Reasoning with Shapes and their Attributes A 2nd Grade Model Mathematics Unit - Curriculum, Instruction, Assessment Driven by the Common Core State Standards

Published by Chicago Teachers Union Quest Center 222 Merchandise Mart Plaza, Suite 400 Chicago, IL 60654 Copyright ©2014 by Chicago Teachers Union Quest Center All Rights Reserved. This book may be reproduced and shared for educational purposes only with credit given to the Chicago Teachers Union Quest Center and the American Federation of Teachers Innovation Fund. Printed in the United States of America Editing and Design: Michael P. Moriarty

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Chicago Teachers Union Officers Karen GJ Lewis, NBCT President Jesse Sharkey Vice President Michael E. Brunson Recording Secretary Kristine Mayle Financial Secretary

Quest Center Staff Lynn Cherkasky-Davis, Director of Professional Learning Michael P. Moriarty, NBCT, Special Projects Facilitator - CCSS Carol Caref, NBCT, Director of Research Jennifer Johnson, Special Projects Facilitator - Teacher Evaluation Walter Taylor, NBCT, Professional Development Facilitator Sarah Hainds, Researcher Pavlyn Jankov, Researcher Deborah Pazera, Support Staff Trisha Raymond, Support Staff Jackson Potter, Staff Coordinator “Designing Curriculum, Instruction, and Assessment from Common Core Standards – Prototype Teaching Units” is a project supported by the American Federation of Teachers Innovation Fund. Published by the Chicago Teachers Union Quest Center. Produced by 45 Chicago Public School teachers.

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Table of Contents

Reasoning with Shapes and their Attributes

The Chicago Teachers Union Quest Center model instructional unit, “Reasoning with Shapes and their Attributes,” is designed specifically for the ease of use by a classroom teacher. This document can be printed for teacher and student copies of all assignments and activities. The electronic version is embedded with links for a teacher to quickly locate and access the desired lesson and resource. This complete unit can be found exclusively at www.sharemylesson.com/ctu. The field test video displaying samples for the unit can be viewed at the CTU Quest Center You Tube Channel or directly at http://youtu.be/wEGU-Czj3aY

Contents Foreword by John T. Baldwin ...........................................................................3

Contributors......................................................................................................5

Unit Overview...................................................................................................6  Rationale  Scope and Sequence

Lesson 1: Attribute Sort.....................................................................................12  Practice Day Lesson

Lesson 2: Attributes – Triangles and Quadrilaterals...........................................26

Lesson 3: Right Angles.......................................................................................34

Lesson 4: Special Quadrilaterals – Rectangles....................................................41  Practice Day Lesson

Lesson 5: Special Quadrilaterals –Squares and Rectangles.................................50  Practice Day Lesson Reasoning with Shapes and their Attributes 2nd Grade www.sharemylesson.com/ctu 1

Table of Contents - 1

Table of Contents

Reasoning with Shapes and their Attributes

Lesson 6: Cubes................................................................................................58

Lesson 7: Sorting 3D Shapes.............................................................................64

Lesson 8 - Assessment Lesson: Attribute Riddles..............................................70

Lesson 9: Making Rectangles............................................................................82  Practice Day Lesson

Lesson 10: How Many Squares Fit?..................................................................91

Lesson 11: Finding the Missing Space...............................................................99

Lesson 12: Fraction of Shapes day 1 (halves and fourths).................................110

Lesson 13: Fractions of Shapes day 2 (halves, fourths, and thirds)....................117

Lesson 14: Fractions of Shapes day 3 (fourths)..................................................129

Lesson 15: Culminating Task – Playground Space Problem................................144

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Table of Contents - 2

Foreword Mathematics John T. Baldwin Professor Emeritus of Mathematics, Statistics and Computer Science University of Illinois at Chicago The traditional cartoon of a lesson plan is a short list, written for the principal, describing each day’s work. The lesson plan usually consisted of either page and problem numbers from a text book or the codes for a set of state standards. There was generally no connection between days. In contrast the high school lesson plans in this model instructional unit entitled, “Quadratic Functions” is organized as 8 lessons encompassing approximately 20 days of instruction. Each lesson spans between 1 and 5 class periods. The 2nd grade lesson plans is the geometry unit entitled, “Reasoning with Shapes and Their Attributes,” is organized as 15 lessons, one for each day. Both units are groups of plans that include a culminating task, resources, EL adaptations, assessments, instructional choices and suggestions, learning progressions, videos, an examination of how each lesson addresses the CCSS, anticipated student/teacher responses, and activities for classroom use. They were written by teachers; I was the consulting mathematician. The 2nd grade unit addresses 3 Common Core Content standards. The 9th grade unit addresses 9 Common Core content standards and embodies the practice standards. What was the role of this unit development project? The mathematical standards of the Common Core emphasize conceptual learning and student involvement; they are organized as a progression of ideas, not a list of techniques; the timing of topics is shifted (some sooner, some later) from previous standards and curriculum. All these changes required the CTU Quest Center teachers to create new adaptations to mathematics instruction and content delivery. So it was a natural step that the Quest Center teachers spent a year learning to understand the progressions across grade levels before developing units of instruction that reflect this continuity. Such detailed lessons provided herein address not only content but the format of the class; what kind of activities take place in the classroom and what adaptations can be made for English Learners. The CTU’s final products are model instructional units of sample lessons developed by teachers who are now better able to lead instructional change. During the first year of unit development, I commented on collections of lessons written for both the second grade math and the Algebra I unit and discussed them with the teacher authors. Working with a group of professionals to design these materials was a rewarding experience. The teachers have the expertise as to what students know and how to present material that they can connect with. I think the most important contribution of a mathematician to this unit development project was the ability to see the big picture – the connections across K-16 mathematics. Ironically, this vision impacts daily teaching practice most emphatically in recognizing how small mistakes in expressing concepts lead to conceptual problems in the future.

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Foreword - 1

Foreword Mathematics The elementary school unit reflects the need to develop student understanding. The first lesson is on classification and attributes. The interactive problems proposed here for understanding the classification of quadrilaterals in 2nd grade will help students learn that classifying mathematical objects is the same process as classifying more mundane objects. Just as all dogs are animals, all squares are rectangles. By providing a standard scheme for organizing each class, the 2nd grade unit provides uniformity conducive to learning. Since time is provided in each class period for student discussion and reasoning about concepts, the students will learn that mathematics makes sense. This unit skillfully introduces the mathematical uses of such words as column and row; then gradually develops the crucial notion for calculating area: the number of unit squares occupying a space. But this is done without spoiling the intuition by introducing area formulas or even the word ‘area’. This unit’s authors know that these notions are reserved for a later grade when students have internalized the basic conceptions such as that different shapes can have the same area (there are many ways to divide a rectangle into 4 equal pieces). This geometric background prepares for studying fractions and instills the notion of ‘unit’ long before the formal study of units and fractions. This is the kind of progression envisioned in the Common Core State Standards. Lesson 5 of the high school Quadratics unit presents two similar but very different problems. The first is a standard problem of relating the height of a ball to the time in flight. The second problem gives a different quadratic relating the time of flight to the distance traveled. The connection between these two problems is given by the transformation d=rt (where r is initial rate in the horizontal direction). This is the kind of understanding expected in Algebra II. The teachers developing this unit were able to move beyond procedure-oriented teaching of the past. One example is the term ‘simplify’: Simplify is usually a meaningless word. It becomes meaningful if translated: transform an expression into one in a specified normal form. Thus one might simplify a quadratic equation to factored form, vertex-normal form or form. Similarly, one of the CCSS changes in high school mathematical curriculum is the study of functions and transformations earlier in high school. A math teacher sees the connections: the language used in describing the effect on the graph of changing the parameters in a quadratic equation should foreshadow (in 9th grade) or echo (in 11 th grade) the discussion of transformations in geometry. These model instructional units then are not a textbook, nor are they scripted (nothing could be further from the Common Core). Rather they recognize that teaching is a dynamic process that changes with each group of students. They emphasize the key ideas of a topic, suggest activities to help students grasp these ideas, and provide the teacher (perhaps new to teaching this topic to this age of student) some typical student responses and suggestions on how to deal with them.

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Foreword - 2

Contributors

Reasoning with Shapes and their Attributes

The 2nd grade model instructional unit, “Reasoning with Shapes and their Attributes,” consisting of 15 lessons to be taught over a 4 week time period was created by a team of classroom teachers, comprised of authors and field test teachers, over the course of two years. Authors – After the unit was designed and field tested, the authors met with the field test teachers to identify how each lesson and culminating task could be improved from the draft version to ensure all students master the standards. They achieved this goal by reviewing observational data, student artifacts, and the results from the units culminating task and assessment. The authors then redesigned particular lessons, sequences, and/or activities per the identified gaps of instruction from the field test. Authors for the “Reasoning with Shapes and their Attributes” unit:  Andrew Friesema  Lori Zaimi  Leslie Monk  Elizabeth Eberle  Sari Freier  Joshua Lerner, NBCT Field Test Teachers - From classrooms throughout Chicago, the field test teachers implemented the original draft of the unit to collect information on the students’ success in mastering the standard(s) by teaching the model instructional unit. The field test teachers reflected during and after every lesson and analyzed data they gathered on the strengths and challenges of the resources and activities developed, and how the students interacted with them as described in the detailed lesson plan. The field test teachers met with the authors to collaborate on what changes need to be addressed for the unit to meet the needs of all our students. Field Test Teachers for the “Reasoning with Shapes and their Attributes” unit:  Vlada Polin  Cheryl Janusz  Elizabeth Eberle  Yezenia Guzman  Leslie Monk CTU Quest Center Unit Development Team:  Project Manager: Michael P. Moriarty, NBCT Special Projects Facilitator  Coordinated by Lynn Cherkasky-Davis, Director for Professional Learning

English Learner Adaptations Team:  Gloria Henllan-Jones, NBCT  Theresa Insalaco-DeCicco, NBCT  Yoni Vallecillo, NBCT  Xiaodong Zhang, NBCT

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Unit Overview

Reasoning with Shapes and their Attributes

Rationale This second grade fifteen lesson unit is meant to be taught over a four week period. “Reasoning with Shapes and Their Attributes” cultivates the students’ conceptual understanding of area and fractions. The students analyze the properties of basic geometric shapes and design a playground in the unit’s culminating task. This unit was designed through careful examination of the Common Core State Standards for second grade mathematics. The unit’s authors reviewed the Common Core Math Standards for Practice and Content in the primary grades, and chose to concentrate this unit on the geometry standards. There were several reasons for this choice. First, these standards appear deceptively simple. However, upon further analysis, it becomes clear that for students to achieve the skills necessary to master these standards, a carefully designed instructional progression is needed. The unit development team believes that teachers may underestimate the complexity of these standards in their planning, and designed the unit to address that concern. These standards are also essential to the development of other skill sets both in second grade and beyond. For example, partitioning of rectangles into squares builds both the concept of arrays in multiplication and the idea of surface area. Partitioning of rectangles and circles develops the concept of fractions as equal portions of wholes. Understanding of fractions is a key concept in developing algebra understanding. Skillfully developing student understanding of these concepts is essential to not only their understanding of geometry, but also, to other areas of mathematical content. In researching developmentally appropriate approaches to teaching geometry to second graders, the unit’s authors looked at the van Hiele levels of geometric thought as articulated in Van De Walle’s Elementary and Middle School Mathematics: Teaching Developmentally (Allyn and Bacon, 2009). The ‘van Hiele levels of geometric thought’ is a hierarchical model of ways of understanding spatial ideas. It begins with visualization, in which students describe shapes based on visual characteristics, or what it “looks like.” The next stage is analysis, in which students are able to talk about classes of shapes rather than just a single shape. Students then generalize about all rectangles rather than only specific rectangles. It is at this level that students begin to move away from what shapes look like, to properties (attributes) of shapes. The lessons in this unit are intended to begin moving students away from just thinking about what shapes look like towards a more specific analysis of properties. Overview of Unit Skills/Content The lesson plan format of this unit is not a typical format that teachers use when planning day to day lessons, but it is one the authors felt best addresses Common Core State Standards for Math and the Standards for Mathematical Practice. In addition to math content, this unit also addresses the pedagogy of communicating with students, appropriate questioning and Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 6

Unit Overview - 1

Unit Overview

Reasoning with Shapes and their Attributes

discussion techniques, student engagement, assessment, and the flexibility and responsiveness of the teacher. This unit asks teachers to think about anticipated student responses and provides suggested teacher supports. This lesson plan format also extensively addresses how students might think about the math content addressed in each lesson. While not an exhaustive list of possible student responses and teacher supports, as each classroom context is different, this resource provides teachers with common misconceptions and understandings about geometry. Because the language of the Standards for Mathematical Practice speak much to the need for have students to “make sense of problems and persevere in solving them” and “use questioning and discussion techniques,” this unit suggests opportunities for this to happen. Through structured problem solving, teachers using this lesson plan format encourage wait time, provide individual problem-solving, and allow students to struggle through the math without “spoon-feeding” them the content/skill. A traditional math classroom might be structured in a way that the teacher reviews the math content at the beginning of a lesson, has the class practice a few problems together, do a few more problems independently, and then ends the lesson by discussing the solutions to each problem. This unit however, encourages a different format to math instruction which the authors feel better addresses the Standards for Mathematical Practice. Assessment The unit’s authors designed a rubric to assess the students’ mastery level of the standards demonstrated in their culminating task. The unit also provides for formative assessment of learning objectives throughout the lessons. Common Core State Standards for second grade Mathematics addressed in this unit:  2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.  2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.  2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.  2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

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Unit Overview - 2

Unit Overview

Reasoning with Shapes and their Attributes

Key Terms in Unit Plan CCSS addressed – The unit’s authors focused on the second grade geometry standards. Each lesson highlights the specific standards of focus, although there may be other standards that could apply to each lesson beyond those that are listed. Standards for Mathematical Practice – The Standards for Mathematical Practice are just as important as the mathematics content standards and are paid close attention to in the plan for instruction of each lesson. While each lesson involves many of the Standards for Mathematical Practice, only one or two these standards are emphasized in any particular lesson for teachers to focus on in developing in their students. Lesson Summary – Each lesson includes an overview of the lesson and a description of how student learning will develop from beginning to end. Lesson Objectives – These are the key ideas that students learn and the instructional outcomes they master by the end of the lesson. Lesson Plan – The individual lesson plans which make up the unit serve as a roadmap for developing student understanding. The unit’s lesson design is different in layout than what has typically been used by teachers. It uses a three column format which flows from Steps and Learning Activities to Anticipated Student Responses to the Analysis of Common Core State Standards Implementation. Steps and Learning Activities – This column shows the major events and flow of the lesson. Within this column is the Introduction, Posing the Problem/Student Work Time, Comparing and Discussing, and Summing Up and/or Reflection/Assessment. The following suggested time distribution allows students to be in control of much of their learning, with the teacher creating an environment which supports student engagement and facilitates their exploration of math content. *for further description of learning activities, see below* Sample Lesson Time Distribution    

Introduction (3-5 minutes) Posing the Problem/Student Work Time (15-20 min) Comparing and Discussing (15-20 min) Summing Up and/or Reflection/Assessment (5-10 min)

Anticipated Student Responses – This column shows additional strategies, questions or statements that the teacher may need to use to help students develop understanding relating to the CCSS and the Standards for Mathematical Practice. Anticipated student responses are classified as “SR” and teacher support is classified as “TR.” When looking at Student Responses, the teacher may see something which looks like S1, S2, S3. This pattern indicates three different student responses: student 1, student 2 and student 3. Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 8

Unit Overview - 3

Unit Overview

Reasoning with Shapes and their Attributes

These responses will indicate three different ways of thinking about the topic. A teacher might have several students in a particular class that share the same thinking as S1, while others share the same thinking as S2. There might be numerous ways to anticipate student thinking within a particular lesson. Analysis of Common Core State Standards Implementation – This column contains points of evaluation for teachers to use as they work with students through each lesson. These points of evaluation can be used to determine if the lesson is effectively addressing the stated objectives, and information on how this part of the lesson addresses the CCSS and Standards for Mathematical Practice. When monitoring student progression through each lesson component (Introduction, Posing the Problem/Student Work Time, Comparing and Discussing, and Summing Up and/or Reflection/Assessment), the teacher will have indicators as to whether or not the lesson’s goals are being met and, if not, how to adjust instruction. Adaptations for ELs – Instructional adaptations for learners of English language are also provided, including content-related word banks and suggested instructional strategies based on student needs. Learning Progressions – This section describes how the students’ previous knowledge, current knowledge, and future knowledge are aligned to the CCSS. *Learning Activities Introduction – Included in some but not all lessons, this section is an opportunity for the teacher to link ideas that were taught in the prior day’s lesson to what will be learned in the current day’s lesson. The teacher can discuss instructional outcomes and goals for the lesson at this time. Posing the Problem – The problem statement is intended to be an open-ended problem that allows for various entry points to the lesson. When presenting the problem statement, the teacher should allow time for students to think about the problem without first explaining how to do the problem. Students then should “struggle” through the problem, making errors and thinking deeply about how to approach the problem during independent student work time. Students should be taught to refer to any prior knowledge they have about the problem to help them in solving it. During the Posing the Problem stage of the lesson plan, the teacher is provided with recommended groupings, deciding if the students should attempt to solve the problem as individuals, partners, or in small groups. As students work through the problem, the teacher should circulate through the room and observe the different methods students are using to solve the problem.

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Unit Overview - 4

Unit Overview

Reasoning with Shapes and their Attributes

Comparing and Discussing – After an appropriate amount of time has been allocated for problem solving, the teacher then brings the class together to discuss their various responses, keeping in mind the different problem solving approaches students used, to help guide the discussion. This section provides a recommended flow for which student responses should be discussed and in what order. In some cases, starting with the wrong answer and having students think about the work of their peers might become the entry point for the discussion. Not all student responses need to be specifically addressed during a discussion as some responses might lead to confusion about the topic. Carefully selecting the responses to compare and discuss, and using them to enhance student learning is important. The discussion should be student led, with the teacher serving as facilitator. Summing Up – After the discussion, the unit provides the teacher various strategies to monitor student progress toward meeting learning objectives. The teacher might make reference to where the class will move next within their learning. Closure should be clear in this wrap up of the lesson. and/or Reflection/Assessment – Each lesson contains some form of a student reflection or assessment. While informal assessment and checks for understanding are embedded throughout the lesson, student reflections and more formal formative assessments become opportunities for the teacher to monitor individual understanding and track progression students are making toward the learning objectives. Chalkboard/Whiteboard Writing – The lessons in this unit are designed in a way so that in the Posing the Problem section, student responses and student thinking are highlighted in writing on the board in a way that shows how student thinking has evolved over the course of the lesson. The unit encourages teachers to use the chalk/whiteboard vs. using an overhead projector or SMART board to help show the progression of ideas from beginning to end. By the end of a lesson, the board should tell a story of how the lesson progressed. Practice Lessons – The practice lessons (which occur on day two of lessons: 1, 4, 5, and 9) were developed following the field test of this unit. The Field Test teachers realized that practice lessons allowed for necessary reinforcement of newly learned concepts and vocabulary, and also provided teachers an opportunity for further assessment of student understanding. Practice lessons follow a similar format as regular lessons: Introduction, Posing the Problem, Comparing and Discussing, and a Summary. However, the time delegated to each portion of the practice lessons may diverge from that of regular lessons, depending on need and level of student understanding. In practice lessons, more weight is typically placed on the independent work portion of the lesson. Here, the teacher can check in with individuals to discern their understanding from the prior day’s comparison and discussion, and their abilities to apply those new understandings. Practice lessons may also be referenced for ideas to provide students with additional supports. Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 10

Unit Overview - 5

Reasoning with Shapes and their Attributes Scope and Sequence Calendar

Day 1

Day 2

Day 3

Day 4

Day 5

Lesson 1 – Attribute Sort Lesson 1 – Attribute Sort Practice Day Lesson

Lesson 2 – Attributes – Triangles and Quadrilaterals

Lesson 3 – Right Angles

Lesson 4 – Special Quadrilaterals – Rectangles

Day 6

Day 7

Day 8

Day 9

Day 10

Lesson 4 – Special Quadrilaterals – Rectangles Practice Day Lesson

Lesson 5 – Special Quadrilaterals –Squares and Rectangles

Lesson 5 – Special Quadrilaterals –Squares and Rectangles Practice Day Lesson

Lesson 6 – Cubes

Lesson 7 – Sorting 3D Shapes

Day 11

Day 12

Day 13

Day 14

Day 15

Lesson 8 – Assessment Lesson 9 – Making Lesson: Attribute Riddles Rectangles

Lesson 9 – Making Rectangles Practice Day Lesson

Lesson 10 – How Many Squares Fit?

