EM2FL26_FPBTEG5_M05_177229 by General

TEACH Grade 5

Module 5

A Story of Units

Fractions Are Numbers ▸ 5 TEACH Module

Module

1 2 3 4 5 6

Place Value Concepts for Multiplication and Division with Whole Numbers

Addition and Subtraction with Fractions

Multiplication and Division with Fractions

Place Value Concepts for Decimal Operations

Addition and Multiplication with Area and Volume

Numerical Patterns, the Coordinate Plane, and Datas

Great Minds® is the creator of Eureka Math®, Wit & Wisdom®, Alexandria Plan™, and PhD Science®. Published by Great Minds PBC. greatminds.org © 2026 Great Minds PBC. All rights reserved. No part of this work may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying or information storage and retrieval systems—without written permission from the copyright holder. Where expressly indicated, teachers may copy pages solely for use by students in their classrooms. Printed in the USA A-Print 1 2 3 4 5 6 7 8 9 10 XXX 35 34 33 32 31 30 29 28 27 26 ISBN 979-8-89417-722-9

A Story of Units®

Fractions Are Numbers ▸ 5 TEACH Module

Addition and Multiplication with Area and Volume Topic A Drawing, Analysis, and Classification of Two-Dimensional and Three-Dimensional Figures Topic B Areas of Rectangular Figures with Fraction and Decimal Side Lengths Topic C Volume Concepts Topic D Volume and the Operations of Multiplication and Addition

EUREKA MATH2 Florida

5 ▸ M5

What’s Included

Grade 5

Module 5 EUREKA MATH² FLORIDA APPLY

LEARN

6/5/25 6:52 PM

Additional Practice

Module 5 EUREKA MATH² FLORIDA

The Teach book empowers teachers with all the tools they need to help students access grade-level content and meet the learning needs of their students through language supports, varied assessments, and effective instructional design.

APPLY

G5 | M5

Grade 5

Module 5

EM2FL26_TEG5_M5CVR_177229.indd 2

LEARN

G5 | M5

TEACH

Name

EM2FL26_SELG5_M5CVR_177588.indd 4

6/5/25 6:25 PM

The Learn book is students’ companion text to the instruction in Teach. It contains all the pages students need for each lesson, including in-class work pages, glossaries, and opportunities for practice.

Name

EM2FL26_SEAG5_M5CVR_177885.indd 4

6/5/25 6:07 PM

The Apply book provides integrative family supports and extensive at-home practice opportunities to help students deepen their understanding.

The digital platform provides seamless digital access to items in Teach, Learn, and Apply. It also includes lesson slides, digital interactives, assignments and assessments, implementation videos, and teaching supports.

EUREKA MATH2 Florida 5 ▸ M5

Assessment The Eureka Math2® assessment system helps teachers understand student learning by generating data from many perspectives. The assessments are designed to monitor student learning and thus provide teachers with feedback that they may use to inform ongoing instruction. The assessments may also be used summatively to produce a grade or report that becomes part of a student, school, or district record. Using the data from a combination of assessments generates a complete picture of student learning. • Exit Tickets at the end of a lesson provide immediate insight into whether students have learned the main concepts and skills in the lesson.

• Pre-Module Assessments in Eureka Math2 EquipTM help teachers identify where students need support by assessing students’ foundational knowledge that is essential to the content of upcoming lessons. Go to the digital platform to access Equip content. • Performance Assessments provide students with multiple means of entry and opportunities to apply a variety of skills to solve complex, real-world problems. This work allows students to demonstrate a deep understanding of content and process standards. Go to the digital platform to access performance assessments.

• Observational Assessment Recording Sheets are checklists in each topic or module that teachers can use to record observations and evidence of student thinking by date or activity.

Eureka Math² Equip Pre-Module Assessment

• Topic Quizzes/Topic Tickets consist of up to 6 items that assess proficiency on major concepts and skills covered in a topic.

Module 1

Eureka Math² Equip Pre-Module Assessment Module 2

Eureka Math² Equip Pre-Module Assessment

Module 3

Module 4

Eureka Math² Equip Pre-Module Assessment Module 5

Module 6

Module Topic B

Topic A L

Topic C L

Topic D L

Module Assessment

• Module Assessments consist of 6–10 items that evaluate proficiency of the major concepts, skills, and applications from the module. Module assessments are available to download or assign digitally on the digital platform.

Daily Exit Tickets Topic A Quiz

Topic B Quiz

Topic C Quiz

Topic D Quiz

EUREKA MATH2 Florida

5 ▸ M5

Lesson Design The Dynamic Middle The Eureka Math2® lessons strategically blend conceptual understanding, procedural fluency, and real-world application. • Conceptual Understanding: Daily lessons include concrete–pictorial–symbolic representations and simple-to-complex progressions to develop understanding, as well as opportunities to identify math relationships and patterns and engage in meaningful discourse to answer Why does this work? Real-World Context

• Procedural Reliability and Fluency: Distributed and interleaved practice through fluency and practice problems provides ample daily opportunities for students to grow through the three stage hierarchy of mathematical fluency. • Application: Real-world problems and collaborative tasks consolidate student thinking through why and how questions, inviting students to draw conclusions and make deep connections among math ideas.

Coherence Following a logical and systematic mathematical progression designed around the mathematical areas of emphasis, Eureka Math2 lessons foster deep and lasting connections.

Tape Diagram

m 40

• A consistent structure—Fluency, Launch, Learn, Land—connects concepts from topic to topic, module to module, and grade to grade. • High-leverage models—such as number bonds, area models, tape diagrams, and number lines—build knowledge within and across grade levels.

Open Number Line

200 − 120 = n 200 − 120 = 100 − 20 = 80

120

200

3 × 40 = m 3 × 40 = 120

200 3 × 40 = m 3 × 40 = 120

100

200 - 120 = n + 50

+ 10 + 10 + 10

120 130 140 150

James needs to save $80 more to buy the art kit.

200

10 + 10 + 10 + 50 = 80 200 − 120 = 80

James needs to save $80 more to buy the art kit.

High-Leverage Models

EUREKA MATH2 Florida 5 ▸ M5

Fairness and Empowerment Eureka Math2 advances fairness and empowers students and teachers. • Lessons provide suggestions for proactive student support, including flexible grouping ideas that leverage students’ home languages and knowledge, academic terminology, and sentence frames to support discourse. • Lessons engage students through a variety of facilitation styles (problem-based learning, direct instruction, guided discovery) to meet all students’ needs. • Authentic problems and videos represent and engage students from different cultures, backgrounds, and abilities.

Talking Tool Share Your Thinking

I know . . . . I did it this way because . . . . The answer is

because . . . .

My drawing shows . . . . I agree because . . . .

Agree or Disagree

That is true because . . . . I disagree because . . . . That is not true because . . . . Do you agree or disagree with

Ask for Reasoning

? Why?

Why did you . . . ? Can you explain . . . ? What can we do first? How is

Say It Again

related to

I heard you say . . . . said . . . . Another way to say that is . . . . What does that mean?

EUREKA MATH2 Florida

5 ▸ M5

Multilingual Learner Support Language supports embedded into instructional design

Multilingual Learner Support

Mathematical content, practices, and language development are interconnected. The curriculum uses a variety of tools to support students’ language needs. Leverage the following supports to customize a plan for your multilingual learners.

Consider using strategic, flexible grouping throughout the module. • Pair students who have different levels of mathematical proficiency.

Amplify rather than simplify language

• Pair students who have different levels of English language proficiency.

Support for interpretive, and expressive language from the WIDA English Language Development Standards Framework, 2020

As applicable, complement any of those groupings by pairing students who speak the same home language.

The following research-informed supports amplify rather than simplify language by drawing students’ attention to important aspects of language, such as language functions, structure, syntax, and morphology. These supports provide multilingual learners with access to sense-making that happens in mathematical discussions. • Strategically pair multilingual learners to leverage every turn and talk and think-pair-share for multilingual learners to clarify in their home language, pause and take time to translate, and orally rehearse responses.

Multilingual Learner Support

Each of the following actions supports language comprehension from written and spoken texts, or interpretive language.

Sé que… Lo hice de esta forma porque… La respuesta es

porque…

En mi dibujo, se ve… Estoy de acuerdo porque…

Estar de acuerdo o en desacuerdo

Talking Tool Share Your Thinking

Eso es verdadero porque… No estoy de acuerdo porque… Eso no es verdadero porque… ¿Estás de acuerdo o en desacuerdo ? ¿Por qué? con

I know . . . . I did it this way because . . . . The answer is

because . . . .

My drawing shows . . . .

Preguntar por el razonamiento

¿Por qué has…? ¿Puedes explicar…? ¿Qué podemos hacer primero? ¿Cómo se relacionan

• Provide varied and flexible opportunities to make meaning. Use different structures such as pairs, small groups, and whole class discussion. Support language used to engage in dialogue.

That is true because . . . . I disagree because . . . .

Decirlo otra vez

That is not true because . . . . Do you agree or disagree with

Ask for Reasoning

? Why?

Te escuché decir que… dijo que… Otra manera de decir lo mismo es… ¿Qué significa eso?

2x 3 = 6 = 11

Why did you . . . ? Can you explain . . . ? What can we do first? How is

Say It Again

related to

I heard you say . . . . said . . . . Another way to say that is . . . . © Great Minds PBC What does that mean?

• Observe multilingual learners’ strengths during discussion. Use those strengths to build toward academic language in student responses. These supports allow multilingual learners to meaningfully participate in mathematical discussions.

I agree because . . . .

Agree or Disagree

218-220_EM2_NL_G5_M6_acknowledgments_Learn_987253_SPA.indd 219

• Support expressive language. Model using the Talking Tool during class discussions, including specific sentence frames and starters.

Consider supporting students in the Always Sometimes Never routine by displaying an equation labeled with words from the statement.

Herramienta para la conversación Compartir tu razonamiento

1/13/2024 11:42:59 AM

Whole Number Fraction

Mixed Number

Fraction Greater Than 1

EUREKA MATH2 Florida

5 ▸ M5

Research-based, multimodal terminology support Lessons provide research-based embedded support for math-specific terms, multiplemeaning terms such as homophones and homographs, academic verbs, and everyday language. Support focuses on what students are expected to know and be able to do by the end of each lesson: • Build or activate background knowledge to honor students’ assets and funds of knowledge • Integrate all four domains of language: reading, writing, speaking, and listening • Use multimodal methods, such as pairing terms with visuals, real items (realia), gestures, or acting out Lessons invite students to engage with concepts through mathematical discussions and experiences, such as digital interactives, manipulatives, or math language routines, before they learn mathematically precise terminology for the concept. This opportunity allows all learners, and especially multilingual learners, to attach new terms to their own lived experience. Lessons contain functional language needed to engage with the mathematics of the grade level. Each grade level explicitly teaches the meaning of a small list of targeted academic verbs, such as combine or approximate, as the verbs take on mathematical meaning. Lessons use synonyms to preview and support meaning before students are expected to use the language.

Math language routines to amplify, assess, and develop language Math language routines are structured but adaptable formats for amplifying, assessing, and developing students’ language. Lesson-embedded routines support sense-making, optimize output, cultivate conversations, and maximize meta-awareness. Math language routines can be used flexibly whenever there are opportunities to support students with interpretive, and expressive language.

EUREKA MATH2 Florida

5 ▸ M5

Universal Design for Learning (UDL) & Differentiation Eureka Math2 has been awarded CAST’s Universal Design for Learning (UDL) Product Certification, demonstrating Great Minds’ commitment to making math accessible for all learners. This certification recognizes that Universal Design for Learning (UDL) is core to the product’s design. UDL suggestions are organized according to the three principles of the framework: engagement, representation, and action & expression. Differentiation suggestions are provided for students who need more individualized support or are ready for a challenge. The chart below summarizes the suggestions provided at point of use in the materials, but teachers are encouraged to apply the suggestions at any time to best meet the needs of their class and individual students.

UDL: Engagement

UDL: Representation

UDL: Action & Expression

Differentiation

Support motivation and enthusiasm

Support perception and comprehension

Support expression of learning and executive function enthusiasm

Offer additional support and provide challenge

• Make real-world connections and relate concepts to students’ interests and experiences • Form cooperative groups with roles to promote participation • Offer choice of, and options within, tasks and activities • Provide action-oriented feedback • Recognize mistakes as opportunities • Support self-regulation by modeling coping strategies for dealing with frustration

• Provide visuals, realia, and hands-on experiences • Activate prior knowledge • Use graphic organizers • Emphasize patterns, key features, and big ideas through labeling, annotations, color-coding, etc. • Chunk information into smaller elements • Pause to provide think time and to emphasize the importance of information

• Provide sentence starters and frames • Offer manipulatives, calculators, or other tools • Provide exemplars, worked examples, checklists, or templates • Model by thinking aloud • Prompt students to plan and strategize • Provide questions that guide self-monitoring and reflection

• Support: Targeted approaches to supporting students with specific challenges of the concepts, procedures, or calculations presented • Challenge: Targeted suggestions for expanding or deepening understanding for students who demonstrate readiness for more

EUREKA MATH2 Florida 5 ▸ M5

Support for Families The resources in the Apply book give students more practice with the concepts learned in class. There are three components in Apply that support students in deepening their understanding of the concepts covered in the daily lesson: Family Math, Practice Partners, and Terminology. Module 6 Topic A

FAMILY MATH Analyze and Generate Numerical Patterns

EUREKA MATH2 Florida B.E.S.T.

3 ▸ M1 ▸ TA ▸ Lesson 1

PRACTICE PARTNER

I can use rules to complete a pattern and write rules to describe a pattern.

Florida B.E.S.T.

EUREKA MATH2 Florida

5 ▸ M6

Terminology

MA.5.AR.3.2

These terms are introduced in the module. Use the blank space to make notes and drawings.

Dear Family, Key Term

Your student is learning to analyze and generate number variable patterns in various forms. They begin by working with two number patterns simultaneously and use the patterns to complete input–output tables. They also identify the relationship between corresponding inputs and outputs and use the relationship to write a rule for finding an output given an input. To extend their understanding of number patterns, students learn to write expressions to represent the output when given an input. They explore how to use a variable to represent any input, and they practice by using the expressions to find outputs when given corresponding inputs. Throughout the topic, students write expressions with addition, subtraction, and multiplication that represent the output when given an input and use the expressions to solve real-world problems. Input, x

Output

Input, x

Output

Input, x

Output

Add 2 to the input

x+2

Students find addition and subtraction relationships between corresponding inputs and outputs when the rules for the inputs and outputs are both to add 1.

Multiply the input by 3

Students find multiplication relationships between corresponding inputs and outputs when the addition rules for the inputs and outputs are different. When the rule for the inputs is to add 1, the number added to the outputs is the coefficient in the multiplication expression.

Date

axes Axes are the pair of perpendicular number lines that create a coordinate system. Axes refers to both the x -axis and the y -axis together.

1. Use the table to complete parts (a) and (b). Input

Output

a. The rule for the inputs is start at 1 and add 1. The rule for the outputs is start at 5 and add 1. Complete the table. b. Write the rule for the corresponding inputs and outputs. Add 4 to each input to get the corresponding output.

(Lesson 7)

I use the rules to find the missing inputs and outputs by adding 1 to find the next terms. I see the output is always greater than the input. I look at some sets of corresponding inputs and outputs. Some corresponding inputs and outputs in this table are 1 and 5, 4 and 8, and 7 and 11 11..

coordinate A coordinate is a number used to identify the location of a point. (Lesson 6)

I must add 4 to any input to find the corresponding output.

Multiply the input by 2, and then add 1

2. Use the table to complete parts (a) and (b).

2x + 1

Students find mixed-operation relationships between corresponding inputs and outputs. They use the addition rules for the inputs and outputs to compare how many times as much is added to the outputs as is added to the inputs. Once students know the factor, they multiply the input by the result. Then they find how much to add or subtract to get the corresponding outputs.

Family Math is a letter to families that describes the major concepts in the current topic.

Name

a. The rule for the inputs is start at 15 and add 1. The rule for the outputs is start at 10 and add 1. Complete the table. b. Write the rule for the corresponding inputs and outputs. Subtract 5 from each input to get the corresponding output.

Input

Output

Practice Partners show the thinking required to approach the problem using steps that are like those named and described in the lesson

coordinate plane A coordinate plane is a plane with a coordinate system used to describe locations of points in the plane. A coordinate plane has perpendicular axes that intersect at 0. (Lesson 7)

105

Terminology includes a list of terms and definitions introduced in the module.

EUREKA MATH2 Florida

5 ▸ M5

Contents Addition and Multiplication with Area and Volume What’s Included . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Lesson Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Lesson 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Classify rectangles and rhombuses based on their properties.

Lesson 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Classify kites and squares based on their properties.

Multilingual Learner Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Lesson 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Universal Design for Learning & Differentiation. . . . . . . . . . . 8

Identify quadrilaterals from given properties.

Support for Families. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Lesson 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Achievement Descriptors: Overview. . . . . . . . . . . . . . . . . . . . . 12 Module Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Why. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Mathematical Language Routines. . . . . . . . . . . . . . . . . . . . . . . . 17 Topic A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Drawing, Analysis, and Classification of Two-Dimensional and Three-Dimensional Figures Lesson 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Analyze hierarchies and identify properties of quadrilaterals.

Lesson 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Identify three-dimensional figures and classify the figures into categories based on defining attributes.

Lesson 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Classify triangles into different categories based on attributes.

Lesson 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Classify trapezoids and parallelograms based on their properties.

Classify quadrilaterals in a hierarchy based on properties.

Topic A Quiz Sample Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Topic A Observational Assessment Recording Sheet . . . . . . . . . . . . . 191

Topic B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Areas of Rectangular Figures with Fraction and Decimal Side Lengths Lesson 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Find areas of square tiles with fraction side lengths by relating the tile to a unit square.

Lesson 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Organize, count, and represent a collection of square tiles.

Lesson 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 Find the area of a rectangle with fraction side lengths by relating the rectangle to a unit square.

Lesson 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Find areas of rectangles with fraction side lengths by using multiplication.

Lesson 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Multiply mixed numbers.

Art

Context Video

Digital Interactive

Math Past © Great Minds PBC

EUREKA MATH2 Florida 5 ▸ M5

Lesson 14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

Lesson 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

Solve real-world problems involving areas of composite figures with mixed-number and decimal side lengths.

Find the volumes of right rectangular prisms by multiplying the edge lengths.

Lesson 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 Solve word problems involving multiplication of mixed numbers.

Topic B Quiz Sample Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Topic B Observational Assessment Recording Sheet . . . . . . . . . . . . . . 335

Topic C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Volume Concepts Lesson 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 Identify attributes and properties of right rectangular prisms.

Lesson 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 Find the volume of right rectangular prisms by packing with unit cubes and counting.

Lesson 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

Lesson 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 Solve word problems involving volumes of right rectangular prisms.

Lesson 25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 Find the volumes of solid figures composed of right rectangular prisms.

Lesson 26. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 Solve word problems involving perimeters of rectangles.

Lesson 27. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 Solve word problems involving perimeter, area, and volume.

Topic D Quiz Sample Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 Topic D Observational Assessment Recording Sheet. . . . . . . . . . . . . . 581

Resources

Find the volume of right rectangular prisms by packing with improvised units.

Spiral Review Sample Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582

Lesson 19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

Standards. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584

Compose and decompose right rectangular prisms to find their volume by using layers.

Assessment to Standards Alignment. . . . . . . . . . . . . . . . . . . . . . . . . . . 588

Lesson 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 Interpret volume as filling.

Lesson 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 Relate volumes of solids and liquid volume.

Topic C Quiz Sample Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

Achievement Descriptors: Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . 589 Achievement Descriptors: Proficiency Indicators. . . . . . . . . . . . . . . . 592 Module Assessment Sample Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 605 Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 Math Past. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

Topic C Observational Assessment Recording Sheet. . . . . . . . . . . . . . 459

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616

Topic D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 Volume and the Operations of Multiplication and Addition

Works Cited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

Lesson 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620

Find the volumes of right rectangular prisms by using the area of the base.

Credits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619

EUREKA MATH2 Florida

5 ▸ M5

FL.5.Mod5.AD5: Classify quadrilaterals into categories based on

attributes and explain the classification.

MA.5.GR.1.1

ADs and their proficiency indicators support teachers with interpreting student work on

FL.5.Mod5.AD6: Classify three-dimensional figures into categories

• informal classroom observations,

FL.5.Mod5.AD7: Find perimeters of rectangles with fraction or

based on attributes.

• data from other lesson-embedded formative assessments,

MA.5.GR.2.1

FL.5.Mod5.AD8: Find and model the area of rectangles with

• Exit Tickets,

fraction side lengths by tiling.

• Topic Quizzes, and

MA.5.GR.2.1

FL.5.Mod5.AD9: Find areas of rectangles with fraction or decimal

• Module Assessments.

side lengths.

This module contains the 13 ADs listed. FL.5.Mod5.AD1: Multiply mixed numbers and whole numbers,

fractions, or mixed numbers.

decimal side lengths.

MA.5.GR.1.2

MA.5.FR.2.2

MA.5.GR.2.1

FL.5.Mod5.AD10: Recognize that volume can be measured by

using unit cubes and that a solid packed without gaps or overlaps by n unit cubes has a volume of n cubic units. MA.5.GR.3.1

FL.5.Mod5.AD2: Model products of fractions and mixed numbers by

FL.5.Mod5.AD11: Measure volumes by counting unit cubes.

using rectangular areas.

MA.5.FR.2.2

MA.5.GR.3.1

FL.5.Mod5.AD3: Solve real-world problems involving multiplication

FL.5.Mod5.AD12: Find volumes of right rectangular prisms with

of mixed numbers.

whole-number edge lengths by using a formula.

MA.5.AR.1.2

MA.5.GR.3.2

FL.5.Mod5.AD4: Classify triangles into categories based on

FL.5.Mod5.AD13: Solve real-world problems involving volumes of

attributes and explain the classification.

right rectangular prisms.

MA.5.GR.1.1

MA.5.GR.3.3

EUREKA MATH2 Florida 5 ▸ M5

Before This Module

Overview

Grade 3 Module 2

Addition and Multiplication with Area and Volume

Students measure liquid volume in metric units and relate metric units and place value relationships.

Grades 3 and 4 Students identify attributes of a variety of shapes, including the number and length of sides, the number and type of angles, and the presence of pairs of parallel sides. They also draw lines of symmetry in two-dimensional figures. In grade 3, students recognize area as an attribute of two-dimensional shapes and find the area of rectangles with whole-number side lengths by tiling and, eventually, by multiplying side lengths.

Grade 5 Module 3 Students use an area model and number lines to multiply fractions by fractions.

Topic A Drawing, Analysis, and Classification of Two-Dimensional and Three-Dimensional Figures Students identify defining attributes and Quadrilaterals classify three-dimensional figures including right rectangular prisms, right triangular Trapezoids prisms, right pyramids, cylinders, cones, and spheres. They sort and classify triangles by using side lengths and types of angles. Parallelograms Students then construct, analyze, and classify trapezoids, kites, parallelograms, rectangles, rhombuses, and squares. They identify Rectangles Rhombuses properties of quadrilaterals that involve pairs of parallel sides, side lengths, diagonals, and lines of symmetry and use these properties Squares to create a hierarchy of quadrilaterals. They use the hierarchy to determine the most specific name of any quadrilateral and all names for the quadrilateral.

Kites

Topic B Areas of Rectangular Figures with Fraction and Decimal Side Lengths Students find areas of rectangles with fraction side lengths, first by tiling with tiles that are squares with unit-fraction side lengths and then by tiling with rectangles with fraction side lengths. Through reasoning, students determine that the area of any rectangle,

EUREKA MATH2 Florida

5 ▸ M5

including one with fraction side lengths, can be found by multiplying the rectangle’s length by its width. Students use an area model to multiply mixed numbers, and then they solve real-world and mathematical problems involving multiplication of mixed numbers and decimals.

Topic C Volume Concepts In their first formal study of volume, students count the number of unit cubes that pack right rectangular prisms. Then they build right rectangular prisms with improvised units to find volume by using something other than unit cubes. They compose and decompose right rectangular prisms into layers in different ways, finding the volume of each layer and multiplying the number of layers by the volume of each layer to find the volume of the right rectangular prism. Students explore conceptual ideas about volume and capacity by differentiating between packing with cubes and filling with a liquid.

After This Module Grade 6 Students draw polygons in all four quadrants of the coordinate plane and use the coordinates of the vertices to solve real-world and mathematical problems, such as finding area and perimeter. They also find the volume of right rectangular prisms with fraction edge lengths by packing them with cubes with unit fraction edge lengths and by applying the formulas for volume.

Topic D Volume and the Operations of Multiplication and Addition Students synthesize the work of topic C by determining that the volume of any right rectangular prism is calculated either by multiplying the area of the base by the height, V = B × h, or by multiplying the three dimensions of the prism, V = l × w × h. They use these two formulas to find volumes and unknown dimensions of right rectangular prisms in both mathematical and real-world problems. Students find the volume of a figure composed of right rectangular prisms by decomposing the figure into right rectangular prisms, finding the volume of each prism, and adding the volumes together. They solve real-world problems involving perimeter, area, and volume.

EUREKA MATH2 Florida 5 ▸ M5

Why Addition and Multiplication with Area and Volume Topic A includes significantly more digital interactives than any other topic in grade 5. Why? The use of digital interactives provides pictorial representation to support the concrete-pictorial-abstract framework and deepen student understanding of quadrilaterals. Each study of a new quadrilateral in topic A introduces students to a concrete representation, such as constructing the quadrilateral by using paper, pencils, right-angle tools, and rulers. A digital interactive provides a pictorial representation that shows various figures that look different than, but are called the same name as, the quadrilateral students constructed. Digital interactives are an ideal way to make comparisons among two-dimensional figures because the measures of angles, sides, and diagonals visually shift as the figure is manipulated. Then students engage with abstract representation in the hierarchy by generalizing properties of the quadrilateral and recording those properties in the correct categories. The repeated sequence for each quadrilateral provides a structure for comparing and manipulating quadrilaterals, inviting students to generalize about the properties in each new digital interactive, which means that topic A has more digital interactives than do other grade 5 topics.

118°

62°

118°

Topic A is about properties of two-dimensional and three-dimensional figures, and topics C and D are about volume of three-dimensional figures. Why is topic B, area of rectangles with fraction and decimal side lengths, included in this module? This module is a study of the attributes and properties of two- and three-dimensional figures. Topic A begins the module with a study of properties and classification of two- and three-dimensional figures. The properties of two-dimensional figures include sides, diagonals, and symmetry. This study of properties and attributes connects to the focus of topic B, area of rectangles with fraction and decimal side lengths, because area is an attribute of two-dimensional figures. Topics C and D extend topic A by focusing on volume, an attribute of three-dimensional figures. © Great Minds PBC

1 unit 6

3 units 5 15

EUREKA MATH2 Florida

5 ▸ M5

Why do students learn to multiply fractions again in module 5 when they already learned it in module 3?

Module 3 Topic B What is 4 × 2 ? 5

A subtle distinction must be made between using an area model to multiply

fractions and finding the area of a rectangle. In module 3, students use an area model to find the product of fractions before learning that for

whole numbers a, b, c, and d, _ a× _ c = __ ac , given that b and d are nonzero, b d bd but students are not asked to find the area of a rectangle with fraction side lengths until module 5 topic B. Even then, they cannot immediately apply

4 5

the familiar formula A = l × w because they have only learned that the formula applies to whole numbers l and w. Thus, students must develop the 2 3

conceptual understanding of what it means to find the area of a rectangle with fraction side lengths. To do this, topic B has students replicate their work with whole-number side lengths from grades 3 and 4: they tile a region with fraction unit squares and count the tiles to see that the area of the region is equal to the number of tiles it takes to cover the region.

Module 5 Topic B

What is the area of the rectangle shown?

Students can then conclude that the area formula can be applied even 2 units 3

when l and w are fractions or decimals.

1 unit 4

Area of 1 rectangle tile: 1 square unit 12 2 units 3

Area of shaded rectangle: 1

1 unit 4

2 x 12 square unit =

2 square units 12

EUREKA MATH2 Florida 5 ▸ M5

Mathematical Language Routines Addition and Multiplication with Area and Volume Routine

Description

Lessons

Always Sometimes Never

This language routine promotes sense-making and justification as students support a claim with examples and nonexamples.

Topic A Lesson 3

Co-construction

This language routine provides structured support for contextualizing and decontextualizing problems, which helps students build abstract reasoning.

Topic B Lesson 15

Critique a Flawed Response

This language routine promotes effective communication techniques for critiquing others’ work, correcting errors, and clarifying meaning.

Topic A Lesson 8

Five Framing Questions

This language routine supports students in analyzing a work sample or solution strategy by providing them with a sequence of questions to frame their thinking.

Topic B Lesson 11

Math Chat

This language routine develops number sense, flexibility, efficiency, and accuracy as students solve a math task and discuss their different solution strategies.

Topic B Lesson 11

Topic A Lesson 4

Topic B Lesson 12

Topic C Lesson 17 Topic D Lesson 23

Numbered Heads

This language routine helps student groups collaborate effectively by encouraging students to reach agreements and holding each student accountable for learning the material.

Topic B Lesson 9

Share, Compare, and Connect

This language routine fosters students’ meta-awareness as they identify, compare, and contrast different mathematical strategies, representations, concepts, and examples.

Topic B Lesson 10

This language routine supports students in making and justifying arguments as well as critiquing the reasoning of others.

Topic A Lesson 3

Take a Stand

Topic B Lesson 14

Topic A Lesson 4 Topic C Lesson 16

Which One Doesn’t Belong?

Think–Pair–Share

This language routine promotes metacognition, reasoning, and mathematical discourse as students use precise language to compare different representations.

Topic A Lesson 5

This language routine supports mathematical discourse and maximizes linguistic output as students mentally rehearse their ideas and discuss with a partner before sharing with the class.

Most lessons in this module contain the Think–Pair–Share routine.

