Chapter Terminology Grade PreK Grade K Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 12 3 12 3 12 6 12 8 13 0 13 3 13 5 13 8 Notes 14 1 Board of Trustees 14 3 Eureka Math Study Guide: A Story of. .. corresponding to all modules Number and Geometry, Measurement 20 days Fractions 20 days | | Eureka Math Study Guide | Grade Grade 20 days Grade M1: Ratios and Unit Rates (35 days) Grade M1: Ratios and Proportional... Figure 2 .1? ?? Grades PreK–5 Year-Long Curriculum Map: A Story of Units Number Major Mathematical Themes in Each Grade? ?Band Grade Grade Grade M1: Properties of Multiplication and Division and Solving