Lesson 11 – Finding the Missing Space

Day 16

Day 17

Day 18

Day 19

Lesson 12 – Fraction of Shapes day 1 (halves and fourths)

Lesson 13 – Fractions of Shapes day 2 (halves, fourths, and thirds)

Lesson 14 – Fractions of Shapes day 3(fourths)

Lesson 15 – Culminating Task – Playground Space Problem

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Lesson 1

Reasoning with Shapes and their Attributes

Attribute Sort Lesson Summary

The sorting activity in this lesson allows students to justify their sorting decisions based on defining attributes (e.g., number of sides, number of angles) as opposed to non-defining attributes (e.g., color, overall size). The sorting activity also requires students to look for and make use of structure. The open nature of the problem is intentionally designed to encourage students to persevere in solving the sorting activity at the center of the lesson, and the comparison and discussion section of the lesson provides students a forum for constructing viable arguments and critiquing the reasoning of others.

Lesson Objectives

Students will be able to:  Organize and sort polygons based on attributes.  Discuss and evaluate the validity of various classification systems.

Common Core State Standards:

Standards for Mathematical Practice:

2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

MP.1: Make sense of problems, and persevere in solving them. MP.3: Construct viable arguments and critique the reasoning of others. MP.7: Look for and make use of structure.

Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials and Attachments List  Lesson Artifacts

Time Allotment - One 60 minute period

Description of Activities: Students will work together in pairs with a set of polygon cards. They will be given a task of classifying them into five groups and communicating their criteria for classification. The teacher will then facilitate a discussion around the criteria and how they support classifying by sides and angles.

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Lesson 1 - 1

Lesson 1

Reasoning with Shapes and their Attributes

Attribute Sort Steps and Learning Activities This column shows the major events and flow of the lesson. All information that is an intended part of the lesson should be included here.

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows additional moves, questions, or statements that the teacher may need to make to support students in meeting the lesson’s objectives. Anticipated student responses are in regular font; teacher support suggestions are in italic.

This column includes analysis of how each part of the lesson addresses CCSS content standards and Standards for Mathematical Practice. The purpose of this information is to make transparent to the reader where and how the development of the standards is apparent in the lesson, and to answer the question, how is this different from what we have done in the past? This portion of the lesson reviews content that the students should have mastered at the end of first grade (distinguishing between defining and non-defining attributes, 1.G.1) and lays the foundation for developing the idea of using defining attributes to classify shapes (2.G.1).

Introduction (10 minutes) The teacher will share that they had a box of shapes which were sorted into five groups, and they dropped them. The shapes need to be put away back into the box. Point out to students that all the shapes are polygons and are the same color and about the same size. Distribute a set of shapes and a sheet of chart paper to each pair of students Posing the Problem (15 minutes) Tell students, “Working with your partner, put the shapes into the five sections on the piece of paper I’ve provided. Be ready to share your thinking about your sort.”

S1: Students sort out all of the triangles first.

T1: Ask students why they chose to put this group of shapes together. Follow up with questions that encourage students to focus on the Pay attention to how students attributes of the shapes. For example, choose to sort their shapes, if students say they put these shapes and whether they use defining together because they are triangles, characteristics such as sides or ask them, how do you know they are angles. triangles? What makes them triangles? How can you use this idea

This problem is designed to draw students’ attention to aspects of the structure of shapes, leaving it up to the students to determine which characteristics they use to sort the shapes. This addresses MP7, look for and make use of structure.

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Lesson 1 - 2

Lesson 1

Reasoning with Shapes and their Attributes

Attribute Sort Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

to sort the rest of the shapes? S2: Students do not have an idea for how to begin sorting. T2: Remind students of other situations in which they have sorted (boys and girls in line, alphabetical order, fiction vs non-fiction, etc.). Ask them, how might they be able to sort the shapes? What do the shapes have in common? What about them is different? S3: Students sort by ‘skinny pointy,’ regular polygons, or some other appearance-based strategy S4: Students sort by # of sides. S5: Students sort by # of vertices. S6: Students identify only convex angles, and do not recognize concave angles. T6: Ask students to count the shapes’ vertices. Comparing and Discussing (25 minutes) Have all of the sorts on the board. Ask students to discuss how they sorted the polygons. Suggested order for students to share out in to illustrate usefulness of sorting by defining attributes:

This question reminds students that there are many real-world situations where we look for patterns or shared characteristics, and that this idea can be applied to the sorting of shapes (MP7)

Asking questions to encourage students to reflect on what they understand and have done with the problem so far, and apply that as a way to continue working through the problem, models a thinking process that will assist students with making sense of problems and persevering in solving them. (MP1)

Ideas for Facilitation: Have audience members ask questions of the presenting group, and share things they find interesting about the presenter’s method.

Qualitative: S3 (skinny & pointy) S1 (shape vs. non shape)

In the “Comparing and Discussing” section, students must articulate why one sorting method is easier and more effective than another. To convince others, they must justify their thinking and critique the thinking of others (e.g.., “This shape goes in this group because it has four sides, and so do all the other shapes in the group.”) (MP3)

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Lesson 1 - 3

Lesson 1

Reasoning with Shapes and their Attributes

Attribute Sort Steps and Learning Activities Quantitative: S4 (number of sides) S5 (number of vertices) Have the group compare and discuss classifying strategies, moving from qualitative to quantitative, from strategies related to how shapes look, to defining attributes. This is intended to show how the sorting strategies based on attributes can be justified based on the number of sides or angles, while strategies based on appearance are more subjective and open to opinion.

Anticipated Student Responses and Teacher Support

If students do not recognize the efficiency of sorting by defining attributes, present a new shape and ask the class which category it would go in based on one qualitative and one quantitative sorting strategy, and why. If they sort it using a non-defining attribute, argue that the shape is not pointy, skinny, etc. and ask students whether/how this can be proven. On the other hand, if they place it in the 5-side category (a defining attribute), that choice can be justified objectively.

Analysis of Common Core State Standards Implementation

Through the sorting activity, comparison of classmates’ strategies, and whole group discussion, students have further developed their understanding of how defining attributes can be used to identify two dimensional shapes and distinguish them from each other (2.G.1).

Summing up (10 minutes) At the end of the class, teacher will compose and students will copy a summary statement into their notebooks. “Today we learned that when sorting shapes, it is easier to sort by something we can check and prove to our friends.”

To reinforce the new learning of the summary and to check for understanding, present a new shape to the class and ask them to put it in a category using one of the ideas from the days comparison and discussion of ideas.

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Lesson 1 - 4

Lesson 1

Reasoning with Shapes and their Attributes

Attribute Sort Resources for English Learners English Language Objective(s):  Students will compare and contrast polygons. Beginning ELs will use tier I and tier II words in phrases or short sentences to explain their reasoning for placing shapes in the specific category. English Learners Unit Vocabulary: Every General Academic Terms (Tier 2) Day Terms/ Phrases (Tier 1) Angles equal attributes Face shapes side sort

Content Specific Terms (Tier 3) polygon triangles number quadrilaterals pentagons hexagons cubes vertices

Learning Progressions Previous Grade 1.G.1 Distinguish between defining attributes versus non-defining attributes; build and draw shapes to possess defining attributes By the end of first grade, students are expected to learn the difference between defining (e.g. number of sides, number of vertices) and non-defining (e.g. color, overall size) attributes, and apply this knowledge to the drawing of shapes Current Grade 2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. By the end of second grade, students will be able to apply their knowledge of defining and non-defining attributes, which they mastered in first grade, to draw shapes based on those defining attributes (e.g., “a triangle is a triangle because it has three sides/angles”) and identify them by name. Next Grade 3.G.1 Understand that shapes in different categories (e.g. rhombuses, rectangles and others) may share attributes (e.g. having four sides), and that the shared attributes can define a larger category (e.g. quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. By the end of third grade, students use defining attributes not only to identify shapes but also to classify shapes into categories and subcategories by their shared attributes.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 16

Lesson 1 - 5

Lesson 1

Reasoning with Shapes and their Attributes

Attribute Sort Materials:  

Attachments:  

chart paper polygon cards

Polygon Cards Attribute Sort Practice Lesson

References: Van De Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 17

Lesson 1 - 6

Lesson 1

Reasoning with Shapes and their Attributes

Attribute Sort Lesson Artifacts:

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Lesson 1 - 7

Lesson 1

Reasoning with Shapes and their Attributes

Attribute Sort Lesson Artifacts:

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Lesson 1 - 8

Lesson 1

Reasoning with Shapes and their Attributes

Attribute Sort Lesson Artifacts:

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Lesson 1: Day 2 Introduction (5 minutes) Review attributes of and define a polygon: -shape with all straight lines -shape is enclosed Posing the Problem (20 minutes of student work time) Teacher will ask students to work with groups to sort cards into two categories: Polygon, Not a Polygon. Students are directed to record their proofs directly on the shapes: All the lines are straight.

One of the lines is curved.

There is an opening.

There are no openings.

For students who are struggling, give them a sheet with the following list duplicated several times. Students can cut and paste a list directly onto each shape and ✓or✗each item. ☐ shape has all straight lines ☐ shape is enclosed ☐ polygon ☐ not a polygon Each group pastes cards with proofs onto a T-chart:

Comparing and Discussing (25 minutes) Terms to use from Lesson Day 1: vertex, sides Student groups bring T-charts to board and share proofs. Teacher will draw attention to a specific shape (highlight with colored marker on each T-chart) on multiple charts that has been categorized as both ‘a polygon’ and ‘not a polygon.’ Teacher will facilitate a discussion about where the card belongs and why. Summary (10 minutes) Teacher will create a larger, anchor T-chart for the class and show class 2 polygons and 2 non-polygons, record proofs directly on cards, and have students paste the four cards in the appropriate columns. Teacher summarizes: “We know a shape is a polygon if it is made up of all straight lines and is enclosed.” Students draw one polygon and one non-polygon in their math notebooks and complete sentence starters below each: I know this is a polygon because ___________. I know this is not a polygon because ___________.

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Lesson 2

Reasoning with Shapes and their Attributes

Attribute – Triangles and Quadrilaterals Lesson Summary

The construction activity in this lesson allows students to use straight edges to construct triangles and quadrilaterals, giving them an opportunity to develop the mathematical practice of using appropriate tools strategically. They are also developing the mathematical practice of looking for and making use of structure by sorting shapes based on a defining attribute, such as the number of sides or vertices that it has. By asking students to come up with their own sorting strategy rather than telling students how to sort the shapes, the task encourages students to make sense of the problem and persevere in solving it. The comparison and discussion section of the lesson allows students to practice constructing viable arguments by explaining how they sorted their shapes using attributes.

Lesson Objectives

Students will be able to:  Construct and sort triangles and quadrilaterals based on attributes.  Discuss and compare their sort and use attributes to classify and name triangles and quadrilaterals.

Common Core State Standards:

Standards for Mathematical Practice:

2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

MP.1: Make sense of problems, and persevere in solving them. MP.3: Construct viable arguments and critique the reasoning of others. MP.5: Use appropriate tools strategically. MP.7: Look for and make use of structure.

Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials and Attachments List  Lesson Artifacts

Time Allotment - One 60 minute period

Description of Activities: Students will connect the dots to enclose an animal inside of either a triangle or a quadrilateral. Students will then cut out the shapes with the enclosed animal and divide them into two groups that look alike. Students will discuss how they divided the shapes into two groups (four sided and three sided) and will begin using the geometric terms “triangle” to identify a 3-sided shape, and “quadrilateral” to describe a 4-sided shape. After identifying and naming the shapes based on the number of sides, students will continue exploring triangles and quadrilaterals by using dot paper to construct triangles and quadrilaterals on their own.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 26

Lesson 2 - 1

Lesson 2

Reasoning with Shapes and their Attributes

Attribute – Triangles and Quadrilaterals Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows the major events and flow of the lesson. All information that is an intended part of the lesson should be included here.

This column shows additional moves, questions, or statements that the teacher may need to make to support students in meeting the lesson’s objectives. Anticipated student responses are in regular font; teacher support suggestions are in italics. Any information that is dependent upon students’ responses to the lesson should be included in this column.

This column includes analysis of how each part of the lesson addresses CCSS content standards and Standards for Mathematical Practice. The purpose of this information is to make transparent to the reader where and how the development of the standards is apparent in the lesson, and to answer the question, how is this different from what we have done in the past?

Posing the Problem (20 minutes) The teacher will explain to students that they are to use a straight edge to connect the dots to enclose each animal.

S1: Students do not have a strategy to sort the shapes

This activity allows students the opportunity to use a straight edge to connect points to form a triangle or quadrilateral, an important early skill designed to lay the groundwork for students to use appropriate tools strategically. Students will not be able to choose tools strategically without this important early introduction to various tools that can be used in problem solving. (MP5)

Steps and Learning Activities

To successfully construct triangles and quadrilaterals around each animal, students need to be told to connect all the dots with the same number without drawing any lines across the animals, only around the animals, for example: “Connect all of the dots around the lion with the number 5.” To make this first task even clearer the teacher may wish to model the first example by using a straight edge to connect all of the dots with a number 1 around the penguin to enclose it in a triangle.

T1: Look carefully at the number of sides and corners and see if that gives you any ideas about how to sort the shapes into two groups. S2: Students sort the shapes into two groups based on number of sides S3: Students sort the shapes into two groups based on the number of angles.

The teacher will ask students to cut out the shapes with the animal enclosed and organize them into two groups that look alike.

Students have to make sense of the attributes of the shapes and use those attributes to sort them into two groups. The openness of this task intentionally challenges students to make sense of the task and to persevere in solving it. Teacher support 1 is intended to encourage students to look at the problem in the context of attributes to help them persevere in solving it. (MP1) Students are looking for and making use of structure by sorting a collection of shapes according to how many sides the shapes have. (MP7)

Teacher will check on students as they work to ensure that students have successfully enclosed the animals in the intended triangle or quadrilateral

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Lesson 2 - 2

Lesson 2

Reasoning with Shapes and their Attributes

Attribute – Triangles and Quadrilaterals Steps and Learning Activities Comparing and Discussing (25 minutes) Have students share sorting technique. Compare and contrast the similarities and differences in the strategy of sorting by the number of sides and the strategy of sorting by the number of corners.

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

Teacher facilitates discussion resulting in students comparing and discussing the differences and similarities in sorting by the number of sides and the number of corners (vertices). Discussion should lead students to realize that both sorting strategies resulted in the same sort because the number of sides and the number of vertices are the same.

This comparison and discussion of sorting techniques is designed to encourage students to develop the ability to construct viable arguments and critique the reasoning of others (MP3). Teachers should encourage students to articulate how they organized the shapes into two categories, three-sided and four-sided shapes or shapes with three vertices and shapes with four vertices. Teachers should also encourage students to explain why the two different strategies result in the same sort.

Summing up (15 minutes) After sorting the shapes into groups of 3 sided shapes and 4 sided shapes or shapes with 3 vertices or 4 vertices, the teacher will summarize the ideas discussed in the comparison and discussion and introduce the terms triangle and quadrilateral. “Today we sorted shapes into two groups based on the number of sides and vertices. The two groups we sorted shapes into were shapes with 3 sides and the 3 vertices, and shapes with 4 sides and 4 vertices. A triangle is a shape that is enclosed by 3 straight lines, and has 3 vertices. A quadrilateral is a shape that is enclosed by 4 straight lines, and has 4 vertices.”

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 28

Lesson 2 - 3

Lesson 2

Reasoning with Shapes and their Attributes

Attribute – Triangles and Quadrilaterals Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Assessment activity: Students use dot paper and a straight edge to construct triangles and quadrilaterals

Analysis of Common Core State Standards Implementation This activity assesses students’ understanding of the attributes of triangles and quadrilaterals and also addresses MP 5, use appropriate tool strategically, by checking to see if they are able to construct triangles and quadrilaterals using a straight edge and dot paper. (MP5)

Resources for English Learners English Language Objective(s):  Students will compare and contrast triangles and quadrilaterals. Beginning ELs will use tier I and tier II words in phrases or short sentences to explain their reasoning for grouping and constructing shapes into the categories of triangle and quadrilateral. English Learners Unit Vocabulary: Every Day Terms/ Phrases (Tier 1) equal Face side sort straight edge

General Academic Terms (Tier 2) angles attributes shapes

Content Specific Terms (Tier 3) polygon triangles number quadrilaterals pentagons hexagons cubes vertices

Learning Progressions Previous Grade 1.G.1 Distinguish between defining attributes versus non-defining attributes; build and draw shapes to possess defining attributes By the end of first grade, students are expected to learn the difference between defining (e.g. number of sides, number of vertices) and non-defining (e.g. color, overall size) attributes, and apply this knowledge to the drawing of shapes Current Grade 2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 29

Lesson 2 - 4

Lesson 2

Reasoning with Shapes and their Attributes

Attribute – Triangles and Quadrilaterals Learning Progressions By the end of second grade, students will be able to apply their knowledge of defining and non-defining attributes, which they mastered in first grade, to draw shapes based on those defining attributes (e.g., “a triangle is a triangle because it has three sides/angles”) and identify them by name. Next Grade 3.G.1 Understand that shapes in different categories (e.g. rhombuses, rectangles and others) may share attributes (e.g. having four sides), and that the shared attributes can define a larger category (e.g. quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. By the end of third grade, students use defining attributes not only to identify shapes but also to classify shapes into categories and subcategories by their shared attributes.

Materials:   

Attachments: 

Connect the dot worksheet Dot paper Ruler or some other straight edge

Connect the dot worksheet

References: Shoseki, T. (2006) Mathematics for Elementary School 2B. Tokyo, Japan: Tokyo Shoseki Co. Ltd. Van De Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 30

Lesson 2 - 5

Lesson 2

Reasoning with Shapes and their Attributes

Attribute – Triangles and Quadrilaterals Lesson Artifacts:

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 31

Lesson 2 - 6

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Lesson 3

Reasoning with Shapes and their Attributes

Right Angles Lesson Summary

The construction of a right angle tool to explore their classroom space to find other right angles using a direct comparison method will allow the students the opportunity to develop the mathematical practice of using appropriate tools strategically. The design of the task as an active investigation of the students’ classroom space is intended to be high interest to encourage students to make sense of the task and persevere in finding right angles throughout their classroom environment. Having students use a direct comparison method to justify that the angle they have discovered is a right angle gives students the opportunity to construct a viable mathematical argument. The comparison and discussion portion of the lesson gives students a framework to develop the mathematical practice of communicating their ideas to others and also to evaluate the ideas of their classmates.