Topic C Lesson 19

5 ▸ M5 ▸ TA

EUREKA MATH2 Florida

Topic A Drawing, Analysis, and Classification of Two-Dimensional and Three-Dimensional Figures In topic A, students identify and classify three-dimensional figures including right rectangular prisms, right triangular prisms, right pyramids, cylinders, cones, and spheres. They construct, analyze, and classify quadrilaterals and identify properties of quadrilaterals that involve angle measures, side lengths, diagonals, pairs of parallel sides, and lines of symmetry. Students then use these properties to create a hierarchy of quadrilaterals. Prior to grade 5, students identify attributes of a variety of shapes. These attributes include the number and length of sides, the number and type of angles, and the presence of pairs of parallel sides. Students identify side lengths, measures of angles, and perpendicular and parallel sides in two-dimensional figures. They also draw lines of symmetry in two-dimensional figures. Topic A introduces students to hierarchies and invites them to interpret how a hierarchy can be useful for identifying properties of objects. Students learn more about classification, understanding that all items in a hierarchy share the properties of the items above them. Students begin to create a hierarchy of quadrilaterals that they continue to develop throughout the topic. Students also classify triangles by their attributes. Students identify the types of angles present and then classify the triangles as acute, obtuse, or right. They also compare side lengths in a triangle to identify the triangles as equilateral, isosceles, or scalene. Over the course of several lessons, students explore properties of trapezoids, parallelograms, rhombuses, rectangles, kites, and squares. They use straightedges and right-angle tools to construct figures. Then they use protractors and rulers to make conclusions about relationships between side lengths, diagonals, and angle measures in their constructions. Students fold figures to make conclusions about lines of symmetry. Through these constructions, digital interactives, class discussions, and activities, students identify properties of each type of quadrilateral. They use the properties to build 18

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA

a hierarchy of quadrilaterals to understand the relationships between the different types of quadrilaterals. Finally, students use the hierarchy of quadrilaterals to complete a variety of tasks. They use properties of quadrilaterals to draw several quadrilaterals that meet a given description. They identify all names of a given quadrilateral and explain why a quadrilateral is generally called by its most specific name. For example, students name a rectangle as a parallelogram, trapezoid, and quadrilateral, but recognize that we call it a rectangle to communicate as much information as possible about the figure. In module 6, students analyze and generate numerical patterns, explore the coordinate plane, and solve problems by collecting and representing decimal data on a line graph and a line plot.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA

Progression of Lessons Lesson 1

Lesson 2

Lesson 3

Analyze hierarchies and identify properties of quadrilaterals.

Identify three-dimensional figures and classify the figures into categories based on defining attributes.

Classify triangles into different categories based on attributes. Z

Figures Two-Dimensional Polygons

Three-Dimensional

Group 1

Group 2

Group 3

Non-Polygons

Triangles Quadrilaterals Pentagons Hexagons

I can interpret and create hierarchies. I know that quadrilaterals are two-dimensional figures and all quadrilaterals are polygons.

Identifying the numbers of edges, vertices, and faces on a three-dimensional figure helps me classify the figure as a right prism, right pyramid, cone, cylinder, or sphere. I can identify the number and shape of faces and bases of a three-dimensional figure. I can use these attributes to classify three-dimensional figures.

I can classify a triangle as an acute, obtuse, or right triangle based on the measures of its angles. Identifying the number of equal side lengths helps me classify a triangle as equilateral, isosceles, or scalene.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA

Lesson 4

Lesson 5

Lesson 6

Classify trapezoids and parallelograms based on their properties.

Classify rectangles and rhombuses based on their properties.

Classify kites and squares based on their properties.

B A

All trapezoids are quadrilaterals. Trapezoids have at least 1 pair of parallel sides. All parallelograms are trapezoids, but not all trapezoids are parallelograms. Parallelograms have 2 pairs of parallel sides, opposite sides with the same length, and diagonals that intersect at their midpoints.

All rectangles are parallelograms and trapezoids, but not all parallelograms and trapezoids are rectangles. All rhombuses are parallelograms and trapezoids, but not all parallelograms and trapezoids are rhombuses. The diagonals of a rectangle are always the same length, but the diagonals of a rhombus are not always the same length. Both rhombuses and rectangles have at least 2 lines of symmetry.

Kites are quadrilaterals with at least 2 pairs of adjacent sides that are the same length. All rhombuses are kites, but not all rectangles are kites. A square is a rectangle, a rhombus, and a kite.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA

Lesson 7

Lesson 8

Identify quadrilaterals from given properties.

Classify quadrilaterals in a hierarchy based on properties.

Description

Quadrilaterals

• 2 pairs of opposite sides that are parallel

Trapezoids

Parallelograms

E I

I can use properties of quadrilaterals to draw and identify them. Some quadrilaterals have properties in common, so sometimes I can draw more than one quadrilateral from the same description.

Rhombuses

Kites

J C

Rectangles G

• 2 pairs of opposite sides that are the same length

Squares B

I can use different types of diagrams to classify quadrilaterals based on their properties. I know that many quadrilaterals have more than one name.

LESSON 1

Analyze hierarchies and identify properties of quadrilaterals. I can analyze hierarchies and identify properties of quadrilaterals.

Lesson at a Glance Students learn how to interpret a hierarchy. They understand that an object can have multiple names, and they understand that the more information they have about an object, the more specifically they can name it. After sorting geometric figures and identifying properties of quadrilaterals, students construct a hierarchy of geometric figures. This lesson formalizes the terms plane and property.

Key Questions • When is a characteristic of a figure a property?

Mathematical Thinking and Reasoning Standard

Language Objectives • Orally and in writing, justify why a statement about a polygon or category of polygons is true or false by using a hierarchy. • In partners, analyze figures and orally classify them based on their properties. • Orally and in writing, name properties of quadrilaterals.

MTR.5.1

• Why do we use hierarchies? • What are properties of quadrilaterals?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 1

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Consider whether to remove Collection of Figures from the student books and cut out one set of figures for each student pair in advance or to have students prepare them during the lesson.

Launch 5 min Learn 35 min • Explore a Hierarchy • Sort Figures • Create a Hierarchy • Problem Set

Land 10 min

• Computer or device* • Projection device* • Teach book*

Students • Collection of Figures (1 per student pair, in the student book) • Scissors • Dry-erase marker* • Learn book* • Pencil* • Personal whiteboard* • Personal whiteboard eraser* * These materials are only listed in lesson 1. Ready these materials for every lesson in this module.

Achievement Descriptors

Florida B.E.S.T. Standards

FL.5.Mod5.AD4 Classify triangles into

MA.5.GR.1.1 Classify triangles or quadrilaterals

categories based on attributes and explain the classification. (MA.5.GR.1.1) FL.5.Mod5.AD5 Classify quadrilaterals into

categories based on attributes and explain the classification. (MA.5.GR.1.1) FL.5.Mod5.AD6 Classify three-dimensional

figures into categories based on attributes. (MA.5.GR.1.2)

into different categories based on shared defining attributes. Explain why a triangle or quadrilateral would or would not belong to a category. MA.5.GR.1.2 Identify and classify

three-dimensional figures into categories based on their defining attributes. Figures are limited to right pyramids, right prisms, right circular cylinders, right circular cones and spheres.

5 ▸ M5 ▸ TA ▸ Lesson 1

Fluency

EUREKA MATH2 Florida

Counting on the Protractor Students count by a unit of 20° on a 180° protractor to prepare for classifying quadrilaterals based on their properties. Display the 180° protractor with a ray pointing to 0° on the outer scale. Look at the outer scale of the protractor. Which tick mark does the ray cross through? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

0° Use the angle on the protractor to count forward by 20°. The first measurement you say is 0°. Ready? Display the angle measure increasing in 20° intervals to 180°.

0°, 20°, … , 160°, 180° Display the picture of the 180° protractor with a ray pointing to 0° on the inner scale. Now look at the inner scale of the protractor. Which tick mark does the ray cross through? Raise your hand when you know.

0° Use the angle on the protractor to count forward by 20°. The first measurement you say is 0°. Ready? Display the angle measure increasing in 20° intervals to 180°.

0°, 20°, … , 160°, 180°

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 1

Choral Response: Classify and Measure Angles Students classify an angle, estimate the angle measure, and determine the angle measure by using a 180° protractor to prepare for classifying quadrilaterals based on their properties. After asking each question, wait until most students raise their hands, and then signal for students to respond. Raise your hand when you know the answer to each question. Wait for my signal to say the answer. Display the acute angle. How would you classify the angle? Acute Display the answer. Estimate the angle measure. Whisper your estimate to your partner.

Acute 50°

Provide time for students to think and share with their partner.

45°. It looks like it’s halfway between 0 and 90. 60°. It looks like it’s more than halfway to 90. Display the protractor. What is the angle measure?

50° Display the angle measure.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 1

Repeat the process with the following sequence:

Obtuse 130°

Right 90°

Obtuse 145°

Acute 35°

Obtuse 135°

Obtuse 157°

Choral Response: Attributes of Polygons Students identify polygons with a specified attribute to prepare for classifying quadrilaterals based on their properties. After asking each question, wait until most students raise their hands, and then signal for students to respond. Raise your hand when you know the answer to each question. Wait for my signal to say the answer. Display the attributes: 3 sides, 3 vertices, and 3 angles. What is the name of a polygon with 3 sides, 3 vertices, and 3 angles? Triangle 28

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 1

Display the answer and then display the triangles. On my signal, say the letter or letters to answer each question.

Attributes: 3 sides, 3 vertices, and 3 angles triangle

Which triangles have at least 2 sides of equal length?

3 in

8 cm

3 in

10 cm

4 cm 3 cm C 5 cm

6 cm

Display triangle A circled. Which triangles have at least 1 right angle?

B and C Display triangles B and C circled. Repeat the process with the following sequence:

Attributes: 4 sides, 4 vertices, and 4 angles quadrilateral

2 cm

6 cm D 6 cm

2 cm

5 cm 3 cm

4 cm E

10 cm

Attributes: 6 sides, 6 vertices, and 6 angles hexagon 2 in

2 in F

2 in 2 in

2 in

G 2 in

2 in 2 in

1 in 3 in 1 in

1 in H

3 in 1 in

3 cm 5 cm

4 cm K

4 cm

5 cm

3 cm

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 1

Launch

Students explore how to use and interpret a hierarchy.

Multilingual Learner Support

Display the diagram of musical instruments.

Consider using strategic, flexible grouping throughout the module.

Musical Instruments

Strings

Winds

• Pair students who have different levels of mathematical proficiency.

Percussion

• Pair students who have different levels of English language proficiency. • Join pairs to form small groups of four.

Strummed

Bowed

Brass

Woodwinds

Guitar

Violin

Trumpet

Clarinet

Drums

As applicable, complement any of these groupings by pairing students who speak the same native language.

Ask students to Think–Pair–Share about what the diagram represents. The diagram shows different types of instruments put into groups. The diagram shows categories of instruments with examples from each category. This diagram represents types of musical instruments. Use the diagram to determine whether the following statements are true or false. Display and read the following statements. Check for student understanding by having students indicate, such as with a thumbs-up or thumbs-down, whether each statement is true or false. If needed, pause to discuss the correct answer with students.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 1

• All musical instruments are string instruments. False. • All percussion instruments are musical instruments. True. • All brass instruments are winds. True. • All wind instruments are brass. False. How do you know from the diagram that not all wind instruments are brass? Wind instruments can also be woodwinds. This diagram represents a hierarchy. A hierarchy places objects into groups based on their characteristics. In a hierarchy, objects may be placed above or below other objects. A trumpet is a type of brass instrument, so the trumpet is listed under brass instruments in the hierarchy. We could call a trumpet a brass instrument, but we call it a trumpet because trumpet is a more specific name. Have students turn and talk about how a hierarchy is helpful for understanding relationships among objects. Transition to the next segment by framing the work. Today, we will interpret and construct hierarchies.

5 ▸ M5 ▸ TA ▸ Lesson 1

Learn

EUREKA MATH2 Florida

Explore a Hierarchy Students explain and interpret the structure of hierarchies. Continue to display the hierarchy of instruments. Point to the word guitar. As you provide the following explanation, gesture to the relevant sections of the hierarchy. According to this hierarchy, the most specific name for a guitar is guitar. But we could also call it a strummed string instrument, a string instrument, or just a musical instrument. According to the hierarchy, what are other names for a clarinet? A woodwind A wind instrument A musical instrument A property of musical instruments is that they make musical sounds. Every object listed below musical instrument in the hierarchy must therefore make musical sounds. String instruments are made of wood. Is a violin made of wood? How do you know? Yes. The violin is made of wood because the violin is a string instrument, and all string instruments are made of wood. If every string instrument is made of wood, then a property of all string instruments is that they are made of wood. A property is something that is true of everything in a category.

Multilingual Learner Support To support understanding of the term property, invite students to share properties of objects with which they are familiar. • A property of water is that it can dissolve many substances. • A property of any object is that it has a size. • A property of all numbers is that they can be added in any order. • A property of rocks is that they are nonliving. As the topic progresses, students build on their understanding of properties of quadrilaterals.

Because being made of wood is a property of string instruments, every string instrument is made of wood, and every instrument in the categories below string instruments is also made of wood. Is every musical instrument made of wood? How do you know? It’s possible that every musical instrument is made of wood, but we don’t know that. We only know that every string instrument is made of wood, and there are musical instruments that are not strings. 32

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 1

Sasha’s violin has a star painted on it. Do you think all violins have stars painted on them? Why? Probably not. It seems unlikely that most people would paint stars on their instruments. Is having a star painted on it a property of violins? How do you know? No. In order for a characteristic to be a property, it must be true about all violins, and we only know that Sasha’s violin has a star on it. Having a star painted on it is not a property of violins. Having a star painted on the violin is an attribute of a particular violin. Having a star painted on it is sometimes, but not always, a characteristic of a violin. Differentiation: Challenge

Invite students to turn and talk about another attribute that a musical instrument might have.

Consider asking students who need additional challenge to create a hierarchy of triangles.

Does this hierarchy tell you everything about musical instruments? Why? No. There are electronic instruments that are not in the hierarchy. No. There are several instruments that are not in the hierarchy. How could this hierarchy be helpful in learning about or choosing musical instruments? If you know you want to play an instrument that doesn’t involve blowing into the instrument, you might choose to play a string instrument or a percussion instrument. If you know you want to play an instrument that is bowed, you could find the section of bowed instruments in the hierarchy. We build hierarchies in mathematics because we have precise definitions and can make statements that are true. For example, what is true about all triangles?

All triangles have 3 sides. All triangles have 3 sides, so a property of triangles is having 3 sides. Display right triangle ABC.

A © Great Minds PBC

C 33

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 1

Invite students to Think–Pair–Share about attributes and properties of the right triangle. Ask them to recall that properties are true of all right triangles and that attributes are characteristics of a particular right triangle. Triangle ABC has a right angle. Having a right angle is a property of right triangles because all right triangles have a right angle. An attribute of triangle ABC is that it is taller than it is wide. This is not a property because it is not true of all triangles. An attribute is that it has vertices named A, B, and C.

Sort Figures Materials—S: Collection of Figures, scissors

Multilingual Learner Support Students are familiar with the names of polygons and solids taught in previous grades (i.e., triangle, quadrilateral, pentagon, trapezoid, parallelogram, rhombus, rectangle, square, cube, sphere, cylinder, cone, rectangular prism, right pyramid). Consider asking what students know about these figures to activate prior knowledge. List the names of familiar figures and discuss the definition of each while displaying an example.

Students sort a collection of figures. Pair students. Invite each pair to remove one Collection of Figures from their books. Direct students to work with their partner to cut out the figures. As you move through the following prompts, allow time for students to sort.

Students are familiar with the symbols used to notate parallel sides from grade 3. Before they sort the figures on Collection of Figures, consider displaying just figure P and brainstorming what the symbols used to notate parallel sides mean.

Universal Design for Learning: Representation

S T

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 1

Sort your figures into two-dimensional figures and three-dimensional figures. How do you know whether a figure is a two-dimensional figure or whether it represents a three-dimensional figure? A two-dimensional figure is flat, and a three-dimensional figure has dashed lines to show the edges that make it a solid. Two-dimensional figures lie in a plane. A plane is a flat, two-dimensional surface that extends forever. The surface of a piece of paper lies in a plane because it is flat. A pencil does not lie in a plane because it is not flat. A pencil is an example of a three-dimensional figure, or a solid. Sort the two-dimensional figures into polygons and non-polygons. How do you know if a figure is a polygon? A polygon is a closed figure. A polygon has 3 or more straight sides.

Multilingual Learner Support The term plane is introduced in this segment. Support students in developing an understanding of the use of the term plane in this module and how it is different from the word plain and other meanings of the word plane. • A piece of paper is like a plane except that it has thickness and does not extend forever in all directions. • Students are likely to think of planes as airplanes. • Something that is plain might be considered ordinary.

Sort the polygons into triangles, quadrilaterals, pentagons, and hexagons. As we sort, what happens to the number of figures we are sorting? The number of figures gets smaller and smaller. We started sorting 20 figures and put aside the 7 that are three-dimensional figures. Then we sorted the 13 two-dimensional figures and put aside the 2 that are not polygons. Then we sorted the 11 polygons into 1 triangle, 7 quadrilaterals, 2 pentagons, and 1 hexagon. Hold up figure N as you make the following statement.

Universal Design for Learning: Representation The object shown in this picture has 4 sides, but it is not a quadrilateral because it is not in one plane.

We sorted our figures into different piles, but some figures fit in more than one group. For example, figure N is in the polygons group and the quadrilaterals group. Invite students to turn and talk about another figure that fits in more than one group.

Create a Hierarchy Students classify figures into a hierarchy. Let’s use our figure sort to create a hierarchy of figures. Write the word Figures to begin the hierarchy.

Consider making a similar object out of toothpicks and clay or similar materials to share this non-example with students.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 1

Invite students to share how they separated the figures, adding to the hierarchy as they explain their sorting. As you create the hierarchy, refer to the cut-out figures from earlier in the lesson or consider taping on figures from each set as examples.

Figures Two-Dimensional Polygons

Three-Dimensional

Non-Polygons

Triangles Quadrilaterals Pentagons Hexagons This hierarchy starts at the top with the general category of figures. Each level underneath shows a more specific category. According to the hierarchy, what are some examples of special types of polygons? Triangles, quadrilaterals, pentagons, and hexagons Every category in the hierarchy shares the names and properties of the types above it. For example, a quadrilateral is a polygon and a two-dimensional figure. Read the following statements. Have students indicate, such as with a thumbs-up or thumbs-down, whether each statement is true or false. After they do, use the hierarchy to explain why. For example, all pentagons must be two-dimensional figures because pentagons are below two-dimensional figures in the hierarchy. All pentagons are polygons, and all polygons are two-dimensional figures, so all pentagons must be two-dimensional figures. • All pentagons are two-dimensional figures. True. • All quadrilaterals are polygons. True. 36

Teacher Note The hierarchy is not intended to show a complete list of all types of polygons. As necessary, ask students to recall that there are other types of polygons not represented here (e.g., octagons and heptagons). In this topic, students focus on two types of polygons: triangles and quadrilaterals.

Promoting Mathematical Thinking and Reasoning When students classify figures into a hierarchy based on the properties of each figure, they are using patterns and structure (MTR.5). Ask the following questions to promote MTR.5: • How are triangles and quadrilaterals related? How can that help you classify them? • What is the same about polygons and non-polygons? • How does what you know about pentagons help you place them in the hierarchy?

Teacher Note In this lesson, students identify three-dimensional figures through the sorting activity. In the next lesson, students identify and classify three-dimensional figures based on defining attributes and add to the three-dimensional side of the hierarchy from this lesson.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 1

• All polygons are quadrilaterals. False. Transition to the next segment by framing the work.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Analyze hierarchies and identify properties of quadrilaterals. Gather the class with their Problem Sets. Facilitate a class discussion about hierarchies and quadrilaterals by using the following prompts. Encourage students to restate or add on to their classmates’ responses. When is a characteristic of a figure a property? Use an example from the Problem Set. A characteristic is a property if it is always true for that type of figure. For example, in problem 2, a property of any rectangle is that it has 4 sides. Why do we use hierarchies? Hierarchies help us classify things into groups. Hierarchies help us see how groups of objects are related. How did the hierarchy of instruments show us more details about a specific instrument, such as a guitar? The hierarchy shows us that the guitar shares properties of the instruments above it, such as strummed instruments and string instruments.

5 ▸ M5 ▸ TA ▸ Lesson 1

EUREKA MATH2 Florida

What are properties of quadrilaterals? Quadrilaterals are polygons with 4 sides.

Exit Ticket 5 min Direct students to the Self-Reflection on the back of the Exit Ticket. Have them read the I can statement to reflect on their learning in this lesson. Invite them to circle the choice that best matches their agreement with the statement. After they complete the Exit Ticket, consider having students confirm whether they still agree with their selection. Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 1

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can analyze hierarchies and identify properties of quadrilaterals.

PRACTICE

Florida B.E.S.T. MA.5.GR.1.1, MA.5.GR.1.2

3. Circle each name that correctly describes the figure.

L Name

73°

Date

1. Consider the two-dimensional figures shown to complete parts (a)–(c).

75°

122° J Two-dimensional figure

Three-dimensional figure

Quadrilateral

K Polygon

Triangle Non-polygon

a. Circle each two-dimensional figure that is a polygon. b. Use blue to color each polygon that is a triangle. c. Use red to color each polygon that is a quadrilateral. 2. Complete the Venn diagram by writing properties of the figures in the correct section. Triangles

3 angles 3 sides

Rectangles

Polygon Two-dimensional figure

4 sides 2 pairs of parallel sides 4 right angles

PROBLEM SET

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 1

4. Leo made a hierarchy to classify some living things.

5. Sasha says that all quadrilaterals are polygons. Ryan says that all polygons are quadrilaterals. Who is correct? Explain.

Living Things

Plants

Trees

Pines

Mosses

Sasha is correct. All quadrilaterals are two-dimensional, closed figures, which makes them all polygons. Not all polygons have exactly 4 sides, so not all polygons are quadrilaterals. Animals

Mammals

Maples

Penguins

Birds

Seagulls

a. Circle whether the statements are true or false. All maples are trees.

True

False

All trees are maples.

True

False

All living things are animals.

True

False

All seagulls are animals.

True

False

All plants use the sun to make energy, so all mosses use the sun to make energy.

True

False

All birds lay eggs, so all penguins lay eggs.

True

False

b. All birds have exactly two legs. Does that mean that all animals have exactly two legs? Explain. No. We only know that having two legs is a property of birds. Because animals are above birds in the hierarchy, properties of birds are not necessarily properties of animals.

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 1

EXIT TICKET

I can analyze hierarchies and identify properties of quadrilaterals.

Florida B.E.S.T. MA.5.GR.1.1, MA.5.GR.1.2

Name

Date

1. Consider the hierarchy shown.

Figures

a. Are all two-dimensional figures non-polygons? No. b. Are all polygons two-dimensional figures? Yes. c. All polygons have at least 3 sides. Does that mean that all two-dimensional figures have at least 3 sides? Explain.

TwoDimensional

Polygons

ThreeDimensional

Non-Polygons

No. We only know that having at least 3 sides is a property of polygons. Two-dimensional figures are above polygons in the hierarchy, so we do not know whether having at least 3 sides is a property of two-dimensional figures.

2. Name two properties of quadrilaterals. All quadrilaterals have 4 sides. All quadrilaterals are polygons.

LESSON 2

Identify three-dimensional figures and classify the figures into categories based on defining attributes. I can use attributes of three-dimensional figures to classify them.

Lesson at a Glance Students explore the defining attributes of right prisms, right pyramids, cones, cylinders, and spheres through interactive and pictorial representations. They determine the numbers of edges, vertices, and faces; identify the shape and numbers of faces and bases of the solids; and then compare the defining attributes of all the solids. Students classify the solids based on defining attributes by using diagrams. This lesson formalizes the terms cone, cylinder, face, right prism, and right pyramid.

Key Questions • What information do you need to classify three-dimensional figures? • In what ways do the defining attributes of right prisms, right pyramids, cones, cylinders, and spheres compare to one another?

Mathematical Thinking and Reasoning Standard MTR.4.1

Language Objectives • Orally compare attributes of right rectangular prisms, right triangular prisms, and right pyramids, and identify them by name both orally and in writing. • Orally compare attributes of cylinders, cones, and spheres, and identify them by name both orally and in writing. • Read prompts in which students make statements about three-dimensional figures. With a partner and as a class, orally assess the accuracy of these statements. • Orally discuss how to add three-dimensional figures to the figures hierarchy, and classify these figures using the hierarchy.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 2

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 5 min Learn 35 min • Right Prisms and Right Pyramids • Cylinders, Cones, and Spheres • Classify Three-Dimensional Figures • Problem Set

• None

Students • None

Land 10 min

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD6 Classify three-dimensional

MA.5.GR.1.2 Identify and classify

figures into categories based on attributes. (MA.5.GR.1.2)

three-dimensional figures into categories based on their defining attributes. Figures are limited to right pyramids, right prisms, right circular cylinders, right circular cones and spheres.

5 ▸ M5 ▸ TA ▸ Lesson 2

Fluency

EUREKA MATH2 Florida

Counting on the Protractor Students count by a unit of 30° on a 180° protractor to develop fluency with classifying quadrilaterals based on their properties. Display the 180° protractor with a ray pointing to 0° on the outer scale. Look at the outer scale of the protractor. Which tick mark does the ray cross through? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

0° Use the angle on the protractor to count forward by 30°. The first measurement you say is 0°. Ready? Display the angle measure increasing in 30° intervals to 180°.

0°, 30°, … , 150°, 180° Display the image of the 180° protractor with a ray pointing to 0° on the inner scale. Now look at the inner scale of the protractor. Which tick mark does the ray cross through? Raise your hand when you know.

0° Use the angle on the protractor to count forward by 30°. The first measurement you say is 0°. Ready? Display the angle measure increasing in 30° intervals to 180°.

0°, 30°, … , 150°, 180°

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 2

Choral Response: Classify and Measure Angles Students classify an angle, estimate the angle measure, and determine the angle measure by using a 180° protractor to develop fluency with classifying quadrilaterals based on their properties. After asking each question, wait until most students raise their hands, and then signal for students to respond. Raise your hand when you know the answer to each question. Wait for my signal to say the answer. Display the acute angle. How would you classify the angle? Acute

Acute 40°

Display the answer. Estimate the angle measure. Whisper your estimate to your partner. Provide time for students to think and share with their partner.

45°. It looks like it’s close to halfway between 0 and 90. 40°. It looks like it’s less than halfway to 90. Display the protractor. What is the angle measure?

40° Display the angle measure.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 2

Repeat the process with the following sequence:

Right 90°

Obtuse 165°

Acute 75°

Straight 180°

Obtuse 112°

Acute 68°

Choral Response: Properties of Polygons Students identify polygons with a specified property to develop fluency with classifying quadrilaterals based on their properties. After asking each question, wait until most students raise their hands, and then signal for students to respond. Raise your hand when you know the answer to each question. Wait for my signal to say the answer. Display the properties: 4 sides, 4 vertices, and 4 angles. 46

Properties: 4 sides, 4 vertices, and 4 angles quadrilateral 2 14 in

1.5 cm 4.5 cm

7.5 cm

3 12 in

5 cm

6.5 cm

9.5 cm

10.5 cm 4 cm

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 2

What is the name of a polygon with 4 sides, 4 vertices, and 4 angles? Quadrilateral Display the answer and then display the quadrilaterals. On my signal, say the letter or letters to answer each question. Which quadrilaterals have at least 2 sides of equal length?

E Display quadrilateral E circled. Which quadrilaterals have at least 1 right angle?

E and F Display quadrilaterals E and F circled. Which quadrilaterals have at least 1 pair of parallel sides?

D, E, and F Display quadrilaterals D, E, and F circled. Repeat the process with the following sequence: Teacher Note Properties: 6 sides, 6 vertices, and 6 angles

Properties: 5 sides, 5 vertices, and 5 angles

hexagon

pentagon

2 cm 5 cm

5 cm

G 5 cm

5 cm 2 cm

2 2 in 1 in 1 in

H 1

2 2 in

1 in 1 in

J 10.3 m

2.5 m 4.1 m

3.4 m 4.5 m

3.8 m

4.1 m

2 2 in 3 2 4 in

7.3 m 2.5 m

1 in

5 cm

2 34 in 1 in

5 cm

M 5 cm

5 cm

In the case of hexagon G, students may notice that there are three arrow marks on one set of parallel sides. Students are familiar with single or double arrow marks from grade 3. Encourage them to infer what three arrow marks represent. Consider explaining that there are three pairs of parallel sides in this particular figure.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 2

Launch

Students identify similar and different attributes among three-dimensional figures in real-world objects. Display the images of the structures. Teacher Note Structure A

Structure B

Anticipate that students may not yet have the vocabulary to precisely describe the defining attributes of the structures that resemble three-dimensional figures. At this point in the lesson, encourage students to recognize and describe the attributes of the structures in their own words.

Structure C

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 2

What do you notice about the structures? The structures look like three-dimensional figures or a combination of three-dimensional figures. Structure A is a right pyramid. Structure B includes two figures: one that is the white cylinder building and one that is the black cone roof. Structure C looks like a lot of different figures with many curved surfaces put together. What do you wonder? I wonder how the structures were designed and built into these shapes. I wonder why people chose to create the structures in these shapes. Which structures have a similar attribute? What is the attribute?

Teacher Note In this topic, the terms three-dimensional figure and solid are used interchangeably. Allow students to similarly use both terms when defining the attributes and classifying the three-dimensional figures.

The tops of structures A and B meet at 1 point. Structures B and C have curved surfaces. Structures B and C are made up of different three-dimensional figures. Which structures have an attribute that the other structures do not have? What is the attribute? Structure A is the only one that has a square as the bottom. Structure B is the only one that has a circle as the bottom. Structure B is the only one made up of a cylinder and a cone. It looks like structure C has more than 2 curved surfaces. Invite students to Think–Pair–Share about how they could sort the structures into categories of similar attributes and to explain why they chose to sort in that way. I could sort structures A and B into a category because the tops of the structures end in 1 point.

Teacher Note

I could sort structures B and C into a category because they both have curved surfaces.

the point above the base of a right pyramid and a cone as the apex. At this point,

I could sort structure B into its own category because it is the only one that includes a cylinder. I could sort structure C into its own category because it is the only one that has more than 2 curved surfaces. © Great Minds PBC

Later in the lesson, students learn to describe

acknowledge descriptions of the apex that use students’ current understanding about vertices and points.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 2

Transition to the next segment by framing the work. Today, we will identify and classify three-dimensional figures into categories based on their defining attributes.

Learn

Right Prisms and Right Pyramids Students identify and compare attributes of right rectangular prisms, right triangular prisms, and right pyramids. Display the Right Rectangular Prism digital interactive. Turn the prism to show students the prism from various perspectives. Then point to one of the prism’s faces.

Multilingual Learner Support Students learn the terms vertex, edge, and face in earlier grades. The definition of face that is formalized in this lesson includes that the flat part must be a polygon. Consider drawing a picture or creating an anchor chart to review familiar terms from earlier grades and include the new information about the term face.

A face is a polygon that is a flat part of the surface of a solid. How many faces do you see?

vertex

I see 6 faces.