Lesson Objectives

Students will be able to:  Construct right angles and use their constructed right angle as a tool to explore their classroom environment for examples and non-examples of right angles.  Organize their collected data.  Share their findings with the rest of the class to create a classroom list of right angles present in the classroom. Common Core State Standards:

Standards for Mathematical Practice:

2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

MP.1: Make sense of problems, and persevere in solving them. MP.3: Construct viable arguments and critique the reasoning of others. MP.5: Use appropriate tools strategically.

Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials and Attachments List

Time Allotment - One 60 minute period

Description of Activities: Students will be introduced to a right angle as the angle formed when they fold a rectangular piece of paper in half. The angle formed by the folded line and the bottom of the piece of paper is a right angle. Students will then work independently to construct a right angle out of an irregularly shaped piece of paper. They will compare the corner they have made to the corner of the paper they folded in the introduction to the lesson and to the corner of a set square. Students will compare the angle they have constructed out of irregularly shaped paper, the right angle they constructed by folding the rectangular piece of paper in the introduction , and the right angle on the set square. The class will come to the consensus that all of these angles match, or are the same. The teacher will share with the class that these angles are called right angles. Students will then use the right angle they constructed to find other right angles in the classroom. Students will record their data on a t chart that includes examples of right angles on one side and nonexamples on the other side. Students will share out their findings, make sure the examples are valid, and compile a comprehensive classroom list of examples of right angles in the classroom.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 34

Lesson 3 - 1

Lesson 3

Reasoning with Shapes and their Attributes

Right Angles Steps and Learning Activities This column shows the major events and flow of the lesson. All information that is an intended part of the lesson should be included here.

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows additional moves, questions, or statements that the teacher may need to make to support students in meeting the lesson’s objectives. Anticipated student responses are in regular font; teacher support suggestions are in italics. Any information that is dependent upon students’ responses to the lesson should be included in this column.

This column includes analysis of how each part of the lesson addresses CCSS content standards and Standards for Mathematical Practice. The purpose of this information is to make transparent to the reader where and how the development of the standards is apparent in the lesson, and to answer the question, how is this different from what we have done in the past?

Introduction (15 minutes) The teacher will distribute rectangular shaped paper to each student and model for the student how to fold it in half. Teacher will then tell the students that the angle formed by the fold and the bottom of the sheet of paper is a right angle. Note to Teacher: Students should form the right angles by folding to give them the experience of creating a right angle by taking a straight line and bisecting it. Teacher will introduce the term ‘angle’ and use the minute and hour hands of a ‘clock’ to show that an angle is the space created between two lines when they meet. Students are then introduced to the term ‘right angle,’ as teacher displays 3 o’clock and says, “’3 o’clock angle’ is another term for ‘right angle.’” The teacher will then manipulate the hour hand to show an angle that is greater than a right angle and will ask students: “Is this a right angle?” “How do you know?” The teacher will then manipulate the hour hand to show an angle that is less than a right angle and will ask students: “Is Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 35

Lesson 3 - 2

Lesson 3

Reasoning with Shapes and their Attributes

Right Angles Steps and Learning Activities this a right angle?” “How do you know?” Students should each be given ‘clocks’ to practice showing right angles/3 o’clock angles and angles that are greater than/less than 3 o’clock. It is not necessary to introduce terms ‘obtuse’ or ‘acute’ at this time. The teacher will tell the students that they will be using their right angles to check their construction of other right angles, and to find other right angles in their classroom.

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

Posing the Problem (5 minutes) The teacher will present each student with an irregularly shaped piece of paper (curved, wavy edges) and present students with the challenge to fold the piece of paper to make a corner that is the same shape as the right angle that they created in the introduction to the lesson. S1: Student does not fold the paper. T1: Check for student understanding to make sure they understand the task. Repeat directions if necessary. If students are still confused, ask them questions about the characteristics of the right angle they made at the beginning of the lesson. What do you notice about the right angle? What does it look like? How could you fold the piece of paper to make it look like the right angle we made at the beginning of the lesson? S2: Student folds paper into corner that is not right angle. T2: The challenge was to create a corner that was the same as the right

Students are not given a specific set of directions to solve the problem; they must come up with their own strategy to successfully fold the irregularly shaped piece of paper. This encourages the development of MP 1, make sense of problems and persevere in solving them. Teacher responses 1 and 2 include possible teacher questions to encourage student perseverance. They are also developing the practice of using appropriate tools strategically by using the right angle they created in the introduction as a model for their construction. Teachers should be informally assessing how accurate students’ constructions are. (MP5)

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Lesson 3 - 3

Lesson 3

Reasoning with Shapes and their Attributes

Right Angles Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

angle they made at the beginning of the lesson. How could you check to see if the corner you constructed is the same as the right angle you made at the beginning of the lesson? (Compare the two by placing the paper on top of the right angle they made at the beginning of the lesson and see if they match up) Do they match up? (Teacher may have to model the use of the student created right angle tool) S3: Student is able to come up with a strategy to successfully fold the irregular shape to create a right angle.

Comparing and Discussing (10 minutes) Have students that were able to successfully construct a right angle using the irregularly shaped paper share their steps for constructing a shape that matches the right angle they made at the beginning of the lesson. Remind students of the term right angle to describe the angle they made at the beginning of the lesson when they folded the rectangular paper.

Ideas for Facilitation Challenge students to explain to the class how they were able to check that their shape was the same as the right angle they made at the beginning of the lesson (matching up their shape with the corner of the right angle they made at the beginning of the lesson to make sure they match).

This comparison and discussion section of the lesson allows students to justify their argument that the angle they created is a right angle using a direct comparison method. (MP3)

Posing the problem (10 minutes) After a discussion of how to determine if an angle is or is not a right angle, students will be directed to compile a list of examples and non-examples of right angles in the classroom. S1: Student is not able to use the right Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 37

Lesson 3 - 4

Lesson 3

Reasoning with Shapes and their Attributes

Right Angles Steps and Learning Activities

Anticipated Student Responses and Teacher Support

angle that they created to identify other examples and non-examples of right angles in the classroom.

T1:Teacher asks student to recall how they evaluated that their original tool was a right angle (compare the corner to the right angle on the cover of their text book and the right angle on the set square) S2. Student is able to successfully evaluate if an angle in the room is or is not a right angle. T2: Model for students how to use the right angle that they created to identify other right angles in the classroom, and how to record the information to be shared later. Comparing and Discussing (20 minutes)

Analysis of Common Core State Standards Implementation

This part of the lesson encourages students to use appropriate tools strategically. Teachers should be informally assessing how students are using the right angle they constructed to evaluate if other objects in the classroom have right angles using a comparison method. (book covers, floor tiles, door frames, window frames, etc.) Are they using the tools or just basing their findings on what shapes “look like”? (MP5)

This comparison and discussion allows students to practice justifying their classification of angles as right angles or not right angles by describing in their own words some type of direct comparison of angles in the classroom to the right angle created at the beginning of the lesson. (MP3) This is an example of how elementary students can construct arguments using concrete referents, for instance, direct comparison of their right angle tool to angles in their classroom.

Students will share out objects that they found in the classroom that they determined had right angles by comparing the angles to the tools they created. Teacher should facilitate this sharing out and discussion by asking students to describe how they know that the items they listed have right angles. Teacher should move students from talking about right angles as right angles just because they look like right angles to describing a method for determining if an angle is a right angle by comparing it to another right angle. Teacher will record students’ findings to create a class chart of right angles in the classroom. Teacher will lead class through a discussion to determine if there is

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 38

Lesson 3 - 5

Lesson 3

Reasoning with Shapes and their Attributes

Right Angles Steps and Learning Activities consensus that all of the items listed are examples of right angles, and if there are any missing; if the list is comprehensive.

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

Summing Up (10 minutes) The class ends with the teacher summarizing the learning that was discussed in the comparing and discussing section of the lesson. This summary is posted for students to record in their notebooks. “Today we learned that we can determine if an angle is a special right angle by comparing it to a right angle tool that we created to see if the angles are the same”

Resources for English Learners English Language Objective(s):  Students will compare and contrast right angles and non-right angles. Beginning ELs will use phrases or short sentences to explain their reasoning for differentiating right angles and non-right angles. English Learners Unit Vocabulary: Every Day Terms/ Phrases (Tier 1) equal face side sort straight edge

General Academic Terms (Tier 2) angles attributes shapes

Content Specific Terms (Tier 3) polygon triangles number quadrilaterals pentagons hexagons cubes vertices right angles

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 39

Lesson 3 - 6

Lesson 3

Reasoning with Shapes and their Attributes

Right Angles Learning Progressions Previous Grade 1.G.1 Distinguish between defining attributes versus non-defining attributes; build and draw shapes to possess defining attributes By the end of first grade, students are expected to learn the difference between defining (e.g. number of sides, number of vertices) and non-defining (e.g. color, overall size) attributes, and apply this knowledge to the drawing of shapes

Current Grade 2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. By the end of second grade, students will be able to apply their knowledge of defining and non-defining attributes, which they mastered in first grade, to draw shapes based on those defining attributes (e.g., “a triangle is a triangle because it has three sides/angles”) and identify them by name. Next Grade 3.G.1 Understand that shapes in different categories (e.g. rhombuses, rectangles and others) may share attributes (e.g. having four sides), and that the shared attributes can define a larger category (e.g. quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. By the end of third grade, students use defining attributes not only to identify shapes but also to classify shapes into categories and subcategories by their shared attributes.

Materials:   

Irregularly shaped pieces of paper for folding 8.5”x11” copy paper Student clocks

References: Shoseki, T. (2006) Mathematics for Elementary School 2B. Tokyo, Japan: Tokyo Shoseki Co. Ltd. Van De Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 40

Lesson 3 - 7

Lesson 4

Reasoning with Shapes and their Attributes

Special Quadrilaterals - Rectangles Lesson Summary

Rather than being given step by step directions as to how to create a rectangle, students are reminded of the right angle they created in the previous lesson, and then given the task of creating a quadrilateral that has 4 right angles. By making the task a problem solving activity rather than a copy design activity, students develop their ability to make sense of problems and persevere in solving them. By then justifying that their constructions are quadrilaterals with 4 right angles by comparing all four angles to the right angle that they created in the previous lesson, students develop the ability to construct viable arguments and justify their ideas with mathematics.

Lesson Objectives

Students will be able to:  Use the techniques they developed in the previous lesson for constructing right angles to create a quadrilateral with four right angles (a rectangle).  Learn that the quadrilateral they have created is a special rectangle called a rectangle. They will learn the attributes that make a quadrilateral a rectangle.

Common Core State Standards:

Standards for Mathematical Practice:

2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

MP.1: Make sense of problems, and persevere in solving them. MP.3: Construct viable arguments and critique the reasoning of others.

Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials and Attachments List

Time Allotment - One 60 minute period

Description of Activities: Students will use the techniques for forming a right angle that they used in the previous lesson to create a quadrilateral with four right angles. They will use the rectangle they create to develop a class list of the attributes of rectangle that will serve as their working definition for a rectangle.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 41

Lesson 4 - 1

Lesson 4

Reasoning with Shapes and their Attributes

Special Quadrilaterals - Rectangles Steps and Learning Activities

This column shows the major events and flow of the lesson. All information that is an intended part of the lesson should be included here.

Posing the Problem (15 minutes) Students will be presented with an irregularly shaped piece of paper similar to the one used in the previous day’s lesson investigating right angles. As a class the teacher will lead the students through a review of how to form a right angle using the irregularly shaped paper. Students will then be given the challenge of using the paper folding technique they used to create a quadrilateral that has 4 right angles.

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows additional moves, questions, or statements that the teacher may need to make to support students in meeting the lesson’s objectives. Anticipated student responses are in regular font; teacher support suggestions are in italics. Any information that is dependent upon students’ responses to the lesson should be included in this column.

This column includes analysis of how each part of the lesson addresses CCSS content standards and Standards for Mathematical Practice. The purpose of this information is to make transparent to the reader where and how the development of the standards is apparent in the lesson, and to answer the question, how is this different from what we have done in the past?

S1: Student does not have a strategy to continue folding the paper to create three additional right angles.

By posing the task as a problem solving challenge rather than having students follow a set of directions for their constructions, the students are allowed to develop their ability to make sense of problems, and the hands on nature of the challenge is an incentive for them to persevere in solving the problem. Teacher response 1 suggests students design their own sketch as a tool to help them with their construction as a way to encourage student perseverance. This will help students conceptualize the

T1: Check to see if students understand the task by asking if they understand what a right angle is from the previous lesson and if they can show you an example. You may also need to ask students if they understand what a quadrilateral is. Ask them to draw what they think a quadrilateral with four right angles might look like. Have them use that sketch to construct the quadrilateral using techniques from the previous days’ lesson. S2: Student folds paper into a quadrilateral

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 42

Lesson 4 - 2

Lesson 4

Reasoning with Shapes and their Attributes

Special Quadrilaterals - Rectangles Steps and Learning Activities

Anticipated Student Responses and Teacher Support

that is not a rectangle

Analysis of Common Core State Standards Implementation

problem. (MP1)

T2: Ask them how they could check their shape to make sure that it is a quadrilateral (check the number of sides) and that it has four right angles. S3: Student is able to use the technique to fold a piece of paper into a right angle to successfully create a rectangle.

Comparing and Discussing (15 minutes) Students should share quadrilaterals that they created using the folding techniques they learned in the previous lesson where they created right angles from irregularly shaped paper. Students should be encouraged to prove to themselves and their classmates that the angles are all right angles using a direct comparison to a known right angle, or by some other method generated by the student.

Ideas for Facilitation: Teacher should facilitate the sharing out so that students are proving to themselves and to their classmates that the quadrilaterals they have formed do indeed have 4 right angles. If students do not know how to prove that they are right angles, remind them of the previous day’s lesson in which they used established right angles such as the corner of their textbook to check if the angles they created were right angles using a direct comparison method. During the comparison and discussion, the teacher should encourage students that are not presenting to listen to students that are sharing their ideas by asking students to restate what ideas or strategies the presenter shared.

This comparison and discussion allows students to develop their ability to construct viable arguments and evaluate the arguments of others by justifying their solutions using mathematics. Students have to prove that their construction is indeed a quadrilateral with 4 right angles and four sides by using their right angle tool to show that the angles on the construction are all right angles. (MP3)

In the next part of the lesson students will be given the task of proving that the opposite sides of their rectangle are the same length. Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 43

Lesson 4 - 3

Lesson 4

Reasoning with Shapes and their Attributes

Special Quadrilaterals - Rectangles Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Posing the problem (10 minutes) Ask students what they notice about the opposite sides of the rectangle. Have students turn and talk to a neighbor to discuss what they notice about the opposite sides of the rectangles they created. Once students have finished with their turn and talk (5 minutes) and the class has confirmed that the opposite sides look the same have students work with their turn and talk partner to figure out how to prove that they really are the same length.

S1: No response.

Comparing and Discussing: (10 minutes) Have students share their findings that the opposite sides of the quadrilaterals that they created look like they are the same length, and have students share how they proved that this is true

Ideas for facilitation: Teacher should facilitate a conversation in which students share their findings that the opposite sides are the same length, and should ask questions that require students to justify their observations to the rest of the class in some way, either through measuring the length of the sides, or by folding the shape along a line of symmetry to meet the opposite side for a direct comparison of length.

Summing Up: (10 minutes) Tell the students that the shape that they made is a special kind of quadrilateral called a rectangle. As a class make a list of the attributes of a rectangle (4 right angles, opposite sides the same length). Have students record the attributes of a rectangle in their notebooks, and add the attributes to the class data pad, or display somewhere in the classroom with other attributes of shapes students have been articulating.

Analysis of Common Core State Standards Implementation

T1: Indicate what are opposite sides by pointing them out to the student and asking them if they notice anything that those sides have in common. S2: They are all the same length. T2: This may be correct if a student has created a square. Squares are addressed in the next lesson. S3: The top and the bottom are the same length and the sides are the same length.

If some students argue that it is a square refer the students to the attributes of a rectangle and ask them and the rest of the class if their square does or does not fit with the attributes of a rectangle. (The idea that a square is a special type of rectangle will be explored further in the following lesson).

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Lesson 4 - 4

Lesson 4

Reasoning with Shapes and their Attributes

Special Quadrilaterals - Rectangles Resources for English Learners English Language Objective(s):  Students will compare and contrast a rectangle with other quadrilaterals. Beginning ELs will use phrases or short sentences to explain how a rectangle is different from other quadrilaterals. English Learners Unit Vocabulary: Every Day Terms/ Phrases (Tier 1) equal Face side sort straight edge

General Academic Terms (Tier 2) angles attributes shapes

Content Specific Terms (Tier 3) polygon triangles number quadrilaterals pentagons hexagons cubes vertices right angles rectangles

Learning Progressions Previous Grade 1.G.1 Distinguish between defining attributes versus non-defining attributes; build and draw shapes to possess defining attributes By the end of first grade, students are expected to learn the difference between defining (e.g. number of sides, number of vertices) and non-defining (e.g. color, overall size) attributes, and apply this knowledge to the drawing of shapes Current Grade 2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. By the end of second grade, students will be able to apply their knowledge of defining and non-defining attributes, which they mastered in first grade, to draw shapes based on those defining attributes (e.g., “a triangle is a triangle because it has three sides/angles”) and identify them by name. Next Grade 3.G.1 Understand that shapes in different categories (e.g. rhombuses, rectangles and others) may share attributes (e.g. having four sides), and that the shared attributes can define a larger category (e.g. quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. By the end of third grade, students use defining attributes not only to identify shapes but also to classify shapes into categories and subcategories by their shared attributes.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 45

Lesson 4 - 5

Lesson 4

Reasoning with Shapes and their Attributes

Special Quadrilaterals - Rectangles Materials:  

Attachments: 

Irregularly shaped pieces of paper for use in folding activities. Right angles from previous day’s lesson

Practice day Lesson

References: Shoseki, T. (2006) Mathematics for Elementary School 2B. Tokyo, Japan: Tokyo Shoseki Co. Ltd. Van De Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 46

Lesson 4 - 6

Lesson 4: Day 2 Introduction (5 minutes) Review attributes of and define a rectangle:

-polygon with 4 right angles -both sets of opposite sides are the same length

Posing the Problem (20 minutes of student work time) Teacher will ask students to work with groups to sort cards into two categories: Rectangle, Not a Rectangle. Students are directed to record their proofs directly on the shapes: It is a polygon. There are only 3 right angles.

There are 4 right angles. The red sides are the same length AND the blue sides are the same length.

For students who are struggling, give them a sheet with the following list duplicated several times. Students can cut and paste a list directly onto each shape and ✓or✗each item. ☐ polygon ☐ 4 right angles ☐ both set of opposite sides are the same length. ☐ rectangle ☐ not a rectangle Each group pastes cards with proofs onto a T-chart:

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Comparing and Discussing (25 minutes) Terms to use from Lesson Day 2: vertex, sides, right angles Student groups bring T-charts to board and share proofs. Teacher will draw attention to a specific shape (highlight with colored marker on each T-chart) on multiple charts that has been categorized as both ‘a rectangle and ‘not a rectangle.’ Teacher will facilitate a discussion about where the card belongs and why. Summary (10 minutes) Teacher will create a large anchor rectangle and have students share out attributes of a rectangle as teacher records them directly on anchor rectangle. Teacher summarizes: “We know a shape is a rectangle if it is a polygon with 4 right angles and both sets of opposite sides are the same length.” Students draw a rectangle in their math notebooks and complete sentence starter: “I know this is a rectangle because ________________.

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Lesson 5

Reasoning with Shapes and their Attributes

Squares and Rectangles Lesson Summary

In this lesson, students will use the learning from previous lessons, as well as new learning to construct and justify their own ideas about the attributes of a square. They are encouraged to persevere in proving their thinking and critique the methods and reasoning of other students through the comparison and discussion section of the lesson. This lesson provides experiences that will give a foundation for 3.G.1, where students will be asked to understand that shapes in different categories may share attributes and that those attributes can define a larger category.

Lesson Objectives

Students will be able to:  Recognize the attributes of a square and how they differ from and are similar to the attributes of a rectangle. These attributes will form a working set of criteria for classifying a shape as a square.

Common Core State Standards:

Standards for Mathematical Practice:

2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

MP.1: Make sense of problems, and persevere in solving them. MP.3: Construct viable arguments and critique the reasoning of others.

Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials and Attachments List

Time Allotment - One class period- 45-60 minutes

Description of Activities: Students will take a rectangular piece of paper, and fold it so that the bottom lies along the left side. They will cut off the excess paper, resulting in a triangular piece of paper that when unfolded is a square. Students will compare the angles of the square to the right angle they created in the previous lesson to determine that all four sides are right angles. They will fold the square in half diagonally, horizontally, and vertically to confirm that all four sides are the same length. They will use these findings to come up with a class list of the attributes of a square; a square is a quadrilateral that has four right angles, and has four sides that are the same length.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 49

Lesson 5 - 1

Lesson 5

Reasoning with Shapes and their Attributes

Squares and Rectangles Steps and Learning Activities This column shows the major events and flow of the lesson

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows additional moves, questions, or statements that the teacher may need to make to help students. Anticipated student responses are in regular font; teacher support suggestions are in italics.

This column contains questions that can be used to determine if the lesson is effectively addressing the stated objectives, and information on how this part of the lesson addresses CCSS content standards and Standards for Mathematical Practice.

Introduction (10 min) Before the students are given a problem, the teacher will lead them through a construction task. First, direct students to create a quadrilateral by taking a paper rectangle, folding the bottom corner up so that the left side is now matched up perfectly with the topside. Instruct students to cut off the excess paper, leaving a square that is folded along a diagonal.

Posing the Task/Student Work Time (15 min.) Ask students to unfold their shapes and ask, “What type of quadrilateral you have made? How do you know it is that shape?”

If students have difficulty folding the rectangle correctly, you may have to model folding the rectangle for them. It is preferable to have the students see the folding of the adjacent sides than to have square pieces of paper provided for them.

This portion of the lesson reviews the previous day’s work of identifying a SR1: It is a rectangle rectangle by its defining attributes, and introduces the notion that other TR1. Ask student to explain how they shapes might require a more careful can prove that this is a rectangle definition. Students will be asked to using the attributes that define a prove their thinking and methods for rectangle that we came up with in identifying attributes of these shapes the previous lesson (opposite sides (MP 3). the same length, four right angles) Anticipated Student Responses

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 50

Lesson 5 - 2

Lesson 5

Reasoning with Shapes and their Attributes

Squares and Rectangles Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

Student 2: It is a square

“Do you remember how we proved a quadrilateral was a rectangle, yesterday? We will use those ideas to see how this new shape is different from other rectangles.” Instruct students to try to prove that their shape is a rectangle or a square. Ask the students to show you the attributes they can prove.

TR 2: Ask students to explain how they know it is a square. If student’s explanation is just that it looks like a square; ask them what attributes make a square a square. (Many students may recognize that the shape is a square but not have a clear definition of what attributes define a square or know how to prove it. The teacher prompts that follow will facilitate their discovery of both.)

SR1: Student will use their folded sheet of paper (right angle) to show that the angles of the shape are all right angles. SR2: Student will fold shape horizontally and vertically, perhaps even diagonally to prove that the sides are the same length.

TR1: teacher should prompt and remind students of yesterdays lesson as necessary to ensure that students are able to prove their thinking

At this point in the lesson, students should be able to manipulate their squares by folding and continuing to check angles against the folded right angle they made in the previous lesson, they are making sense and persevering in solving this problem, using previous knowledge (MP3).

Comparing and Discussing (20 min.) Have students share their new learning about this new shape- Teacher will affirm that this shape is a “square”.  You should guide the discussion toward the goal of a class consensus on the attributes of a

By asking questions and guiding the discussion of characteristics the teacher models a way to continue working through the problem and a thinking process that will assist students with Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 51

Lesson 5 - 3

Lesson 5

Reasoning with Shapes and their Attributes

Squares and Rectangles Steps and Learning Activities square, making sure that these student ideas are shared out during the discussion:

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

making sense of problems and persevering in solving them. (MP1)

SR1: I checked with the folded piece of paper and saw that all four angles were square angles. SR2: I folded my shape and lined up the edges to show that all four sides were the same length.

Students are working on constructing viable arguments to explain to their classmates how they discovered that all the angles are right angles, and that all four sides are the same length. They are analyzing the arguments of others, by listening and participating in the discussion (MP3).

SR3: When we made the rectangle the sides were not all the same length. SR4: In the rectangle all of the angles were square angles. Say, “What can we say about squares now that we have proved they are different than the rectangles we looked at yesterday?” Ask students to write a list of characteristics of a square as a class using all of their ideas and the characteristics of a rectangle that were discussed the day before. Summing up (5 min.) Have students record the characteristics of a square in their notebooks as well as methods that they used to prove that the shape is a square. A square is a rectangle where all four sides are the same length.

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Lesson 5 - 4

Lesson 5

Reasoning with Shapes and their Attributes

Squares and Rectangles Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Student Reflection (5 min) Have students reflect on the lesson in their math notebooks. Possible prompts: 

A question I have now is...



What new thing did you learn today?



Next time, what could you try to solve a problem?

Analysis of Common Core State Standards Implementation

Through the comparison activity, discussion of methods for proving their thinking, and whole group discussion, students have further developed their understanding of how defining attributes can be used to identify two dimensional shapes and distinguish them from each other (2.G.1).

Resources for English Learners English Language Objective(s): Students will compare and contrast a square with a rectangle. Beginning ELs will use phrases or short sentences to explain the similarities and differences between a square and a rectangle. English Learners Unit Vocabulary: Every Day Terms/ Phrases (Tier 1) equal face side sort straight edge

General Academic Terms (Tier 2) angles attributes shapes

Content Specific Terms (Tier 3) polygon triangles number quadrilaterals pentagons hexagons cubes vertices right angles rectangles squares

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 53

Lesson 5 - 5

Lesson 5

Reasoning with Shapes and their Attributes

Squares and Rectangles Learning Progressions Previous Grade 1.G.1 Distinguish between defining attributes versus non-defining attributes; build and draw shapes to possess defining attributes By the end of first grade, students are expected to learn the difference between defining (e.g. number of sides, number of vertices) and non-defining (e.g. color, overall size) attributes, and apply this knowledge to the drawing of shapes Current Grade 2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. By the end of second grade, students will be able to apply their knowledge of defining and non-defining attributes, which they mastered in first grade, to draw shapes based on those defining attributes (e.g., “a triangle is a triangle because it has three sides/angles”) and identify them by name. Next Grade 3.G.1 Understand that shapes in different categories (e.g. rhombuses, rectangles and others) may share attributes (e.g. having four sides), and that the shared attributes can define a larger category (e.g. quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. By the end of third grade, students use defining attributes not only to identify shapes but also to classify shapes into categories and subcategories by their shared attributes.

Materials: 

Attachments: 

Sheets of construction paper- rectangular, Note: teacher may want to create some squares in the same method as the class will construct theirs, for use by those students who struggle with the construction.

Practice Day Lesson

References: Shoseki, T. (2006) Mathematics for Elementary School 2B. Tokyo, Japan: Tokyo Shoseki Co. Ltd. Van De Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 54

Lesson 5 - 6

Lesson 5: Day 2 Introduction (5 minutes) Review attributes of and define a square: -polygon with four right angles -has 4 sides that are the same length Posing the Problem (20 minutes of student work time) Teacher will ask students to work with groups to sort cards into two categories: Square, Not a Square. Students are directed to record their proofs directly on the shapes: It is a polygon. There are 4 right angles.

Not a polygon.

All the sides are not the same length

All sides are the same length

For students who are struggling, give them a sheet with the following list duplicated several times. Students can cut and paste a list directly onto each shape and ✓or✗each item. ☐ polygon ☐ 4 right angles ☐ all sides are the same length. ☐ square ☐ not a square Each group pastes cards with proofs onto a T-chart:

Square

Not a Square

Comparing and Discussing (25 minutes) Terms to use from Lesson Day 1: vertex, sides, right angles Student groups bring T-charts to board and share proofs. Teacher will draw attention to a specific shape (highlight with colored marker on each T-chart) on multiple charts that has been categorized as both ‘a square and ‘not a square.’ Teacher will facilitate a discussion about where the card belongs and why.

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Summary (10 minutes) Teacher will create a large anchor square and have students share out attributes of a square as teacher records them directly on anchor square. Teacher summarizes: “We know a shape is a square if it is a polygon with 4 right angles and has 4 sides that are the same length.” Students draw a square in their math notebooks and complete sentence starter: “I know this is a square because ________________.

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Lesson 6

Reasoning with Shapes and their Attributes

Cube Lesson Summary

This lesson introduces students to the cube; students will use toothpicks and gum drops to construct a cube, identifying the number of faces, edges and corners that the cube has.

Lesson Objectives

Students will be able to:  Identify the properties of a cube.  Construct a 3D cube using toothpicks and gum drops.

Common Core State Standards:

Standards for Mathematical Practice:

2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision.

Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials and Attachments List

Time Allotment - One class period 45-60 minutes

Description of Activities: Students will use their prior knowledge of 2D and 3D shapes to discuss the attributes of a cube. Students will then create a 3D cube using toothpicks and gum drops.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 57

Lesson 6 - 1

Lesson 6

Reasoning with Shapes and their Attributes

Cube Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows the major events and flow of the lesson

This column shows additional moves, questions, or statements that the teacher may need to make to help students. Anticipated student responses are in regular font; teacher support suggestions are in italics.

This column contains questions that can be used to determine if the lesson is effectively addressing the stated objectives, and information on how this part of the lesson addresses CCSS content standards and Standards for Mathematical Practice.

Introduction: Teacher will ask students to recall the attributes of shapes discussed in prior day’s lessons.

Make sure that students recall the attributes of a square as this will lead into the cube – if square is not discussed, ask students to think about the properties of a square.

Students recall attributes of shapes from prior days lesson, noting sides, faces, corners

Posing the Problem: Today you will create a cube using toothpicks and gum drops – teacher shows a 3D cube to the class.

For the initial part of the lesson, do not give students a 3 dimensional shape of a cube; have them think about the properties before constructing.

Steps and Learning Activities

“How many toothpicks and gum drops will you need to create your cube?”

Have students think about their answer, write their answer down and then discuss with a partner. Teacher During this stage of the lesson, focus will then have students share out their on student predictions and thoughts predictions, but do not come to a around the properties of a cube. conclusion on the correct answer at this time. “Once you have your prediction, get your materials and begin creating your cube” After constructing their cube, have students write down the attributes of a cube.

S1: Is able to create a cube using 8 gum drops and 12 toothpicks S2: Student is unable to decide how many toothpicks and gum drops are needed. T2: Have student examine a 3D cube, let them count the number of sides and corners.

Students will persevere in creation of cube, students will identify appropriate tools needed to construct cube Students will identify the properties of a cube – corners, faces and edges.

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Lesson 6 - 2

Lesson 6

Reasoning with Shapes and their Attributes

Cube Steps and Learning Activities Comparing & Discussing This discussion will focus on the properties of a cube – open the discussion with the original question “how many toothpicks and gum drops did you need to create your cube” Discuss student’s responses – come to agreement on 12 toothpicks and 8 gum drops.

Anticipated Student Responses and Teacher Support

Ask students to describe the shapes that they see in their cube – can they say that there are 6 squares?

Summing Up Today we learned that a cube has 6 square faces, 8 corners and 12 edges.

Analysis of Common Core State Standards Implementation

Students will critique the reasoning of their peers – come to consensus that 12 toothpicks and 8 gum drops are needed to construct a cube and that the cube is created with 6 squares.

Students are able to determine that John’s statement is incorrect – that a cube has 8 corners and 6 faces, in 2D format you are unable to see the hidden faces.

Evaluation: John said that the cube has 3 faces and 7 corners. Do you agree or disagree with John’s statement, why? Please write down your answer.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 59

Lesson 6 - 3

Lesson 6

Reasoning with Shapes and their Attributes

Cube Resources for English Learners English Language Objective(s):  Students will use words, phrases, or short sentences to describe the properties of a cube. English Learners Unit Vocabulary: Every Day Terms/ Phrases (Tier 1) equal face side sort straightedge area rows columns size fit tile horizontal

General Academic Terms (Tier 2) angles attributes shapes riddles overlap

Content Specific Terms (Tier 3) polygon triangles number quadrilaterals pentagons hexagons cubes vertices right angles rectangles squares 3 Dimensional (3D)

Learning Progressions Previous Grade 1.G.A.2 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. By the end of first grade, students are expected to be able to create composite shapes (both two-dimensional and threedimensional) by using various two-dimensional shapes. As pertains to this lesson, students in first grade may have already composed shapes using cubes, but have not created a cube itself. Current Grade 2.G.A.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. By the end of second grade, students will be able to identify the properties of a cube – a cube has 6 equal faces, 12 edges and 8 corners. Next Grade 3.G.A.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. By the end of third grade, students are able to apply attributes of shapes to a broader category – that quadrilaterals encompass squares, rectangles, rhombuses. Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 60

Lesson 6 - 4

Lesson 6

Reasoning with Shapes and their Attributes

Cube Attachments:  Cube Graphic Organizer

Materials:   

3D Cube Toothpicks Gum Drops

References: Shoseki, T. (2006) Mathematics for Elementary School 2B. Tokyo, Japan: Tokyo Shoseki Co. Ltd. Van De Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 61

Lesson 6 - 5

Name: Lesson: Cube 1. How many gum drops and toothpicks will you need to create your cube? Gum Drops: __________ Toothpicks:__________

2. Now create your cube using the number of gum drops and toothpicks you think you will need. 3. How many gum drops and toothpicks did you use in creating your cube? Gum Drops: __________ Toothpicks:__________

4. What are the attributes of a cube?

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Name: Lesson: Cube John said that a cube has 3 faces and 7 corners. Do you agree or disagree with John’s statement, why? Please write down your answer.

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Lesson 7

Reasoning with Shapes and their Attributes

Sorting 3D Shapes Lesson Summary

This lesson provides the necessary connection between identifying and drawing shapes based on specific attributes that are two-dimensional and those that are three-dimensional. Again, the problem is designed to allow students to justify their sorting decisions, look for and make use of structure as they sort, and construct viable arguments and critique the reasoning of others through their small group discussions and whole class justifications. As they work through the problem in their small groups, the open nature of the problem also encourages students to persevere in solving it.

Lesson Objectives

Students will be able to:  Recognize and classify 3D shapes using their attributes and justify their groupings based on attributes of 3D shapes. Common Core State Standards:

Standards for Mathematical Practice:

2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

MP2: Reason abstractly and quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP7: Look for and make use of structure.

Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials and Attachments List

Time Allotment - One class period 45-60 minutes

Description of Activities: Students will decompose 3D shapes into 2D shapes and then will sort 3D models using attributes of the whole shape as well as those of their component parts. Students will then use those component parts to prove their classification of each 3D shape.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 64

Lesson 7 - 1

Lesson 7

Reasoning with Shapes and their Attributes

Sorting 3D Shapes Steps and Learning Activities This column shows the major events and flow of the lesson

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows additional moves, questions, or statements that the teacher may need to make to help students. Anticipated student responses are in regular font; teacher support suggestions are in italics.

This column contains questions that can be used to determine if the lesson is effectively addressing the stated objectives, and information on how this part of the lesson addresses CCSS content standards and Standards for Mathematical Practice.

SR1: Student names corners or angles as part of their shape and possibly counts them.

Students are being asked to recall attributes of 2D shapes from previous lessons and use these terms to describe the new 3D shape, making use of what they know about the structure of 2D shapes to sort 3D shapes (MP7). By deconstructing the three-dimensional shapes into their 2D components, the students are reasoning abstractly, and then using that abstraction later in the context of the sorting activity (MP2).

Introduction: (10 min.) Give each student one 3D shape to work on individually.

Ask students to look at their shapes and see if they can name any parts of these shapes. *Students may use imprecise mathematical language. This is to be expected. You should consistently supply the correct mathematical terms for any attributes that students name while acknowledging students correct mathematical observations.

TR1: Supply the term “vertex” or “vertices” as needed to describe this attribute. SR2: Student describes the faces of their shape as a 2D shape, e.g., a rectangular prism has several rectangular faces, a cylinder has a circle as a base, etc. Students may call these “sides.” TR2: Apply the terms “face” or “base” to these shapes as needed. SR3: Student notices edges where the faces meet and possibly counts them. Student may call these “sides.” TR3: Apply the term “edge” to this attribute as needed.

Students are being asked to recall attributes of 2D shapes from previous lessons and use these terms to describe the new 3D shape,

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 65

Lesson 7 - 2

Lesson 7

Reasoning with Shapes and their Attributes

Sorting 3D Shapes Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Posing the Problem: (15 minanticipated student work time) Remind students of the first lesson’s problem where the teacher dropped the box full of shapes and needed to figure out how they had been sorted. Students will not necessarily describe these sorts in mathematically precise terms, which is to be expected. Teacher should supply the terms as in the introduction. Remind the students of the problem that occurred in the first lesson. Remind them of how student groups sorted those 2D shapes. (Sorting 2D shapes) Ask the students how they would sort our new group of shapes (3D models)?

Tell students to work in small groups to sort these 3D shapes.

SR1: Student will not have any ideas on how to sort shapes. TR1: Remind the students, using math notebook, of the ways they sorted the polygons through the first (few) lessons.

SR2: Students sort shapes based on straight edges and curved edges. SR3: Students sort shapes based on the shape of their faces (squares, rectangles, triangles, circles, etc.)

Analysis of Common Core State Standards Implementation making use of what they know about the structure of 2D shapes to sort 3D shapes (MP7). By deconstructing the three-dimensional shapes into their 2D components, the students are reasoning abstractly, and then using that abstraction later in the context of the sorting activity (MP2).

As students sort the shapes, the teacher asks them to justify why they sorted the shapes into certain categories according to defining attributes. By building their arguments based on information from the first sorting activity, students are constructing a logical progression of thinking. (MP3).

SR4: Students sort shapes based on the presence or absence of an apex.

In this section of the lesson the students are justifying their sorting, therefore constructing viable arguments for those sorts (MP3).

Comparing and Discussion: (20 min.) Ask student groups to share out their categorization of the shapes with the class. It is not necessary for each sort to be shared. It is important for the sorts shared to be based on the goals of the teacher for the comparison and discussion.

As they share, describe, and listen to the justifications of their peers they are participating in critiquing the reasoning of others (MP3).

Ask students in the audience to describe which attributes led the presenting groups to sort the shapes Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 66

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Lesson 7

Reasoning with Shapes and their Attributes

Sorting 3D Shapes Steps and Learning Activities the way they did

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

As each group presents, ask the audience: How is the way this group sorted the same or different from the way you sorted? Additional questions for facilitation: ■

Do you think this group used the 2D shapes of the faces?