Term

A line segment where 2 faces meet is called an edge. How many edges do you see? I see 12 edges. Point to one of the prism’s vertices. A point where the edges meet is called a vertex. How many vertices do you see? I see 8 vertices. Use the slider in the interactive to move the faces away from the prism. This view helps us see the faces of the solid more clearly. What shape are the faces of the solid?

edge

face

Point to two adjacent faces. Then trace the edge where they meet.

Face Edge

a polygon that is a ﬂat part of the surface of a solid a line segment where two faces of a solid meet

a point where the sides meet Vertex in a two-dimensional shape and a point where the edges meet in a three-dimensional solid

Its faces are rectangles. 50

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 2

This solid is called a right prism. A right prism is a solid with 2 identical polygonal faces, called bases, connected by rectangular faces. What do you notice about the faces of this right prism? I notice the faces opposite each other appear to be identical rectangles. Which faces do you think are the bases of this right prism? Why? I think any of the two faces opposite each other can be the bases because the faces opposite each other are identical polygons and the rest of the faces are rectangular. Direct students to problem 1 in their books. 1. Use the figures shown to complete the table.

By definition, a polygon is the outline of the figure and not the interior. However, when describing a face of a three-dimensional figure, sometimes the face is referred to by the name of the polygon. For example, the face of a solid may be described by saying the faces are triangles, but the faces really includes the interior as well, not just the outline. While this definition restricts faces to polygons, some curricula allow faces to be circular.

Differentiation: Support

Figure

Solid

Number of Edges

Number of Vertices

Number of Faces

Type of Faces

Type of Bases

Right rectangular prism

Rectangles

Right triangular prism

Right pyramid

Triangles

Rectangles Triangles

Rectangle

Throughout this lesson, consider providing manipulatives so students can rotate and study the solids in physical form. Printed versions of the solids from various views can also support students who need more time to analyze the defining attributes.

Triangles Rectangle

Which figures in problem 1 are right prisms? How do you know? Figures A and B are right prisms because they have 2 identical polygonal faces that are connected by rectangular faces.

Teacher Note

Consider directing students’ attention to the color-coding on the interactive showing the three pairs of 2 identical rectangles that are on opposites sides of the prism.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 2

We can name a right prism by the type of base it has. Figure Ais a right rectangular prism. Why do you think we call it a right rectangular prism? The bases are rectangles, so the prism includes rectangular in its name. Rotate the prism in the interactive so different rectangles are the base. With each rotation, invite a student to point to a base. Then complete the figure Arow of the table as a class. Display the Right Triangular Prism digital interactive. Turn the prism to show students the prism from various perspectives.

Universal Design for Learning: Representation As students study the right rectangular prism, right triangular prism, and right pyramid, have them highlight the edges and write a V near the vertices on the prisms and right pyramid.

Is this solid also a right prism? How do you know? Yes. The solid has 2 identical polygonal faces that are connected by rectangular faces.

V V

V V V

V V V B

V V V

V C

How is this right prism different from the one we saw before? The bases are triangles instead of rectangles. Figure B is a right triangular prism. Why do you think we call it a right triangular prism? The bases are triangles, so the prism includes triangular in its name. Can any of the faces be the bases of a right triangular prism? Why? No. The bases must be the triangles because the triangles are the only 2 identical polygonal faces on the right triangular prism connected by rectangular faces. Guide students to count the numbers of edges, vertices, and faces of the right triangular prism by using the digital interactive. Consider having a volunteer point at each attribute as the class counts, and then switch students for the next attribute. Then complete the figure Brow of the table as a class. Invite students to Think–Pair–Share about how the numbers of edges, vertices, and faces differ for the triangular prism and the rectangular prism.

Differentiation: Challenge To challenge students to think further about the properties of right rectangular prisms and their relationship to cubes, pose the following questions: • If all the faces of this right rectangular prism were identical, what would the figure look like? • What makes a cube different from a right rectangular prism? • Is it possible to have a right rectangular prism with all different-size faces? Why?

The right triangular prism has fewer edges, vertices, and faces than the right rectangular prism. The triangular prism has 9 edges, 6 vertices, and 5 faces, while the rectangular prism has 12 edges, 8 vertices, and 6 faces. 52

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 2

Display the Right Pyramid digital interactive. Rotate the right pyramid to show students the solid from various perspectives. Use the interactive to guide students to count the numbers of vertices, edges, and faces. Use the slider in the interactive to move the faces away from the right pyramid. What shapes are the faces of the solid? How many of each shape are there? There are 4 triangular faces and 1 rectangular face. How is this solid different from the two right prisms we saw? There are not 2 identical faces connected by rectangular faces. There are 4 triangular faces that meet at a point. This solid is called a right pyramid. A right pyramid is a solid with a polygonal face, called the base, connected to triangular faces that meet at a vertex, called the apex. In this right pyramid, what shape is the base? How do you know? The base is a rectangle. I know because the base has 4 sides and 4 right angles and the base is connected to the 4 triangular faces. Figure C is a right pyramid. Why is this figure a right pyramid? Figure C is a right pyramid because it has a polygonal base connected to triangular faces that meet at a point. Direct students to work with a partner to complete the figure C row of the table. Then have students compare the right pyramid and right triangular prism by asking the following questions. How are the faces of the right pyramid similar to the faces of the right triangular prism? The faces of both solids include rectangles and triangles. At least 1 of the faces is a rectangle for both solids. Both solids have 5 faces. How are the faces of the right pyramid different from the faces of the right triangular prism?

Multilingual Learner Support Students are informally introduced to right rectangular prisms and right triangular prisms in this lesson as they explore defining attributes of solids. Students may describe the solids as right prisms with rectangular bases and right prisms with triangular bases instead of by using the adjectives rectangular and triangular to name and describe the solids. In topic C, students formally define a right rectangular prism.

The right pyramid has 4 triangular faces. The right triangular prism has 2 triangular faces. © Great Minds PBC

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 2

The triangular faces of the right pyramid meet at an apex. The triangular faces of the right triangular prism do not meet. In the right pyramid, the faces that are not the base are triangles. In the right triangular prism, the faces that are not bases are rectangles. Invite students to turn and talk about which defining attributes make a three-dimensional figure a right pyramid or a right prism.

Differentiation: Challenge Consider challenging students to name right pyramids with bases that are not rectangles. Display right pyramids with nonrectangular bases.

Cylinders, Cones, and Spheres Students identify and compare attributes of cones, cylinders, and spheres. Direct students to problem 2.

Ask the following questions:

2. Use the figures shown to complete the table.

• What shape is the base of each of these right pyramids? How do you know? • We can name a right pyramid by the type of base it has. For example, a right pyramid with a square base is a square pyramid. What names could you give the right pyramids shown? Explain.

Figure

Solid

Shape of Bases

Cylinder

Circles

Cone

Circle

Sphere

What do the figures in problem 2 have in common? They all have curved surfaces.

As time allows, consider this activity for the whole class to have students apply what they know about attributes of right rectangular pyramids to right pyramids with different bases.

Teacher Note Some students may confuse a right triangular prism with a right pyramid because they are bolth solids with triangular and rectangular faces and both have 5 faces. Encourage students to recognize the difference by relating how the bases and faces of prisms and right pyramids differ.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 2

What is different about the figures? Figures Dand Ehave circular surfaces, but figure Fdoes not.

Teacher Note

Figure Ehas a sharp point, but figures Dand Fdo not.

Students may say all the figures are circular. Remind them that three-dimensional figures cannot be circular because circles are two-dimensional figures. Then have students identify which of the three figures have circular surfaces.

Figure Flooks like it has no flat surfaces, but figures Dand E have at least 1 flat surface. Place students into groups of three. Give each student in the group a letter representing one of the figures from problem 2. Give students 1 minute to count the numbers of vertices, edges, and faces on their figure. Anticipate that students may not know or may not respond with numbers that are accurate. Encourage students to use what they know about vertices, edges, and faces from analyzing right prisms and right pyramids, and tell students that they can change their minds throughout the discussion. When students have finished, or the struggle is no longer productive, invite them to share the numbers of vertices, edges, and faces with their group. Have students share their reasoning, and encourage the other students in the group to ask clarifying questions if they disagree. When students are finished, regroup as a class to confirm the attributes of the figures with the following questions. How many faces do you see in figure D? How do you know? Figure D does not have any faces. A face must be a polygon and a flat part of the surface of a solid. Figure Ddoes not have any surfaces like that. Point to the curved surface around the middle of figure D. Why is the curved surface between the 2 circular bases not considered a face? It is not considered a face because faces are flat, and this part of the figure is not flat.

Differentiation: Support Consider gathering several examples of cones, cylinders, and spheres of various sizes, such as a basketball, a baseball, a party hat, canned foods, and pictures or drawings. Invite students to turn each figure in various ways to see the numbers of edges, vertices, and faces and how they do not change with different orientations or sizes of the solids. Then transition to pictures and drawings of the solids in a variety of orientations and with a variety of heights and diameters.

Figure D is a cylinder. A cylinder is a solid with 2 flat, parallel, circular surfaces that are identical, called bases, and a curved surface connecting the 2 bases. These experiences can help students confirm that figures keep the same attributes regardless of how the figures are oriented.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 2

Invite students to Think–Pair–Share about the following prompt. Kelly and Ryan look at the same cylinder. Kelly says the cylinder has no edges or vertices. Ryan looks at the sides of the cylinder and says the cylinder has 2 edges and no vertices. Who do you agree with? Why? I agree with Kelly. An edge is a line segment where 2 faces meet. Since there are no faces, there cannot be any edges. That also means there are no vertices because there are no points where the edges meet. Consider inviting students to compare the table in problem 2 with the table in problem 1. Ask students why they think the table in problem 2 does not have columns for the numbers of vertices, edges, and faces or the types of faces. As a class, complete the figure Drow in the table for problem 2. Then direct students to consider figures Dand E. What is the same or different about the attributes of figure D and figure E? Figure Ehas a sharp point, but figure Ddoes not. Both figures have circular surfaces, but figure Ehas 1 circular surface and figure Dhas 2 circular surfaces. Both figures have curved surfaces. Figure E is a cone. A cone is a solid with a flat, circular surface called the base and a curved surface that connects the base to a point called the apex. Invite students to Think–Pair–Share about why figure Edoes not have faces, edges, or vertices. Figure Edoes not have any flat surfaces that are polygons, so it does not have any faces. Since it does not have any faces, it cannot have any edges or vertices. Leo says that cones have 2 faces because he can see a circle as the base and a curved surface around the base that meets at the apex. Do you agree with Leo? Why? I disagree. The curved surface of the cone is not flat, so it cannot be a face. The circle is not a face because it is not a polygon.

Teacher Note The definition of cone in later grades specifies that the apex is not a point within the base or a point in the same plane as the base. If this were the case, there would be a flat surface and not a cone. This concept is taken out of the given definition of cone in this lesson to instead specify that there is a curved surface around the cone.

Direct students to work as a group to complete the figure E row in the table.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 2

Figure F is a sphere. What is different about the sphere compared to the other solids we have seen? The sphere only has a curved surface. It does not have any flat surfaces. A sphere is a round solid that does not have faces, edges, or vertices. Direct groups to complete the figure Frow in the table. Have students draw an X in the table to indicate that there are no bases on a sphere. Invite students to Think–Pair–Share about the similar and different attributes of cones, cylinders, and spheres. Cylinders and cones have at least 1 circular surface. Spheres do not have any flat surfaces. Cylinders, cones, and spheres do not have edges. Cones have an apex. Cylinders and spheres do not have an apex. All three solids have curved surfaces.

Classify Three-Dimensional Figures Students analyze and sort three-dimensional figures. Display the chart. Invite students to Think–Pair–Share about how the three-dimensional figures are classified and explain how they know by using the figures’ attributes.

Group 1

Group 2

Group 3

The figures are classified by the number of bases they have. I know because the figures in group 1 all have 2 bases, the figures in group 2 have 1 base, and the figure in group 3 does not have any bases.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 2

Display the hierarchy of figures from lesson 1.

Figures Two-Dimensional Polygons

Three-Dimensional

Non-Polygons

Teacher Note

Triangles Quadrilaterals Pentagons Hexagons In the previous lesson, we created a hierarchy of figures. We classified figures as two-dimensional or three-dimensional and then classified the two-dimensional figures into polygons and non-polygons. Invite students to Think–Pair–Share about how they could classify the three-dimensional figures from this lesson on the hierarchy.

The categories students use to classify the three-dimensional figures from this lesson may not be comprehensive for all three-dimensional figures. Allow students to select categories that are appropriate for the figures from this lesson without ensuring they are appropriate for all threedimensional figures.

We could classify the figures by whether they have faces. Promoting Mathematical Thinking and Reasoning

We could classify them by the number of bases. We could classify them by whether they have an apex. Direct students to problem 3. Have students work with a partner to classify the three-dimensional figures on the hierarchy under Three-Dimensional. 3. Choose an attribute that you can use to classify three-dimensional figures. Then use that attribute to complete the hierarchy to classify the figures listed in the word bank. Word Bank Cones

Right prisms

Right pyramids

Cylinders

Spheres

Students engage in discussions that reflect on mathematical thinking (MTR.4) as they discuss how to organize their hierarchy of three-dimensional figures. Ask the following questions to promote MTR.4: • What evidence do you have to support your thinking? • Which solid has this attribute? How do you know for sure?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 2 Sample: Figures

Two-Dimensional

Polygons

Triangles Quadrilaterals

Non-Polygons

Three-Dimensional

Apex

Pentagons Hexagons Cones

Right Pyramids

No Apex

Spheres Right Cylinders Prisms

When students are finished, invite several pairs to share how they classified the three-dimensional figures on the hierarchy. Encourage others to ask clarifying questions or to explain whether they disagree with any of the classifications.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Identify three-dimensional figures and classify the figures into categories based on defining attributes. Facilitate a class discussion about identifying and classifying three-dimensional figures into categories based on defining attributes by using the following prompts. Encourage students to restate or add to their classmates’ responses. © Great Minds PBC

5 ▸ M5 ▸ TA ▸ Lesson 2

EUREKA MATH2 Florida

What information do you need to classify three-dimensional figures? We need to know the defining attributes of the figures, which can include the numbers of edges, vertices, and faces, as well as the types of faces, the types of bases, whether the figure has an apex, and whether the figure has curved surfaces or flat surfaces. In what ways do the defining attributes of right prisms, right pyramids, cones, cylinders, and spheres compare to one another? Right prisms and right pyramids have edges. Cones, cylinders, and spheres do not. Spheres, cones, and cylinders do not have vertices, faces, or edges. Spheres do not have a base. Cylinders, cones, and spheres have curved surfaces. Right prisms and right pyramids have only flat surfaces. Cylinders and cones have at least 1 circular base. Right prisms and right pyramids have polygonal bases. Cones and right pyramids have an apex, while right prisms, cylinders, and spheres do not. All right pyramids have some triangular faces, and all right prisms have some rectangular faces.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 2

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can use attributes of three-dimensional figures to classify them.

PRACTICE

Florida B.E.S.T. MA.5.GR.1.2

Three-Dimensional Figure Name

Use the word bank to write the name for each three-dimensional figure shown. Then list the defining attributes of each figure by describing the number of faces,

•

the number of edges,

•

the number of vertices, and

•

the shape of the base if the figure has any bases.

Cone

Right rectangular prism

0 faces 0 edges 0 vertices

Circular base

Word Bank Cone

Defining Attributes

Date

•

Name

Right pyramid

Cylinder

Sphere

Right triangular prism

Cylinder

0 faces 0 edges 0 vertices

Circular bases Three-Dimensional Figure

Name

Defining Attributes 5.

Sphere

0 faces 0 edges 0 vertices

Right pyramid

Rectangular base

Right rectangular prism

5 faces 8 edges 5 vertices

6 faces 12 edges 8 vertices

Right triangular prism

5 faces 9 edges 6 vertices

Triangular bases

Rectangular bases

PROBLEM SET

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 2

EXIT TICKET

I can use attributes of three-dimensional figures to classify them.

Florida B.E.S.T. MA.5.GR.1.2

7. Mark each statement as true or false. If the statement is false, explain why. Statement

True

A sphere has 1 face.

A cone and a right pyramid have an apex. A right prism can have hexagonal bases.

Explain A sphere does not have faces, it has a curved surface.

Name

Date

Write the name of each three-dimensional figure shown. Then list the defining attributes of each figure by describing

A cylinder has 2 edges. A cone, a cylinder, and a sphere have curved surfaces. All right prisms have the same numbers of faces, edges, and vertices.

False

A cylinder has no edges because it has no faces.

Right prisms have different numbers of faces, edges, and vertices depending on the shape of the bases.

•

the number of faces,

•

the number of edges,

•

the number of vertices, and

•

the shape of the base if the figure has any bases. Three-Dimensional Figure

8. Eddie classified the three-dimensional figures in the table below. What two mistakes did Eddie make? Explain how you know. Circular Bases

Rectangular Bases

Triangular Bases

Name

Defining Attributes

Cylinder

0 faces 0 edges 0 vertices

Circular bases

No Bases 2.

Right rectangular pyramid

4 faces 6 edges 4 vertices

Rectangular base

Eddie classified the right triangular prism as having rectangular bases, but it has triangular bases. He classified the right pyramid as having a triangular base, but the right pyramid has a rectangular base, and the other faces are triangles.

PROBLEM SET

LESSON 3

Classify triangles into different categories based on attributes. I can classify triangles based on their attributes.

Lesson at a Glance Students construct a triangle and classify triangles based on their attributes. They use rulers, right-angle tools, and symbols in drawings to help them. This lesson formalizes the terms acute triangle, obtuse triangle, right triangle, equilateral triangle, isosceles triangle, and scalene triangle.

Key Questions • What information is needed to classify a triangle?

Mathematical Thinking and Reasoning Standard MTR.4.1

Language Objectives • Orally discuss attributes of triangles relating to angle measurements and side lengths, and discuss their classifications based on these attributes. • Read statements about equilateral triangles. Orally discuss whether the statements are always, sometimes, or never true.

• What tools or symbols are helpful when classifying triangles?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 3

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

• Prepare seven signs—Acute and Equilateral, Acute and Scalene, Acute and Isosceles, Right and Scalene, Right and Isosceles, Obtuse and Scalene, and Obtuse and Isosceles. Hang the signs up in seven different locations in the classroom. • Consider whether to remove Assorted Triangles from the student books and cut out one set per student in advance or have students prepare them during the lesson.

Launch 5 min Learn 35 min • Construct a Triangle • Classify Triangles • Equilateral Triangles • Classify Triangles and Explain Reasoning • Problem Set

• Assorted Triangles (in the teacher edition) • Prepared signs

Students • Assorted Triangles (in the student book) • Index card • Ruler • Toothpicks (3 per student pair)

Land 10 min

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD4 Classify triangles into

MA.5.GR.1.1 Classify triangles or quadrilaterals

categories based on attributes and explain the classification. (MA.5.GR.1.1)

into different categories based on shared defining attributes. Explain why a triangle or quadrilateral would or would not belong to a category.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 3

Fluency

Whiteboard Exchange: Attributes of Three-Dimensional Figures Students identify three-dimensional figures with specified attributes to build fluency classifying three-dimensional figures. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the attributes and the three figures labeled with letters. Which of the three-dimensional figures has 5 faces, 9 edges, 6 vertices, and triangular bases? Write the letter.

Attributes: 5 faces, 9 edges, 6 vertices, and triangular bases

Display a circle around figure B. Write the name of figure B.

Display the answer.

right triangular prism

Repeat the process with the following sequence:

Attributes: 5 faces, 8 edges, 5 vertices, and rectangular base

right pyramid

Attributes: 0 faces, 0 edges, 0 vertices, and circular bases

cylinder

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 3

Counting on the Number Line by Grams and Kilograms Students count by a unit of 500 grams and rename grams as kilograms to build fluency with converting measurements from module 4. Display the number line. Use the number line to count forward by 500 grams to 3,000 grams. The first measurement you say is 0 grams. Ready? Display each measurement one at a time on the number line as students count.

0 g, 500 g, … , 2,500 g, 3,000 g

500 g

1,000 g 1,500 g 2,000 g 2,500 g 3,000 g

0 kg

500 g

1 kg

0 kg

500 g

1 kg 1 kg 500 g 2 kg 2 kg 500 g 3 kg

1,500 g

2 kg

2,500 g

3 kg

Now count forward by 500 grams again. This time rename every 1,000 grams as a number of kilograms. The first measurement you say is 0 kilograms. Ready? Display each measurement one at a time on the number line as students count.

0 g, 500 g, … , 2,500 g, 3 kg Now count forward by 500 grams again. This time use mixed units, kilograms and grams, when possible. The first measurement you say is 0 kilograms. Ready? Display each measurement one at a time on the number line as students count.

0 g, 500 g, … , 2 kg 500 g, 3 kg

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 3

Choral Response: Classify and Measure Angles Students classify an angle and use a 180° protractor to determine the angle measure to prepare for classifying triangles. After asking each question, wait until most students raise their hands, and then signal for students to respond. Raise your hand when you know the answer to each question. Wait for my signal to say the answer. Display the acute angle. How would you classify the angle? Acute Display the answer. Estimate the angle measure. Whisper your estimate to your partner.

Acute 60°

Provide time for students to think and share with their partner.

45°. It looks like it’s halfway between 0° and 90°. 70°. It looks like it’s more than halfway to 90°. Display the protractor. What is the angle measure?

60°

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 3

Show the angle measure. Repeat the process with the following sequence:

Right 90°

Obtuse 135°

Obtuse 147°

Acute 52°

Launch

Acute 45°

Materials—S: Assorted Triangles

Students analyze and sort triangles into categories. Direct students to gather their six assorted triangles and to place the triangles in front of them. Prompt students to turn and talk about what they notice about the triangles. Then invite students to sort the triangles into at least two categories. After students finish sorting, direct them to look at their partner’s sort and see if they can determine how the triangles are sorted. © Great Minds PBC

Teacher Note If students are unsure of how to begin sorting the triangles, invite them to look for different types of angles and the presence of perpendicular line segments.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 3

How did your partner sort the triangles? My partner made three groups: triangles with a right angle, triangles with an obtuse angle, and triangles with only acute angles. My partner made a group of triangles with perpendicular sides and a group without perpendicular sides. Transition to the next segment by framing the work. Today, we will learn new ways to classify triangles.

Learn

Teacher Note

Construct a Triangle Materials—S: Triangles, ruler

Students construct a triangle by drawing three points and connecting them with line segments. A

Draw three points on your personal whiteboard that do not lie on the same line and label them A, B, and C. ¯, BC  Use your ruler to draw  AB    ¯, and AC    ¯. What shape have you created?

Invite students to show their triangles by holding up their whiteboards. Invite them to Think–Pair–Share about what they notice about each of the triangles and the attributes that the triangles share.

Students may question why the three points they draw cannot share the same line. Draw three collinear points and connect the points by drawing line segments to demonstrate what happens when the three points share the same line (i.e., they do not form a triangle).

A B

Each triangle looks a little different than the others. Each triangle has three sides, three angles, and three corners. The sides and angles are different sizes, but each shape is still a triangle.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 3

Triangles are made up of three points that do not lie on the same line and the line segments connecting them. Each side of the triangle is a line segment. Each corner of the triangle is a vertex. It is where two sides of the triangle intersect to form an angle. How many vertices does a triangle have?

Universal Design for Learning: Representation

Invite students to examine the triangles they sorted. Direct them to count the number of sides, vertices, and angles of each triangle. Invite them to trace each angle by moving their fingers along one side of the triangle to the vertex and then along the adjacent side.

Consider creating a bubble map graphic organizer that shows ways to classify triangles. Label the center of the bubble map: Ways to Classify Triangles. As the lesson progresses, add the bubbles By Angle Type and By Side Length, as well as their related bubbles.

Classify Triangles Materials—S: Triangles, index card, ruler

Right

Students sort and classify triangles using side length and angle measure. Acute

Direct students to locate the triangle with vertices P, Q, and R. Then display the notation △PQR.

By Angle Type

One way to name this triangle is to use the triangle symbol and its three vertices. We can call it triangle PQR. What other names do you think this triangle has? △QPR

Ways to Classify Triangles

△RPQ

We can name the triangle by beginning with any vertex.

Direct students to locate △GHI.

What do you notice about the angles in △GHI? It looks like ∠G is a right angle.

Angles H and I look like acute angles.

Equilateral

By Side Length Scalene

Isosceles

at least 2 sides of equal length D

Invite students to use their index card as a right-angle tool to confirm whether each angle is acute, obtuse, or right. Some of you sorted the triangles by the type of angles they have. We can use angle measure to classify triangles. Because ∠G is a right angle, we can classify △GHI as a right triangle.

Obtuse

Include a description and a visual to accompany each type of triangle. For example, in the isosceles bubble, sketch a triangle with at least 2 sides of equal length, using tick marks to indicate that the lengths are equal.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 3

Invite students to look for other triangles that have a right angle. Is △GHI the only right triangle?

No, △ABC is also a right triangle because ∠A is a right angle.

Direct students to problem 1 in their books.

Teacher Note

1. Complete the table for each triangle.

Triangle

Attributes

△GHI

H △PQR R P Q

Angle

Side Length

acute triangle

scalene triangle

No sides have the same length.

obtuse triangle

isosceles triangle

right triangle

equilateral triangle

No sides have the same length.

acute triangle

scalene triangle

obtuse triangle

isosceles triangle

right triangle

equilateral triangle

∠G is a right angle.

Classification

∠P is an

obtuse angle.

The sides of a triangle are line segments, which means each side can be named by using the letters that also represent the line ¯. segment. For example, side JL or  JL

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 3

Triangle

Attributes

Angle

Side Length

acute triangle

scalene triangle

obtuse triangle

isosceles triangle

All angles are acute.

right triangle

equilateral triangle

All 3 sides have the same length.

acute triangle

scalene triangle

obtuse triangle

isosceles triangle

right triangle

equilateral triangle

right triangle

isosceles triangle

△ JKL K

Sides JL and KL have the same length.

L △MNO O

N △ABC

All angles are acute.

∠A is a right angle.

Classification

Sides AB and AC have the same length.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 3

Triangle

Attributes

△DEF

∠E is obtuse.

Sides ED and EF have the same length.

Classification Angle

Side Length

obtuse triangle

isosceles triangle

Direct students to classify △GHI as a right triangle by circling right triangle in the Classification column. How do we know that △GHI is a right triangle? It is a right triangle because ∠G is a right angle.

In the Attributes column for △GHI, write the following: ∠G is a right angle. Direct students to do the same. We can also classify a triangle based on the length of its sides, or line segments.

Invite students to use their rulers to measure each side of △GHI to see if any two sides have the same length. Do any two sides have the same length? No.

△GHI is also a scalene triangle. No sides of a scalene triangle have the same length.

Circle scalene triangle in the Classification column. In the Attributes column, write the following: No sides have the same length. Direct students to do the same.

Teacher Note When measuring, students can measure each side length to the nearest sixteenth of an inch or to the nearest tenth of a centimeter.

Which other triangle appears to be scalene? △PQR

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 3

Invite students to measure each side length to confirm that △PQR is scalene.

Direct students to circle scalene triangle for △PQR in the Classification column. In the Attributes column, write the following: No sides have the same length.

R P

Is △PQR also a right triangle, like △GHI?

No. △PQR doesn’t have any right angles.

Invite students to use their right-angle tools to determine the type of angles in △PQR. What types of angles are in △PQR?

∠P is an obtuse angle.

Acute and obtuse angles

Direct students to look at the choices for classifying a triangle by its angles. Then invite students to Think–Pair–Share about whether △PQR should be classified as an acute, an obtuse, or a right triangle. It’s not a right triangle because there are no right angles. It could be an obtuse triangle because it has an obtuse angle. It has two acute angles and one obtuse angle, but I don’t know if it is an acute or obtuse triangle. △PQR is an obtuse triangle. An obtuse triangle has one obtuse angle.

Circle obtuse triangle in the Classification column. In the Attributes column, write the following: ∠P is an obtuse angle. Direct students to do the same.

Invite students to turn and talk about how a right scalene triangle is similar to and different from an obtuse scalene triangle.

Multilingual Learner Support As students record the defining attributes of each triangle, consider providing sentence frames and key terminology as a scaffold for students to choose from when completing the table. • All 3 angles are acute. (acute triangle) • ∠ is an obtuse angle. (obtuse triangle) • ∠

is a right angle. (right triangle)

• Sides and length. (isosceles triangle)

are equal in

• All 3 sides have the same length. (equilateral triangle) • No sides have the same length. (scalene triangle)

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 3

Direct students to locate △JKL.

What do you notice about the sides of △ JKL?

It looks like 2 of the sides have the same length.

Prompt students to fold △JKL by folding ∠J on top of ∠K.

What do you notice about the two sides that line up? They fit perfectly on top of one another. They match. They have the same length. What do you notice about the two angles that line up? They match. They have the same size.

Direct students to open the triangle back up.

△JKL is an isosceles triangle. An isosceles triangle has at least two sides

of equal length.

Circle isosceles triangle in the Classification column. In the Attributes column, write the following: Sides JL and KL have the same length. Direct students to do the same. Invite students to Think–Pair–Share about how to classify △JKL based on its angles. It’s not a right triangle because it does not have any right angles.

It’s not an obtuse triangle because it does not have any obtuse angles.

Teacher Note Triangles can be named by angle type (e.g., obtuse triangle), side lengths (e.g., scalene triangle), or more specifically by using both angle type and side lengths (e.g., obtuse scalene triangle).

It must be an acute triangle because all three of its angles are acute.

An acute triangle has only acute angles. △ JKL is an acute triangle.

Circle acute triangle in the Classification column. In the Attributes column, write the following: All angles are acute. Direct students to do the same. Invite students to turn and talk about what they notice about the fold line they created within △JKL. Listen for students to notice that the fold line shows the line of symmetry of the triangle.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 3

Direct students to locate △MNO.

What do you notice about the sides of △MNO? It looks like they all have the same length.

Prompt students to fold △MNO along a line of symmetry by folding ∠M on top of ∠N.

What do you notice about the two sides that line up? They match. They have the same length.

It’s like what happened when we folded △JKL.

What do you notice about the two angles that line up? They match. They have the same size. Direct students to open the triangle back up and to look for other lines of symmetry. Invite students to confirm the lines of symmetry by folding other angles on top of one another (i.e., ∠M onto ∠O and ∠N onto ∠O). How many lines of symmetry does △MNO have?

What did you notice about the sides? They all fit perfectly on top of one another. They all have the same length. Prompt students to look at the Side Length column within the Classification column. What do you think we call a triangle when all three sides have the same length? An equilateral triangle

An equilateral triangle has three sides of equal length. △MNO is an equilateral triangle.