■

Did they sort by counting parts of the 3D shapes?

Are there any shapes that do not fit into a category? Why do you think that shape was hard to fit into a category? Aid the comparing and discussing of the lesson by providing the vocabulary face, vertex, and edge. Also remind students of previous 2D relevant vocabulary: quadrilateral, triangle, rectangle, square, and right angle. These vocabulary words, an illustration, and accompanying attributes could be collected on a class data pad or anchor chart.

As they share, describe, and listen to the justifications of their peers they are participating in critiquing the reasoning of others (MP3).

Summing Up (5 min) Have students describe their sorts using words or pictures. Students should record their own thinking as well as the thinking of their

At the end of this lesson, students have expanded their use of defining attributes to describe threedimensional shapes as well as twoReasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 67

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Lesson 7

Reasoning with Shapes and their Attributes

Sorting 3D Shapes Steps and Learning Activities classmates, especially where they differed.

Anticipated Student Responses and Teacher Support

Student Reflection (5 min)

Analysis of Common Core State Standards Implementation dimensional shapes. They have created a structure and method in which to recognize these shapes (2.G.1.).

Students will record one new thing they learned in their math notebook. If needed, provide students with Reflection Sentence Starters: ● I learned…. ● Now I know… ● I liked… ● Next time I will try….because…

Resources for English Learners English Language Objective(s):  Students will compare and contrast 3D shapes and 2D shapes. Beginning ELs will use phrases or short sentences to explain their reasoning for identifying 3D shapes. English Learners Unit Vocabulary: Every Day Terms/ Phrases (Tier 1) equal face side straight edge sort number

General Academic Terms (Tier 2) angles attributes shapes

Content Specific Terms (Tier 3) polygon triangles quadrilaterals pentagons hexagons cubes vertices right angles rectangles squares three dimensional (3D)

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Lesson 7

Reasoning with Shapes and their Attributes

Sorting 3D Shapes Learning Progressions Previous Grade 1.G.1 Distinguish between defining attributes versus non-defining attributes; build and draw shapes to possess defining attributes By the end of first grade, students are expected to learn the difference between defining (e.g. number of sides, number of vertices) and non-defining (e.g. color, overall size) attributes, and apply this knowledge to the drawing of shapes. Current Grade 2.G.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. By the end of second grade, students will be able to apply their knowledge of defining and non-defining attributes, which they mastered in first grade, to draw shapes based on those defining attributes (e.g., “a triangle is a triangle because it has three sides/angles”) and identify them by name. Next Grade 3.G.1 Understand that shapes in different categories (e.g. rhombuses, rectangles and others) may share attributes (e.g. having four sides), and that the shared attributes can define a larger category (e.g. quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. By the end of third grade, students use defining attributes not only to identify shapes but also to classify shapes into categories and subcategories by their shared attributes.

Materials: 

Set of 3D geo-models for each small group of students; cylinder, cube, rectangular prism, triangular prism, triangular pyramid, square pyramid, cone, octagonal prism, and pentagonal prism. *As many 3D shapes as possible should be provided. It is necessary to have at least a cylinder, cube, cone, prism and pyramid represented.

References: Van de Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon. Olson, Joanne K. Levis, John M. Vann, Roberta Bruna, Katherine Richardson. “Enhancing Science for ELLs: Science Strategies for English Language Learners That Benefit All Students” Science and Children, Vol. 46, No. 5, January 2009.

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Lesson 8

Reasoning with Shapes and their Attributes

Attribute Riddles: Assessment Lesson Lesson Summary

This lesson will assess student mastery of the CCSS standard 2.G.1 by having students read riddles and determine, based on the characteristics described, which shape to draw. This will require students to recognize the defining attributes of these shapes, and look for and use the structure of those shapes to decide on an answer to the riddle. The open-ended nature of the riddles will encourage multiple responses and student discussion, thereby encouraging students to construct a viable argument and critique the reasoning of others.

Lesson Objectives

Students will be able to:  Correctly identify and draw various shapes based on specific attribute rules by applying all of their previous experiences with 2D and 3D shapes. Common Core State Standards:

Standards for Mathematical Practice:

2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

MP.2: Reason abstractly and quantitatively. MP.3: Construct viable arguments and critique the reasoning of others. MP.5: Use appropriate tools strategically.

Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials and Attachments List  Lesson Artifacts

Time Allotment - One class period 45-60 minutes

Description of Activities: Students will solve attribute riddles in pairs, recording their responses by drawing the shapes and the reasoning behind each response in writing. This is preparation for the assessment portion of the lesson where students will solve attribute riddles individually to demonstrate their understanding of the attributes of 2D and 3D shapes. The intent of this lesson is to assess student understanding of attributes and their relation to the classification of shapes, which has been the focus of the last six lessons.

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Reasoning with Shapes and their Attributes

Attribute Riddles: Assessment Lesson Steps and Learning Activities This column shows the major events and flow of the lesson

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows additional moves, questions, or statements that the teacher may need to make to help students. Anticipated student responses are in regular font; teacher support suggestions are in italics.

This column contains questions that can be used to determine if the lesson is effectively addressing the stated objectives, and information on how this part of the lesson addresses CCSS content standards and Standards for Mathematical Practice.

Posing the Problem (15 min – anticipated student work time) Have students gather in pairs and hand out centimeter grid paper, straightedges, and the partner attribute riddles. Remind students to read the riddles carefully and talk to each other to solve the riddles. Remind students that the answer must follow all of the rules in the riddle.

See attached attribute riddles and answer sheet.

SR1: Student has no idea how to solve a particular riddle. TR1: Have students draw out each clue as they read the riddles. Once students have a solution, have them check to make sure the solution follows all the rules of the riddle.

In this portion of the lesson, students are asked to identify and draw the answers to open ended riddles. Several riddles have more than one correct answer; this is intentional to encourage student discourse. As they discuss the solutions to the riddles students are encouraged to justify their own thinking and critique the thinking of their partners (MP3).

Each pair should have an answer for all of the 5 riddles. Students will draw and label their responses on the centimeter grid paper. Student pairs should then discuss and record their reasons for choosing that answer. *Teacher Note: The first column of the attribute riddle rubric applies to this portion of the lesson. You will need to use information from your observations during this section to complete the rubric for each student.

Because this is an assessment lesson, students are expected to complete these tasks with minimal teacher support or redirection.

Ideas for Facilitation: ■

Teacher should make sure that students are carefully Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 71

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Attribute Riddles: Assessment Lesson Steps and Learning Activities reading the riddles.1 ■

Anticipated Student Responses and Teacher Support

Teachers should encourage the pairs to talk through their answers. Ex. Can you explain to your partner why you think the answer is a square?

Analysis of Common Core State Standards Implementation

In the comparing and discussing section of the lesson, students should justify their answers and the possible answers of other students. Teachers should informally assess students’ ability to justify their answers and critique the reasoning of other students at this point in the lesson (MP3), as well as student mastery of mathematical vocabulary and reasoning (2.G.1.).

Comparing and discussing (20 min) Have student pairs share out their answers and rationales to the class. Correct answers to each riddle are included on the answer key. While the number of answers given in a particular class may differ, any correct answer should be encouraged and discussed, with the teacher providing correct mathematical vocabulary when necessary. Questions for Facilitation: ■

What part of the riddle helped you decide on that answer?

■

Is there another shape that could be the answer to that riddle?

■

What did you and your partner talk about when you made that decision?

All possible answers for each riddle need not be discussed.

1

Teacher could have ELs highlight key words in the riddles to help them understand the riddles better. According to Haynes, it works especially well for math classes (Haynes 89). Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 72

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Lesson 8

Reasoning with Shapes and their Attributes

Attribute Riddles: Assessment Lesson Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

Summing up (5 min) Give all of the possible answers to each riddle at this point, if students have not come up with all possibilities on their own. Assessment (20-25 min) Give each student a copy of the assessment riddles, a straightedge, and centimeter grid paper. Tell the students; this is the time where you want to know what they have learned about solving attribute riddles.

Students should work individually on the riddles at this point. The artifacts they produce should be scored using the Attribute Riddle Rubric at the end of the lesson to assess student mastery of 2.G.1. as well as progress toward the mathematical practice standards.

They will need to work on their own to solve, draw, and explain their own answers.

Students will individually read riddles and determine, based on the characteristics described, which shape to draw. This will require students to recognize the defining attributes of these shapes, and look for and use the structure of those shapes to decide on an answer to the riddle. The open-ended nature of the riddles will encourage multiple responses and student explanations. (2.G.1.)

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Attribute Riddles: Assessment Lesson Resources for English Learners English Language Objective(s):  Students will compare and contrast the shapes they have learned. They will draw the shapes, write the names of the shapes or name them verbally. They will use phrases or short sentences to explain their reasoning for identifying the shapes. English Learners Unit Vocabulary: Every Day Terms/ Phrases (Tier 1) equal face side sort straightedge

General Academic Terms (Tier 2) angles attributes shapes riddles

Content Specific Terms (Tier 3) polygon triangles number quadrilaterals pentagons hexagons cubes vertices right angles rectangles squares three dimensional 3D

Learning Progressions Previous Grade 1.G.1 Distinguish between defining attributes versus non-defining attributes; build and draw shapes to possess defining attributes By the end of first grade, students are expected to learn the difference between defining (e.g. number of sides, number of vertices) and non-defining (e.g. color, overall size) attributes, and apply this knowledge to the drawing of shapes. Current Grade 2.G.1 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. By the end of second grade, students will be able to apply their knowledge of defining and non-defining attributes, which they mastered in first grade, to draw shapes based on those defining attributes (e.g., “a triangle is a triangle because it has three sides/angles”) and identify them by name. Next Grade 3.G.1 Understand that shapes in different categories (e.g. rhombuses, rectangles and others) may share attributes (e.g. having four sides), and that the shared attributes can define a larger category (e.g. quadrilaterals). Recognize rhombuses, Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 74

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Reasoning with Shapes and their Attributes

Attribute Riddles: Assessment Lesson Learning Progressions

rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. By the end of third grade, students use defining attributes not only to identify shapes but also to classify shapes into categories and subcategories by their shared attributes.

Materials:    

Attachments:  Partner Attribute Riddles  Partner Attribute Riddles- KEY  Assessment Attribute Riddles  Assessment Attribute Riddles - KEY  Attribute Riddle Rubric

Copies of attribute riddle cards Centimeter grid paper Rulers or straightedges for drawing shapes Attribute Riddle Rubric

References: Van de Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon. Haynes, Judie. Getting Started with English Language Learners: How Educators Can Meet the Challenge Association for Supervision and Curriculum Development 2007.

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Lesson 8

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Attribute Riddles: Assessment Lesson Artifacts

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Lesson 8

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Attribute Riddles: Assessment Lesson Artifacts

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Partner Attribute Riddles 1

2

I am a polygon.

I am a polygon.

I have four sides.

I have three vertices.

I have no right angles.

What shape am I?

What shape am I?

3

4

I am a polygon.

I am a polygon.

I have four sides.

All of my sides are the same length.

I have two sides the same length.

What shape am I?

What shape am I?

78

Partner Attribute Riddles- answer key P1

P2

I am a polygon. I have four sides. I have no right angles. What shape am I?

I am a polygon. I have three vertices. What shape am I?

Possible answers: rhombus, parallelogram, trapezoid, quadrilateral. (note: any polygon with four equal sides is a rhombus; only a non-square rhombus meets the criteria in this riddle)

Answer: triangle

P3

P4

I am a polygon. I have four sides. I have two sides the same length. What shape am I?

I am a polygon.

Possible answers: rectangle, square,(parallelogram, trapezoid) these are unlikely but correct

79

All of my sides are the same length. What shape am I? Possible answers: rhombus, square (regular pentagon, hexagon, equilateral triangle) these are unlikely but correct

Assessment Attribute Riddles A1

A2 I am a polygon.

I am a polygon.

I have four right angles.

My opposite sides are the same size.

All of my sides are the same length.

I have exactly four sides.

What shape am I?

What shape am I?

Assessment Attribute Riddles- Answer Key A1

A2 I am a polygon.

I am a polygon. I have four right angles.

My opposite sides are the same length.

All of my sides are the same length.

I have exactly four sides.

What shape am I?

What shape am I?

Answer: square

Answer: rectangle (square, rhombus) these are unlikely but correct

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Attribute Riddle Rubric

4

Working with others Student was an engaged partner, listening to suggestions of partner and working cooperatively throughout lesson.

Use of Tools Student draws all shapes accurately while using tools appropriately and precisely.

3

Student cooperated with partner, but needed prompting to stay on-task.

Student uses tools to draw shapes but makes mistakes

2

Student was an engaged partner but had trouble listening to others and/or working cooperatively. Student did not work effectively with partner.

Student uses tools to draw shapes but cannot use tools to help accurately draw. Student uses tools inappropriately or inaccurately.

1

Mathematical Concepts Student uses defining attributes to correctly identifies and draws the correct shape that is the answer to the riddle, and draws more than one correct answer to those riddles with many answers. Student correctly identifies the answer to the riddle and draws the shape correctly. Student correctly identifies an answer to the riddle but does not draw it correctly. Student does not correctly identify any answer to the riddle or draw the correct shape.

81

Mathematical reasoning Student clearly and thoughtfully explains their reasoning for each answer and uses correct mathematical vocabulary in their writing.

Student explains their reasoning clearly and attempts to use mathematical vocabulary in their explanations. Student explains their thinking but some answers are incomplete or unclear. Student explanation may or may not include mathematical vocabulary. Student attempts to explain their thinking incomplete, but the explanation is unclear and incomplete.

Lesson 9

Reasoning with Shapes and their Attributes

Making Rectangles Lesson Summary

When students create rectangles using same-size squares in this lesson, they are transitioning from composing twodimensional shapes to form composite shapes (which they learned to do in 1st grade) and laying a foundation for partitioning rectangles into same-size squares. This lesson asks students to look for and make use of structure by paying attention to the rows and columns of squares that form their rectangles. The students will also model with mathematics by noticing how two different arrangements of squares represent two different ways of counting, even though both arrangements have the same total number of squares.

Lesson Objectives

Students will be able to:  Find the area of various sized rectangles, identifying the number of rows and columns in each along with the total area. Common Core State Standards:

Standards for Mathematical Practice:

2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

MP.4: Model with mathematics MP.7: Look for and make use of structure

Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials and Attachments List  Lesson Artifacts

Time Allotment - One class period 45-60 minutes

Description of Activities: Students will be asked to create as many rectangles as they can using only six tiles, students will then discuss the different arrangements that are possible as well as misconceptions around what a rectangle is (some students might choose to make non-rectangular figures, others may not include a square). Students will then be asked to create as many rectangles as they can, identifying the number of rows and columns in each arrangement using 12 tiles. This activity will develop students’ ability to compose rectangles. Decomposition of rectangles will take place in the next lesson.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 82

Lesson 9 - 1

Lesson 9

Reasoning with Shapes and their Attributes

Making Rectangles Steps and Learning Activities This column shows the major events and flow of the lesson

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows additional moves, questions, or statements that the teacher may need to make to help students. Anticipated student responses are in regular font; teacher support suggestions are in italics.

This column contains questions that can be used to determine if the lesson is effectively addressing the stated objectives, and information on how this part of the lesson addresses CCSS content standards and Standards for Mathematical Practice.

Introduction (8 min) The teacher will pose an introductory task: “Using only six tiles, create a rectangle and trace it on your paper. How many rows and columns does your rectangle have?”

If students create shapes that are not rectangles, review the attributes of rectangles taught during previous lessons.

Teacher should make sure to clarify the following before children begin: ● ● ●

The introductory task is a review of content the student should have mastered in 1st grade (composing twodimensional shapes to create a composite shape, 1.G.2). The task lays the foundation for counting same-size squares that make up a rectangle (2.G.2).

Rows are horizontal groups of tiles. Columns are vertical groups of tiles. The shape must be a rectangle.

Posing the Problem (10 min anticipated student work time) The teacher poses the problem: “How many different rectangles can you make using exactly 12 tiles? Please trace each rectangle and record the number of rows and columns in each. Describe your rectangle using the words rows and columns.” Tell the students that the shape must be completely filled. There can be no spaces or gaps between tiles. Then give students exactly 12 tiles.

S1: Students are able to correctly create and write descriptions for the following rectangles: ● ● ● ● ● ●

1 row of 12 1 column of 12 6 rows of 2 6 columns of 2 3 rows of 4 3 columns of 4

S2: Students create non-rectangular shapes. T2: Remind students the characteristics of a rectangle. A rectangle has two sets of parallel sides. Opposite sides are the same

This portion of the lesson lays the foundation for standards 3.MD.5, 3.MD.6, and 3.MD.7, which address area of plane figures. Emphasizing counting by rows and columns provides students with experience with the concept of area and prepares them to eventually derive the formula for area of a rectangle (length times width) in third grade. Encouraging students to consider alternate counting strategies emphasizes the relationship between the unit squares and the larger rectangle, drawing their attention to

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Reasoning with Shapes and their Attributes

Making Rectangles Steps and Learning Activities

Anticipated Student Responses and Teacher Support length. S3: Students create rectangles but do not describe their formation using rows of tiles. For example, a “3 rows of 4” can be split in half and counted as “2 groups of 6” or “6 + 6”) T3: Affirm students’ counting strategies, as they may correctly have counted the number of square tiles. Then ask: “How can you count this by thinking about rows and columns?”

Comparing and Discussing (15 min) Invite students to share rectangles until all six arrangements have been shown. Use enlarged square tiles on the board to make the formations visible. Facilitate discussion around the following idea: How many different rectangles were we able to make using the 12 tiles? In the discussion, students should compare the different arrangements of rectangles, describe how they are similar or different, and explain why identifying the rows and columns can make counting the squares easier and more efficient. Ask students to try to use a number model to describe the area of the rectangle.

If students have difficulty beginning the conversation, prompt them with questions such as: How are the rectangles similar? How are the rectangles different? If students continue to count squares by one by one, ask the class how we can find a quicker and easier way to count squares.

Analysis of Common Core State Standards Implementation

the structure of the shape and how it can be used to find the area. Many students may be content to count in simple ways that are most comfortable for them, but with this extra teacher support students will begin to recognize how rows and columns provide a structure for making counting efficient and accurate (MP7).

The difference between “4 rows of 3” and “3 rows of 4” is more than just whether the rectangle is presented vertically or horizontally. The two different arrangements represent two different models of how the squares might be counted (3 + 3 + 3 + 3 = 12 or 4 + 4 + 4 = 12). Encouraging students to see this difference helps them model with mathematics by creating a number sentence that represents how they counted squares to find the area (MP4).

If students confuse horizontal and vertical arrangements of rectangles, ask the class: What is the difference between 4 rows of 3 and 4 columns of 3? Does it matter? Students might use repeated addition, in various forms to represent the area of the rectangle.

Through creating rectangles of samesize squares, and comparing and contrasting the rectangle arrangements, students will have laid a solid foundation for partitioning rectangles into same-size squares (2.G.2), which is the content of the following lesson.

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Reasoning with Shapes and their Attributes

Making Rectangles Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

Summing up (5 min) Write a summary of today’s learning for students to record in their notebooks. Rectangles can be formed by using rows and columns of squares. Paying attention to these rows and columns helps us count the area ,or space the rectangle takes up, in an easier way. Student Reflection (8 min) Give students two more square tiles and instruct them to create a new rectangle. Instruct students to write a reflection about their new rectangle, describing it in terms of rows or columns, and explaining how these rows and columns help them count the squares.