Circle equilateral triangle in the Classification column for △MNO. In the Attributes column, write the following: All 3 sides have the same length. Direct students to do the same.

Differentiation: Challenge Prompt students who are ready for a challenge to determine if it is possible for a triangle to have 2 obtuse angles or to have 2 right angles. Have students explain their reasoning. Consider asking students to reason about what kind of triangles always have at least 1 line of symmetry (i.e., isosceles and equilateral) and which triangles can never have a line of symmetry (i.e., scalene).

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 3

Invite students to Think–Pair–Share about how to classify △MNO based on its angles. It’s not a right triangle because it does not have any right angles.

It’s not an obtuse triangle because it does not have any obtuse angles. It’s an acute triangle because all three of its angles are acute. Circle acute triangle in the Classification column. In the Attributes column, write the following: All angles are acute. Direct students to do the same. We said that an isosceles triangle is a triangle with at least two sides of equal length. What does that tell you about an equilateral triangle? An equilateral triangle must also be an isosceles triangle because it has three sides of equal length and that is more than two. Circle isosceles triangle in the Classification column. Direct students to do the same.

Direct students to locate △ABC and △DEF. Invite students to read the listed attributes for each triangle and confirm they are accurate. Then prompt students to write the correct classification for each triangle in the table.

Teacher Note The term isosceles triangle has two different meanings. • Exclusive definition: An isosceles triangle is a triangle with exactly two sides of equal length. • Inclusive definition: An isosceles triangle is a triangle with at least two sides of equal length. The inclusive definition is used in all grades. Therefore, an equilateral triangle is a special type of isosceles triangle. Similar to the relationship between squares and rectangles, all equilateral triangles are also isosceles, but not all isosceles triangles are equilateral.

E Teacher Note

Invite students to Think–Pair–Share about how they classified △ABC and △DEF.

△ABC is a right and isosceles triangle because it has one right angle and it has

two sides that have the same length.

△DEF is an obtuse and isosceles triangle because it has one obtuse angle and it has

two sides that have the same length. 78

The Triangle Classification interactive presents triangles students can classify by their angles and sides. Consider using the tool to generate additional triangles for the whole class to classify or allowing students to generate triangles for their own practice.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 3

Invite students to turn and talk about how equilateral triangles and isosceles triangles are similar and different.

Equilateral Triangles Materials—S: Toothpicks

Students construct a model of an equilateral triangle and reason about its attributes. Partner students and distribute three toothpicks to each pair of students. Invite students to confirm that the toothpicks have the same length. Build a triangle by using the three toothpicks. Each toothpick represents one side of the triangle. What type of triangle did you make? How do you know? An equilateral triangle because each side is equal in length

Teacher Note Consider using other readily available classroom materials instead of toothpicks, such as rulers, straws, or unsharpened pencils. The objects must be linear and have the same length.

Present the following statement: An equilateral triangle can also be a scalene triangle. Use the Always Sometimes Never routine to engage students in constructing meaning and discuss their ideas. Give students 1 minute of silent think time to evaluate whether the statement is always, sometimes, or never true. Have students discuss their thinking with their partner. Circulate and listen as they talk. Identify a few students to share their thinking. Then facilitate a class discussion. Invite students to share their thinking with the whole group. Encourage them to provide examples and nonexamples or use their toothpicks to support their claim. Conclude by coming to the consensus that the statement is never true because a scalene triangle has no sides of equal length and an equilateral triangle has three sides that have the same length.

Multilingual Learner Support To support students in sharing their thinking with a partner, direct students’ attention to the Agree or Disagree portion of the Talking Tool.

5 ▸ M5 ▸ TA ▸ Lesson 3

EUREKA MATH2 Florida

As time allows, repeat the Always Sometimes Never routine for the following statements: • An equilateral triangle can also be an isosceles triangle. This statement is always true. An isosceles triangle has at least two sides that have the same length. All three sides of an equilateral triangle have the same length and that is more than two sides. • An equilateral triangle has side lengths that measure 3 inches. This statement is sometimes true. The side lengths of an equilateral triangle can be any length, but they must all be equal to one another. • An equilateral triangle can also be an obtuse triangle. This statement is never true. When you make an obtuse angle with two sides of the same length, the third side is not long enough to complete the triangle if it is the same length as the other two sides. • An equilateral triangle can also be a right triangle. This statement is never true. When you make a right angle with two sides of the same length, the third side is not long enough to complete the triangle if it is the same length as the other two sides. • An equilateral triangle is an acute triangle. This statement is always true. If you cannot make an equilateral triangle with an obtuse angle or a right angle, all the angles must be acute. Invite students to turn and talk about how equilateral triangles are different from the other types of triangles.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 3

Classify Triangles and Explain Reasoning

Materials—T: Signs

Students classify triangles based on visible attributes and explain their reasoning. Display △XYZ.

What do you notice about △ XYZ?

Promoting Mathematical Thinking and Reasoning

When students explain their reasoning and analyze their classmates’ reasoning for how they classified their triangles, they are engaging in discussions that reflect on mathematical thinking (MTR.4).

I see a right angle and I think two sides might be the same length. There are marks on the two sides that look like they have the same length. Tell students that the tick marks on sides XY and XZ are used to indicate that those sides are equal in length. Then invite students to think about how to classify △XYZ by using its angles and side lengths.

Ask the following questions to promote MTR.4:

Then display △TUV and △HAT one at a time and invite students to think about how to classify them by their angles and side lengths. Introduce the Take a Stand routine to the class. Draw students’ attention to the seven signs hanging in the classroom. Divide the students into three groups and assign each group one of the triangles. Invite students to stand beside the sign that best describes their triangle. When all students are standing near a sign, provide time for groups to discuss the reasons why they chose that sign. Then call on each group to share their reasons for their selection. Each group should tell which triangle they classified before giving their reasons for the classification. Invite students who change their minds during the discussion to join a different group.

• Why is your classification correct? Convince your classmates. • What questions can you ask your classmates to make sure you understand their reasoning?

Universal Design for Learning: Action & Expression To support the explanations for how to classify each triangle, consider providing the following sentence frames: It is a an angle that is It is a has same length.

triangle because it has . triangle because it sides that have the

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 3

H U

Have students return to their seats and invite them to turn and talk about their strategies for classifying triangles.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Classify triangles into different categories based on attributes. Use the following prompts to guide a discussion about how to classify triangles: What information do we need to classify a triangle? We need to know whether the angles are acute, obtuse, or right. We need to know if any of the sides have equal lengths. 82

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 3

What tools or symbols are helpful when classifying triangles? A ruler and a right-angle tool are helpful for measuring. Tick marks are helpful because they show which sides have equal length. Lines of symmetry show that some sides have equal length. The right-angle symbol is helpful because it tells us that there is a right angle in the triangle. Explain the differences between acute, obtuse, and right triangles. Obtuse triangles have one obtuse angle. Right triangles have one right angle. Acute triangles have only acute angles. Explain the differences between scalene, isosceles, and equilateral triangles. Equilateral triangles have three sides that are equal in length. An isosceles triangle has at least two sides that are equal in length. Scalene triangles have no sides that are the same length.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 3

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET PRACTICE

I can classify triangles based on their attributes.

5. △STU has 1 line of symmetry. What does this tell you about the measures of ∠S and ∠T ?

Florida B.E.S.T. MA.5.GR.1.1

Name

Date

Use the triangles to complete problems 1–4.

U D

∠S and ∠T have the same angle measure.

1. Classify each triangle by its side lengths. Then classify each triangle by its angle measures. Equilateral

Isosceles

Scalene

Acute

Right

Obtuse

B, A, D, E

C, F

A, D

C, E

B, F

6. Use a ruler to connect 3 points to make a triangle. Sample:

A D

2. Which, if any, triangles are acute isosceles? Triangles A and D are acute isosceles triangles.

Triangle F is an obtuse scalene triangle.

△BCE

b. Classify your triangle based on its angles and sides.

3. Which, if any, triangles are obtuse scalene?

a. Use the points you connected to name your triangle.

△BCE is a right scalene triangle.

c. Oka connects points A, B, and C. She says, “I made △ABC.” Do you agree with Oka? Why or why not? I disagree with Oka. Points A, B, and C form a straight line, not a triangle.

4. Casey says that triangle C is a right equilateral triangle. Do you agree with Casey? Why? I disagree with Casey. Triangle C is not a right equilateral triangle because all of the sides are not equal in length. Triangle C is a right scalene triangle.

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 3

EXIT TICKET

I can classify triangles based on their attributes.

Florida B.E.S.T. MA.5.GR.1.1

7. Can a triangle have 2 right angles? How do you know? No, a triangle cannot have 2 right angles. If there are 2 right angles, there is no way for the 3 sides to connect. Name

Date

Complete the table for each triangle.

Triangle

PROBLEM SET

Classification Angle Type

Side Length

Acute

Isosceles

Obtuse

Equilateral

Right

Scalene

Acute

Isosceles

Obtuse

Equilateral

Right

Scalene

Acute

Isosceles

Obtuse

Equilateral

Right

Scalene

EUREKA MATH2 Florida

This page may be reproduced for classroom use only.

5 ▸ M5 ▸ TA ▸ Lesson 3 ▸ Assorted Triangles

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 3 ▸ Assorted Triangles

This page may be reproduced for classroom use only.

LESSON 4

Classify trapezoids and parallelograms based on their properties. I can classify trapezoids and parallelograms.

Lesson at a Glance Students learn that trapezoids and parallelograms are special types of quadrilaterals. After each student constructs their own trapezoid and parallelogram, the class uses the constructions to identify attributes and properties of the figures. Students explore side length, diagonals, and lines of symmetry to describe properties of trapezoids and parallelograms. Then they begin to construct a hierarchy of quadrilaterals. This lesson formalizes the term midpoint.

Key Questions • What are some properties of trapezoids? • Why is trapezoid a more specific name for a shape than quadrilateral?

Mathematical Thinking and Reasoning Standard MTR.5.1

Language Objectives • Orally discuss attributes of trapezoids. • Orally discuss attributes of parallelograms. • Orally discuss how to add trapezoids and parallelograms to the figures hierarchy. • Read statements about trapezoids and parallelograms. Orally discuss whether the statements are always, sometimes, or never true.

• What additional information does the name parallelogram give us about a quadrilateral that the name trapezoid does not?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 4

Agenda

Materials

Lesson Preparation

Fluency 5 min

Teacher

• Prepare three signs on paper. Label one sign Strongly Agree, another sign Strongly Disagree, and the third sign Undecided. Hang the signs in different locations in the classroom. • Ensure students have access to a ruler for the Problem Set.

Launch 5 min Learn 40 min • Construct a Trapezoid • Construct a Parallelogram • Explore Diagonals and Symmetry • Hierarchy of Quadrilaterals • Problem Set

Land 10 min

• Blank paper (4 sheets) • Ruler • Index card

Students • Blank paper • Ruler • Index card • Scissors

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD5 Classify quadrilaterals into

MA.5.GR.1.1 Classify triangles or quadrilaterals

categories based on attributes and explain the classification. (MA.5.GR.1.1)

into different categories based on shared defining attributes. Explain why a triangle or quadrilateral would or would not belong to a category.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 4

Fluency

Choral Response: Identify and Classify Three-Dimensional Figures Students use knowledge of defining attributes to build fluency with identifying and classifying three-dimensional figures. After asking each question, wait until most students raise their hands, and then signal for students to respond. Raise your hand when you know the answer to each question. Wait for my signal to say the answer. Display the attribute and the three figures labeled with letters. Which of these figures has a rectangular base? Figure Aand figure B

Attribute: The ﬁgure has at least 1 rectangular base.

Show figures A and B circled. What is the name of figure A? Right pyramid

A right pyramid

right rectangular prism

What is the name of figure B? Right rectangular prism

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 4

Repeat the process with the following sequence: Attribute: The ﬁgure has at least 1 triangular face.

right pyramid

right triangular prism

right pyramid

Attribute: The ﬁgure has at least 1 circular base.

cylinder

cone

Launch

Attribute: The ﬁgure has 8 edges.

Attribute: The ﬁgure has 8 vertices.

Attribute: The ﬁgure has no faces or bases.

right rectangular prism

Attribute: The ﬁgure has an apex.

sphere

right pyramid

cone

Materials—T: Signs

Students make a conjecture about whether all parallelograms are trapezoids. Introduce the Take a Stand routine to the class. Direct students’ attention to the signs hanging in the classroom: Strongly Agree, Strongly Disagree, and Undecided. Read the following statement. All parallelograms are quadrilaterals, but not all parallelograms are trapezoids. Invite students to stand beside the sign that best describes their thinking. © Great Minds PBC

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 4

When all students are standing near a sign, allow 2 minutes for groups to discuss the reasons why they chose that sign. Then call on each group to share reasons for their selection. Invite students who change their minds during the discussion to join a different group. Have students return to their seats. If the statement had only said that all parallelograms are quadrilaterals, choosing whether to agree or disagree might have been more straightforward. All parallelograms have 4 sides, so all parallelograms are quadrilaterals. Transition to the next segment by framing the work. Today, we will construct and classify trapezoids and parallelograms and compare their proprieties to the properties of other quadrilaterals.

Learn

Construct a Trapezoid Materials—T/S: Blank paper, ruler, index card

Students construct a trapezoid and identify its properties.

Universal Design for Learning: Action & Expression Consider offering isometric grid paper as students construct the trapezoid. Students can use a straightedge to trace the lines on the grid to create parallel sides.

Let’s construct trapezoids to identify properties of trapezoids. What must a trapezoid include? How do you know? It must have 4 sides because all quadrilaterals have 4 sides. It must have at least 1 pair of parallel sides because that is the definition of a trapezoid. What tools can we use to help us draw straight and parallel sides? We can use a straightedge, such as a ruler, and an index card as a right-angle tool. Guide students in the construction and direct them to complete each part of the construction as it is modeled.

Teacher Note Encourage students to follow the directions for the constructions but to use their own creativity when determining how far apart the parallel lines should be and where to put the third and fourth sides. It is important that students construct a variety of trapezoids.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 4

• Use a straightedge to draw a line segment. • Align one side of the right-angle tool, or index card, to the line segment. • Align the straightedge to the right-angle tool so that the straightedge is perpendicular to the line segment. • Slide the right-angle tool along the straightedge to the desired location of the parallel line segment. • Mark the parallel line segment by drawing along the side of the right-angle tool. • Remove the right-angle tool and extend the second line segment with the straightedge, if necessary. • Complete the trapezoid by drawing a third and a fourth side. Use notation to mark any pairs of parallel sides.

Teacher Note Students may create right angles, but it is not necessary that they do so. To avoid creating a right angle, slide the straightedge away from the end of the first line segment before sliding the rightangle tool up or down to create the second parallel line segment. Encourage students to do the same.

Multilingual Learner Support Consider asking the following questions while circulating:

Compare your trapezoid with your partner’s. Turn and talk about what is similar and what is different. Circulate as students talk. Anticipate the following responses:

• How many pairs of parallel sides does your trapezoid have? Where are they? • Does your trapezoid have any right angles? If so, where are they?

• Our trapezoids both have 1 pair of parallel sides. • My trapezoid is taller than my partner’s.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 4

• My partner’s trapezoid is smaller than mine. • My trapezoid has 2 pairs of parallel sides, but my partner’s trapezoid has only 1 pair of parallel sides. As students share, look for three to five student samples to collect. Select a variety of trapezoids (e.g., a trapezoid with 2 pairs of parallel sides, trapezoids with long and short side lengths, trapezoids that include right angles). Then display the student samples. If students do not create trapezoids with right angles or 2 pairs of parallel sides, consider displaying the following figures:

Universal Design for Learning: Representation As students discuss characteristics of quadrilaterals in this topic, consider creating a quadrilaterals chart to display each figure’s properties. Add to the chart as students learn properties of other figures.

Figure Trapezoid

Properties For all trapezoids:

At least 1 pair of parallel sides

We all drew trapezoids. There are some things about our shapes that are the same and some that are different. What property do they all have in common that makes them quadrilaterals? They all have 4 sides and 4 angles. What property do the quadrilaterals all have that helps us classify them as trapezoids? They all have at least 1 pair of parallel sides. Identify attributes of various displayed trapezoids. For example, one might have right angles, and one might have 2 pairs of parallel sides. Invite students to turn and talk about the differences between attributes of a specific trapezoid and properties of all trapezoids. 94

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 4

Construct a Parallelogram Materials—T/S: Blank paper, ruler, index card

Students construct a parallelogram and identify its properties. Let’s construct parallelograms to identify properties of parallelograms. What must the figures include? How do you know? The figures must include 2 pairs of parallel sides. Opposite sides in a parallelogram are parallel. Guide students in the construction and direct them to complete each part of the construction as it is modeled. • Use a straightedge to draw a line segment. • Align one side of the right-angle tool, or index card, to the line segment. • Align the straightedge to the right-angle tool so that the straightedge is perpendicular to the line segment. • Slide the right-angle tool along the straightedge to the desired location of the parallel line segment. • Mark the parallel line segment by drawing along the side of the right-angle tool. • Remove the right-angle tool and extend the second line segment with the straightedge, if necessary. • Draw another pair of parallel sides that cross the first pair. • Use notation to mark pairs of parallel sides. Teacher Note To construct a variety of parallelograms, encourage students to use their own creativity when determining how far apart the parallel lines should be and where to put the third and fourth sides.

5 ▸ M5 ▸ TA ▸ Lesson 4

EUREKA MATH2 Florida

Circulate as students work, checking that their constructions are correct. When students are finished with the construction, use the following prompts. Some of the quadrilaterals we constructed earlier have only 1 pair of parallel sides, and some have 2 pairs of parallel sides. But we classified them all as trapezoids. Now, the quadrilaterals we constructed all have 2 pairs of parallel sides, so they can be classified as parallelograms. When can a trapezoid also be classified as a parallelogram? A trapezoid can also be called a parallelogram if it has 2 pairs of parallel sides. Measure the sides of your parallelogram. Label the side lengths on the parallelogram. Pair students. Invite students to compare parallelograms with their partner and check each other’s measurements. Then have students Think–Pair–Share to determine what is similar and what is different about the sizes and side lengths of the parallelograms. Our parallelograms are not the same size. My parallelogram is wider than my partner’s. My parallel sides are closer together than my partner’s. My parallelogram is a rectangle, but my partner’s parallelogram is not. My parallelogram has 4 sides that are the same length, and my partner’s parallelogram has 2 different pairs of sides that are the same length. In both of our parallelograms, opposite sides have the same length. Choose three to five student samples to collect. Select a variety of parallelograms (e.g., a parallelogram with 4 equal side lengths and a parallelogram that includes right angles). Then display the student samples. Or consider showing the following samples.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 4 2 in 1 in

1 in 2 in

3 in

6 cm

5 cm

3 in

6 cm

3 in

We all drew a parallelogram. There are some things about our figures that are the same and some things that are different. What property do they all have in common that makes them parallelograms? They have 2 pairs of parallel sides. What do you notice about the side lengths of the parallelograms? Opposite sides have the same length. We think that the opposite sides of a parallelogram always have the same length, but let’s look at some more examples. Open and display the Geometry World: Side Length Explorer digital interactive. Create a parallelogram. Invite students to Think–Pair–Share about what they notice about the side lengths of a parallelogram. I notice that opposite sides have the same length. © Great Minds PBC

Teacher Note Topic A uses several digital interactives to show properties of quadrilaterals and three-dimensional figures. To familiarize yourself with how the digital interactives function, consider taking some time to explore the Geometry World: Quadrilateral Explorer digital interactive. Display the digital interactive and allow students to direct movement of the vertices to create quadrilaterals that interest them.

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 4

Display several parallelograms to show that opposite sides always have the same length. Create rectangles as well. Discuss that rectangles are parallelograms and show that the opposite sides of rectangles also have the same length. Write the following statement: All parallelograms have opposite sides that are the same length. Does this statement describe an attribute of some parallelograms or a property of all parallelograms? How do you know? It is a property. I know because it is true for all parallelograms. We know that opposite sides in a parallelogram have the same length. Let’s look at angle measures in parallelograms. Display the four figures and ask the following question.

Universal Design for Learning: Representation If a quadrilaterals chart was created earlier in the lesson, consider adding examples of specific parallelograms, including their properties.

Figure Trapezoid

For all trapezoids:

Parallelogram

If a parallelogram has right angles, is that a feature of the parallelogram or a property of all parallelograms? Why? It is a feature of the parallelogram. Not all parallelograms have right angles. Invite students to turn and talk about the differences between attributes of a specific parallelogram and properties of all parallelograms.

5 cm

Properties

At least 1 pair of parallel sides

For all parallelograms:

6 cm

2 pairs of parallel sides

Opposite sides have the same length.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 4

Explore Diagonals and Symmetry Materials—T/S: Ruler, scissors

Students discover that the diagonals of a parallelogram intersect at their midpoints. What is a diagonal of a quadrilateral? A line segment connecting opposite corners of the quadrilateral Ask students to draw the diagonals on the parallelograms they constructed. Let’s look at special properties of the diagonals of parallelograms. Open and display the Geometry World: Diagonal Explorer digital interactive. Display a quadrilateral and point to the diagonals.

Differentiation: Challenge Display parallelogram ABCD and present the statement: All parallelograms have opposite angles with the same measure.

The diagonals look like they are always different lengths. The diagonals always cross somewhere, but the place that they cross moves around as the vertex moves. Create a parallelogram and adjust the vertices. Ask students what they notice about the diagonals of the parallelogram. The diagonals are not the same length. One diagonal is longer than the other. Invite students to Think–Pair–Share about the segments that make up the diagonals. Some of the diagonal segments look like they are the same length. Two of the segments are the same length, and the other two segments are the same length. It looks like the diagonals cut each other in half.

63°

117°

These are the diagonals of a quadrilateral. The diagonals of a quadrilateral are the line segments that connect opposite vertices. Adjust the vertices of the quadrilateral. Invite students to Think–Pair–Share about what they notice about the diagonals of a quadrilateral.

63° D

117° C

Then ask students the following questions. • Is the statement true or false? How do you know? • Is it a property or an attribute of parallelograms to have opposite angles with the same measure? How do you know? Consider using the Geometry World: Angle Explorer digital interactive to create a parallelogram and show opposite angle measures. As needed, encourage students to measure and label the angle measures of other parallelograms to justify their reasoning.

5 ▸ M5 ▸ TA ▸ Lesson 4

EUREKA MATH2 Florida

Display several different parallelograms, including squares and rectangles. Guide students to notice the diagonals of each new parallelogram, and then invite them to turn and talk about whether their observations about the diagonals in the first parallelogram seem to be true for all parallelograms. Each diagonal splits the other diagonal into 2 equal-length segments. We can say that the diagonals of a parallelogram intersect at their midpoints. A midpoint is the point on a line segment that divides a line segment into 2 line segments of equal length. Write the following statement: The diagonals of parallelograms intersect at their midpoints. Does this statement describe a property of parallelograms? How do you know? It is a property. I know because it is true for all parallelograms. Do diagonals of trapezoids intersect at their midpoints? Explain. Not necessarily. It depends on the trapezoid. Use the digital interactive to show trapezoids with diagonals that do not intersect at their midpoints. Having diagonals that intersect at their midpoints is a property of parallelograms because it is true for all parallelograms, but it is not a property of trapezoids because it is not true for all trapezoids.

Multilingual Learner Support Consider relating the new term midpoint to students’ real-world experiences. For example, ask students to identify any of the following: • The midpoint between a desk and the door • The midpoint of the length of the wall • The midpoint of a ruler

Are diagonals of parallelograms the same length? Explain. Not necessarily. If the parallelogram is a square or rectangle, then the diagonals look like they are the same length.

100

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 4

Use the digital interactive to show a parallelogram with diagonals that are the same length and a parallelogram with diagonals that are not the same length.

Ask students to cut out the parallelograms they constructed. Direct students to fold their parallelograms to determine whether they have any lines of symmetry. Then have students turn and talk about what lines of symmetry, if any, they found. Invite students to share. Neither my parallelogram nor my partner’s parallelogram has any lines of symmetry. My parallelogram has many lines of symmetry because I made a square. There are lines of symmetry along the diagonals and through the midpoints of the sides. My parallelogram has 2 lines of symmetry because I made a rectangle. Open and display the Geometry World: Symmetry Explorer digital interactive. Create a parallelogram. Then find lines of symmetry in the figure. Display several parallelograms, including at least one that does not have a line of symmetry. Then invite students to turn and talk about what they notice regarding the lines of symmetry. Can we say that it is a property of parallelograms to have at least 1 line of symmetry? Why? No. Not all parallelograms have a line of symmetry. Invite pairs to discuss and sketch a parallelogram that does have a line of symmetry.

101

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 4

Hierarchy of Quadrilaterals Students identify trapezoids and parallelograms and begin to classify quadrilaterals into a hierarchy. Display the hierarchy of figures.

Figures Two-Dimensional Polygons

Three-Dimensional

Non-Polygons

Triangles Quadrilaterals Pentagons Hexagons In a previous lesson, we created a hierarchy of figures. On the side for two-dimensional figures, we could expand on any of these polygon categories, but we are studying quadrilaterals. So today, we will start a hierarchy of quadrilaterals—one that fits into the hierarchy of figures. Start the hierarchy of quadrilaterals by writing Quadrilaterals. Invite students to help add the properties of quadrilaterals to the hierarchy. Because trapezoids are quadrilaterals, we can classify them below quadrilaterals in the hierarchy. That means a property of trapezoids is that they are 4-sided polygons. What are additional properties of trapezoids? A property of trapezoids is that they have at least 1 pair of parallel sides.

102

Quadrilaterals • Polygon with 4 sides

Trapezoids • At least 1 pair of parallel sides

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 4

Add trapezoids and their properties to the class hierarchy. Use the hierarchy to determine whether the following statements are true or false. Display and read the following statements. Check for student understanding by having students indicate, such as with a thumbs-up or thumbs-down, whether each statement is true or false. • All trapezoids are quadrilaterals. True. • All quadrilaterals can be classified as trapezoids. False. • All trapezoids have exactly 1 pair of parallel sides. False. Take time to address misconceptions and ensure students understand why each statement is true or false. Then present the following statement: A parallelogram is a trapezoid. Use the Always Sometimes Never routine to engage students in constructing meaning and discussing their ideas.

Promoting Mathematical Thinking and Reasoning Students use patterns and structure (MTR.5) when they apply what they know about hierarchies to determine whether statements about trapezoids, parallelograms, and quadrilaterals are true. Ask the following questions to promote MTR.5: • How are trapezoids and quadrilaterals related? How can knowing how they are related help you determine properties of trapezoids? • How are parallelograms and trapezoids related? How can knowing how they are related help you determine properties of parallelograms?

Give students a few moments of silent think time to evaluate whether the statement is always, sometimes, or never true. Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Then facilitate a class discussion. Invite students to share their thinking with the whole group. Encourage them to provide examples and nonexamples to support their claim. Conclude by coming to the consensus that the statement is always true because a parallelogram is a special type of trapezoid with parallel opposite sides. Because parallelograms are trapezoids, we can classify them below trapezoids in the hierarchy. Add parallelograms and the property that opposite sides are parallel to the class hierarchy. © Great Minds PBC

Teacher Note Anticipate that when interpreting the hierarchy of quadrilaterals, some students may be confused by the idea that a parallelogram is a trapezoid. Students typically want to give a shape a single name rather than recognize that many shapes can be classified in several ways. A parallelogram can be classified as a polygon, a quadrilateral, and a trapezoid, but its most specific name is parallelogram.

103

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 4

Invite students to summarize the properties of parallelograms that they learned in today’s lesson. Then add those properties to the hierarchy.

Quadrilaterals

• Opposite sides that are parallel • Opposite sides that have the same length

• Polygon with 4 sides

• Diagonals intersecting at midpoints As time allows, repeat the Always Sometimes Never routine by using the following statements. Parallelograms have 4 right angles. This is sometimes true. Some parallelograms are also rectangles, and rectangles have 4 right angles. Parallelograms have only 1 pair of parallel opposite sides. This is never true. All parallelograms have 2 pairs of parallel opposite sides.

Trapezoids • At least 1 pair of parallel sides

Parallelograms • Opposite sides that are parallel • Opposite sides that have the same length • Diagonals intersecting at midpoints

Parallelograms have diagonals that are the same length. This is sometimes true. Some parallelograms are also rectangles, and rectangles have diagonals that are the same length. We showed that the diagonals of parallelograms intersect at their midpoints. How does the hierarchy show you that this is true for all parallelograms but not for all trapezoids? Parallelograms are listed below trapezoids in the hierarchy. Because the property is listed as a property of parallelograms, it does not always apply to the quadrilaterals above it in the hierarchy. Are all properties of parallelograms properties of trapezoids? How do you know? No. Parallelograms are classified as trapezoids in the hierarchy, so all properties of trapezoids are properties of parallelograms but not all properties of parallelograms are properties of trapezoids.

104

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 4

Read the following statement from Launch. All parallelograms are quadrilaterals, but not all parallelograms are trapezoids. Invite students to turn and talk about their original conjecture, whether they would change it and why.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Classify trapezoids and parallelograms based on their properties. Facilitate a class discussion about classifying trapezoids and parallelograms based on their properties by using the following prompts. Encourage students to restate or add to their classmates’ responses. Continue to display the hierarchy of quadrilaterals. When is a shape classified as a quadrilateral but not as a trapezoid? If a shape has 4 sides but no pairs of parallel sides, then it is a quadrilateral but not a trapezoid. What are some properties of trapezoids? A trapezoid has exactly 4 straight sides and at least 1 pair of parallel sides. What makes a trapezoid a special type of quadrilateral? A trapezoid is a 4-sided polygon, which is a quadrilateral, that has at least 1 pair of parallel sides. © Great Minds PBC

105

5 ▸ M5 ▸ TA ▸ Lesson 4

EUREKA MATH2 Florida

Why is trapezoid a more specific name for a shape than quadrilateral ? Trapezoid is a more specific name because it gives us more information than the name quadrilateral. Quadrilateral only tells us the shape has 4 straight sides, but trapezoid tells us that the shape also has at least 1 pair of parallel sides. What makes a parallelogram a special type of trapezoid? A parallelogram is a trapezoid that has 2 pairs of parallel sides. What additional information does the name parallelogram give us about a quadrilateral that the name trapezoid does not? When a figure is classified as a parallelogram, we know that its opposite sides are parallel. When a figure is classified as a parallelogram, we know that its opposite sides are the same length. When a figure is classified as a parallelogram, we know that its diagonals intersect at their midpoints.