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Making Rectangles Resources for English Learners English Language Objective(s):  Students will use words, phrases, or short sentences to explain how they find the area of the rectangles. English Learners Unit Vocabulary: Every Day Terms/ Phrases (Tier 1) equal face side sort straightedge area rows columns size

General Academic Terms (Tier 2) angles attributes shapes riddles

Content Specific Terms (Tier 3) polygon Triangles number quadrilaterals pentagons hexagons cubes vertices right angles rectangles squares three dimensional 3D

Learning Progressions Previous Grade 1.G.2 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. By the end of first grade, students are expected to be able to create composite shapes (both two-dimensional and threedimensional) by using various two-dimensional shapes. As pertains to this lesson, students in first grade may have already composed rectangles from squares. Current Grade 2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. By the end of second grade, students will understand how a rectangle can be seen to be composed of a number of samesize squares. The students may notice a way of simplifying their counting by recognizing rows and columns. Students do not yet learn the connection to multiplication of side and length. Next Grade 3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. 3.MD.6 Measure areas by counting units squares (square cm, square m, square in, square ft, and improvised units). Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 86

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Making Rectangles Learning Progressions

3.MD.7 Relate area to the operations of multiplication and addition. By the end of third grade, students identify the same-size squares that cover a plane figure as square units. The number of square units covering a plane figure determines the area of the plane figure. Students identify various methods of counting square units, including multiplying side and length. Note to Teacher: The Geometry standards from 1st and 2nd grade are precursors to the study of area, which is listed as a Measurement and Data domain in 3rd grade.

Attachments:  Practice Day Lesson

Materials:   

12 square unit tiles per student (or pair) 2 additional tiles per student (or pair) Enlarged tiles for blackboard

References: Van de Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon.

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Lesson 9

Reasoning with Shapes and their Attributes

Making Rectangles Artifacts:

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Making Rectangles Artifacts:

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Introduction (2 minutes) Review meaning of ‘row’ and ‘column’ and display visual:

Posing the Problem (15 minutes of student work time) Materials: grid paper, square units (enough for each student to have 20), magnetic square units (150 for board use), rulers (available for use as straight edge) Teacher will write task on board and make square units available for student use: •

Draw a rectangle using 20 squares on grid paper. Record the number of rows and number of columns. Write a number sentence to represent your rectangle.

•

Draw a rectangle with 8 rows and 2 columns. Record total squares used. Write a number sentence to represent your rectangle.

Comparing and Discussing (30 minutes) Students share out various rectangle configurations using 20 squares, along with number of rows/columns and number sentences. Students are encouraged to describe rectangles in terms of rows and columns. Students are then asked to display a rectangle using 8 rows and 2 columns. Summary (10 minutes) Teacher summarizes: “We can describe rectangles by referring to the number of rows and number of columns they are made up of.” Students fill out grid paper-exit slip: Draw a rectangle using 10 squares. _____ rows _____ columns

Number sentence: ________________________________

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Lesson 10

Reasoning with Shapes and their Attributes

How Many Squares Fit? Lesson Summary

As part of lesson 8, students used same-size squares to compose rectangles. This lesson places student learning wholly in the domain of standard 2.G.2 because, rather than composing a rectangle, students are partitioning a rectangle into same-size squares and then counting the squares to find the total number. They continue to see these square arrangements as a series of rows or columns because this allows them to use repeated addition as a counting method. Students develop their ability to use tools strategically by using only one tile and a pencil to cover the rectangle’s area. Finally, in the comparison and discussion sections, students make viable mathematical arguments about the accuracy and efficiency of various tiling and counting strategies.

Lesson Objectives

Students will be able to:  Find how many unit squares fit inside a rectangle. Describing this as the area of the rectangle.

Common Core State Standards:

Standards for Mathematical Practice:

2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

MP.3: Construct viable arguments and critique the reasoning of others MP.5: Use appropriate tools strategically MP.7: Look for and make use of structure

Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials List  Lesson Artifacts

Time Allotment - One class period 45-60 minutes

Description of Activities: Students will be asked to determine how many unit squares will fit inside a rectangle. First, students will be shown the rectangle and the size of the unit square. They will share with each other how they might go about determining how many squares would fit inside. Then they are given a rectangle on paper. The rectangle will have tic marks evenly spaced around the perimeter as a guide. Using a square tile, students will find the number of squares that fit inside the rectangle. The goal is for students to realize that the tic marks are a helpful guide because they show that rows and columns are uniformly spaced.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 91

Lesson 10 - 1

Lesson 10

Reasoning with Shapes and their Attributes

How Many Squares Fit? Steps and Learning Activities This column shows the major events and flow of the lesson

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows additional moves, questions, or statements that the teacher may need to make to help students. Anticipated student responses are in regular font; teacher support suggestions are in italics.

This column contains questions that can be used to determine if the lesson is effectively addressing the stated objectives, and information on how this part of the lesson addresses CCSS content standards and Standards for Mathematical Practice. This introduction helps students make the transition from the previous lesson into the 2nd grade content of this lesson. Previously, students composed same-size squares to form a rectangle. Now, with the teacher’s guidance, students begin to consider how they might partition a rectangle into same-size squares (2.G.2).

Introduction (5 min) Show students an enlarged rectangle and an enlarged unit square tile. Ask the students to think about how many unit squares might fit inside the rectangle. Elicit ideas of how we might go about finding this. Teacher should clarify the following terms during the introduction.  

Rows are horizontal groups of squares. Columns are vertical groups of squares. [2]

Posing the Problem/ Student Work Time (10 min) Pose the problem: “Today, you will receive a rectangle and a square tile. Let’s find a quick and easy way to find how many unit squares fit inside the rectangle.” Give students the sheet with a 4-inch by 5-inch rectangle, one square inch pattern block, and a pencil.

The responses below represent a continuum of counting strategies, ranging from least to most mathematically developed. S1: Student does not understand how to tile with access to only one tile. T1: How can we use this tile to think about how many more tiles would fit? Can you use your pencil to hold your place? Your finger? S2: Student does not tile squares so they are adjacent. T2: Is it important that the tiles touch? Or not touch? Why do you think this?

S1 has trouble conceiving of his unit square as a kind of counting tool. This may be because S1 has only one tile. The teacher’s response helps the student think beyond the one tile. In this way, it is similar to how a paper clip can be iterated to find length. Although this connection may not be made explicit, it is a starting point for helping the student use appropriate tools strategically (MP5).

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 92

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Lesson 10

Reasoning with Shapes and their Attributes

How Many Squares Fit? Steps and Learning Activities

Anticipated Student Responses and Teacher Support

S3: Student overlays tracing of the tiles. T3: Why did you choose to overlap the tiled squares? Would another student overlap his tile in the same way you did? Would we be able to agree on an answer? S4: Student finds correct number of squares, but traces them in a nontraditional order (e.g., in a spiral starting at the perimeter and working its way in). T4: Why did you choose to draw your tiles in that design? Does that help you find the total number? Is there a simpler way? S5: Student traces the tiles in rows or columns systematically, but continues to draw squares after a pattern should be evident. T5: Can you think of a quicker way to find the total number of squares? Do you have to draw all the squares? S6: Student draws the first row or column, and then iterates the pattern without drawing to find the total number of squares. Student may use repeated addition or multiplication while doing this. T6: How can you be sure that this method works without drawing all the squares? What pattern do you see that helps you here?

Comparing and Discussing (25 min) Invite students to share their methods for finding the number of squares that fit inside the rectangle. Make student thinking visible by modeling each student’s work using a few enlarged rectangles and the unit square.

If students are unsure of how to begin the comparison and discussion, prompt the class with a question such as: Which method looks similar to the way you counted? If students arrived at different

Analysis of Common Core State Standards Implementation

Students represented by S4 and S5 are counting in ways that are comfortable for them, but these methods are not mathematically developed. The teacher’s responses help these students notice the structure of the rectangle and how patterns of rows and columns provide a more efficient and accurate way to find the total number of squares than the methods they are using. S6 reflects a welldeveloped understanding of making use of structure in this context (MP7).

This section of the comparison and discussion helps students move beyond the idea of “correctness.” The teacher’s questions help students go deeper. They must make arguments about which tiling and counting methods are the most efficient, and

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 93

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Lesson 10

Reasoning with Shapes and their Attributes

How Many Squares Fit? Steps and Learning Activities Remind students that the discussion should not be about the answer, but instead about the different solution methods. [3] ● ● ● ●

Anticipated Student Responses and Teacher Support

numbers of squares as their answer, create tension by asking questions such as: Did we all count the same number of squares? Why not? Which do you agree with? Disagree? Why? Why are some methods accurate and others not? Is it okay to overlap the squares? To leave gaps? Why? Why not?

Analysis of Common Core State Standards Implementation

how paying attention to rows and columns might help in this process. Thus, students are paying attention to structure (MP7) and constructing and critiquing arguments using this knowledge (MP3).

Facilitate discussion around the following idea: What is the quickest and easiest way to find how many squares fit inside the rectangle? When discussing different solutions that arrived at the correct number of total squares, focus on the students’ tiling or counting methods: 

Are some methods quicker in finding the total number of squares than others? What makes them quicker?



What patterns do you see in the way the student counted?



How many squares are in each row? How many rows are there? How does this help us find the total number of squares?

Following the prior lesson, this objective asks students to partition a rectangle into same-size squares, and consider how these squares might be counted (2.G.2). Students begin to see the benefit of paying attention to rows and columns of squares when counting. They also recognize the inaccuracy of overlapping squares or leaving gaps. Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 94

Lesson 10 - 4

Lesson 10

Reasoning with Shapes and their Attributes

How Many Squares Fit? Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

Summing up (5 min) Write a summary of today’s learning for students to record in their notebooks. When we partition a rectangle into same-size squares, the squares should touch exactly at their sides and not overlap. We can count rows and columns of squares using repeated addition.

Student Reflection (10 min) Show students a 5 by 9 rectangle. Ask students how they would find the number of squares that fit inside in a quick and accurate way. They should write their thinking down in their journal.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 95

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How Many Squares Fit? Resources for English Learners English Language Objective(s):  Students will use words, phrases, or short sentences to explain how they find out how many unit squares fit inside a rectangle English Learners Unit Vocabulary: Every General Academic Terms (Tier 2) Day Terms/ Phrases (Tier 1) Angles Equal Face Attributes Side Shapes Sort Riddles Straightedge Overlap Area Rows Columns Size Fit Tile Horizontal Vertical

Content Specific Terms (Tier 3) Polygon Triangles Number Quadrilaterals Pentagons Hexagons Cubes Vertices Right angles Rectangles Squares 3 dimensional (3D)

Learning Progressions Previous Grade 1.G.2 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. By the end of first grade, students are expected to be able to create composite shapes (both two-dimensional and threedimensional) by using various two-dimensional shapes. As pertains to this lesson, students in first grade may have already composed rectangles from squares. Current Grade 2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. By the end of second grade, students will understand how a rectangle can be seen to be composed of a number of samesize squares. The students may notice a way of simplifying their counting by recognizing rows and columns. Students do not yet learn the connection to multiplication of side and length.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 96

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Reasoning with Shapes and their Attributes

How Many Squares Fit? Learning Progressions Next Grade 3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. 3.MD.6 Measure areas by counting units squares (square cm, square m, square in, square ft, and improvised units). 3.MD.7 Relate area to the operations of multiplication and addition. By the end of third grade, students identify the same-size squares that cover a plane figure as square units. The number of square units covering a plane figure determines the area of the plane figure. Students identify various methods of counting square units, including multiplying side and length. Note to Teacher: The Geometry standards from 1st and 2nd grade are precursors to the study of area, which is listed as a Measurement and Data domain in 3rd grade.

Materials:   

Sheet with 4” by 5” rectangle (1 copy per student) Enlarged rectangles for class presentation 1 unit square tile for each student

References: When students first begin to learn area concepts, they progress through stages of cognitive development that start with concrete and move to abstract. Whereas this unit has already exposed students to various polygons, now the students are being asked to quantify how much space a rectangle covers based on how many squares can fit inside it. It is important that the terms “area” and “units” are not taught directly here, as that would be developmentally inappropriate for 2nd grade (nor are those terms required by the Common Core Standards). In fact, in Japan, these terms would not be introduced until the 4th grade. Rather, this lesson helps students begin to explore the concept of area by simply investigating the size of a shape. They are aided at their developmental level by the use of concrete tools such as the square tile. More abstract ideas, such as the multiplication algorithm for area, should be avoided. Van De Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 97

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Reasoning with Shapes and their Attributes

How Many Squares Fit? Artifacts:

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Lesson 11

Reasoning with Shapes and their Attributes

Finding the Missing Space Lesson Summary

In this lesson, students make use of skills from 2.G.2 but do so with added complexity. They pay attention to rows and columns of squares, learned from the previous lesson, but must also model a missing number of same-size squares by using a number sentence. As in the previous lesson, students use tools strategically and attend to precision. When comparing and discussion solution strategies, students develop flexibility in their understanding of how the real-life situation can be modeled using more than one number sentence.

Lesson Objectives

Students will be able to:  Find the number of tiles needed to complete tiling a given area. Common Core State Standards:

Standards for Mathematical Practice:

2.G.2 Partition a rectangle into rows and columns of samesize squares and count to find the total number of them.

MP.4: Model with mathematics MP.5: Use appropriate tools strategically.

2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Time Allotment - One class period 60 minutes Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials and Attachments List  Lesson Artifacts

Description of Activities: Students will be asked to use their knowledge of rows and columns to help them find the number of tiles that are missing to finish tiling a given area. Students will discover that they can add rows or columns repeatedly to find the total area and that they can subtract out the given area to find the missing area.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 99

Lesson 11 - 1

Lesson 11

Reasoning with Shapes and their Attributes

Finding the Missing Space Steps and Learning Activities This column shows the major events and flow of the lesson

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows additional moves, questions, or statements that the teacher may need to make to help students. Anticipated student responses are in regular font; teacher support suggestions are in italics.

This column contains questions that can be used to determine if the lesson is effectively addressing the stated objectives, and information on how this part of the lesson addresses CCSS content standards and Standards for Mathematical Practice.

S1: Student cannot identify the number of missing tiles and is not sure how to use one tile to find the number of squares that fit in the uncovered space.

S1 has trouble conceiving of his unit square as a kind of counting tool. This may be because S1 has only one tile. The teacher’s response helps the student think beyond the one tile. In this way, it is similar to how a paper clip can be iterated to find length. Although this connection may not be made explicit, it is a starting point for helping the student use appropriate tools strategically (MP5).

Posing the Problem/Student Work Time (15 minutes) Pose the following problem to the students: “I began creating my quilt by laying square tiles over a larger sheet of cloth. But I did not have enough tiles to complete my quilt. Write a number sentence to show how many more tiles I will need to complete my quilt.”

T1: How can you use this tile to determine the number of tiles that would fit in the uncovered space? S2: Student identifies the number of missing tiles but cannot write a number sentence to express this. T2: What is the total number of tiles that make up the rectangle? How do you know?

Allow students access to paper with image above, as well as one square unit tile and a pencil for tracing. Explain what is expected by “number sentence” and give an example (i.e., a story problem that can be represented using an addition number sentence).

S3: Student incorrectly or inaccurately tiles the open space. T3: Let’s pay attention to being accurate and precise. Is that the actual size of each tile? Is that the same size that the other students will draw on their papers? Will your answers be the same? S4: Student doubles the area because they think the open space is the same as the tiled space based on appearance. T4: How can you be sure that the open space is the same as the tiled space? Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 100

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Lesson 11

Reasoning with Shapes and their Attributes

Finding the Missing Space Anticipated Student Responses and Teacher Support

Steps and Learning Activities

S5: Student writes the following addition or subtraction number sentences to represent how many more tiles are needed: ● ● ●

9 + 6 = 15 9 + ☐= 15 15 – 9 = 6

Comparing and Discussing (20 min) Invite students to share solutions that show different number sentences.1 Facilitate discussion around the following ideas: ●

●

How can we use the tiled space to help us find the missing number of tiles? What are some good number sentences to use to show the total number of tiles that make up the rectangle?

Have a number of enlarged quilt images to model students’ solution methods. Make student thinking visible by writing ideas and number sentences under the quilt image.

To help students think about the methods used by their classmates, ask questions such as: ● How did ______ find the missing space? ● Was _____ thinking about rows or columns when finding the missing space? How do you know?

Analysis of Common Core State Standards Implementation

These questions ask students to go further than simply representing the situation using a number sentence. Students must think carefully about what the numbers represent in each equation, and why addition and subtraction might both work. They also must be flexible in their thinking (i.e., in 9 + ☐= 15, ☐represents the number of tiles to fill the uncovered space; in 15 – 9 = 6, we find that number by subtracting the total number of squares from the number of known squares). These are all ways of helping students model with mathematics (MP4).

If students have difficulty considering what the number sentences mean, and which ones would be correct, ask the following: ● How are these number sentences similar? ● How are these number sentences different? ● What does the ___ (e.g., 9, 6, 15) represent in this number sentence? ● Do both addition and subtraction number sentences

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 101

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Lesson 11

Reasoning with Shapes and their Attributes

Finding the Missing Space Steps and Learning Activities

Anticipated Student Responses and Teacher Support work for this situation? Why?

If students are “stuck” believing there is only one correct answer, consider asking the following: ● Is there a “best” number sentence to use to show the total number of tiles that make up the rectangle? Can more than one number sentence be correct?

Analysis of Common Core State Standards Implementation

This lesson moves past the previous lesson’s goal of partitioning a rectangle into same-size squares by asking students to use a number sentence to represent a missing number of squares (2.OA.4). Students continue to make use of rows and columns when finding the number of squares (2.G.2).

Summing up (5 min) Write a summary of today’s learning for students to record in their notebooks. To save time, you may want to have a photocopied summary available to have students paste into their notebooks. We can find the number of tiles in an open space by thinking about how many tiles would fit in one row and then counting the number of rows. We can add the number of tiles we already have to the number of tiles that would fit in the uncovered space to find the total number of tiles in the rectangle. Student Reflection (5 min) Choose one student’s number sentence and explain in writing what each number represents in the rectangle.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 102

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Finding the Missing Space Resources for English Learners English Language Objective(s):  Students will use words, phrases, or short sentences to explain how they find out how many tiles needed to complete a given area. English Learners Unit Vocabulary: Every Day Terms/ Phrases (Tier 1) equal face side sort straightedge area rows columns size fit tile horizontal vertical space partition

General Academic Terms (Tier 2) angles attributes shapes riddles overlap

Content Specific Terms (Tier 3) polygon triangles number quadrilaterals pentagons hexagons cubes vertices right angles rectangles squares 3 Dimensional (3D)

Learning Progressions Previous Grade 1.G.2 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. By the end of first grade, students are expected to be able to create composite shapes (both two-dimensional and threedimensional) by using various two-dimensional shapes. As pertains to this lesson, students in first grade may have already composed rectangles from squares. Current Grade 2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. By the end of second grade, students will understand how a rectangle can be seen to be composed of a number of samesize squares. The students may notice a way of simplifying their counting by recognizing rows and columns. Students do not yet learn the connection to multiplication of side and length.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 103

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Lesson 11

Reasoning with Shapes and their Attributes

Finding the Missing Space Learning Progressions Next Grade 3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement. 3.MD.6 Measure areas by counting units squares (square cm, square m, square in, square ft, and improvised units). 3.MD.7 Relate area to the operations of multiplication and addition. By the end of third grade, students identify the same-size squares that cover a plane figure as square units. The number of square units covering a plane figure determines the area of the plane figure. Students identify various methods of counting square units, including multiplying side and length. Note to Teacher: The Geometry standards from 1st and 2nd grade are precursors to the study of area, which is listed as a Measurement and Data domain in 3rd grade.

Materials:   

Attachments: 

Rectangle worksheet Large paper for class presentation Square unit tile for each student

Rectangle Worksheet

References: Van De Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 104

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Lesson 11

Reasoning with Shapes and their Attributes

Finding the Missing Space Artifacts: Math Notebook

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Lesson 11

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Finding the Missing Space Artifacts: Student Artifact – Math Notebook

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Reasoning with Shapes and their Attributes

Finding the Missing Space Artifacts: Student Artifact – Rectangle Worksheet

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Finding the Missing Space Artifacts:

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Rectangle Worksheet Name: _______________________________________________

Number Sentence: __________________________________

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Reasoning with Shapes and their Attributes

Fractions of Shapes (1/2 and 1/4) Lesson Summary

The partitioning of squares into halves and fourths, as well as the idea that equal shares of the same whole need not be the same shape, are the central tenets of standard 2.G.3. When proving that two shares of different shapes are indeed equal, students must reason abstractly, perhaps making use of visual models. While reasoning abstractly, students will construct arguments using these visual models.

Lesson Objectives

Students will be able to:  Partition rectangles into two and four equal pieces, noticing that equal shares do not need to be the same shape. Common Core State Standards:

Standards for Mathematical Practice:

2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

MP2: Reason abstractly and quantitatively. MP3: Construct viable arguments and critique the reasoning of others. Time Allotment - One class period 45-60 minutes Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials List  Lesson Artifacts

Description of Activities: Using dot paper or geo-boards, students will partition rectangles into two and four equal pieces. Students will create as many representations as possible to partition each shape, noticing that there is more than one or two ways to partition each shape.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 110

Lesson 12 - 1

Lesson 12

Reasoning with Shapes and their Attributes

Fractions of Shapes (1/2 and 1/4) Steps and Learning Activities This column shows the major events and flow of the lesson

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows additional moves, questions, or statements that the teacher may need to make to help students. Anticipated student responses are in regular font; teacher support suggestions are in italics.

This column contains questions that can be used to determine if the lesson is effectively addressing the stated objectives, and information on how this part of the lesson addresses CCSS content standards and Standards for Mathematical Practice.

Posing the Problem/ Independent Work Time (15 min) Hold up a paper square and ask students to consider what half of the square might look like. Then ask the students:

S1 (Two Equal Pieces):

How many different ways can you show two equal pieces of the whole square?

T1: Great. Are there more possibilities? How do you know?

How many different ways can you show four equal pieces of the whole square?

S2 (Four Equal Pieces): Student thinks fourths means the square has to be folded four times (creating eight equal parts).

This is the first lesson in the unit that addresses 2.G.3. Students have already composed squares into a composite rectangle, as well as partitioned a rectangle into same-size squares. A crucial aspect of this lesson is recognizing that equal shares of identical wholes do not need to have the same shape. Bridging from the previous lessons, in this lesson students can break apart and recompose shapes in order to prove that two different shapes are equal shares of the same whole.

Remind the students that they can use T2: Does this create four equal pieces? as many paper squares as they need Does the number of folds always equal to show their solutions. They should the number of equal pieces? keep in mind: all of the squares are the same size whole. S3 (Four Equal Pieces): Give students access to same-sized paper squares, which they can fold to create fractional parts.

Students should outline their folds with a marker to make their solutions visible.

T3: Great. Are there more possibilities? Do all four parts have to be the same shape?

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Reasoning with Shapes and their Attributes

Fractions of Shapes (1/2 and 1/4) Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

Comparing and Discussing (20 min) Ask students how many different ways they were able to find. Create tension, as students will want to know all the ways when they hear their peers have found more than they have. Invite students to show the different possibilities for two equal pieces. Keep these visible on the board. Invite students to show the different possibilities for four equal pieces. Keep these visible on the board. Remind students that all of the squares are the same size whole.

To begin the comparison and discussion, use open-ended prompts such as: How many equal pieces does each solution show? If students do not trust that shares of different shapes are equal sizes, ask questions such as: ● How can we be sure the pieces are equal size? ● How can we prove that these four equal pieces are equal, even though they are different shapes? ● Do two pieces have to be the same shape in order to take up the same amount of space? How can we be sure? In response, students may respond with comments such as: ● “Even though two of the pieces are squares and two of the pieces are rectangles, they are still the same. If you cut one of the squares in half and put it end to end, it would be the same size and shape as the rectangle.”

Questions such as these help students see beyond the shapes of each share. Students begin to reason abstractly by thinking of how two different shapes might represent the same amount of space, even though they appear on the surface to be different (SMP2).

When students are asked how they can be sure that two different shapes can take up the same amount of space, they are developing their ability to make a viable mathematical argument. Students must think about how to communicate their reasoning to others, such as by cutting or rotating two halves of a square to compose a rectangle. The next time students make a mathematical argument, they may be more likely to represent their ideas visually in order to communicate more effectively (SMP3).

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Reasoning with Shapes and their Attributes

Fractions of Shapes (1/2 and 1/4) Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

Summing up (5 min) Write a summary of today’s learning for students to record in their notebooks. To save time, you may want to have a photocopied summary available to have students paste into their notebooks. Today we saw that there are several ways to partition a square into two and four equal pieces. Shapes that are divided into two equal pieces are “cut into halves”. Each piece represents one half of the whole. Shapes that are divided into four equal pieces are “cut into fourths.” Each piece represents one fourth of the whole. Halves and fourths are not always the same shape. They just have to take up an equal amount of space. 1 An equal part of a shape is a fraction of the shape. Student Reflection (10 min) Have students compare and contrast the following arrangements and explain in writing why both show halves even though the halves are different shapes.

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Fractions of Shapes (1/2 and 1/4) Resources for English Learners English Language Objective(s):  Students will use words, phrases, or short sentences to explain how they partition rectangles into two and four equal parts and the equal shares may not be the same shape. English Learners Unit Vocabulary: Every Day Terms/ Phrases (Tier 1) equal face side sort straightedge area rows columns size fit tile horizontal vertical space partition whole

General Academic Terms (Tier 2) angles attributes shapes riddles overlap

Content Specific Terms (Tier 3) polygon triangles number quadrilaterals pentagons hexagons cubes vertices right angles rectangles squares 3 Dimensional (3D) fraction one half one fourth one third

Learning Progressions Previous Grade 1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. By the end of first grade, students can partition circles and rectangles into equal shares and use vocabulary terminology to describe these shares. Note that these terms (e.g., half, fourth) are not expressed as fractions with numerator and denominator. Students must also understand that, as the shape is partitioned into more shares, the shares become smaller. Current Grade 2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. Standard 2.G.3 adds on to the previous year’s standard in two ways. For one, 2nd grade students also understand thirds in their knowledge of fractional parts. Secondly, students can explain that two halves of the same-size whole may not be the same shape (e.g., one may be a triangle and the other a rectangle). Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 114

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Lesson 12

Reasoning with Shapes and their Attributes

Fractions of Shapes (1/2 and 1/4) Learning Progressions Next Grade 3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. In 3rd grade, students understand the shares created by partitioning shapes as having a certain area. The areas of these shapes can be expressed as a unit fraction. This theoretical foundation is in conjunction with other area standards developed in 3rd grade.

Materials:   

Multiple paper squares available to each student Markers (for tracing fold marks) Enlarged paper squares

References: Van De Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 115

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Lesson 12

Reasoning with Shapes and their Attributes

Fractions of Shapes (1/2 and 1/4) Artifacts:

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 116

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Lesson 13

Reasoning with Shapes and their Attributes

Fractions of Shapes (half, fourth, and third) Lesson Summary

The partitioning activities in this lesson require students to use the same skill developed in the previous lesson, but with a different whole, a circle. The use of a circle allows students to find halves, fourths, and thirds of a different shape. Students will begin to see that there are many ways to partition a shape, but that the parts must always be equal in space to be considered a fraction of the whole. The task is presented in an open ended way to encourage students to persevere in problem solving by exploring different methods of partitioning the circle into equal parts. During the comparison and discussion section of the lesson, students will share their methods and solutions with the class. Students will share and compare methods to construct viable arguments that prove they have partitioned the circles into equal parts of the whole.

Lesson Objectives

Students will be able to:  Determine how to partition circles into two, three, and four equal pieces to recognize and explain that shapes other than a square and rectangle can also be divided into equal shares. Common Core State Standards:

Standards for Mathematical Practice:

2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

MP.1: Make sense of problems and persevere in solving them. MP.3: Construct viable arguments and critique the reasoning of others. MP.5: Use appropriate tools strategically.

Time Allotment - One class period 45-60 minutes

Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials and Attachments List  Lesson Artifacts

Description of Activities: Using circular dot paper or circular geo-boards, students will partition different sized circles into two, three, and four equal pieces. Students will use strategies including direct comparison to justify that the pieces they have divided the circle into are equal pieces. Students will share and discuss different methods used to partition the circle into equal parts and explain how they know the parts are equal. Through the comparison and discussion portion of the lesson, students will understand how to justify to their classmates that the circle can be divided into two, three, or four equal pieces.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 117

Lesson 13 - 1

Lesson 13

Reasoning with Shapes and their Attributes

Fractions of Shapes (half, fourth, and third) Steps and Learning Activities This column shows the major events and flow of the lesson

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows additional moves, questions, or statements that the teacher may need to make to help students. Anticipated student responses are in regular font; teacher support suggestions are in italics.

This column contains questions that can be used to determine if the lesson is effectively addressing the stated objectives, and information on how this part of the lesson addresses CCSS content standards and Standards for Mathematical Practice.

Introduction (5 minutes) Note to teacher: Have images of rectangle from previous day displayed and the circle dot paper image for class to see. Refer to the images while students compare wholes.

It is important for the teacher to draw attention to the different “whole” being used in this lesson. This portion of the lesson requires students to compare attributes of the different wholes/shapes using knowledge gained earlier in the unit. Reviewing this information also allows the teacher to emphasize the circle as a “whole” shape before the students begin to partition the whole into equal parts. (2.G.1)

Tell students, Yesterday we saw the different ways we can divide a square into 2 (halves) or 4 (fourths) equal parts of the whole. Today we are going to look at a new whole. Display image of circle dot paper for class to see. Ask students, What is the whole that we are working with today? Ask, How is today’s whole, a circle, different than a rectangle?

SR1: Circle SR2: No Response TR2: (While displaying a rectangle.) Look at the shapes, how are they different? SR3: Round or not straight TR3: Right, a rectangle has straight sides and a circle is round.

Ask, What do you notice about the dots around the outside of the circle on the dot paper?

SR1: There are 12. TR1: There are 12. Do we notice anything else about them? SR2: They are spaced out/not touching. Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 118

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Reasoning with Shapes and their Attributes

Fractions of Shapes (half, fourth, and third) Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

TR2: Do they look like they are equal distances apart? SR2: Yes or about the same. TR2: The dots on the outside of the paper are equal distances apart.

*See attachments for images of possible student work.

Posing the Problem/ Student Work Time (15 minutes) Tell students, today we are working with a new whole, the circle. Like other shapes, a circle can also be divided into equal pieces. Tell students, How many different ways can you show two, three, and four equal pieces of the whole - a circle? Use the circle dot paper (attached) to help you solve this problem. Be ready to show how you partitioned the circles and explain how you know the pieces are equal parts of the whole. Make multiple copies of circle dot paper available for students to partition using a straightedge or cut and fold (direct comparison). Students may select the tool they think will work best. While students work circulate around the room and observe student work. Engage students in conversations about their work by using the Teacher Responses. Encourage students to share work with one another. *see attached Circular paper

SR1: Student shows only one representation. TR1: Can you find other ways to find 2, 3, or 4 equal parts of the whole? How can you use the dots to help you? SR2: Student partitions the circle by counting the number of dots in between each line segment. TR2: Why are you counting dots? Will that help you find other equal parts? How will you be able to prove the parts are the same size? (prompt to cut and compare if needed) SR3: Student has difficulty showing thirds (finding a method or not creating equal pieces). TR3: What does it mean to have three equal parts of a whole? How does that compare to the work we did yesterday? How can you use the dots to help you?

The problem is designed to encourage students to find multiple ways to partition the circle. It is left open to allow students to determine their own entry point to the problem. In the allotted amount of time, students should be able to find more than one way to partition the circles. Those struggling will be involved in a conversation with the teacher or a classmate which will guide them in their problem solving. (MP1)

The problem is made accessible by giving students multiple tools to solve it. Students need to determine the tool that best works for them. (MP5) The circular dot paper allows students to use their knowledge of equal groups and direct comparison to partition the circle. Using the dots with a fold or a straightedge to draw line segments shows that they understand their purpose, but the teacher will need to prompt them to compare and prove they are equal sizes to meet the standard. Folding paper allows students to use the

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 119

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Fractions of Shapes (half, fourth, and third) Steps and Learning Activities

Anticipated Student Responses and Teacher Support

SR4: Student draws line segments in between the dots (not using dots as the line segment endpoints). TR4: How do you know if these parts

are equal parts of the whole? Can you prove these parts are equal? How might you use the dots on your paper to help you?

Analysis of Common Core State Standards Implementation

direct comparison method to prove equal parts. Students just “eyeballing” (e.g. they look the same size) are not using the tools correctly and will be unable to construct a viable argument to prove the parts are equal. (MP 3, 5)

SR5: Student does not run all line segments through the center of the circle. TR5: How do you know that these parts are equal parts of the whole? How can the dots on the paper help you solve the problem? Why do you think there is a dot in the center of the circle? SR6: Student does not draw straight line segments. TR6: I notice you are drawing line segments. What do you know about line segments? SR6: They are straight or they have points or no response. TR6: (If no response from S) Point out examples of line segments in the room. How are they different than the lines you drew? (Guide them to see that lines must be straight. Prompt students to use straightedge). Comparing and Discussing (20-30 minutes) Have several copies of dot paper on the board for students to show the different ways that each circle can be partitioned. Begin by asking, how many different way did you find the partition the

Ideas for Facilitation: Why is the center of the circle important in partitioning into equal pieces? How did you use the center dot to help you find equal parts?

Through the partitioning activities (Lessons 11 & 12), students have now created 2, 3, and 4 equal shares of a circle and a rectangle. The discussion in this lesson requires students to use “halves”, “fourths”, “one fourth”, “a half”, etc. to describe the parts of the whole. This lesson reinforces the idea of halves and fourths and introduces

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 120

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Reasoning with Shapes and their Attributes

Fractions of Shapes (half, fourth, and third) Steps and Learning Activities

Anticipated Student Responses and Teacher Support

circles into 2, 3, or 4 equal shares? Students will share strategies used to partition the circles into equal parts and explain how they know parts are equal. Students not presenting will listen and ask questions about the work. They compare the different strategies used. Strategies/Solutions to include in discussion: Student that does not partition using the center of the circle. Student who showed two (halves) or four (fourths) equal pieces. (Show both direct comparison method and using the dots method.) Student work that shows three unequal parts. Students that partitioned into three equal pieces (thirds).

How did you divide your circle into halves or fourths? How did you know the pieces are equal? Is there a different way to show the pieces are equal?

Analysis of Common Core State Standards Implementation

the “third” as an additional way of creating equal parts of a whole. (2.G.3)

Why are these three parts of the circle not thirds? How did you divide your circle into three equal pieces or thirds? How do you know that your pieces are equal sizes? To engage students in conversation with one or another use prompts such as: Why do you _______ partitioned his circles this way? How is your work different from what _____ did?

Use student work and questions in Ideas for Facilitation to guide discussion. During discussion remind students that the parts are “part of the whole” (the circle). Summing up (5 minutes) Yesterday we learned about dividing rectangles into halves and fourths. Today we saw that we can also divide circles into halves and fourths as well as thirds (or 3 equal pieces). Have on board so that students may copy into Math Notebooks. When a whole is divided into three equal pieces, it is “cut into thirds.” Each piece represents a third of the whole. Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 121

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Reasoning with Shapes and their Attributes

Fractions of Shapes (half, fourth, and third) Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

A fraction of a shape is an equal part. Student Reflection/ Assessment (5 minutes) Students will record one new thing they learned in their Math Notebook or on an Exit Slip. If needed, provide students with Reflection Sentence Starters:  I learned….  Now I know…  I liked…  Next time I will try….because….

Reflecting upon their learning allows students to put their newly gained knowledge into their own words. Taking the time to do this at end of a lesson will benefit future discussions of similar concepts. It also allows for students to acknowledge personal challenges and successes in the lesson. (MP3)

Resources for English Learners English Language Objective(s):  Students will use words, phrases, or short sentences to explain how they partition circles into two and four equal parts and the equal shares may not be the same shape. English Learners Unit Vocabulary: Every Day Terms/ Phrases (Tier 1) equal face side sort straightedge area rows columns size fit tile horizontal vertical space partition whole circle

General Academic Terms (Tier 2) angles attributes shapes riddles overlap

Content Specific Terms (Tier 3) polygon triangles number quadrilaterals pentagons hexagons cubes vertices right angles rectangles squares 3 Dimensional (3D) fraction one half one fourth one third

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 122

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Reasoning with Shapes and their Attributes

Fractions of Shapes (half, fourth, and third)

Learning Progressions Previous Grade 1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. By the end of first grade, students can partition circles and rectangles into equal shares and use terminology to describe these shares. Note that these terms (e.g., half, fourth) are not expressed as fractions with numerator and denominator. Students must also understand that, as the shape is partitioned into more shares, the shares become smaller. Current Grade 2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. Standard 2.G.3 adds on to the previous year’s standard in two ways. For one, 2nd grade students also understand thirds in their knowledge of fractional parts. Secondly, students can explain that two halves of the same-size whole may not be the same shape (e.g., one may be a triangle and the other a rectangle). Next Grade 3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. In 3rd grade, students understand the shares created by partitioning shapes as having a certain area. The areas of these shapes can be expressed as a unit fraction. This theoretical foundation is in conjunction with other area standards developed in 3rd grade.

Materials:

Attachments:

 circle dot paper (several copies of each size for each student)  straightedge  scissors

 

4 Circle Dot Paper 1 Circle Dot Paper

References: Van De Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 123

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Fractions of Shapes (half, fourth, and third) Artifacts:

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Halves

Fourths

Thirds

125

Attachments – Anticipated Student Responses: Incorrect Student Responses Not Using Center Dot

Difficulty Making Thirds

126

Attachments – Circle Dot Paper

127

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Lesson 14

Reasoning with Shapes and their Attributes

Fractions of Shapes (fourths) Lesson Summary

In this lesson, students use methods from previous lessons to prove that equal parts of different shapes are equal. This lesson provides students with a variety of shapes divided into four parts, so that students can focus on proving if the parts are equal sizes regardless of their shape within identical wholes. It provides them with familiar models encountered earlier in the unit to aid in their explanations and new models that will require them to prove the equal parts can be different shapes within identical wholes as stated in the standard. While not as open as other problems in the unit, the problem still allows students to choose their methods and tools and requires them to justify their arguments.