106

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 4

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET PRACTICE

I can classify trapezoids and parallelograms.

Florida B.E.S.T. MA.5.GR.1.1

3. Mark each statement as always true, sometimes true, or never true. Statement

Name

Always True

Sometimes True

Never True

Date

A trapezoid has 2 pairs of parallel sides.

1. Consider the polygons shown.

A trapezoid can have more than 4 sides.

A trapezoid is a quadrilateral.

A quadrilateral is a trapezoid.

4. Mark each statement as true or false. If the statement is false, sketch an example that shows why.

a. Circle each trapezoid. b. Use red to color each parallelogram.

Statement

True

False

Sketch

2. Consider the trapezoid. A parallelogram has only 1 pair of parallel sides.

A parallelogram cannot have any lines of symmetry.

126°

54° a. How many right angles are in the trapezoid?

b. Is this true for all trapezoids? Explain. No. Having right angles is not a property of trapezoids.

PROBLEM SET

107

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 4

Statement

The diagonals of a parallelogram intersect at their midpoints.

A parallelogram has side lengths that are all equal.

True

False

6. Sana says that because all parallelograms are trapezoids, all trapezoids must also be parallelograms. Is she correct? Explain your reasoning with a picture and words.

Sketch

Sana is incorrect. All parallelograms are trapezoids, but not all trapezoids are parallelograms. Some trapezoids have only 1 pair of parallel sides, so they are not parallelograms. Parallelograms have 2 pairs of parallel sides.

This drawing shows a trapezoid that is not a parallelogram.

5. Consider the trapezoid.

a. Kelly says that a property of trapezoids is that they have 1 pair of opposite sides with equal length. Toby disagrees. He says that Kelly’s statement describes a feature of this trapezoid, not a property of all trapezoids. Who is correct? Explain. Toby is correct. A property of trapezoids must be true for all trapezoids. Because not all trapezoids have 1 pair of opposite sides with equal length, Kelly’s statement describes a feature of this trapezoid, not a property of all trapezoids. b. Sketch a trapezoid that does not have 1 pair of opposite sides with equal length. Sample:

108

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 4

EXIT TICKET

I can classify trapezoids and parallelograms.

Florida B.E.S.T. MA.5.GR.1.1

Name

Date

1. Consider the polygons shown.

T T

T P

a. Circle each polygon that is a quadrilateral. b. Write the letter T in each polygon that is a trapezoid. c. Write the letter P in each polygon that is a parallelogram. 2. When can a trapezoid be classified as a parallelogram? A trapezoid can be classified as a parallelogram if both pairs of opposite sides are parallel.

109

LESSON 5

Classify rectangles and rhombuses based on their properties. I can classify rectangles and rhombuses based on their properties.

Lesson at a Glance Students learn that rectangles and rhombuses are special types of parallelograms. They explore side length, diagonals, and lines of symmetry to describe properties of rectangles and rhombuses. Students continue their work with the quadrilateral hierarchy, classifying certain parallelograms as rectangles or rhombuses.

Key Questions • What makes a rectangle a special type of parallelogram? • What additional information does the name rhombus give us about a quadrilateral that the name parallelogram does not?

Mathematical Thinking and Reasoning Standard MTR.4.1

Language Objectives • Listen to statements that describe a quadrilateral to determine if they are true for rectangles, rhombuses, or both. • Orally justify thinking by using properties related to sides, angles, and lines of symmetry. • Orally and in writing, describe why a figure is classified in one or more categories by using the properties of the figure and the relationships among other categories in the hierarchy.

• How are the diagonals of rectangles and rhombuses similar and different?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 5

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Prepare four signs on paper—two labeled Rectangles and two labeled Rhombuses. Hang the signs up in different locations in the room.

Launch 5 min Learn 35 min • Construct a Rhombus • Analyze a Rectangle • Hierarchy of Quadrilaterals • Problem Set

Land 10 min

• Blank paper (5 sheets) • Ruler • Index card • 4ʺ protractor • Scissors

Students • Blank paper • Ruler • Index card • 4ʺ protractor • Scissors

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD5 Classify quadrilaterals into

MA.5.GR.1.1 Classify triangles or quadrilaterals

categories based on attributes and explain the classification. (MA.5.GR.1.1)

into different categories based on shared defining attributes. Explain why a triangle or quadrilateral would or would not belong to a category.

111

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 5

Fluency

Whiteboard Exchange: Round Decimals Students round a number to the nearest ten and one to build fluency with rounding decimals from module 4. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display 8.7 ≈

Round 8.7 to the nearest ten.

Display the rounded value and then display 8.7 ≈

Round 8.7 to the nearest one.

Display the rounded value.

8.7 ≈

Repeat the process with the following sequence:

112

12.1 ≈

35.06 ≈

164.83 ≈

160

12.1 ≈

35.06 ≈

164.83 ≈

165

Teacher Note Consider displaying or prompting students to draw a vertical number line for each unit they round to as a visual aid to support understanding.

10 8.7

9 8.7

8.5

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 5

Counting on the Number Line by Milliliters and Liters Students count by a unit of 500 milliliters and rename milliliters as liters to build fluency with converting measurements from module 4. Display the number line. Use the number line to count forward by 500 milliliters to 3,000 milliliters. The first measurement you say is 0 milliliters. Ready? Display each measurement one at a time on the number line as students count.

0 mL, 500 mL, … , 2,500 mL, 3,000 mL Now count forward by 500 milliliters again. This time rename every 1,000 milliliters as a number of liters. The first measurement you say is 0 liters. Ready?

0 mL

500 mL 1,000 mL 1,500 mL 2,000 mL 2,500 mL 3,000 mL

500 mL

1,500 mL

2,500 mL

500 mL

1 L 500 mL

2 L 500 mL

Display each measurement one at a time on the number line as students count.

0 L, 500 mL, … , 2,500 mL, 3 L Now count forward by 500 milliliters again. This time use mixed units, liters and milliliters, when possible. The first measurement you say is 0 liters. Ready? Display each measurement one at a time on the number line as students count.

0 L, 500 mL, … , 2 L 500 mL, 3 L

113

5 ▸ M5 ▸ TA ▸ Lesson 5

EUREKA MATH2 Florida

Choral Response: Lines of Symmetry Students determine whether a given line is a line of symmetry for a figure to develop fluency with classifying quadrilaterals based on their properties. After asking each question, wait until most students raise their hands, and then signal for students to respond. Raise your hand when you know the answer to each question. Wait for my signal to say the answer. Display the figure. Is the line shown a line of symmetry for the figure? Yes. Display the line disappearing. How many total lines of symmetry does the figure have?

2 Display all lines of symmetry. Repeat the process with the following sequence:

114

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 5

Launch

Students examine and compare four quadrilaterals. Introduce the Which One Doesn’t Belong? routine. Display the picture of the four figures. A.

Invite students to study the figures. Give students 1 minute to find a category in which three of the items belong but a fourth item does not. When time is up, invite students to explain their chosen categories and to justify why one item does not fit. © Great Minds PBC

115

5 ▸ M5 ▸ TA ▸ Lesson 5

EUREKA MATH2 Florida

Highlight responses that emphasize reasoning about properties of quadrilaterals and the hierarchy of quadrilaterals. Figure A doesn’t belong because it has 4 sides of equal length. Figure B doesn’t belong because it has 4 right angles. Figure C doesn’t belong because it has 2 right angles. Figure D doesn’t belong because it is not a trapezoid. If students suggest that figure C doesn’t belong because it is a trapezoid, remind them that figures A and B are also trapezoids. Ask questions that invite students to make connections between trapezoids and parallelograms. • Which of the figures are quadrilaterals? How do you know? • Which of the figures are trapezoids? How do you know? • Which of the figures are parallelograms? How do you know? Ask questions that invite students to use precise language, make connections, and ask questions of their own. Are any of the figures rectangles? How do you know? Yes. Figure B is a rectangle. I know because it has 4 right angles. Are any of the figures rhombuses? How do you know? Yes. Figure A is a rhombus. I know because it has 4 sides that are the same length. Transition to the next segment by framing the work. Today, we will learn about the properties of two special types of parallelograms, rectangles and rhombuses.

116

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 5

Learn

Construct a Rhombus Materials—T/S: Blank paper, ruler, index card, protractor, scissors

Students construct a rhombus and identify its properties. Let’s construct rhombuses to identify properties of rhombuses. What must these figures include? The figures need to have opposite sides that are parallel and 4 sides of equal length.

Teacher Note The plural forms of the word rhombus are rhombuses, which is the form used here, and rhombi, which is also correct.

Guide students in the construction. As you complete each part of the construction, direct students to do the same. • Draw an angle that is not a straight angle. • Use your ruler to mark points on the rays of the angle that are the same distance from the endpoint. These points will help you form two of the sides of your rhombus that are of equal length. • Label the vertex of the angle as B and the points marked on the rays as A and C. These three points will be three of the vertices of the rhombus. • Use a straightedge and right-angle tool to draw a line that is parallel to one of the sides of the angle, making sure it passes through the point marked on the other ray. • Repeat for the other side of the original angle. • Label the last vertex D. • Label parallel sides with arrow marks. • Measure the sides and angles and write the measurements on the construction. Differentiation: Challenge Invite students who need additional challenge to write different construction instructions for a rhombus that use the same tools.

117

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 5

A 3 in D 119˚ 61˚

3 in

61˚

119˚ 3 in C

Circulate as students work, checking that their constructions are correct. When students are finished with the construction, use the following prompts. In the previous lesson, some of the parallelograms that we created had 4 sides with the same length, and others did not. But we classified them all as parallelograms. Today, all the parallelograms we constructed have 4 sides that have the same length. When can a parallelogram also be classified as a rhombus? A parallelogram can also be called a rhombus if it has 4 sides that have the same length. Choose three to five student samples to collect. Select a variety of rhombuses. Then display the student samples. Or consider showing the following samples.

5 cm 5 cm

5 cm

5 cm 5 cm

5 cm

Teacher Note Ensure that a square is among the sample constructions collected for display. Consider preparing a construction of a square to include in the samples if a square is not constructed by a student.

5 cm

Are rhombuses parallelograms? How do you know? Yes. Rhombuses have 2 pairs of parallel sides. What makes a rhombus different from other parallelograms? All 4 sides must have the same length.

118

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 5

Have students cut out their rhombuses. Then invite students to fold their rhombus to determine how many lines of symmetry, if any, it has. Direct students to turn and talk about how many lines of symmetry their rhombus has and where the lines of symmetry are. The 2 diagonals in a rhombus are lines of symmetry. Are the diagonals in all parallelograms lines of symmetry? Not necessarily. It depends on what the parallelogram looks like. We think that the 2 diagonals in a rhombus are lines of symmetry, but let’s look at some more examples. Open and display the Geometry World: Symmetry Explorer digital interactive. Create a rhombus and locate a line of symmetry. This line of symmetry passes through the opposite vertices of this rhombus. That means that the diagonal of this rhombus is a line of symmetry. Display several rhombuses to show that the diagonals of the rhombus are lines of symmetry. Can we say that it’s a property of rhombuses to have at least 2 lines of symmetry? Why?

Universal Design for Learning: Representation If you used a quadrilaterals chart in the previous lessons, consider adding examples of specific rhombuses and rectangles, including their properties. Figure Parallelogram

5 cm

Rhombus

Write the following statement: All rhombuses have at least 2 lines of symmetry. Does this statement describe a property of rhombuses? How do you know?

Rectangle

6 cm

• 2 pairs of parallel sides

• Opposite sides have the same length.

• 4 sides have the same length. • At least 2 lines of symmetry For all rectangles: • 4 right angles

Yes. It is a property because it’s true of all rhombuses. Direct students to Think–Pair–Share about the following question.

For all parallelograms:

For all rhombuses:

Yes. All rhombuses have 2 diagonals, and both are lines of symmetry. Create parallelograms that are not rhombuses to show that the diagonals of parallelograms that are not rhombuses are not lines of symmetry.

Properties

• At least 2 lines of symmetry

Do the diagonals in your rhombus intersect at their midpoints? How do you know? Yes. I measured them. Yes. The diagonals of all parallelograms intersect at their midpoints.

119

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 5

Ask students to measure the four angles around the intersection point of the diagonals. Invite them to turn and talk about what they notice. Are the diagonals of a rhombus perpendicular? How do you know? Yes. The diagonals intersect at right angles. Invite students to turn and talk about properties of rhombuses that differentiate them from other parallelograms.

Analyze a Rectangle Materials—S: Ruler, protractor

Students analyze a rectangle and identify its properties. Display the rectangle from problem 1.

Is this figure a rectangle? How do you know? Yes. It’s a quadrilateral with 4 right angles. Are rectangles parallelograms? How do you know? Yes. Opposite sides in rectangles are parallel.

We determined that rhombuses are special types of parallelograms with diagonals that are lines of symmetry. Because rectangles are also parallelograms, let’s see what we notice about lines of symmetry in rectangles.

120

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 5

Pair students. Direct students to problem 1 in their books. Ask students to complete parts (a)–(e) of problem 1 with their partner. 1. Consider rectangle ABCD.

60°

120° 120°

60°

a. Use a ruler to draw the diagonals of the rectangle. b. Measure the diagonals and write their lengths.

5 _ 1 2

inches

5 _ 1 2

inches

c. What do you notice about the diagonal lengths of the rectangle? The diagonals are the same length. d. Measure the four angles around the intersection point of the diagonals. Record the measurements on the figure. e. Are the diagonals perpendicular? How do you know? No. They do not intersect at right angles.

121

5 ▸ M5 ▸ TA ▸ Lesson 5

EUREKA MATH2 Florida

When students are finished, bring the class together. The rectangle in problem 1 has diagonals that are the same length. What does a parallelogram with diagonals of different lengths look like? It is not a rectangle. It does not have any right angles. Ask students to turn and talk about whether they think the 2 diagonals in a rectangle always have the same length. Open and display the Geometry World: Diagonal Explorer digital interactive. Create a rectangle. Invite students to Think–Pair– Share about what they notice about the diagonals of the rectangle. The diagonals of a rectangle are the same length. Display several different rectangles to show that the diagonals in a rectangle always have the same length. Having diagonals that are the same length is a property of rectangles because it is true for all rectangles, but it is not a property of parallelograms because it is not true for all parallelograms.

122

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 5

Ask students to complete parts (a) and (b) of problem 2 with their partner. 2. Another copy of rectangle ABCD is shown. a. Use your ruler to draw the rectangle’s lines of symmetry.

b. Sketch a parallelogram without lines of symmetry.

123

5 ▸ M5 ▸ TA ▸ Lesson 5

EUREKA MATH2 Florida

When students are finished, bring the class together. How many lines of symmetry does your rectangle have? Explain.

2. The lines of symmetry are the lines through the midpoints of opposite sides. The 2 lines through the midpoints of opposite sides of a rectangle are lines of symmetry. Do all parallelograms have 2 lines of symmetry through the midpoints of opposite sides? Not necessarily. It depends on what the parallelogram looks like. We think that every rectangle has at least 2 lines of symmetry and that 2 of those lines of symmetry are through the midpoints of opposite sides. Let’s look at some more examples. Open and display the Geometry World: Symmetry Explorer digital interactive. Create a rectangle. Then use the lines to find lines of symmetry in the figure. Invite students to Think–Pair–Share about what they notice about the lines of symmetry. A rectangle has at least 2 lines of symmetry. There are 2 lines of symmetry in a rectangle. The lines of symmetry cross through the midpoints of opposite sides of the rectangle. Display several different rectangles to show that all rectangles have at least 2 lines of symmetry. Display a square as well to show an example of a rectangle with more than 2 lines of symmetry. Write the following statement: All rectangles have at least 2 lines of symmetry. Does this statement describe an attribute or property of rectangles? How do you know? The statement describes a property. I know because it’s true of all rectangles. Invite students to turn and talk about what makes a rectangle a special type of parallelogram.

124

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 5

Hierarchy of Quadrilaterals

Multilingual Learner Support

Materials—T: Signs

Students use the hierarchy of quadrilaterals to classify and analyze statements about rectangles and rhombuses. Quadrilaterals

Display the hierarchy of quadrilaterals that students created in the previous lesson.

• Polygon with 4 sides

Are all rhombuses parallelograms? How do you know? Yes. Opposite sides of a rhombus are parallel. Are all rectangles parallelograms? How do you know?

Trapezoids • At least 1 pair of parallel sides

Yes. Opposite sides of a rectangle are parallel. Are all rectangles rhombuses? How do you know? No. Rectangles do not necessarily have 4 sides that are the same length.

If you created a quadrilaterals chart in previous lessons, consider reviewing familiar terms (parallel sides, midpoint, right angles, and lines of symmetry) while gesturing to the appropriate visuals before beginning the discussion.

Parallelograms • Opposite sides that are parallel • Opposite sides that have the same length • Diagonals intersecting at midpoints

Are all rhombuses rectangles? How do you know? No. Not all rhombuses have right angles. Because rhombuses and rectangles are parallelograms, we can add them both below parallelograms in the hierarchy. But because rhombuses are not always rectangles, we cannot place rhombuses below rectangles. Similarly, rectangles are not always rhombuses, so we cannot place rectangles below rhombuses.

125

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 5

Add rectangles and rhombuses to the class hierarchy. Invite students to summarize the properties of rectangles that they learned in this lesson. Then add those properties to the hierarchy. • 4 right angles • Diagonals that have the same length • At least 2 lines of symmetry Invite students to summarize the properties of rhombuses that they learned in this lesson. Then add those properties to the hierarchy. • 4 sides that have the same length • At least 2 lines of symmetry

Quadrilaterals • Polygon with 4 sides

Promoting Mathematical Thinking and Reasoning Students engage in discussions that reflect on mathematical thinking (MTR.4) as they communicate about the definitions for categories of figures by using their properties and their relationships to other figures in the quadrilateral hierarchy. Ask the following questions to promote MTR.4:

Trapezoids

• How can we describe a special type of parallelogram by using the word rhombus?

• At least 1 pair of parallel sides

• How are you using the hierarchy when you determine properties of rectangles?

Parallelograms • Opposite sides that are parallel • Opposite sides that have the same length • Diagonals intersecting at midpoints

Rectangles • 4 right angles • Diagonals that have the same length • At least 2 lines of symmetry

Rhombuses • 4 sides that have the same length • At least 2 lines of symmetry

Direct students to the rhombus and rectangle signs around the room. I will read and display a statement. If the statement describes all rectangles, stand near a sign that reads rectangles. If the statement describes all rhombuses, stand near a sign that reads rhombuses. If the statement describes all rectangles and rhombuses, stand up near your seat. If the statement describes neither all rectangles nor all rhombuses, sit in your seat.

126

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 5

Give students a minute to ask questions and clarify the directions. Then read and display as many of the following statements as time allows. If students disagree about any of the statements, ask them to turn and talk. Encourage them to use the hierarchy to justify their thinking. • I have 4 sides. (Indicates both) • All my sides are the same length. (Indicates rhombuses) • All my angles have the same measure. (Indicates rectangles) • I have opposite sides that are the same length. (Indicates both) • I have at least 2 lines of symmetry. (Indicates both) • All my sides are the same length, and all my angles have the same measure. (Indicates neither)

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

127

5 ▸ M5 ▸ TA ▸ Lesson 5

Land

EUREKA MATH2 Florida

Debrief 5 min Objective: Classify rectangles and rhombuses based on their properties. Facilitate a class discussion about properties of parallelograms by using the following prompts. Encourage students to restate or add to their classmates’ responses. Continue to display the hierarchy of quadrilaterals. In a previous lesson, we showed that the diagonals of parallelograms intersect at their midpoints. Is that true for rhombuses? Is that true for rectangles? How do you know? Yes. I can see from the hierarchy that rhombuses and rectangles are special types of parallelograms. That means all properties of parallelograms are also properties of rhombuses and rectangles. How are the diagonals of rectangles and rhombuses similar and different? The diagonals of both rectangles and rhombuses intersect at their midpoints. The diagonals of rectangles have the same length, but do not intersect at right angles. The diagonals of rhombuses do not have the same length, but do intersect at right angles. The diagonals of rhombuses are lines of symmetry but the diagonals of rectangles are not lines of symmetry. What makes a rectangle a special type of parallelogram? A rectangle is a parallelogram that has 4 right angles. What additional information does the name rhombus give us about a quadrilateral that the name parallelogram does not? When a quadrilateral is classified as a rhombus, we know that the quadrilateral has 4 sides that are the same length. Invite students to turn and talk about what a shape that is both a rhombus and a rectangle might look like.

128

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 5

129

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 5

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can classify rectangles and rhombuses based on their properties.

PRACTICE

Florida B.E.S.T. MA.5.GR.1.1

Name

2. Identify each statement as a property of rhombuses only, rectangles only, or both rhombuses and rectangles. Statement

Date

1. Consider the parallelograms shown.

Rhombus Only

Rectangle Only

Opposite sides are parallel. All sides are the same length.

X X

There are at least

2 lines of symmetry.

a. Circle each rhombus.

130

Diagonals are the same length.

There are 4 right angles.

Diagonals are lines of symmetry.

b. Use red to color each rectangle.

PROBLEM SET

Rhombus and Rectangle

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 5

EXIT TICKET

I can classify rectangles and rhombuses based on their properties.

Florida B.E.S.T. MA.5.GR.1.1

For problems 3–6, draw a figure with the properties listed if it is possible. If it is not possible, explain why. 3. Draw a rectangle with 4 sides of the same length.

4. Draw a rectangle that is not a parallelogram.

Name

This is not possible. All rectangles have opposite sides of equal length and 2 pairs of parallel sides. All rectangles are also parallelograms.

5. Draw a rhombus with exactly 1 pair of parallel sides.

Date

1. Consider the quadrilaterals shown. a. Circle each quadrilateral that is a rhombus. b. Draw an X on each quadrilateral that is a rectangle.

6. Draw a rhombus that is also a rectangle.

This is not possible. All rhombuses are parallelograms with 2 pairs of parallel sides.

2. When can a parallelogram be classified as a rhombus?

A parallelogram can be classified as a rhombus if it has 4 sides of equal length.

3. When can a parallelogram be classified as a rectangle?

A parallelogram can be classified as a rectangle if it has 4 right angles.

PROBLEM SET

131

LESSON 6

Classify kites and squares based on their properties. I can classify kites and squares based on their properties.

Lesson at a Glance Students construct a kite and describe its properties. They use the properties of kites to determine that rhombuses are kites, but rectangles are not. When they construct a quadrilateral that is both a rectangle and a rhombus, students discover that the quadrilateral is a square and that squares are kites. Students add squares and kites to the hierarchy of quadrilaterals. They find that kites are unique because kites can only be classified as trapezoids when also classified as squares or rhombuses. This lesson introduces the term kite.

Key Questions • When is a quadrilateral classified as a kite? • Why is a square also a rectangle, a rhombus, and a kite?

Mathematical Thinking and Reasoning Standard MTR.4.1

Language Objectives • Orally and in writing, describe the properties shared by quadrilaterals and kites, properties shared by rhombuses and squares, and properties shared by rectangles and squares. • Analyze a set of quadrilaterals and orally justify why a given quadrilateral belongs to a category by using the quadrilateral’s properties and the relationships among other figures in the hierarchy.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 6

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 5 min Learn 35 min • Construct a Kite • Construct a Square • Hierarchy of Quadrilaterals • Problem Set

Land 10 min

• Blank paper • Ruler • 4ʺ protractor • Scissors

Students • Blank paper (2 sheets) • Ruler • 4ʺ protractor • Scissors • Index card

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD5 Classify quadrilaterals into

MA.5.GR.1.1 Classify triangles or quadrilaterals

categories based on attributes and explain the classification. (MA.5.GR.1.1)

into different categories based on shared defining attributes. Explain why a triangle or quadrilateral would or would not belong to a category.

133

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 6

Fluency

Whiteboard Exchange: Multiply Multi-Digit Whole Numbers Students multiply a two-digit number by a three-digit number to build procedural fluency with multiplying multi-digit whole numbers by using the standard algorithm. 45 × 367 = 16,515 Display 45 × 367 =

Write and complete the equation by using the standard algorithm.

367 45 3 3

1 835 2 2 + 14 680 1 1 1 6,5 1 5

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the product and the recording of the standard algorithm in vertical form. Repeat the process with the following problem: 35 × 602 =

21,070

Whiteboard Exchange: Round Decimals Students round a number to the nearest one and tenth to build fluency with rounding decimals from module 4. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display 2.93 ≈

Round 2.93 to the nearest one.

Display the rounded value and then display 2.93 ≈

2.93 ≈

2.9

Round 2.93 to the nearest tenth.

134

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 6

Display the rounded value. Repeat the process with the following sequence:

4.167 ≈

15.55 ≈

4.167 ≈

4.2

15.55 ≈ 15.6

30.458 ≈

30.458 ≈ 30.5

Teacher Note

Launch

Students sketch a kite. Direct students to problem 1 in their books. Sketch what comes to your mind when I say the word kite.

Students in grade 5 are not expected to master kites. Kites are included because their properties contrast with the quadrilaterals students are expected to master. It’s the contrast that supports students with identifying properties and attributes that help them classify and categorize figures with accuracy.

Give students about 1 minute to sketch a kite. 1. Sketch a kite. Sample:

Multilingual Learner Support The word kite has multiple meanings. The mathematical definition of the word kite will be formalized as the lesson progresses. To support the use of the word kite in this lesson, point out different meanings of the word: • A kite is a quadrilateral with at least 2 pairs of adjacent sides that have the same length. • A kite is a toy that is flown on a windy day. • A kite is a type of bird.

135

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 6

When time is up, invite students to compare their kites to another student’s kite and discuss any similarities and differences. Then display the picture of children launching a kite. Invite students to turn and talk about any similarities and differences they notice between the kites they drew and the kite in the picture. Not all kites that we fly look like the kite in the picture. Some look like birds, airplanes, or other shapes. Describe the shape of the kite in this picture. It has 4 sides. It has 4 angles. It is shaped like a diamond. It is made up of 4 triangles.

Universal Design for Learning: Action & Expression Consider offering an alternative means for constructing the kite. To minimize fine motor demands, provide a template such as the following.

Have students Think–Pair–Share about how the shape of the kite in the picture is similar to and different from the polygons classified in previous lessons.

I know this kite is a quadrilateral because it has 4 sides. It looks like this kite has a line of symmetry from the top corner to the bottom corner. The top two sides look like they are the same length, and the bottom two sides look like they are the same length.

The kite doesn’t have any parallel sides. In mathematics, there is a type of quadrilateral called a kite. If a kite is a quadrilateral, what must be true about a kite? A kite must have 4 sides. Transition to the next segment by framing the work. Today, we will learn the properties of a kite and compare its properties to the properties of other quadrilaterals.

136

I Direct students to use a straightedge to connect point I to points K and T, forming an angle. Then direct them to draw a fourth point, E, somewhere on the dashed line. Lastly, have students connect points K and T to point E to make a kite.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 6

Learn

Construct a Kite Materials—T/S: Blank paper, ruler, protractor, scissors

Students construct a kite and identify its properties. Let’s construct a kite. Guide students in the construction. As you complete each part of the construction, direct students to do the same. • Use a ruler to draw an acute, obtuse, or right angle. Mark a point on each ray to create two sides that are the same length, somewhere between 3 and 4 inches long for easiest cutting. • Label the vertex of the angle as I and the endpoints of the line segments as K and T. Label the points inside the angle so they are visible when the angle is cut out. • Use scissors to cut along the sides of the angle.

Teacher Note To ensure that students construct a kite and not a triangle, direct students not to make points K, T, and E collinear.

• Fold the angle in half, matching point K to point T. • Unfold the angle. Mark a point on the fold and label it E.

Universal Design for Learning: Representation The digital interactive Kite Construction helps students visualize many kites efficiently and accurately. They can see an angle of any measure, fold the angle in half, and move points along the fold to see different kites. Consider allowing students to experiment with the tool individually or demonstrating the activity for the whole class.

137

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 6

Complete the construction of the kite, thinking aloud as you work. I can draw the other two sides by using my ruler to connect points K and T to point E. I draw the line from the point on the edge to the point on the fold.

Direct students to complete their drawings. Then cut out the kite and have students do the same. Ask students to compare the kite they constructed to their sketch of a kite from earlier. Invite them to share any similarities and differences. Then invite students to measure the side lengths of their kite and Think–Pair–Share about what they find.

Teacher Note When marking a point on the fold line, students are likely to place it in a location similar to where you model it. However, some students may mark a point that will create a shape called a dart. This is also an accurate construction of a kite.

It is not necessary to name the dart for students, but recognize that the shape is a kite because it has two pairs of adjacent sides that are equal in measure.

Two of the sides are the same length, and the other two sides are also the same length. The same-length sides are the two sides that are next to each other, not the opposite sides like in a parallelogram.

Teacher Note If students construct a kite that is also a rhombus, they may find more than 2 pairs of adjacent sides that are the same length by grouping the sides in different ways. Because a rhombus is also a kite, we say that a kite has at least 2 pairs of adjacent sides that are the same length.

138

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 6

Yes, a kite has at least 2 pairs of adjacent sides that are the same length. Adjacent means “next to.” Adjacent sides are sides that are next to each other. Compare your kite to your partner’s kite. What do you notice about the symmetry of each kite? Each kite has at least 1 line of symmetry. A kite has 1 line of symmetry through a diagonal. Invite students to turn and talk about how kites are similar to and different from other quadrilaterals. Multilingual Learner Support

Construct a Square Materials—S: Blank paper, index card, protractor, ruler

Students construct a square and identify properties of a square. Let’s see how the kite relates to other quadrilaterals in our hierarchy. Display the rhombus and rectangle.

Consider supporting the term kite by having students tape or glue the kites they construct beside problem 1 in their books. Have them label their kite with the word kite. Direct students to use a different-color highlighter to highlight each pair of adjacent sides that are the same length and to sketch and label the line of symmetry.

139

5 ▸ M5 ▸ TA ▸ Lesson 6

EUREKA MATH2 Florida

Ask students to recall the properties of quadrilaterals by directing them to the hierarchy of quadrilaterals they created in previous lessons. Create a list as students answer the following questions. How can you classify the quadrilateral on the left? It can be classified as a trapezoid, parallelogram, and rectangle. What is the most specific name of the quadrilateral on the left? How do you know? Rectangle. It has 4 right angles. Write rectangle to identify it as the most specific name of the quadrilateral on the left. Then ask the following questions. Is the quadrilateral on the right also a trapezoid? Is it a parallelogram? Is it a rectangle? It can be classified as a trapezoid and as a parallelogram but not as a rectangle. What is the most specific name of the quadrilateral on the right? How do you know? Rhombus. All its sides are the same length. Write rhombus to identify it as the most specific name of the quadrilateral on the right. Then continue the discussion. Describe the symmetry of rhombuses. Rhombuses have 2 lines of symmetry through their diagonals. Describe the symmetry of rectangles. Rectangles have 2 lines of symmetry through the midpoints of their sides. Have students Think–Pair–Share about the following questions. Are either of these polygons also kites? That is, do either of them have at least 2 pairs of adjacent sides that are the same length and at least 1 line of symmetry through a diagonal? Yes, the quadrilateral on the right is also a kite.

140

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 6

How does knowing the properties of rectangles and rhombuses help you decide whether these figures are kites? All rhombuses have at least 2 pairs of adjacent sides that are the same length and at least 1 line of symmetry through a diagonal. All rhombuses are also kites. Not all rectangles have adjacent sides that are the same length. Not all rectangles have a line of symmetry through a diagonal. So not all rectangles are kites. Invite students to turn and talk about the following two questions. Can a rectangle ever be a rhombus? Can a rhombus ever be a rectangle? Pair students. Ask pairs to construct a quadrilateral that is both a rectangle and a rhombus by using tools such as a straightedge, ruler, protractor, and right-angle tool. Circulate as students work. As necessary, support students by asking the following questions: • What are the properties of rectangles? • What are the properties of rhombuses? • Does your quadrilateral have 2 sets of opposite sides that are parallel? • Does your quadrilateral have 4 right angles? • Are all 4 sides of your quadrilateral the same length? • Are the diagonals of your quadrilateral the same length? When students have drawn a quadrilateral that can be classified as both a rectangle and a rhombus, ask the following questions. What is the most specific name of the quadrilateral you constructed? How do you know? A square. All its sides are the same length, and all its angles are right angles. Squares are quadrilaterals that are both rectangles and rhombuses.

141

5 ▸ M5 ▸ TA ▸ Lesson 6

EUREKA MATH2 Florida

If squares are quadrilaterals that are both rectangles and rhombuses, how many lines of symmetry do squares have? Why? Squares have 4 lines of symmetry. 2 of the lines of symmetry go through the midpoints of the sides of the square because all squares are rectangles. 2 lines of symmetry are the diagonals of the square because all squares are rhombuses. Consider having students check that the square they constructed has 2 lines of symmetry through the midpoints of its sides and 2 lines of symmetry through its diagonals. Are all squares kites? Why? Yes. Squares have at least 2 pairs of adjacent sides that are the same length. Yes. Squares are rhombuses, and rhombuses are kites. So squares are kites. Have students turn and talk about how squares are similar to and different from other quadrilaterals.

142

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 6

Hierarchy of Quadrilaterals Students reason about where squares and kites belong in the hierarchy of quadrilaterals. Quadrilaterals

Display the hierarchy of quadrilaterals that students created in the previous lesson. Invite students to turn and talk about where to put kites in the hierarchy. Are all kites trapezoids? Why?

• Polygon with 4 sides

Trapezoids

Kites

• At least 1 pair of parallel sides

• At least 2 pairs of adjacent sides that have the same length • At least 1 line of symmetry

Parallelograms • Opposite sides that are parallel • Opposite sides that have the same length • Diagonals intersecting at midpoints

Not all kites have at least Rectangles Rhombuses 1 pair of parallel sides. A kite • 4 sides that have the same length • 4 right angles doesn’t have • Diagonals that have the same length • At least 2 lines of symmetry any parallel • At least 2 lines of symmetry sides unless it is a rhombus or a square. Squares Not all kites are trapezoids, so we can’t place kites under trapezoids.

• 4 lines of symmetry

143

5 ▸ M5 ▸ TA ▸ Lesson 6

EUREKA MATH2 Florida

All kites have at least 2 pairs of adjacent sides that are the same length. That makes them a special type of quadrilateral. Add kites and their properties to the hierarchy branching off from quadrilaterals. Are there any other quadrilaterals that can be classified as kites? Which ones and why? Yes. All rhombuses are kites because they have at least 2 pairs of adjacent sides that are the same length. Connect kites to rhombuses in the hierarchy. Invite students to Think–Pair–Share about where to put squares in the hierarchy. Squares are rectangles and rhombuses, so squares should go under rectangles and rhombuses. Are all squares parallelograms? How do you know? Yes. I know because opposite sides of a square are parallel. Yes. I know because squares are rectangles and rectangles are parallelograms. So squares are parallelograms. Yes. I know because squares are rhombuses and rhombuses are parallelograms. So squares are parallelograms. Are all squares trapezoids? Why? Yes. Squares have at least 1 pair of parallel sides. Yes. Squares are parallelograms, and parallelograms are trapezoids. So squares are trapezoids. Are all squares kites? Why? Yes. Squares have at least 2 pairs of adjacent sides that are the same length. What other properties of squares should we list that are not already listed in the hierarchy? Squares have 4 lines of symmetry: 2 lines of symmetry through the midpoints of their sides and 2 lines of symmetry through their diagonals.

144

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 6

Add squares and their property to the hierarchy under rectangles and rhombuses. Then display the four quadrilaterals. Have students turn and talk about how they could use the hierarchy of quadrilaterals to help them identify the names of the quadrilaterals. Say the following statements. Have students give a thumbs-up if they agree with the statement or a thumbs-down if they disagree. Invite volunteers to share their reasoning for why they agree or disagree with each statement. Quadrilaterals B and C are parallelograms.

Promoting Mathematical Thinking and Reasoning

Students engage in discussions that reflect on mathematical thinking (MTR.4) when they use the definitions and properties of types of quadrilaterals to determine where to place kites and squares in the hierarchy. Ask the following questions to promote MTR.4:

I agree. Both quadrilaterals have opposite sides that are parallel. Quadrilateral A is a kite because it has 2 angles that have the same measure.

• What details are important to think about when placing kites in the hierarchy? • How are you using the definition of a square to determine where to place squares in the hierarchy?

I disagree. Quadrilateral A doesn’t have 2 pairs of adjacent sides that are the same length.

Universal Design for Learning: Action & Expression

I disagree. Quadrilateral A is a trapezoid.

Consider providing time for students to reflect on questions such as the following. Students can turn and talk or jot ideas on a sticky note. • What did you learn? • What were you surprised to learn? • What do you still have questions about?

145

5 ▸ M5 ▸ TA ▸ Lesson 6

EUREKA MATH2 Florida

Quadrilateral B is not a kite. I disagree. Quadrilateral B has at least 2 pairs of adjacent sides that are the same length. I disagree. Quadrilateral B is a square, and all squares are kites. Quadrilateral D is not a trapezoid. I agree. Quadrilateral D does not have at least 1 pair of parallel sides. I agree. Quadrilateral D is a kite, and kites are only trapezoids if they are rhombuses or squares.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Classify kites and squares based on their properties. Facilitate a class discussion about properties of kites and squares by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Direct students to the hierarchy of quadrilaterals. When is a quadrilateral classified as a kite? A quadrilateral is a kite when it has at least 2 pairs of adjacent sides that are the same length.

146

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 6

Why are all squares also rectangles, rhombuses, and kites? Squares are rectangles because squares have 4 right angles. Squares are rhombuses because squares have 4 sides that are the same length. Squares are kites because squares have at least 2 pairs of adjacent sides that are the same length. When can a kite be classified as a trapezoid? If all 4 sides of a kite are the same length, then it is a rhombus. All rhombuses are trapezoids because rhombuses have at least 1 pair of parallel sides. If all 4 sides of a kite are the same length and it has 4 right angles, then it is a square. Squares are trapezoids because squares have at least 1 pair of parallel sides.

147

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 6

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can classify kites and squares based on their properties.

PRACTICE

Florida B.E.S.T.

148

Kite Rhombus

Rectangle

Sample:

3. Kite that is not a trapezoid

2. Kite with 4 equal side lengths and no right angles

Sketch the shape as described.

Parallelogram

b. Circle each square.

Trapezoid

a. Use red to color each kite.

Quadrilateral

Date

1. Consider the quadrilaterals shown.

Polygon

Name

4. Consider the polygons shown. Mark each name that can be used to classify the polygon. More than one name may be marked.

MA.5.GR.1.1

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 6

EXIT TICKET

I can classify kites and squares based on their properties.

Florida B.E.S.T. MA.5.GR.1.1

5. Scott knows that all rhombuses and squares are also kites. Because all rhombuses and squares are also trapezoids, Scott thinks that all kites must be trapezoids as well. Is he correct? Explain. Scott is incorrect. A kite can have parallel sides, but having parallel sides is not a property of kites. Because having at least 1 pair of parallel sides is a property of trapezoids but not a property of kites, not all kites are trapezoids.

Name

Date

1. Consider the quadrilaterals shown.

a. Circle each quadrilateral that can be classified as a kite. b. Write the letter S in each quadrilateral that can be classified as a square.

2. When can a rhombus be classified as a square? A rhombus can be classified as a square when all the angles are right angles.

PROBLEM SET

149

LESSON 7

Identify quadrilaterals from given properties. I can use properties to identify quadrilaterals.

Lesson at a Glance Students compare three quadrilaterals that look similar and attempt to identify them. When they see the same three quadrilaterals with markings that show right angles and same side lengths, they realize that markings help them identify properties of and classify quadrilaterals. Working in pairs, students sketch quadrilaterals based on given descriptions. They identify common properties of the quadrilaterals they sketch and of quadrilaterals sketched by classmates. Students identify how they can classify a group of quadrilaterals based on what the quadrilaterals have in common.

Key Questions • Why can we draw different types of quadrilaterals based on the same description?

Mathematical Thinking and Reasoning Standard

Language Objectives • Read descriptions of different quadrilaterals to draw quadrilaterals with the given properties and attributes. In partners, analyze sketches of quadrilaterals. • Orally and in writing, identify the attributes the quadrilaterals have in common and classify the quadrilaterals based on their common properties.

MTR.1.1

• What properties help us classify quadrilaterals?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 7

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

• Print or copy Quadrilateral Cards and cut out the cards. Prepare enough cards for each student pair to have the cards for Quadrilaterals 1–6 or Quadrilaterals 7–12. • Prepare 12 posters on paper titled Quadrilateral 1, Quadrilateral 2, Quadrilateral 3, … , Quadrilateral 12. Hang them around the room. • Provide tools for sketching quadrilaterals, such as rulers, straightedges, and index cards (right-angle tools), for students to self-select.

Launch 5 min Learn 35 min • Sketch Quadrilaterals • Classify Quadrilaterals • Problem Set

Land 10 min

• Quadrilateral Cards (in the teacher edition) • Paper (12 sheets)

Students • Sticky notes (6 per student pair) • Quadrilateral Cards (6 cards per student pair) • Index card • Ruler

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD5 Classify quadrilaterals into

MA.5.GR.1.1 Classify triangles or quadrilaterals

categories based on attributes and explain the classification. (MA.5.GR.1.1)

into different categories based on shared defining attributes. Explain why a triangle or quadrilateral would or would not belong to a category.

151

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 7

Fluency

Write and complete the equation by using the standard algorithm. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

43 × 809 = 34,787 ×

809 43 2

2427 3 + 32360 3 4,7 8 7

Display the product and the recording of the standard algorithm in vertical form. Repeat the process with the following problem: 54 × 760 = 41,040 .

152

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 7

Display 3.84 ≈

Round 3.84 to the nearest one.

Display the rounded value and then display 3.84 ≈ Round 3.84 to the nearest tenth.

3.84 ≈ 4 3.84 ≈ 3.8

Display the rounded value. Repeat the process with the following sequence:

Launch

5.279 ≈ 5

18.55 ≈ 19

50.456 ≈ 50

5.279 ≈ 5.3

18.55 ≈ 18.6

50.456 ≈ 50.5

Students reason about the most specific names of quadrilaterals. Display the three quadrilaterals A, B, and C.

These three quadrilaterals are slightly different in size and shape. Identified by their most specific names, one is a square, one is a rectangle, and one is a rhombus. © Great Minds PBC

153

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 7

Invite students to turn and talk about which quadrilateral is a square, which is a rectangle, and which is a rhombus. Have students write which quadrilateral they think is which on their whiteboards. Multilingual Learner Support

What would help you identify which quadrilateral is which?

To support students with naming each quadrilateral and explaining their rationale, consider providing the following sentence frame:

We need to know which quadrilaterals have right angles. We need to know which quadrilaterals have the same side lengths. If you only know which quadrilaterals have 4 right angles, can you identify the square? No. We can identify the square and the rectangle, but we would still need to know whether their sides are the same length to know which is which.

I know quadrilateral is a because .

If you only know which quadrilaterals have 4 sides that are the same length, can you identify the square? No. We can identify the square and the rhombus, but we would still need to know whether angles are right angles to know which is which. Display the three quadrilaterals with markings to show right angles and side lengths that are the same.

154

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 7

Invite students to turn and talk about which quadrilateral is a square, which is a rectangle, and which is a rhombus now that the quadrilaterals are marked to show right angles and equal side lengths. Have students compare what they think now to what they wrote on their whiteboards. Identify each of these quadrilaterals by its most specific name. How do you know which is which? I know quadrilateral A is a square because it has 4 right angles and 4 sides that are the same length. I know quadrilateral B is a rhombus because it has 4 sides that are the same length. I know quadrilateral C is a rectangle because it has 4 right angles. When you draw a quadrilateral, what should you include to be sure someone can identify what type of quadrilateral it is? We should include angle measures and mark right angles. We should measure and label side lengths. We should mark which sides are the same length and which sides are parallel. Transition to the next segment by framing the work. Today, we will sketch and identify quadrilaterals from given properties.

155

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 7

Learn

Sketch Quadrilaterals Materials—T: Posters; S: Sticky notes, Quadrilateral Cards, index card, ruler

Students sketch quadrilaterals from given descriptions. Display the description and quadrilaterals:

Description

Quadrilaterals

• parallelogram • 2 different side lengths

Invite students to Think–Pair–Share about which of quadrilaterals A–F can be drawn from the description and which cannot. Encourage students to share their reasoning. Quadrilaterals C and D can be drawn from the description. Both are parallelograms with 2 different side lengths. Quadrilaterals A and B cannot be drawn from the description. They are not parallelograms. Quadrilaterals E and F cannot be drawn from the description. They do not have 2 different side lengths.

156

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 7

Display quadrilaterals C and D. Can we classify both quadrilateral C and quadrilateral D as rectangles? Why?

No. Quadrilateral D is a rectangle, but quadrilateral C is not a rectangle. Quadrilateral C does not have 4 right angles. Although one of these quadrilaterals is a rectangle, we can’t classify them both as rectangles because one of the two quadrilaterals is not a rectangle. Can we classify both quadrilateral C and quadrilateral D as parallelograms? Why? Yes. Quadrilaterals C and D are both parallelograms. They both have opposite sides that are parallel and are the same length. Is there any other type of quadrilateral that we can classify quadrilaterals C and D as? Explain. Yes. Both quadrilaterals C and D are trapezoids. Why can we classify these quadrilaterals as trapezoids even though not all trapezoids are parallelograms and not all trapezoids have 2 different side lengths? These quadrilaterals are parallelograms, and all parallelograms are trapezoids. Because both quadrilaterals C and D are parallelograms and both are trapezoids, we can classify these quadrilaterals as parallelograms and classify them as trapezoids.

157

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 7

Pair students. Give each pair 6 sticky notes and a set of 6 Quadrilateral Cards, either a set of cards for quadrilaterals 1–6 or a set of cards for quadrilaterals 7–12. Quadrilateral 1

Quadrilateral 2

Quadrilateral 3

Quadrilateral 4

• 4 right angles

• 2 pairs of opposite sides that are parallel

• a parallelogram

• 2 pairs of opposite sides that are parallel

• 4 sides that are the same length

• at least 2 right angles

• 2 pairs of opposite sides that are the same length

Quadrilateral 5

Quadrilateral 6

Quadrilateral 7

Quadrilateral 8

• at least 2 pairs of adjacent sides that are the same length

• a trapezoid

• at least 1 pair of parallel sides

• a rhombus

Quadrilateral 9

Quadrilateral 10

Quadrilateral 11

Quadrilateral 12

• 4 right angles

• 4 sides that are the same length

• a parallelogram

• at least 2 pairs of sides that are the same length

• 2 pairs of opposite sides that are the same length

• at least 1 right angle

• no right angles

• 4 angles that have the same measure

• no parallel sides

Universal Design for Learning: Action & Expression Encourage students to refer to the hierarchy of quadrilaterals from previous lessons to help them sketch, label, and identify types of quadrilaterals.

Universal Design for Learning: Engagement Students may make assumptions about the given attributes and properties and become frustrated when their assumptions do not work or they encounter an unfamiliar property. For example, they may assume that the statement at least 2 right angles for quadrilateral 3 means the quadrilateral can be drawn with exactly 2 right angles, but a parallelogram with at least 2 right angles actually requires 4 right angles. The descriptions for quadrilaterals 11 and 12 include properties that have not previously been discussed. Consider discussing strategies such as the following for persevering and dealing with frustration: • Use self-talk with statements such as “I can do this.”

Invite students to discuss the types of quadrilaterals described by the properties and attributes on their assigned set of Quadrilateral Cards. Then direct pairs to sketch one quadrilateral for each sticky note. Instruct them to mark right angles, sides that are the same length, and parallel sides as necessary based on the quadrilateral’s description. Invite students to self-select materials such as rulers, straightedges, and right-angle tools to support the accuracy of their sketches. 158

• Have a growth mindset; instead of thinking “I don’t get it,” think “I don’t get it yet.” • Pause to take deep breaths and calm down before working again. • Choose a different approach. • Ask a classmate or the teacher a clarifying question.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 7

Circulate as pairs work. Check that students sketch the quadrilaterals described and include the appropriate markings on their sketches. Make sure students have one quadrilateral sketched on each sticky note and that they do not write any names of the quadrilateral on the sticky note. When pairs finish their sketches, have them put each sticky note on the poster that shows that quadrilateral’s card number. As students place their sticky notes, invite pairs to turn and talk about similarities and differences they notice in the other sketches on the poster.

Differentiation: Support To support students, have them first write names of quadrilaterals that have the properties or attributes listed on the card before attempting to sketch the quadrilateral.

The table shows the types of quadrilaterals that can be sketched for each description. If one of the quadrilaterals shown is not already on a particular poster, sketch the quadrilateral on a sticky note and add it to the appropriate poster. Quadrilateral 1

Quadrilateral 2

Quadrilateral 3

Quadrilateral 4

Quadrilateral 5

Quadrilateral 6

Quadrilateral 7

Quadrilateral 8

Quadrilateral 9

Quadrilateral 10 Quadrilateral 11 Quadrilateral 12

Then have students write the names of quadrilaterals that all the figures on the poster can be classified as on their corresponding Quadrilateral Cards. © Great Minds PBC

Teacher Note As students classify the quadrilaterals on the posters, remind them to select the names that would be correct for all the figures on the card. For example, if the description on the card says 4 right angles and 2 pairs of opposite sides that are the same length, the poster should include sketches of rectangles and squares. However, squares would not describe all quadrilaterals on the poster. All the quadrilaterals on the poster for that description could be classified as rectangles, parallelograms, and trapezoids, so students should write those names on the card.

159

5 ▸ M5 ▸ TA ▸ Lesson 7

Classify Quadrilaterals Materials—T: Posters

Students identify quadrilaterals based on common properties. Invite each pair of students to go to a poster for a quadrilateral that they did not sketch. For example, if students drew quadrilaterals 1–6, have them go to a poster for one of quadrilaterals 7–12. Invite students to turn and talk about similarities and differences among the sketches. Direct students to problem 1 in their books. Have them use the recording sheet in problem 1 to list 1–3 things that all the sketches on the poster have in common. Direct students to write the types of quadrilaterals that describe all quadrilaterals on the poster based on what the sketches have in common. Then have students move to the poster for another quadrilateral that they did not sketch. Allow students 2 or 3 minutes at each poster before having them move to another.

EUREKA MATH2 Florida

Promoting Mathematical Thinking and Reasoning When students sketch quadrilaterals from given properties and attributes and classify groups of quadrilaterals based on what the quadrilaterals have in common, they are actively participating in effortful learning (MTR.1). Ask the following questions to promote MTR.1: • What can you figure out about the properties of these quadrilaterals by looking at all the sketches? • What information or facts do you need to classify a group of quadrilaterals?

Circulate as students work. Ask the following questions to support student thinking: • What differences did you notice in the sketches on this poster? • What do all the sketches on the poster have in common? • How can you classify all the quadrilaterals on this poster?

Universal Design for Learning: Action & Expression Instead of having students list what the sketches have in common, consider inviting them to identify the quadrilaterals their classmates sketched. Make a second set of Quadrilateral Cards but cut off the description number. Give Quadrilateral Cards 7–12 to the students who sketched quadrilaterals 1–6, and give Quadrilateral Cards 1–6 to the students who drew quadrilaterals 7–12. Have students match each Quadrilateral Card to the correct poster by looking for the sketches of quadrilaterals that all share the properties and attributes listed on the card. Students should still write the types of quadrilaterals that describe all quadrilaterals on the card.

160

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 7

1. For each quadrilateral: • Write the quadrilateral number. • Write 1–3 things all the sketches have in common. • Write the names of quadrilaterals that describe all quadrilaterals on the card. Quadrilateral number: Things the sketches have in common: 1. 2. 3. Names that describe all quadrilaterals on the card:

Quadrilateral number: Things the sketches have in common: 1. 2. 3. Names that describe all quadrilaterals on the card:

161

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 7

When most students have visited 6 posters, have them return to their seats. Consider choosing several posters and inviting students to share what all the sketches on each poster have in common and the names that describe all the quadrilaterals on the poster.

Quadrilateral 1

Quadrilateral 2

Quadrilateral 3

Quadrilateral 4

• Squares

• Rhombuses

• Rectangles

• Parallelograms

• Rectangles

• Parallelograms

• Trapezoids

• Rhombuses

• Trapezoids

• Parallelograms

• Kites

• Trapezoids • Kites Quadrilateral 5

Quadrilateral 6

Quadrilateral 7

Quadrilateral 8

• Kites

• Trapezoids

• Squares • Rectangles • Rhombuses • Parallelograms • Trapezoids • Kites

Quadrilateral 9

Quadrilateral 10

Quadrilateral 11

Quadrilateral 12

• Rectangles

• Rhombuses

• Parallelograms

• Kites

• Parallelograms

• Trapezoids

• Trapezoids • Kites

162

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 7

Have students Think–Pair–Share about the following questions. Why can we sketch more than one type of quadrilateral from the same description? Different types of quadrilaterals have some of the same properties. Why are there so many different types of quadrilaterals on poster 6, but we can only classify the quadrilaterals on poster 6 as trapezoids? The only properties that all the quadrilaterals on poster 6 have in common are the properties of trapezoids. Why can we classify the quadrilaterals on poster 1 as so many types of quadrilaterals even though all the quadrilaterals on poster 1 are squares? Squares have the properties of rectangles, rhombuses, parallelograms, trapezoids, and kites, so we can classify squares as all those types of quadrilaterals.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

163

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 7

Land

Debrief 5 min Objective: Identify quadrilaterals from given properties. Use the following prompts to facilitate a class discussion about identifying quadrilaterals from given properties. Encourage students to restate or add on to their classmates’ responses. Display the description from Quadrilateral Card 4 and the quadrilaterals.

Description

Quadrilaterals

• 2 pairs of opposite sides that are parallel • 2 pairs of opposite sides that are the same length

Why can we draw different types of quadrilaterals based on the same description? Different types of quadrilaterals can have some of the same properties, such as right angles, parallel sides, and sides that are the same length. Describe any strategies you used when classifying quadrilaterals based on your classmates’ sketches. We looked for things that all the quadrilaterals on the poster had in common. We looked for sides that were marked as being the same length or that were marked as parallel. We looked for right angles.

164

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 7

The four quadrilaterals in the table each have 2 pairs of opposite sides that are parallel and that are the same length. Why can’t we classify all the quadrilaterals as rectangles because one of them is a rectangle? Not all the quadrilaterals are rectangles, so we can’t classify them as rectangles. Not all rhombuses and not all parallelograms are rectangles, so we can’t classify them as rectangles. Why can we classify these quadrilaterals as trapezoids even though not all trapezoids have 2 pairs of opposite sides that are parallel and that are the same length? All these quadrilaterals are parallelograms, and all parallelograms are trapezoids.

165

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 7

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET PRACTICE

I can use properties to identify quadrilaterals.

Florida B.E.S.T. MA.5.GR.1.1

Name

2. Write all the names from the word bank that can be used to classify each figure shown. Word Bank

Date

1. Mark each shape with the given property. Property At least 1 pair of opposite sides are parallel.

Trapezoid X

Parallelogram X

Rectangle X

Rhombus X

Square

Quadrilateral

Trapezoid

Parallelogram

Square

Rhombus

Kite b.

Quadrilateral

Trapezoid

Parallelogram

2 pairs of opposite sides are parallel.

Opposite sides are the same length.

Rectangle

Square Rhombus Kite

Quadrilateral

Quadrilateral Diagonals intersect at their midpoints.

Diagonals intersect at a right angle.

All sides are the same length.

All angles are right angles.

Rectangle

Trapezoid

f. Quadrilateral

Quadrilateral

Trapezoid

Kite

Parallelogram

Rhombus Kite

166

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 7

EXIT TICKET

I can use properties to identify quadrilaterals.

Florida B.E.S.T. MA.5.GR.1.1

3. Consider the quadrilaterals shown.

Name

Date

Write all the names for each figure shown. a. What properties do the figures have in common? Sample:

Quadrilateral

Opposite sides are parallel.

Trapezoid

All sides have the same length.

Parallelogram

Diagonals intersect at their midpoints.

Rectangle Kite Rhombus

b. What is the most specific name that could describe all three figures?

Square

Rhombus

Quadrilateral Trapezoid Parallelogram

PROBLEM SET

167

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 7 ▸ Quadrilateral Cards

Quadrilateral 1

Quadrilateral 2

Quadrilateral 3

Quadrilateral 4

Quadrilateral 5

Quadrilateral 6

• 4 right angles

• a parallelogram • at least 2 right angles

• 2 pairs of opposite sides that are parallel

• at least 2 pairs of adjacent sides that are the same length

• a trapezoid

• 4 sides that are the same length

• 2 pairs of opposite sides that are parallel

Quadrilateral 7

Quadrilateral 8

Quadrilateral 9

Quadrilateral 10

Quadrilateral 11

Quadrilateral 12

• at least 1 pair of parallel sides

• a rhombus

• 4 right angles

• a parallelogram

• 4 angles that have the same measure

• 2 pairs of opposite sides that are the same length

• 4 sides that are the same length

• at least 2 pairs of sides that are the same length

168

• 4 sides that are the same length

This page may be reproduced for classroom use only.

• 2 pairs of opposite sides that are the same length

• no right angles

• at least 1 right angle

• no parallel sides

LESSON 8

Classify quadrilaterals in a hierarchy based on properties. I can classify quadrilaterals.

Language Objectives

Lesson at a Glance Students analyze a triangle Venn diagram and recognize how triangles are sorted into categories based on their properties. After analyzing a similar Venn diagram for quadrilaterals, students compare the diagram to the hierarchy of quadrilaterals and sort a set of quadrilaterals and non-quadrilaterals based on their properties. Through the Critique a Flawed Response routine, students realize that although quadrilaterals are often identified by their most specific names, many can be classified as other types of quadrilaterals as well. There is no Problem Set for this lesson. Instead, there are activities for students to complete in pairs.

Key Questions • How do diagrams help us classify quadrilaterals based on their properties?

Mathematical Thinking and Reasoning Standard

• Interpret a Venn diagram for classifying quadrilaterals by orally describing the properties of each category in the Venn diagram and the relationships between the categories. • Orally justify the placement of a polygon in the quadrilateral Venn diagram by using the properties of quadrilaterals and the relationships between categories.

MTR.5.1

• If a quadrilateral can have more than one name, how do we know what to call it?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 8

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

• Consider tearing out the Sprint pages in advance of the lesson. • Consider whether to remove Quadrilateral Venn Diagram from the student books in advance or to have students remove them during the lesson. • Consider whether to remove Polygons from the student books and cut them out in advance or to have students prepare them during the lesson.

Launch 10 min Learn 30 min • Quadrilateral Venn Diagram • Sort Quadrilaterals • Classify Quadrilaterals

Land 10 min

• None

Students • Round to the Nearest One Sprint (in the student book) • Quadrilateral Venn Diagram (in the student book) • Polygons (in the student book) • Scissors • Glue

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD5 Classify quadrilaterals into

MA.5.GR.1.1 Classify triangles or quadrilaterals

categories based on attributes and explain the classification. (MA.5.GR.1.1)

into different categories based on shared defining attributes. Explain why a triangle or quadrilateral would or would not belong to a category.

171

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 8

Fluency

Sprint: Round Decimals Materials—S: Round to the Nearest One Sprint

Students round a number to the nearest one to build fluency with rounding decimal numbers from module 4. Have students read the instructions and complete the sample problems.

Round to the nearest one. 1. 4.7

≈

2. 24.19 ≈

Direct students to Sprint A. Frame the task: I do not expect you to finish. Do as many problems as you can, your personal best. Take your mark. Get set. Think! Time students for 1 minute on Sprint A. Stop! Underline the last problem you did. I’m going to read the answers. As I read the answers, call out “Yes!” if you got it correct. If you made a mistake, circle the answer. Read the answers to Sprint A quickly and energetically. Count the number you got correct and write the number at the top of the page. This is your personal goal for Sprint B. Celebrate students’ effort and success. Provide about 2 minutes to allow students to complete more problems or to analyze and discuss patterns in Sprint A. If students are provided time to complete more problems on Sprint A, reread the answers but do not have them alter their personal goals. 172

Teacher Note Consider asking the following questions to discuss the patterns in Sprint A: • What do you notice about problems 1–5? 12–22? • How did your strategy for rounding in problem 5 compare to your strategy for rounding in problem 6? Did your strategy change as the number of place values increased throughout the Sprint?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 8

Lead students in one fast-paced and one slow-paced counting activity, each with a stretch or physical movement. Point to the number you got correct on Sprint A. Remember, this is your personal goal for Sprint B. Direct students to Sprint B.

Teacher Note Count forward by 5 tenths in decimal form from 0 to 5 for the fast-paced counting activity. Count backward by 5 tenths in decimal form from 5 to 0 for the slow-paced counting activity.

Take your mark. Get set. Improve! Time students for 1 minute on Sprint B. Stop! Underline the last problem you did. I’m going to read the answers. As I read the answers, call out “Yes!” if you got it correct. If you made a mistake, circle the answer. Read the answers to Sprint B quickly and energetically. Count the number you got correct and write the number at the top of the page. Determine your improvement score and write the number at the top of the page. Celebrate students’ improvement.