Lesson Objectives

Students will be able to:  Analyze various shapes to determine whether or not they are partitioned into fourths. Students will explain their thinking and how they determined the parts are equal. Common Core State Standards:

Standards for Mathematical Practice:

2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

MP.1: Make sense of problems and persevere in solving them. MP.3: Construct viable arguments and critique the reasoning of others. MP.5: Use appropriate tools strategically

Time Allotment - One class period 45-60 minutes (assessment may require additional time)

Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials and Attachments List  Lesson Artifacts

Description of Activities: Students will analyze various geometric shapes and determine if they are correctly partitioned into fourths. Students will use a direct comparison method or recreating the shapes on dot paper to determine equal areas in each shape design to determine if particular shapes are divided into four equal parts or fourths. Students will be asked to explain how they know different shapes and different sized pieces can be equal parts of the same whole.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 129

Lesson 14 - 1

Lesson 14

Reasoning with Shapes and their Attributes

Fractions of Shapes (fourths) Steps and Learning Activities This column shows the major events and flow of the lesson

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows additional moves, questions, or statements that the teacher may need to make to help students. Anticipated student responses are in regular font; teacher support suggestions are in italics.

This column contains questions that can be used to determine if the lesson is effectively addressing the stated objectives, and information on how this part of the lesson addresses CCSS content standards and Standards for Mathematical Practice.

Introduction (5 minutes) Tell students, Yesterday we saw that we can divide a circle into two, three, and four equal parts. Today we are going to look at different shapes divided into four parts to determine and prove if the parts are fourths. What does it mean when a whole is divided into fourths?

S1: Each of the four parts is equal. T1: How do we know when parts are equal? S1: The parts are the same size or take up the same space. S2: It is divided into four parts. T2: What needs to be true about the parts to call them fourths? S2: The parts much be equal in size.

The lesson begins by reviewing and reemphasizing the importance of equal parts of a whole being equal sizes. This is essential in the students’ developing understanding of fractions. (1.G.3, 2.G.3) Here the teacher does not tell the students this, but uses questions to allow students to define this in student friendly language.(MP3)

(h)

Posing the Problem/Student Work Time (15 minutes) Ask, Which of these shapes is divided into four equal pieces? How do you know? Use the handout to determine and prove which of the shapes are divided into fourths or four equal parts. Be ready to explain and/or how you know the parts are equal parts of the whole.

S1: Shapes (a) and (f) are divided into four equal pieces – all parts are equal size and same shape pieces of the whole T1: How do you know those are equal pieces? How could you prove to a friend these pieces are equal?

The problem requires students to choose a method and tools they think will best help them justify their argument. (MP5) The problem provides examples and non-examples so that students must justify their argument for each shape whether the parts are equal or unequal. (MP3)

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 130

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Reasoning with Shapes and their Attributes

Fractions of Shapes (fourths) Steps and Learning Activities While students work circulate around the room observing student work and engaging them in conversations about their work. Encourage students to explain their methods and thinking to you and to others in their groups.

Anticipated Student Responses and Teacher Support

S2: Shapes (e) and (h) are divided into four equal pieces – different shapes, same size. T2: How do you know those are equal pieces? How could you show or prove to a friend these pieces are equal?

Analysis of Common Core State Standards Implementation

S3: Shapes (b) and (c) are not divided into four equal pieces – different shape and different sizes. T3: How do you know they are not equal pieces of the whole (fourths)? How could you show/ prove to a friend they are not equal? S4: Shape (d) is not divided into four equal pieces. T4: How many parts does it have? S4: Three S5: Shape (c), (d), and (b) are not divided into four equal pieces. T5: How could prove the pieces are equal sizes? How could you show a friend the parts are equal if you cut them out? What does it mean when pieces are equal? S6: Shape (g) is not divided into fourths. T6: How is this whole divided different from the others? How many parts does it have? S7: Shape (g) is divided into equal parts. T7: Remind students the question says fourths. Is this shape divided into fourths? How many parts does it have? Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 131

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Reasoning with Shapes and their Attributes

Fractions of Shapes (fourths) Anticipated Student Responses and Teacher Support

Steps and Learning Activities

S8: Student having difficulty with proving equality can practice subdividing squares and rectangles to determine equality of parts. T8: (Provide S with precut square and rectangles to practice folding) Can you divide a rectangle into

Comparing and Discussing (20-30 minutes) Have several copies of larger shapes on dot paper on the board so that students can reference the pieces as they explain their thinking and show their work. Begin by discussing what it means to have four equal pieces. What is a fourth of a whole? Call students to the board to share their work and explain how they know the parts are equal or unequal. Include the following comparisons in student share out and discussion: Discuss shape g (not fourths). Discuss shapes a and f (same shape, same size). Discuss shapes b, c and d (same shape, different sizes). Discuss shapes e and h (different shapes, same sizes). If needed, use questions from Ideas for Facilitation to guide discussion.[3]

Analysis of Common Core State Standards Implementation

halves or two equal parts? How do you know these parts are equal? How can you use this to help you decide which shapes have four equal parts?

Ideas for Facilitation:  How do you know when parts of the same whole are equal in size? How can you prove the parts are equal in each different shape?  How do we know shape g is not divided into fourths?  How do we know that the parts are equal or fourths of the whole? How could you show or prove it?  How do you know these parts are not fourths or unequal parts? To engage students in conversation with one or another use prompts such as:  Can you show _____ how you proved the parts were equal?  How do you think _____ was able to prove the parts are

In the Comparison and Discussion section, students will share their explanations and methods for proving the parts of the different wholes are equal or unequal. Students will need to model how they used their chosen tools to justify their arguments. Students will need to explain their thinking, to justify their arguments, and the thinking of others to build knowledge about equal parts. (MP3/MP5) To further address the standard, it is important for the teacher to prompt students to use the fourths, a fourth of, one fourth of, etc. language in their arguments. Since this lesson focuses specifically on fourths and provides examples and non-examples, it will further help the teacher determine if students have mastered their understanding of fourths and equal parts of a whole. (2.G.3/MP3)

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 132

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Lesson 14

Reasoning with Shapes and their Attributes

Fractions of Shapes (fourths) Anticipated Student Responses and Teacher Support

Steps and Learning Activities

Analysis of Common Core State Standards Implementation

equal? Show him/her what you think they did.  Why do you think _____ chose to use this tool for this shape?  What questions do you have about how _____ work? Summing up (5 minutes) Write on the board and have students record in math notebooks: Today we looked more at what it means for different wholes to be divided into fourths – we learned that in some shapes, the equal parts may have the same size and the same shape, while other equal parts might be of equal size but different shapes. Assessment Students solve in their Notebooks or on an Exit Slip. (See attachment for assessment page.) There are two sandwiches that are the same size square. Each sandwich is cut into fourths, but the sandwiches are cut differently. What are two different ways the sandwich can be cut? Using the dot paper draw the two sandwiches and show two different ways to cut into fourths. There are two ribbons that are the same size rectangle. What are two different ways the ribbons can be divided into thirds? Using the dot paper draw the two pieces of ribbon and show 2 different ways to cut into thirds. Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 133

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Reasoning with Shapes and their Attributes

Fractions of Shapes (fourths)

Resources for English Learners English Language Objective(s):  Students will use words, phrases, or short sentences to explain how they know if a shape is partitioned into fourths and how they know the parts are equal. English Learners Unit Vocabulary: Every Day Terms/ Phrases (Tier 1) equal face side sort straightedge area rows columns size fit tile horizontal vertical space partition whole circle equal

General Academic Terms (Tier 2) angles attributes shapes riddles overlap

Content Specific Terms (Tier 3) polygon triangles number quadrilaterals pentagons hexagons cubes vertices right angles rectangles squares 3 Dimensional (3D) fraction one half one fourth one third

Learning Progressions Previous Grade 1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. By the end of first grade students are expected to partition circles and rectangles into halves and fourths. They should be able to describe the equal parts/shares using halves, fourths, and quarters. They should also understand that a half is more than a fourth of the same shape because there are less equal parts of the same whole. Current Grade 2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 134

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Lesson 14

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Fractions of Shapes (fourths) Learning Progressions By the end of second grade, students will have built upon their knowledge of fractions from first grade by partitioning circles and rectangles into halves, fourths, and thirds. Students should be able to describe an equal share of a shape using fraction language such as half, third, half of, a fourth of, etc., but not fraction notation. Students should be able to explain and demonstrate how they know shares are equal even if they are not the same shape. Next Grade 3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For examples, partition a shape into 4 parts with equal areas, and describe the area of each part as ¼ of the area of the shape. By the end of third grade, students will build upon their knowledge of halves, fourths, and thirds to partition shapes into equal shares. Students will be able to describe each part as a unit fraction of the whole. Students will be expected to turn their fraction language mastered in second grade into fraction notation (1/4) to describe an equal part of a whole.

Materials:      

Attachments:  

Printout of shapes (two per student, 1 to cut and 1 to compare and extras) scissors dot paper straightedge large copies of shapes (for Comparing and Discussing) Student Assessment (one per student)

Printout of shapes Student Assessment

References: Van De Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 135

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Fractions of Shapes (fourths) Artifact: Math Notebook

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Fractions of Shapes (fourths) Artifact: Student Work

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Fractions of Shapes (fourths) Artifact:

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Reasoning with Shapes and their Attributes

Fractions of Shapes (fourths) Student Artifacts - Assessment

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 139

Lesson 14 - 11

Lesson 14

Reasoning with Shapes and their Attributes

Fractions of Shapes (fourths) Student Artifact - Assessment

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 140

Lesson 14 - 12

Lesson 14

Reasoning with Shapes and their Attributes

Fractions of Shapes (fourths) Student Artifact - Assessment

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 141

Lesson 14 - 13

Attachment – Shapes

(h) 142

Name: ______________________________________ Student Assessment 1.

There are two sandwiches that are the same size square. Each sandwich is cut into fourths, but the sandwiches are cut differently. What are two different ways the sandwich can be cut?

Using the dot paper draw the two sandwiches and show two different ways to cut into fourths.

2.

There are two ribbons that are the same size rectangle. What are two different ways the ribbons can be divided into thirds?

Using the dot paper draw the two pieces of ribbon and show 2 different ways to cut into thirds.

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Lesson 15

Reasoning with Shapes and their Attributes

Culminating Task Lesson Summary

This lesson is an assessment of all lessons in the unit, with focus standards of 2.G.2 and 2.G.3.Students will apply their knowledge of partitioning, paper folding, tiling or another appropriate strategy to determine if a playground space has been divided equally into thirds. Students will use appropriate tools and strategies to help justify their thinking on whether or not a playground space has been divided fairly into thirds.

Lesson Objectives

Students will be able to:  Determine whether or not various shapes have equal shares.  Partition a rectangle into thirds, where each one-third piece is a different shape.

Common Core State Standards:

Standards for Mathematical Practice:

2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

MP.3: Construct viable arguments and critique the reasoning of others. MP.4: Model with mathematics. MP.5: Use appropriate tools strategically. MP.6: Attend to precision.

2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. Time Allotment - One class period 45-60 minutes (assessment may require additional time)

Lesson Resources:  EL Objectives and Vocabulary  Learning Progressions  Materials and Attachments List  Lesson Artifacts

Description of Activities: Students will determine whether or not two different rectangular playground spaces have been sub-divided into three equal play spaces, they will then be asked to partition a third rectangular space into three equal playground spaces.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 144

Lesson 15 - 1

Lesson 15

Reasoning with Shapes and their Attributes

Culminating Task Steps and Learning Activities This column shows the major events and flow of the lesson

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

This column shows additional moves, questions, or statements that the teacher may need to make to help students. Anticipated student responses are in regular font; teacher support suggestions are in italics.

This column contains questions that can be used to determine if the lesson is effectively addressing the stated objectives, and information on how this part of the lesson addresses CCSS content standards and Standards for Mathematical Practice.

1. Posing the Problem Because this is the final assessment, students will work on this problem individually; the teacher will use the lessons rubric to identify student’s progression within the lesson. Write the statement below on the board. For recess today, we will have three grade levels on the playground and we need to make sure that each grade level has the same amount of space to play. Each grade level has been asked to share one way that the space might be shared equally – below are the two examples that the other grade levels have shared. Are their examples fair? Explain your thinking.

S1: Student has difficulty tracing tiles in space without creating any gaps or overlaps. T1: Let’s pay attention to being accurate and precise. Is that the actual size of each tile? Is that the same size that the other students will draw on their papers? Will your answers be the same?

S2: Can correctly identify the number of tiles needed for each space but cannot justify thinking. T2: What method did you use to find your answer? Write more about what Once you determine whether or not you did to try to convince the other each is fair, come up with a different grade levels that theirs is/is not fair. way that you might separate the playground space so that each S3: Student says that 1st grade and 3rd grade level has the same amount of grade options are both fair space to play on and explain why T3: Can the same number of tiles fit in you think your way is best. each part? How do you know? 1st Grade Option: A

C

B Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 145

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Lesson 15

Reasoning with Shapes and their Attributes

Culminating Task Steps and Learning Activities

Anticipated Student Responses and Teacher Support

Analysis of Common Core State Standards Implementation

*Dimensions of rectangle A is 3 X 2 *Dimensions of rectangle B is 3 X 2 *Dimensions of rectangle C is 3 X 4 3rd Grade Option: A

B

C

*Dimensions of rectangles A,B and C are each (2 X 4) 2nd Grade Option:

Allow students access to paper with images mentioned above, as well as four square unit tiles and a pencil for tracing.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 146

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Lesson 15

Reasoning with Shapes and their Attributes

Culminating Task Resources for English Learners English Language Objective(s):  Students will use words, phrases, or short sentences to explain how they determine shapes have equal shares. They will explain how they partition a rectangle into thirds and each one-third piece is a different shape. English Learners Unit Vocabulary: Every Day Terms/ Phrases (Tier 1) equal face side sort straightedge area rows columns size fit tile horizontal vertical space partition whole circle equal

General Academic Terms (Tier 2) angles attributes shapes riddles overlap

Content Specific Terms (Tier 3) polygon triangles number quadrilaterals pentagons hexagons cubes vertices right angles rectangles squares 3 Dimensional (3D) fraction one half one fourth one third

Learning Progressions Previous Grade 1.G.2 Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. By the end of first grade, students are expected to be able to create composite shapes (both two-dimensional and threedimensional) by using various two-dimensional shapes. As pertains to this lesson, students in first grade may have already composed rectangles from squares. Current Grade 2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. By the end of second grade, students will know how to identify and create different shapes that have the same area. Students will be able to identify halves and thirds of a whole within different shapes. Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 147

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Lesson 15

Reasoning with Shapes and their Attributes

Culminating Task Learning Progressions Next Grade 3.MD.7a Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. By the end of third grade, students will be able to add areas to find the total area of a figure. In the 2nd grade learning progression, students are adding areas within a non-rectilinear figure to find the total area; in 3rd grade they will be able to find the total area of a rectilinear figure.

Materials:   

Attachments:  

Paper with 3 rectangles, two partitioned differently and one that has not been partitioned 4 Square unit tiles for each student Scissors

Assessment sheet – student worksheet (3 pages) Student Rubric

References: When examining the teaching of area to students, the research team looked at the Van De Walle text’s sections on teaching area along with the Van Hiele levels of geometric thought. On page 392 of Van De Walle, it says “research suggests that it is a significant leap for students to move from counting squares inside of a rectangle to a conceptual development of a formula. Battista (2003) found that students often try to fill in empty rectangles with drawings of squares and then count the result one square at a time.” This is extremely important for our research team when beginning to examine area at the beginning stages – our lessons focus on the partitioning of rectangles, using tiles – students being able to count the tiles initially. We are not yet focused on students understanding the formula for area, as we feel that this is premature. Van De Walle later says “an important concept to review is the meaning of multiplication as seen in arrays. Show students rows and columns of objects or squares and discuss why multiplication tells the total amount…” In our lessons, we are more focused on the students understanding of the amount of space being occupied, and less interested in the formula for area. However, we would like for students to notice patterns in being able to find area – for instance, do students recognize repeated addition of rows or columns, are students able to notice the formula without being taught the formula. We would like these ideas to be explored to better support students understanding of area. Van De Walle, J.A., Karp, K.S., Bay – Williams, J.M. (2010) Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Allyn & Bacon.

Reasoning with Shapes and their Attributes – 2nd Grade www.sharemylesson.com/ctu 148

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Name: _____________________________

Date: ___________

First Grade’s Idea: Is this playground space separated equally into three parts? How do you know? A

C

B

_____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________

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Third Grade’s Idea: Is this playground space separated equally into three parts? How do you know?

A

B

C

_____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________

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Second Grade’s Idea: Design a different way to separate the playground space into three equal parts. Explain how you know that the playground is separated equally.

_____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________ _____________________________________________________

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Student Name: Playground Space Rubric Persevere in Problem Solving (MP1)

Use of Tools (MP5)

Mathematical Concepts (MP7, 2.G.2, 2.G.3)

Mathematical Reasoning (MP3)

4

Student worked to solve all three problems, worked through any challenges and was able to answer all three problems, regardless of accuracy

Student accurately separates each of the three play spaces in all three scenarios, while using tools appropriately and precisely

Student correctly determines which playground spaces are equal/not-equal and is able to create a third design that is separated into three equal parts, each having a different shape

Student explains reasoning for each of the three playground spaces and uses correct mathematical vocabulary in writing whenever appropriate

3

Student needed some encouragement to work through problems, was able to answer to all three problems, regardless of accuracy

Student uses tools to separate the playground spaces, but makes mistakes in one scenario

Student is able to correctly determine which playground spaces are equal/not-equal but is unable to design a third space that is separated into three equal parts, each having a different shape

Student explains reasoning for two of the playground spaces and mostly uses correct mathematical vocabulary in writing whenever appropriate

2

Student needed a lot of encouragement to work through problems, was able to come up with an answer to at least one problem, regardless of accuracy

Student uses tools to separate the playground spaces, but makes mistakes in two scenarios

Student is unable to determine which playground spaces are equal/not-equal and is unable to design a third space that is separated into three equal parts, each having a different shape

Student explains reasoning for one of the playground spaces and mostly uses correct mathematical vocabulary in writing whenever appropriate

1

Student needed a lot of encouragement to work through problems, was unable to solve any of the problems, regardless of accuracy

Student uses tools inappropriately or inaccurately

Student shows an attempt, but does not correctly determine equality of the three playground spaces.

Student is unable to explain reasoning for the three playground spaces

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Acknowledgements

Amy Andrews, NBCT John T. Baldwin Ava Patricia Belisle-Chatterjee John Boggs, NBCT Genevieve Boyle Renee Butala, NBCT Brenda Cerda, NBCT Allison Dodson, NBCT Elizabeth Eberle Sari Freiel Katie Friedl, NBCT Andrew Friesema Daniel Greenfield, NBCT Yezenia Guzman Ned Harkness Bonita Harris Gloria Henllan-Jones, NBCT Julie Hines-Lyman, NBCT Theresa Insalaco-DeCicco, NBCT Cheryl Janusz Earl Johnson, NBCT Meghan Landers, NBCT Joshua Lerner, NBCT Kelly McCann, NBCT

Tanya Mead, NBCT Keshanna Milsap, NBCT Paul Mirek, NBCT Leslie Monk, NBCT Georgia Anne Moore, NBCT Michael P. Moriarty, NBCT Michelle Nash, NBCT Sharon Newman, NBCT Chris Nho Erin O'Brien, NBCT Sarah O'Brien Freeda Pirillis, NBCT Vlada Polin Shirley Roberson, NBCT Humberto Rodriguez Lisa Roule, NBCT Kelly Rueda, NBCT Suzanne Schaefer, NBCT Timothy Shanahan Yoni Vallecillo, NBCT Daniel Washco, NBCT Shakura Weathers Lori Zaimi, NBCT Xiaodong Zhang, NBCT

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