Launch

Consider activating prior knowledge about triangles by reviewing types of triangles, with visuals of each type, before presenting the triangle Venn diagram.

• A right triangle is a triangle that has a right angle.

Students reason about triangles that are organized in a Venn diagram.

• An acute triangle is a triangle that has all acute angles.

Display triangle E. Invite students to Think–Pair–Share about how they would classify the figure. It is a polygon. It is a triangle. It is a right triangle. It is an isosceles triangle.

Multilingual Learner Support

• An obtuse triangle is a triangle that has an obtuse angle.

• An isosceles triangle is a triangle with at least 2 sides of equal length. Note that an equilateral triangle is also an isosceles triangle. • An equilateral triangle is a triangle with all sides of equal length.

173

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 8

Display the triangle Venn diagram.

Triangles

Right Isosceles

Acute

D E

A Equilateral

Obtuse

Ask students what they notice and wonder. I notice triangle E is in the section labeled Right and in the section labeled Isosceles. I notice this has a lot of sections. I notice all the triangles are labeled with a letter and sorted into categories. I notice some of the triangles are marked to show right angles and sides that are the same length. I notice the categories are different colors.

174

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 8

I notice that a triangle can be inside more than one section because it can be classified as more than one type. I wonder why the triangles are sorted this way. I wonder if all triangles can be sorted into at least one of the sections. I wonder if we can sort quadrilaterals by using a similar diagram. Point to the section labeled Right. Triangles B and E are both right triangles. How is triangle E different from triangle B? Triangle E is an isosceles triangle. It is also in the section labeled Isosceles. Triangle B is not an isosceles triangle. Triangle E has 2 sides that are the same length and triangle B does not. Point to triangle F. What can you say about triangle F? Triangle F is an isosceles triangle. Triangle F is an obtuse triangle. Triangle F has 2 sides that are the same length. Triangle F has an obtuse angle. Point to triangle G. What type of triangle is triangle G? How do you know? Triangle G is an equilateral triangle. I know because it is in the section labeled Equilateral. Triangle G is an equilateral triangle. I know because all its sides are the same length. Can we say triangle G is isosceles? Why? Yes. The section for equilateral triangles is inside the section for isosceles triangles. Yes. Equilateral triangles have at least 2 sides that are the same length.

175

5 ▸ M5 ▸ TA ▸ Lesson 8

EUREKA MATH2 Florida

Can we say triangle G is acute? Why? Yes. The section for equilateral triangles is inside the section for acute triangles. Yes. All angles of an equilateral triangle have measures less than 90°. Are all isosceles triangles equilateral triangles? There are isosceles triangles that are not in the section labeled Equilateral, so not all isosceles triangles are equilateral triangles. The section labeled Equilateral is completely inside the section labeled Isosceles, so all equilateral triangles are isosceles triangles, but only some isosceles triangles are equilateral triangles. Would it make sense to put a quadrilateral in this diagram? Why? No. A quadrilateral is not a triangle, so it does not go inside the diagram. Have students turn and talk about whether they think we can use a similar diagram to sort quadrilaterals. Transition to the next segment by framing the work. Today, we will classify and organize quadrilaterals in a Venn diagram.

176

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 8

Learn

Quadrilateral Venn Diagram Students analyze a quadrilateral Venn diagram and compare it to the hierarchy of quadrilaterals. Display the Venn diagram for quadrilaterals. Universal Design for Learning: Representation

Quadrilaterals

Consider analyzing a portion of the quadrilateral Venn diagram before showing the entire diagram. Ask students to turn and talk about which statement is true: All parallelograms are rectangles or All rectangles are parallelograms. Display the portion of the diagram with sections for parallelograms and rectangles.

Trapezoids

Parallelograms Rhombuses

Rectangles

Kites

Parallelograms Rectangles

Squares Ask how the diagram shows that all rectangles are parallelograms. Invite students to sketch a quadrilateral that would be in the section for rectangles and a quadrilateral that would be in the section for parallelograms but not in the section for rectangles.

177

5 ▸ M5 ▸ TA ▸ Lesson 8

EUREKA MATH2 Florida

What do you notice about this diagram? I notice that this Venn diagram is for quadrilaterals. I notice this diagram is big. I notice that the section labeled Squares is inside the sections labeled Rectangles, Rhombuses, and Kites. I notice that part of the section labeled Kites is inside the section labeled Parallelograms and part is not. Direct students to the hierarchy of quadrilaterals created in lesson 6. Invite students to turn and talk to compare the quadrilateral Venn diagram to the hierarchy of quadrilaterals. Point to the section for trapezoids on the displayed quadrilateral Venn diagram. Facilitate a discussion about the diagram by using the following prompts. Is the section for trapezoids inside or outside the section for quadrilaterals? Inside The section for trapezoids is inside the section for quadrilaterals because all trapezoids are quadrilaterals. Does that mean that all quadrilaterals are trapezoids? Why?

Universal Design for Learning: Representation To connect the diagram for quadrilaterals and the hierarchy of quadrilaterals, consider having students write each quadrilateral’s properties on Quadrilateral Venn Diagram. Or provide students with a copy of Quadrilateral Venn Diagram with the properties already written in each section.

No. Some quadrilaterals do not have any parallel sides. Point to the section for parallelograms. All parallelograms are quadrilaterals, but not all quadrilaterals are parallelograms. How does the diagram show that? The section for parallelograms is inside the section for quadrilaterals. There is space outside the parallelogram section for quadrilaterals that are not parallelograms. Why is the section for parallelograms inside the section for trapezoids? All parallelograms are trapezoids. What sections are completely inside the section for parallelograms? Why? Squares, rectangles, and rhombuses are all inside the section for parallelograms because they are all parallelograms.

178

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 8

Point to the section for rhombuses. Have students turn and talk about what they notice about the section for rhombuses. Are all rhombuses squares? How can you tell by looking at the diagram? No. Some rhombuses are squares, but some are not. Only part of the section for rhombuses includes squares. Why is the section for squares inside both the section for rectangles and the section for rhombuses? Squares are both rectangles and rhombuses. Have students Think–Pair–Share about why part of the section for kites is inside the section for parallelograms and part of it is not. All rhombuses are kites, and all rhombuses are parallelograms. So kites that are rhombuses are also parallelograms. Not all kites are rhombuses. So not all kites are parallelograms. If a polygon is in the section labeled Rectangles but not in the section labeled Squares, then the most specific name for the polygon is rectangle. What else can we classify a rectangle as? How can you tell from the diagram? Rectangles are parallelograms, trapezoids, and quadrilaterals. The section for rectangles is inside the sections for parallelograms, trapezoids, and quadrilaterals. It is not possible to be in the section for rectangles without being in the section for parallelograms, which is inside the section for trapezoids, which is inside the section for quadrilaterals. Invite students to turn and talk about how the quadrilateral Venn diagram can help them classify quadrilaterals.

179

5 ▸ M5 ▸ TA ▸ Lesson 8

EUREKA MATH2 Florida

Sort Quadrilaterals Materials—S: Quadrilateral Venn Diagram, Polygons, scissors, glue

Students sort quadrilaterals and non-quadrilaterals. Pair students. Direct them to remove Quadrilateral Venn Diagram and Polygons from their books. Have pairs cut out each polygon and work together to place each polygon in the correct section of Quadrilateral Venn Diagram. Circulate as students work. Encourage students to use the properties listed in the hierarchy of quadrilaterals to help them sort the polygons as needed. When pairs are finished, invite them to share the section in which they placed each polygon. As students share, consider asking clarifying questions such as the following: • Why did you place the polygon in that section? • How did you know the polygon is a parallelogram but not a rhombus?

Universal Design for Learning: Action & Expression The polygons do not need to be precisely cut out, but they will fit in Quadrilateral Venn Diagram better if they are cut close to the picture. Consider offering an alternative method for completing Quadrilateral Venn Diagram to reduce the fine motor demands of the task. Have students write the letter of each polygon in the correct section on the diagram.

• How did you know the polygon is not a kite? • How did you know the polygon is not a quadrilateral? If students disagree about the placement of a polygon, prompt them to justify their reasoning by using the properties of quadrilaterals. Once students identify the correct section for each polygon, have them glue it to the Venn diagram. What did you consider when you thought about where to place a polygon? I thought about the attributes of the polygon.

Multilingual Learner Support Consider inviting students to use the Share Your Thinking section of the Talking Tool to explain their reasoning for placing the polygons in the various sections of Quadrilateral Venn Diagram.

I thought about the properties of quadrilaterals. I thought about how many pairs of parallel sides a quadrilateral has or whether it has right angles. I noticed which sides of a quadrilateral are the same length. Have students turn and talk about how they can use Quadrilateral Venn Diagram to identify all the names of a quadrilateral and the most specific name of a quadrilateral.

180

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 8

Classify Quadrilaterals Students name all possible classifications for given quadrilaterals. Display the quadrilateral Venn diagram with sorted quadrilaterals. Promoting Mathematical Thinking and Reasoning

Quadrilaterals D

Trapezoids

Parallelograms

E I

Rhombuses

Kites

J C

Rectangles G

Squares B

When students apply their understanding of the hierarchy of quadrilaterals to help them place polygons in the quadrilateral Venn diagram, they are using patterns and structure (MTR.5). Ask the following questions to promote MTR.5: • How are the quadrilateral Venn diagram and hierarchy of quadrilaterals related? How could that help you organize the polygons? • How can you use what these quadrilaterals have in common to help you classify them?

Introduce the Critique a Flawed Response routine and present the following prompt: Tyler says that there are two trapezoids in the Venn diagram: polygons A and E. Give students 1 minute to identify the error or ambiguity. Invite students to share. Tyler thinks that a polygon can only be one kind of quadrilateral. Tyler might think that a trapezoid can only have 1 pair of parallel sides. Give students 2 minutes to correct Tyler’s statement based on their own understanding. Circulate and identify a few students to share their thinking. Purposefully choose work that allows for rich discussion about the properties of quadrilaterals.

181

5 ▸ M5 ▸ TA ▸ Lesson 8

EUREKA MATH2 Florida

A trapezoid is a quadrilateral with at least 1 pair of parallel sides. Parallelograms, rhombuses, rectangles, and squares all have 2 pairs of parallel sides, so they are all trapezoids. There are seven trapezoids in the diagram. Parallelograms, rhombuses, rectangles, and squares are also trapezoids because they are quadrilaterals with at least 1 pair of parallel sides. The trapezoids in the diagram are A, E, H, I, J, G, and B. Then facilitate a class discussion. Invite students to share their solutions with the whole group. Lead the class to consensus about how best to correct the flawed response. There are seven trapezoids in the diagram. They are polygons A, E, H, I, J, G, and B. Direct students to problem 1 in their books. Have them remain in pairs to complete the problem. Circulate as students work. Ask the following questions to support students as needed: • How can you tell whether a quadrilateral can be classified as something else by using the quadrilateral Venn diagram? • How can you tell whether a quadrilateral can be classified as something else by using the hierarchy of quadrilaterals? • How can you tell a quadrilateral’s most specific name by using the quadrilateral Venn diagram? • How can you tell a quadrilateral’s most specific name by using the hierarchy of quadrilaterals?

182

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 8

Use the quadrilateral Venn diagram with sorted quadrilaterals A–L to complete problem 1. 1. Mark each name of the polygon with X. Then circle the X for the most specific name of the polygon. Polygon

Quadrilateral

Trapezoid

Parallelogram

Rectangle

Rhombus

Square

Kite

Invite pairs to compare and discuss their answers with another pair. Then ask the following questions. Which quadrilateral can be classified as any type of quadrilateral? Where is it in the quadrilateral Venn diagram? Polygon B can be classified as any type of quadrilateral. Polygon B is in the section for squares, which is inside all the other sections. Why does it make sense that a square is every other type of quadrilateral? It makes sense because a square has the properties of all the other quadrilaterals. If a quadrilateral is in a section that is inside another section, then it has the names of both sections. Invite students to turn and talk about whether they prefer the quadrilateral Venn diagram or hierarchy of quadrilaterals to help them classify quadrilaterals. © Great Minds PBC

183

5 ▸ M5 ▸ TA ▸ Lesson 8

Land

EUREKA MATH2 Florida

Debrief 5 min Objective: Classify quadrilaterals in a hierarchy based on properties. Facilitate a class discussion about classifying quadrilaterals by using the following prompts. Encourage students to restate or add on to their classmates’ responses. What properties help us classify quadrilaterals? The sides, if any, that are the same length The number of pairs of parallel sides The number of right angles Where diagonals intersect and whether they are the same length The number of lines of symmetry How do diagrams help us classify quadrilaterals based on their properties? Give an example. A quadrilateral can have properties of more than one type of quadrilateral, and a diagram shows which properties of quadrilaterals are included in the name of another quadrilateral. All parallelograms are trapezoids, so parallelograms are under trapezoids in the hierarchy of quadrilaterals and inside the section for trapezoids in the Venn diagram. If a quadrilateral can have more than one name, how do we know what to call it? We usually call a quadrilateral by its most specific name. That way, we can identify all the quadrilateral’s properties. We call any polygon by its most specific name so it is clear which properties the polygon has without us having to use lots of words to describe them.

184

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 8

185

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA ▸ Lesson 8

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Number Correct:

1. 2.9

≈

2. 4.9

≈

3. 3.1

≈

4. 6.1

≈

5. 5.4

≈

7. 7.62 ≈ 8. 8.62 ≈ 9. 3.57 ≈ 10. 5.57 ≈ 11. 9.57 ≈ 12. 14.2 ≈ 13. 14.4 ≈ 14. 26.8 ≈ 15. 26.6 ≈ 16. 61.6 ≈ 17. 61.62 ≈ 18. 37.28 ≈ 19. 37.45 ≈ 20. 49.36 ≈ 21. 49.91 ≈ 22. 49.55 ≈

186

Improvement:

Round to the nearest one.

6. 5.48 ≈

Number Correct:

5 8 9 4 6 10 14 14 27 27 62 62 37 37 49 50 50

23. 3.816

≈

24. 13.168 ≈ 25. 7.092

≈

26. 57.902 ≈ 27. 6.457

≈

28. 76.574 ≈ 29. 4.803

≈

30. 64.308 ≈ 31. 3.006

≈

32. 30.016 ≈ 33. 80.601 ≈

4 13 7 58 6 77

≈

36. 55.529 ≈ 37. 255.259 ≈ 38. 79.809 ≈ 39. 579.980 ≈ 40. 99.479 ≈ 41. 199.794 ≈ 42. 499.506 ≈ 43. 699.005 ≈ 44. 999.609 ≈

3. 2.1

≈

4. 5.1

≈

5. 4.4

≈

10. 4.57 ≈

35. 7.2

≈

9. 2.57 ≈

2. 3.9

8. 7.62 ≈

≈

7. 6.62 ≈

34. 2.7

≈

6. 4.48 ≈

1. 1.9

11. 9.57 ≈ 12. 13.2 ≈ 13. 13.4 ≈

14. 25.8 ≈

255

15. 25.6 ≈

16. 51.6 ≈

580

17. 51.62 ≈

18. 27.28 ≈

200

19. 27.45 ≈

500

20. 39.36 ≈

699

21. 39.91 ≈

1,000

22. 39.55 ≈

4 7 8 3 5 10 13 13 26 26 52 52 27 27 39 40 40

23. 2.816

≈

24. 12.168 ≈ 25. 6.092

≈

26. 56.902 ≈ 27. 5.457

≈

28. 75.574 ≈ 29. 3.803

≈

30. 63.308 ≈ 31. 2.006

≈

32. 20.016 ≈ 33. 70.601 ≈

3 12 6 57 5 76 4 63 2 20 71

34. 1.7

≈

35. 7.1

≈

36. 55.529 ≈ 37. 155.259 ≈ 38. 79.809 ≈ 39. 479.980 ≈ 40. 99.479 ≈ 41. 199.794 ≈ 42. 399.506 ≈ 43. 599.005 ≈ 44. 999.609 ≈

56 155 80 480 99 200 400 599 1,000

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA ▸ Lesson 8

EXIT TICKET

I can classify quadrilaterals.

Florida B.E.S.T. MA.5.GR.1.1

Name

Date

Write all names for each quadrilateral shown. Circle the most specific name for each figure.

Trapezoid Parallelogram Rectangle Square Kite Rhombus

Trapezoid

187

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA

Topic A Quiz Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

A N SW E R K E Y ▸ TO PI C Q U I Z A-1 Quadrilateral 1. Circle all the names for each figure.

Trapezoid Parallelogram Rhombus Quadrilateral

Rectangle

Trapezoid

Square

Parallelogram Rhombus

Quadrilateral

Rectangle

Trapezoid

Square

Parallelogram Rhombus Rectangle Square

Quadrilateral Trapezoid Parallelogram Rhombus Rectangle Square

188

221

222

A N S W E R K E Y ▸ TO P I C Q U I Z A-1

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA

4. Indicate all the quadrilaterals that have the properties listed.

2. Which statements correctly describe a relationship between types of quadrilaterals? Choose all that apply.

Property

A. All rectangles are parallelograms.

Trapezoid

Parallelogram

Rectangle

Rhombus

Square

B. All trapezoids are rectangles. C. All parallelograms are rectangles.

4 right

D. All quadrilaterals are trapezoids.

angles

E. All squares are parallelograms.

Distractor rationale: A. Correct.

2 pairs of

B. Incorrect. The student may not have recognized that all rectangles have 4 right angles but not all trapezoids have 4 right angles.

opposite sides that have the same length

C. Incorrect. The student may not have recognized that all rectangles have 4 right angles but not all parallelograms have 4 right angles.

D. Incorrect. The student may not have recognized that not all quadrilaterals have at least 1 pair of parallel sides.

E. Correct.

At least

2 lines of

3. Circle an answer choice from each list to make the statements true. All

(A)

symmetry

have 2 pairs of parallel sides.

Because they have at least 1 pair of parallel sides, they are also all A

parallelograms

quadrilaterals

trapezoids

rectangles

(B)

rhombuses squares trapezoids

A N S W E R K E Y ▸ TO P I C Q U I Z A-1

223

224

A N S W E R K E Y ▸ TO P I C Q U I Z A-1

189

EUREKA MATH2 Florida

5 ▸ M5 ▸ TA

6. Indicate all the three-dimensional figures that have the properties listed.

5. Classify the triangle shown by its angle measures and side lengths.

Circle an answer choice from each list to make each statement true. The triangle is

(A)

because it has

(B)

90°.

The triangle is

(C)

because it has

(D)

sides of equal length.

an acute triangle

3 angles less than

an acute triangle

an equilateral triangle

1 angle equal to

an equilateral triangle

an isosceles triangle

1 angle greater than

an isosceles triangle

an obtuse triangle

a right triangle

a scalene triangle

190

A N S W E R K E Y ▸ TO P I C Q U I Z A-1

Cone

Cylinder

Curved surface

2 bases

an obtuse triangle

Property

An apex

225

226

Right Pyramid

A N S W E R K E Y ▸ TO P I C Q U I Z A-1

Right Triangular Prism

EUREKA MATH2 Florida 5 ▸ M5 ▸ TA

Observational Assessment Recording Sheet Grade 5 Module 5 Topic A

Student Name

Drawing, Analysis, and Classification of Two-Dimensional and Three-Dimensional Figures Standards and Achievement Descriptors

MA.5.GR.1.1

FL.5.Mod5.AD4 Classify triangles into categories based on attributes and explain the classification.

MA.5.GR.1.1

FL.5.Mod5.AD5 Classify quadrilaterals into categories based on attributes and explain the classification.

MA.5.GR.1.2

FL.5.Mod5.AD6 Classify three-dimensional figures into categories based on attributes.

Dates and Details of Observations

PP Partially Proficient P Proficient HP Highly Proficient

This page may be reproduced for classroom use only.

191

5 ▸ M5 ▸ TB

EUREKA MATH2 Florida

Topic B Areas of Rectangular Figures with Fraction and Decimal Side Lengths In topic B, students find areas of rectangles with fraction and decimal side lengths, first by tiling and then by multiplying a rectangle’s length by its width. Students multiply mixed numbers and solve both mathematical and real-world problems involving mixed numbers and decimals. In grade 3, students recognize area as an attribute of plane figures. They find areas of rectangles with whole-number side lengths by tiling, multiplying side lengths, and using the area formula. They partition rectangles into parts with equal areas and express the area of each part as a fraction of the whole. In grade 4, students apply the area formula for rectangles to solve real-world and mathematical problems. In grade 5 module 3, students multiply a fraction or whole number by a fraction by using models such as number lines and area models. In the first part of topic B, students find areas of rectangles with side lengths that are fractions less than 1 and side lengths that are fractions greater than 1. Students reason about the number of tiles with unit-fraction side lengths that fit in a unit square. They tile a unit square with squares and then rectangles and identify the area of 1 tile as a fraction of 1 square unit. Building on their tiling work from earlier grades, students realize that a rectangle’s area is the product of the number of tiles used to cover it and the area of 1 tile. Students find that the area of the rectangle is the same regardless of the size of the tile used. Recognizing that they can determine areas of rectangles with fraction side lengths in the same way they do for rectangles with whole-number side lengths, students multiply a rectangle’s length by its width to determine its area. In the second part of the topic, students multiply mixed numbers. They begin by decomposing mixed-number factors into their whole-number and fractional parts. They create an area model to represent and add the partial products. After making connections between the area model and the distributive property, students transition to multiplying mixed numbers without a visual model. They deepen their understanding

192

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB

of mixed-number and decimal multiplication when they use and compare a variety of methods to determine the areas of composite figures and solve other mathematical and real-world problems. Students compare solution strategies and use estimation to determine whether answers are reasonable. In topic C, students extend their understanding of quantifying the amount of space an object takes up to include a third dimension, height. They also find the volume of solids, with an emphasis on right rectangular prisms with whole-number side lengths.

193

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB

Progression of Lessons Lesson 9

Lesson 10

Lesson 11

Find areas of square tiles with fraction side lengths by relating the tile to a unit square.

Organize, count, and represent a collection of square tiles.

Find the area of a rectangle with fraction side lengths by relating the rectangle to a unit square.

Area of 1 rectangle tile:

1 square unit 12 2 unit 3

I can tile a unit square with squares that have unit-fraction side lengths. The area of each tile is 1 out of the total number of tiles it takes to cover the unit square.

Area of shaded rectangle: 1

1 unit 4

1 unit

Area of 1 tile:

1 square unit 4

34 total tiles 1

2 x 12 square unit = 12 square units

I can tile a unit square with rectangular tiles that have unit-fraction side lengths. I can use the area of the rectangular tiles to determine the area of a rectangle with one side length that is a unit fraction and one side length that is a fraction less than 1.

34 × 4 = 4 = 8 4 2

Area of all tiles: 8 4 square units

I can tile a rectangle with square tiles that have fraction side lengths. The area of the rectangle is the product of the area of each square tile and the number of square tiles.

194

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB

Lesson 12

Lesson 13

Lesson 14

Find areas of rectangles with fraction side lengths by using multiplication.

Multiply mixed numbers.

Solve real-world problems involving areas of composite figures with mixed-number and decimal side lengths.

7 units 3

2 3

3 units 2

7 3 21 × = 3 2 6

12 1 ft

21 square units 6

I can determine the area of any rectangle, whether it has whole-number side lengths or fraction side lengths, by multiplying the rectangle’s length and its width.

20 ft

4 3

Garage

Island

3 4

6 12

Garden

Living Room and Kitchen

Bedroom

16 1 ft

12 ft

Rectangles can have side lengths that are mixed numbers. I can multiply mixed numbers by using an area model and the distributive property or by converting mixed numbers to fractions greater than one.

8 1 ft 6 1 ft

Bath

24 ft

3 4

Driveway

11 3 ft 4

I can use a variety of strategies to solve real-world problems that involve area of composite figures with mixed-number side lengths.

195

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB

Lesson 15 Solve word problems involving multiplication of mixed numbers. Kayla

Sana Tyler I can solve word problems involving area and multiplication with mixed numbers. I can use estimation to determine whether my answers are reasonable.

196

LESSON 9

Find areas of square tiles with fraction side lengths by relating the tile to a unit square. I can find the areas of squares that have fraction side lengths.

Lesson at a Glance Students reason about the number of square tiles with fraction side lengths that are needed to cover unit squares. They recognize patterns in the numbers of tiles needed, and they find the areas of squares with unitfraction side lengths. Students conclude that the area of a unit square is the same regardless of the size of the tile used to cover the unit square.

Key Questions • How can you find the area of a square tile with unit-fraction side lengths by tiling a unit square?

Mathematical Thinking and Reasoning Standard MTR.5.1

Language Objectives • Orally describe patterns between the side length of a square tile, the area of a square tile, and the number of square tiles needed to cover a unit square. • In partners and as a class, orally describe how to find the area of a square tile by using the relationship between the sizes of the unit square and the square tile.

• Does the size of the tile you use to cover a unit square affect the area of the unit square? Explain.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 9

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Prepare 2ʺ x 2ʺ squares by folding and cutting a piece of 4ʺ x 4ʺ patty paper vertically and horizontally. Prepare enough for each student group of three to have at least 8 small squares.

Launch 10 min Learn 30 min • Areas of Square Tiles with Fraction Side Lengths • Tile Unit Squares • Problem Set

• None

Students • Patty paper, 4ʺ x 4ʺ (1 per student group) • Patty paper, 2ʺ x 2ʺ (8 per student group)

Land 10 min

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD8 Find and model the area

MA.5.GR.2.1 Find the perimeter and area of a

of rectangles with fraction side lengths by tiling. (MA.5.GR.2.1)

rectangle with fractional or decimal side lengths using visual models and formulas.

199

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 9

Fluency

Counting on the Number Line by Centimeters and Meters Students count by a unit of 50 centimeters and rename centimeters as meters to build fluency with converting measurements from module 4. Display the number line. Use the number line to count forward by 50 centimeters to 300 centimeters. The first measurement you say is 0 centimeters. Ready? Display each measurement one at a time on the number line as students count.

0 cm, 50 cm, … , 250 cm, 300 cm Now count forward by 50 centimeters again. This time rename every 100 centimeters as a number of meters. The first measurement you say is 0 meters. Ready?

0 cm

50 cm 100 cm 150 cm 200 cm 250 cm 300 cm

50 cm

1 m 1 m 50 cm 2 m 2 m 50 cm 3 m

150 cm

250 cm

Display each measurement one at a time on the number line as students count.

0 m, 50 cm, … , 250 cm, 3 m Now count forward by 50 centimeters again. This time use mixed units, meters and centimeters, when possible. The first measurement you say is 0 meters. Ready? Display each measurement one at a time on the number line as students count.

0 m, 50 cm, … , 2 m 50 cm, 3 m

200

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 9

Whiteboard Exchange: Add Decimals Students determine the sum to develop procedural fluency with adding decimals from module 4. Display 0.63 + 0.29 =

Write and complete the equation. Show your method. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

0.63 + 0.29 = 0.92 0.62 0.01

Display the sum and sample student work. Repeat the process with the following sequence:

0.47 + 0.378 = 0.848

2.7 + 9.63 = 12.33

7.142 + 4.876 = 12.018

201

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 9

Whiteboard Exchange: Multiply a Fraction and a Whole Number Students determine the product to prepare for using multiplication to find areas of rectangles beginning in lesson 11. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display 3 × _  1 = 2

Write and complete the equation. Write the product as a fraction.

3 × 2 = 2 = 12

Display the product. Rewrite the product as whole or mixed number. Display the whole or mixed number. Repeat the process with the following sequence:

4 × 2 = 2 = 2 4 × 3 = 3 = 1 3 7 × 3 = 3 = 4 3 9 × 4 = 4 = 6 4 10 × 5 = 5 = 6

202

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 9

Launch

Students identify a need to find the area of squares with fraction side lengths by partitioning a unit square. Display the picture of different types of flooring. The picture shows samples of carpet and wood floors. Imagine that you get to choose the flooring for a bedroom in a new home. What would you choose? Why would you choose that type of flooring? I would choose carpet because it comes in fun colors. I would choose carpet because it feels better on my feet than wood does. I would choose wood because I can slide across the floor. What other flooring materials might you use in a bedroom that are not shown in this picture? Tile, stone, concrete, bamboo, or linoleum What might influence your decision about which flooring to use? How much the flooring costs What color the carpet is What the flooring looks like How large the area that needs to be covered with flooring is Which flooring lasts the longest

203

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 9

Display the site plan of the home without measurements.

Garage

What do you notice? What do you wonder? I notice that the drawing looks like different rooms in a home. I notice that the home has one bedroom and one bathroom. I notice that there aren’t any measurements on the drawing.

Driveway

Island Bathroom Garden

Living Room and Kitchen

Bedroom

I notice that the rooms look like rectangles. I wonder how big the rooms are. Play the Floor Tiles video. If necessary, replay the video and ask students to note any details. Give students 1 minute to turn and talk about what they noticed. Engage students in a brief conversation about the video. Discuss student observations and any relevant questions they have. Consider the following possible sequence of questions. What did you notice? I noticed she wanted to tile the bathroom. I noticed that the length and the width of the bathroom are mixed numbers. I noticed that a whole tile wouldn’t fit in the corner. What do you wonder? I wonder what the area of the bathroom is.

Multilingual Learner Support The lessons in this topic use the word tile both as a noun and as a verb. Consider pausing the video and pointing to a tile as you say, “We are using the word tile in two different ways: the owner of the home is tiling the floor, and she is covering the floor with tiles.” Discuss how the verbs tile and cover can be used interchangeably.

I wonder what the area of the small square is. I wonder how many small squares make up one whole tile. Invite students to turn and talk about how they could find the area of the square with side lengths of _  1  unit. 2

Transition to the next segment by framing the work. Today, we will find the area of squares with fraction side lengths. 204

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 9

Learn

Areas of Square Tiles with Fraction Side Lengths Materials—S: Patty paper

Students reason about the number of square tiles with fraction side lengths that are needed to tile larger squares. Display the picture of the unit square. The side lengths are all 1 unit.

Multilingual Learner Support

1 unit

How do you know that the side lengths are all 1 unit? A square has 4 sides of equal length, so if one side is 1 unit, then the other three sides must each also be 1 unit. Because the side lengths are each 1 unit, we can call this a unit square. What do you know about the area of a unit square? The area of a unit square is 1 square unit.

Consider providing a labeled visual to support the distinction between unit square and square unit. Students should recall from previous grades that a unit square is a square with sides that are each 1 unit. A square unit is a unit of measurement that describes the space covered by a unit square.

Unit Square

Display the square partitioned into fourths.

1 unit

Each part has the same area. Each part is _  1 of the square. 4

Area:

1 unit

Square Unit 1 unit

1 unit

Each part is the same size.

1 unit

What can we say about each part of the square?

1 unit 1 unit

What do you notice about this square?

We know that area is the amount of flat space a shape takes up. Why is area measured in square units? When we measure area, we fill up the space inside the shape with squares. When we measure area, we want to find how many square tiles cover the shape without gaps or overlaps.

205

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 9

Point to one fourth of the square as you say the following.

If each of these smaller squares is _  1 of the unit square, I wonder what the length 4

of one side of each smaller square is.

Display the picture of the square with side lengths of _  1 unit and the square with side lengths 2 of 1 unit.

1 unit 2

1 unit

Invite students to Think–Pair–Share about how many squares with side lengths of _  1  unit 2 they think will fit into or cover the unit square with no gaps or overlaps. I know at least 1 smaller square will fit because it is smaller than the unit square.

Teacher Note

I think 2 smaller squares will fit because the side length of the smaller square is half the side length of the unit square.

For this lesson, the size of one piece of patty

I think 4 smaller squares will cover the unit square because there will be 2 smaller  1 + _ squares on each half and _  1 = 1.

Hamburger patty paper (5   inches by

Divide students into groups of three. Give each group both sizes of patty paper. Hold up one of the larger squares of patty paper.

paper (4 inches by 4 inches) is one unit.

__1

5 1  inches), available in boxes of 1,000, is 2

also an option for this lesson. Any paper may be used if patty paper is not available.

This is a square with side lengths of 1 unit. What is the area of this square?

1 square unit Hold up one of the smaller squares of patty paper.

This is a square with side lengths of _  1 unit. Let’s call this smaller square a tile because 2

Universal Design for Learning: Representation

we are using it to tile the unit square.

Consider using the Tiling digital interactive

Let’s find how many tiles we need to cover 1 unit square without gaps or overlaps.

to tile the unit square to confirm that 4 tiles

Invite students to work with their group to find the number of tiles they need to cover 1 unit square. How many tiles do you need to cover 1 unit square?

__1

with side lengths of    unit are needed to 2

cover 1 square unit. Students will explore more with the digital interactive in the next segment.

4 206

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 9

If it takes 4 tiles to cover 1 unit square, or to make 1 square unit, what is the area of each tile? How do you know?

_ 1 square unit. Each tile is 1 equal part out of the 4 parts that make up the square unit. 4 The unit square has an area of 1 square unit, so each tile has an area of _  1  of 1 square unit, 4 or _  1 square unit. 4 _ 1 square unit. I know because 1 square unit ÷ 4 = _ 1 square unit. 4

Use the Numbered Heads routine to continue the discussion. Organize students into groups of three and assign each student a number, 1 through 3. Display the following problem.

How many square tiles with side lengths of _  1 unit do you need to cover 2 square units? 2 Draw to show your thinking.

Give students 2 minutes to answer the question as a group. Invite students to record their thinking on their whiteboards. If necessary, provide students with patty paper to help them find the number of square tiles they need. Help students recall that any one of them could be the spokesperson for the group, so they should be prepared to answer. Call a number, 1 through 3. Have the students assigned that number share their group’s findings.

1 unit

We found that we need 8 square tiles with

side lengths of _  1 unit to cover 2 square units. 2

We knew that 1 unit square has an area of

1 square unit, so we drew 2 unit squares to show an area of 2 square units. We partitioned each unit square to show 4 square

1 unit

1 unit 2 1 unit 2

tiles with side lengths of _  1 unit. Then we counted the total number of square tiles.

to cover 1 square unit, so we multiplied 2 and 4 to find the number of tiles we need to cover 2 square units. 2 × 4 = 8, so we need

8 square tiles with side lengths of _ 1 unit to cover 2 square units. 2

Students use patterns and structure (MTR.5) when they use the relationship between a unit square and smaller square tiles with fraction side lengths to determine the area of the tiles. Ask the following questions to promote MTR.5: • How are the square tiles and the unit square related? How can that help you find the area of each tile? • How can what you know about the area of a unit square help you determine the areas of the square tiles?

1 unit

We knew that it takes 4 square tiles with side lengths of _  1  unit

Promoting Mathematical Thinking and Reasoning

1 unit 2 x 4 = 8 207

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 9

As time allows, repeat the process with the following problem.

How many square tiles with side lengths of _  1 unit do you need to cover 3 square units? 2 Draw to show your thinking.

Invite students to turn and talk about any patterns they notice.

Tile Unit Squares Students determine the areas of squares with fraction side lengths by tiling a unit square. Open and display the Tiling digital interactive. Tile the unit square with square tiles that have side lengths of _  1  unit. 2

How many square tiles with side lengths of _  1 unit did we need 2 to cover 1 unit square?

1 2

1 1 2

4 Invite students to think about whether it will take more or fewer tiles that have side lengths

of _  1 unit to cover the unit square than it took to cover it with tiles with side lengths of _  1  unit. 3

Let’s tile the unit square to see how many tiles with side lengths of _ 1 unit it takes to cover the unit square. Count with me. 3

Tile the unit square with tiles that have side lengths of _  1 unit. Count aloud 3 the number of tiles that are needed. How many square tiles with side lengths of _  1unit did we need 3 to cover the unit square?

1 3

1 1 3

To tile 1 square unit, why does it take more tiles with side lengths of _  1 unit than tiles 3 1 _ with side lengths of     unit? 2

Tiles with side lengths of _  1 unit are smaller than tiles with side lengths of _  1 unit, so it 3

takes more of the tiles with side lengths of _  1 unit to cover the same amount of space. 3

208

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 9

Point to one of the tiles with side lengths of _  1 unit. Invite students to Think–Pair–Share 3

to find the area of a square tile with side lengths of _  1 unit and to explain how they know. 3 _ 1 square unit. Each of the square tiles with side lengths of _  1 unit is 1 part out of the 3

9 parts that make up the square unit.

The unit square has an area of 1 square unit, so each part has an area of _  1  of 1 square unit, 9

or _  1 square unit. 9

_ 1 square unit. I know because 1 square unit ÷ 9 = _ 1 square unit. 9

Use the Numbered Heads routine to continue the discussion. Organize students into groups of three and assign each student a number, 1 through 3. Teacher Note

Display the following problem.

How many square tiles with side lengths of _  1 unit do you need to cover the unit square? 4

Give students 2 minutes to answer the question as a group. Invite students to sketch their thinking on their whiteboards. Help students recall that any one of them could be the spokesperson for the group, so they should be prepared to answer.

Some students may recognize that they can multiply the side lengths to find the area of the tile. Although that method is correct, highlight tiling with different-size squares with unit-fraction side lengths to draw connections to the meaning of area.

1 unit

Call a number, 1 through 3. Have the students assigned that number share their group’s findings. We found that we need 16 tiles. Each side of the unit square is 1 unit.

  1 + _ Because _  1 + _ 1 + _ 1 = 1, we knew that 4 square tiles with side lengths 4

of _  1 unit fit along the length and 4 fit along the width. 4 × 4 = 16. 4

If necessary, confirm the answer by using the digital interactive to tile

the unit square with square tiles that have side lengths of _  1  unit. 4

What is the area of a square with side lengths of _  1 unit? How do you know? 4 1 __    square unit. Each of the square tiles is 1 part out of the 16 parts that make up the 16

square unit.

The unit square has an area of 1 square unit, so each part has an area of __  1   of

1 square unit, or __   1  square unit.

209

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 9

Repeat this process with square tiles that have side lengths of _  1  unit. 5

When students have finished, invite them to Think–Pair–Share about any patterns they notice. As the side length of the square tiles decreases, I need more tiles to cover the unit square. The number of tiles that I need to cover the unit square is equal to the product of the denominators. The area of one square tile is equal to the product of its side lengths.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

210

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 9

Land

Debrief 5 min Objective: Find areas of square tiles with fraction side lengths by relating the tile to a unit square. Facilitate a class discussion about areas of squares with fraction side lengths by using the following prompts. Encourage students to restate or add on to their classmates’ responses. How can you find the area of a square tile with unit-fraction side lengths by tiling a unit square? We can find the number of square tiles we need to cover a unit square and then divide 1 square unit by that number. If it takes 9 equal-size tiles to cover 1 square unit of area, then each tile has an area of

1 square unit ÷ 9, or _ 1 square unit. 9

Does the size of the tile you use to cover a unit square affect the area of the unit square? Explain. No, a unit square always has an area of 1 square unit. No. I can use tiles with side lengths that are different unit fractions to cover a unit square. Even though the number of tiles changes, the area of the unit square is the same no matter what size tile is used to cover it. No. When the tiles are smaller, it takes more of them to cover the unit square, but the area of the unit square is still 1 square unit.

211

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 9

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can find the areas of squares that have fraction side lengths.

PRACTICE

Florida B.E.S.T. MA.5.GR.2.1

Name

2. Draw a line to match each square tile to the picture that shows 1 square unit covered with that tile.

Date

1 Square Unit

Square Tile

1. Use the picture of the unit square to complete parts (a) and (b).

1 unit

212

1 unit

1 unit 6

1 unit

square tiles are used to partition the unit square into ninths.

b. Each square tile has a side length of

1 unit 5

_1 3

unit.

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 9

3. Use the picture of the square tile and the unit square to answer parts (a) and (b).

Use the Read–Draw–Write process to solve the problem.

5. Julie decorates a table by tiling it with square tiles with side lengths of _1 unit. The area of the 8 table is 1 square unit. a. How many square tiles does Julie use to tile the table? Julie uses 64 square tiles to tile the table.

1 unit

1 unit 10

1 unit are needed to tile the unit square. 100 square tiles with side lengths of __ 10

1 unit? b. What is the area of one square with side lengths of __ 10

1 square unit ___ 100

4. Use the picture of the unit square to answer parts (a) and (b).

b. What is the area of one square tile with side lengths of _1 unit?

1 square unit ÷ 64 = __ square unit 64 1

__1 square unit 64

1 unit

a. 49 squares with side lengths of

_1 7

unit are needed to tile a unit square.

b. What is the area of one of the square tiles used to cover this 1 square unit?

__1 square unit 49

PROBLEM SET

213

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 9

EXIT TICKET

I can find the areas of squares that have fraction side lengths.

Florida B.E.S.T. MA.5.GR.2.1

Name

Date

_1 What is the area of a square tile with side lengths of unit? Sketch to show how you know. 5

1 unit

__1

The area is 25 square unit.

214

LESSON 10

Organize, count, and represent a collection of square tiles. I can use square tiles to find area.

Lesson at a Glance Students determine how to cover a rectangular area with square tiles that have fraction side lengths. They continue their work with area by counting a collection of square tiles. Students decide how to organize, count, and represent the tiles. Then they analyze the work of others and discuss efficient organizing and counting strategies as a class. Use classroom observations and classwork to analyze student thinking after the lesson. The Exit Ticket for this lesson serves as an opportunity for students to reflect on their counting strategies.

Key Questions • How can you find the total area of a rectangle when you know the number of square tiles that it takes to cover the rectangle? • What strategies can you use to help you count your collection?

Mathematical Thinking and Reasoning Standard MTR.3.1

Language Objectives • Write or draw to record how to organize and count a collection of square tiles to determine an estimate of the total area covered by the collection, the actual area covered by the collection, and an equation that describes how to find the total area. • Listen to and orally describe counting strategies including partitioning, creating unit squares, and finding the area of one tile and multiplying— to accurately count and compare and connect strategies.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 10

Agenda

Materials

Lesson Preparation

Fluency 5 min

Teacher

• Tear out and cut apart Tiles Counting Collection from the student books and place one tile set in each envelope. Prepare enough collections for each student pair to have one. • Provide tools for students to choose from to help organize their counts. Tools may include cups, whiteboards, or bags.

Launch 5 min Learn 40 min • Tile Rectangles • Organize, Count, and Record • Share, Compare, and Connect • Problem Set

• Envelopes (12) • Tiles Counting Collection (1 per student pair, in the student book)

Students • Organizational tools • Envelope with Tiles Counting Collection (1 per student pair)

Land 10 min

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD8 Find and model the area

MA.5.GR.2.1 Find the perimeter and area of a

of rectangles with fraction side lengths by tiling. (MA.5.GR.2.1)

rectangle with fractional or decimal side lengths using visual models and formulas.

217

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 10

Fluency

Counting on the Number Line by Centiliters and Liters Students count by a unit of 50 centiliters and rename centiliters as liters to build fluency with converting measurements from module 4. Display the number line. Use the number line to count forward by 50 centiliters to 300 centiliters. The first measurement you say is 0 centiliters. Ready? Display each measurement one at a time on the number line as students count.

0 cL, 50 cL, … , 250 cL, 300 cL Now count forward by 50 centiliters again. This time rename every 100 centiliters as a number of liters. The first measurement you say is 0 liters. Ready?

0 cL

50 cL 100 cL 150 cL 200 cL 250 cL 300 cL

50 cL

150 cL

50 cL

1L 50 cL

2 L 2 L 50 cL 3 L

250 cL

Display each measurement one at a time on the number line as students count.

0 L, 50 cL, … , 250 cL, 3 L Now count forward by 50 centiliters again. This time use mixed units, liters and centiliters, when possible. The first measurement you say is 0 liters. Ready? Display each measurement one at a time on the number line as students count.

0 L, 50 cL, … , 2 L 50 cL, 3 L

218

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 10

Whiteboard Exchange: Multiply a Fraction and a Whole Number Students determine the product to prepare for using multiplication to find areas of rectangles with fraction side lengths beginning in lesson 11. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display 4 × _   = 1 3

Write and complete the equation. Write the product as a fraction. Display the product. Rewrite the product as whole or mixed number.

4 × 1 = 4 = 11 3

Display the whole or mixed number. Repeat the process with the following sequence: Teacher Note Validate all correct responses that may not

3× 2 =6 =2 3 3

5 × 1 = 5 = 11 4 4 4

7 × 3 = 21 = 5 1 4 4 4

10 × 4 = 40 = 8 5 5

9 × 5 = 45 = 7 3 6 6 6

be displayed. For example, when given the

__ 6 3 1 write 7 __ , or 7 __ .

problem 9 × 5 , a student may choose to 6

219

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 10

Launch

Students reason about how to find the number of square tiles with fraction side lengths that are needed to cover a rectangular area. Display the site plan of the home. The owner of the home wants to put tiles on the top of the island in the kitchen.

Garage

Invite students to turn and talk about what an island is and what it is used for in a home. What information do we need before we can find how many tiles it will take to cover the top of the island?

Driveway

Island Bathroom Garden

Living Room and Kitchen

Bedroom

We need to know how big the tiles are. We need to know the area of the island. We need to know the length and the width of the island. Display the following statement: The island in the kitchen measures 3 feet by 7 feet. Now do we have enough information to find how many tiles will cover the island? Why? No. We still need to know the length and width of the tiles.

1 foot

Display the square tile with side lengths of 1 foot. What if the owner of the home used these tiles? How many of these tiles would she need to cover the island? How do you know?

1 foot

She would need 21 tiles. The area of the island is 3 × 7, or 21 square feet. Each of those tiles is 1 square foot, and 21 ÷ 1 = 21. 220

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 10

Display the square tile with side lengths of _ 1 foot.

1 foot 2

Invite students to turn and talk about how they could find the number

of square tiles with side lengths of _ 1foot that would be needed to cover 2

the island.

1 foot 2

Transition to the next segment by framing the work. Today, we will use square tiles to tile a region and find its area.

Learn

Tile Rectangles Students determine the number of square tiles with fraction side lengths that are needed to cover a rectangle. Direct students to problem 1 in their books. 1. How many square tiles with side lengths of _ 1unit are needed to cover the rectangle without gaps or overlaps?

4 units 2 units

32 square tiles

221

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 10

Invite students to Think–Pair–Share about how they could find the number of square tiles with side lengths of _ 1unit that are needed to cover the rectangle. 2

We could draw lines to partition the rectangle. We could partition the rectangle into unit squares and then partition each unit square into squares with side lengths of _ 1 unit. 2

How can we partition the rectangle to show unit squares? Because the length is 4 units, we need to draw 4 unit squares that fit along the length. Because the width is 2 units, we need to draw 2 unit squares to fit along the width. We can draw to make a 2 by 4 array inside the rectangle.

4 units

Partition the rectangle to show unit squares. Direct students to do the same. Can we use what we have drawn so far to find how many square tiles with side lengths of

_1unit we need to cover the rectangle? How

2 units

do you know? No. We’ve drawn only enough squares to show how many tiles with side lengths of 1 unit we need to cover the rectangle. Yes. I know that 4 squares with side lengths of _ 1unit are needed to tile one unit square, 2

so I can multiply the number of square units by 4. Invite students to work with a partner to find the number of square tiles with side lengths of _ 1unit they need to cover the rectangle. 2

When students have finished, discuss their methods for finding the solution.

How many square tiles with side lengths of _ 1unit do you need to cover the rectangle?

Universal Design for Learning: Representation Consider using the Tiling digital interactive to confirm the number of tiles needed to cover the rectangles in this segment.

1 2 1 2

222

2 4

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 10

Display the site plan and the problem from Launch.

Garage

Driveway

If the island in the kitchen measures 3 feet by 7 feet, how many square tiles with side lengths of _ 1foot does the

Island

owner need to tile the island? We know that 21 square tiles are needed if the tiles have side lengths of 1 foot. Does the owner need more than or fewer than 21 tiles with side

Bathroom Garden

Living Room and Kitchen

Bedroom

lengths of _ 1foot to tile the island? 2

How do you know?

She needs more than 21 tiles. Square tiles with side lengths of _ 1foot take up less space 2

than square tiles with side lengths of 1 foot, so it takes more tiles with side lengths of

_1foot to cover the island. 2

Invite students to work with a partner to find the number of tiles needed to tile the island. As students work, circulate and provide guidance as needed. When students have finished, discuss the solution.

How many square tiles with side lengths of _ 1foot does the owner of the home need to tile the island? How do you know?

The owner needs 84 tiles. The area of the island is 21 square feet. Each square foot can be covered by 4 square tiles with side lengths of _ 1foot, and 21 × 4 = 84. 2

The owner needs 84 tiles. I know that the area of 1 square tile with side lengths of _ 1 foot 2

is _ 1square unit. The area of the island is 21 square feet, so I thought about how many 4

one-fourths are in 21. 2 1÷_ 1= 84. 4

223

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 10

What if the owner uses square tiles that have side lengths of _ 1foot? Would she need 3

more than or fewer than 84 tiles? How do you know?

She would need more than 84 tiles. Square tiles with side lengths of _ 1foot take up less 3

space than square tiles with side lengths of _ 1foot, so it takes more tiles with side lengths of _ 1foot to cover the island.

Invite students to turn and talk about how they can find the number of squares with fraction side lengths that fit into a rectangle.

Organize, Count, and Record Materials—S: Envelope of Tiles Counting Collection, organizational tools

Students use self-selected strategies to organize and count a collection and to record their process. Partner students and distribute a counting collection to each pair. Direct students to the recording page in their books. Briefly orient students to the materials and procedure for the counting collection activity: • Partners will collaborate to count a collection.

• The collection consists of square tiles with side lengths of _ 1unit. Students must count 2

to find the total area that can be covered by their collection. • Partners will make their own recordings to show how they counted. • Partners may use organizational tools. Organizational tools may include readily available classroom items such as cups and whiteboards. Before they begin to count, invite partners to work together to estimate the total area covered by the tiles in their collection. Have partners write their estimates. Then encourage them to talk about how they will organize their collection to count. Invite students to select organizational tools they would like to use, with the understanding that the tools may be exchanged as students refine their plans. Ask partners to begin counting their collections. Circulate and notice how students engage in the following behaviors. 224

Promoting Mathematical Thinking and Reasoning Students complete tasks with mathematical fluency (MTR.3) when they self-select strategies with which to organize and count the total area of their collection of square tiles. Ask the following questions to promote MTR.3: • What strategies can help you count your collection efficiently? • Why did you choose to organize your collection this way? Did that work well?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 10

Organizing: Strategies may include grouping tiles, creating 1 unit square from 4 tiles, and writing expressions or equations. Students may also organize their collections by using attributes that do not support counting efficiently, such as skip-counting by fourths. Counting: Students may count subgroups and then add to find the total. Or they may use a combination of multiplication and addition to find the total. Recording: Recordings may include drawings, numbers, expressions, equations, and written explanations. Circulate and use questions and prompts such as the following to assess and advance student thinking: • Show and tell me what you did. • How can you organize your collection to make it easier for you to count? • How does the way you organized your collection make it easier for you to count? • How did you keep track of what you already counted and what you still needed to count? • How did you know how to write your total? • How close was your estimate to your actual area? Select two or three pairs of students to share their counting work in the next segment. If possible, take pictures to show the class in the next segment. As partners share, consider displaying their recordings alongside the counting collections so students can see the written representation that corresponds to each collection. The samples show possible strategies. They demonstrate • partitioning; • creating unit squares; and • finding the area of 1 tile and multiplying.

Teacher Note As students count, they exhibit different levels of sophistication in their counting strategies. During the sharing of work, students with less sophisticated counting strategies have an opportunity to hear new ideas. If time allows, add or remove some tiles from the collection and invite students to count the new collection by using a strategy they heard from another group. Ensure that the number of tiles in the new collection is not divisible by 4 so the total area is not a whole number.

Teacher Note Plan what students should do when they finish counting their collections and recording how they counted. For example, they could • try another way to organize and count, or • explain their recording method to another pair.

225

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 10

Leo

For this counting collection, I am partners with Partitioning

1 1 2 2 For this counting collection, I am partners with Leo . __ 1 5 square units We think they cover__1 aunit totalbyarea of tiles 21 unit We are counting . 2 Count the collection. Record your unitwork. by unit tiles

We are counting

. . .

We think theywe cover a total area 15 square units . This is how organized andofcounted the collection: This is how we organized and counted the collection:

1 square 1 square 1 square 1 square 1 square 1 square 1 square 1 square 1 square 2 unit unit unit unit unit unit unit unit unit

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 = 82 Area: 8 1 square units 2

We counted a total area of

8__21 square units 8

altogether.

1 2 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + __ 21 = 8__21 . This is an equation that describes how we counted.

square units This is an equation describes We counted a totalthat area of 8 how we counted.altogether. 1

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 = 82

226

2 2 MATH2 Florida EUREKA EUREKA 5 ▸ ▸M5 ▸TBM5 ▸ ▸Lesson ▸ TB 10 ▸10Lesson 10 EUREKAMATH MATHFlorida Florida 5 M5▸5▸TB Lesson

Creating Creating Unit Squares Unitcounting Squares collection, I am partners with Creating Unit Squares For this

Julie

1 work. 1 work. Count Count collection. the collection. Record Record your Countthe the collection. Recordyour your work. unit by unit tiles

We are counting

For this counting For thiscollection, counting collection, I am partners amwith partners Julie with. Julie Julie . . For this counting collection, I amIpartners with 11 1 7 square units unit unit unit by unit tiles unitby by unit tiles We think cover a21 total area oftiles We are counting Wethey are counting . We are counting .. 22 2

. . .

We think they cover acover total area of square units . units .. 7 square 7 square We think We think cover they aorganized total areaa of total of 7units This isthey how we andarea counted the collection: This is how we organized and counted the collection:

This is how This we is organized how we organized and counted and the counted collection: the collection:

1 square 1 square 1 square 1 square unit unit unit unit 1 square 1 1square square 1 1square square 1 1square square 1 square unit

unit unit

unit

1 square square 1 1square square1 square 1 1square square 1 square1 1square 1 square unit unit unit unit unit unit unit unit unitunit EUREKA MATH2 Florida

square 11square unit unit

5 ▸ M5 ▸ TB ▸ Lesson 1

8 ×1= 8 8 ×1= 8 1

2 Florida 2 Florida EUREKA EUREKA MATHMATH

square 1 square 1 square 2 2 2 unit unit unit

8 ×1 1 =1 8

8+ = 88 + =8 Creating Unit Squares 2 8 + 21 2 8 1 2

1 2 Area: 8 1 Area: square82units square units 2 2 Record your work. Count the collection. 1

Area: 8

5 ▸ M5 5 ▸▸ M5 TB ▸▸ TB Lesson ▸ Lesson 10 10

square units

For this counting collection, Creating Creating Unit Squares Unit Squares 12 1I am partners with Julie . square8units square units We counted We a total counted areaa of total8 area of altogether. altogether. 2 2 1 1 unit by unit tiles 8 square units Count Count the collection. the collection. Record Record your work. your work. Weaare counting . 12 2 We counted total area of altogether. square units

8 2we counted. This is an equation This isaan that equation describes that how describes how we counted. We total area of . . Forcounted this Forcounting this counting collection, collection, I am partners I 1am partners with with Julie altogether. Julie 1 1 1 This is anWe equation that describes how . 7 square units think they 8× 1 + cover 8= ×a8 1total + area = 8ofwe counted. 2 1 2 1 21 21 . . unit by tiles unit tiles 8 ×that 1 + describes = 8 byunitunit This isWe an equation We are counting are counting . 2 2 2 how 2 we counted. .. This is how we organized and counted the collection: 8×1+

8 2 7 square . units units We think We they thinkcover they cover a totalaarea totalof 2area of7 square 1 square 1 square 1 square 1 square unit unit unit This isThis howiswe how organized we organized and counted and counted the collection: the unit collection: 1 square 1 square 1 square 1 square 1 square 1 square 1 square 1 square unit unit unit unit unit unit unit unit

1 square unit

1 square 2 unit

1 square 1 unit 1 square square 2 2 unit unit

227 227

227

M5 TB 10 Lesson 10 10 ▸ M5 ▸55TB ▸▸ M5 ▸ Lesson ▸▸ TB ▸▸ Lesson

For this counting collection, I am partners with Ryan .. For this counting collection, I am partners with Ryan 1 For this counting collection,11 Iunit am by partners with Ryan . 1 unit tiles Finding the1Area Area of 11Multiplying Tile andcounting Multiplying Finding the Finding Area the of of Tile and Tile Multiplying We and are . unit by unit tiles 2 2 We are counting . 21 21 unit by unit tiles We are counting . 5 ▸ M5 ▸ TB ▸ Lesson 10 2 work. Count the collection. collection. Record your work.2 Count theCount collection. the Record your Record work. your . 7 square units We think they cover a total area of . square. units Wethis think they collection, cover a total area of with. 7Ryan For this counting For collection, counting I am partners I am with partners Ryan For this counting collection, I am partners 7with Ryan . . square units We think they cover a1 total area of Finding the Area of 1 Tile and Multiplying 1 1 1 This is how we organized and counted the collection: unit by unit unit tiles by unit tiles unit by unit tiles 5 ▸ M5 ▸ TB ▸ Lesson 10 We are counting We Weare are counting . . 22 This iscounting how we2 organized and2counted. the collection: thewe collection. Record your work. ThisCount is how organized and counted the collection: Wethink thinkthey cover a total area square units . . 7 square units 7 square7. units We think they We cover athey total cover areaa of total area of of counting collection, I am partners with Ryan . Finding the Area of 1 Tile For andthis Multiplying

10 we organized and counted the collection: This is how

1 10we organized This is how This we is organized how and counted and the counted collection: the1 collection: unit by unit tiles Wethe are collection. 2 Count 10 counting Record2your work.

10 10 For this counting collection, I am partners with Ryan . 7 square units We think they cover a total area of 10 1010 1 1 10 unit by unit tiles 10 We are counting . 2 2 This is how we organized and counted the collection: 10 1010 4 . 7 square units 4 10 We think 10 they cover a total area of 4 10

2 Florida EUREKA MATH22 Florida Florida EUREKA MATH EUREKA MATH

EUREKA MATH2 Florida

. .

1 unit the collection: 4 This is how 4 we organized and counted 1 unit 10 10 10

1 unit

1 unit 1 unit 10 4 1 unit1 unit 1 unit 10

1 unit

Area of 1 1tile: 11 square unit 1 4 Area of 1 tile: unit 4 square Area of 1 tile: Area of square 1 tile:unit square 41 unit 4 4 Area of 1 tile: square unit 1 unit 34 total 4 tiles 34 total tiles 34 total tiles 341 unit total tiles 34 total 34 1 12 341 34 2tiles 2 34 × = 34=×834=× 1 = =8 34 = 8 2 4 44 × 441 = 44 = 8 4 4 1 tile:41 square 1Area unit of34 4 unit42 34 4 34 × =8 2 2 = Area of all Area tiles:of tiles: 8units 4 square 8 4allsquare 42 units 4 4 tiles 34 tiles: total Area of all 2 square units 8 Area of all tiles: 8 4 square units 42 2 1 34 Area all41×tiles: units Area of 1of tile: square 8unit square units 34 =48square 2 2 =

We counted a total area 8 of4 square altogether. 4 units 4 square units 8 area We counted We a counted total areaa of total of altogether. altogether. 4 4 2 34 total tiles82 2 square units This is an equation that howunits weunits counted.altogether. all describes tiles: We counted a Area totalof area of 884 4square square We counted a total of counted. altogether. This is an This equation is anthat equation describes thatarea how describes how 2 we counted. 1 we 34 24

228 228

34 ×area = = 288 square units . We counted a 1total altogether. 2 4 1 of 4 44 = =describes × × 34 34 8 8 42 This is an equation that how. we counted. . 4 4that 4 describes This is an equation how we counted. units 8 4 square We counted aoftotal area8of2 square altogether. all tiles: units 1 2 This is an Area equation that 34 describes ×4 1 = 8 2how we counted. . 34 × 4 = 8how 4 . This is an equation that describes 41 42 we counted. 34 2× = 8 . 14 2 4 = units 348 × square We counted a total area of

4 4

altogether.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 10

Count the collection. Record your work. For this counting collection, I am partners with

We are counting

We think they cover a total area of

This is how we organized and counted the collection:

We counted a total area of

altogether.

This is an equation that describes how we counted. .

Self-Reflection Write one thing that worked well for you and your partner. Explain why it worked well. Making 2 by 2 squares with 4 tiles was helpful because we knew that each 2 by 2 square was equal to 1 unit square. Then we could count all our unit squares to find the total area.

229

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 10

Write one challenge you had. How did you work through the challenge? We were not sure what to do with the 2 extra tiles at first. We put them together to make part of a square and figured out what fraction of the square they represent.

Multilingual Learner Support Consider inviting students to use the Talking Tool as they share their work, ask questions about their classmates’ work, and compare their work to their classmates’.

Share, Compare, and Connect Students discuss strategies for organizing and compare their efficiency.

• The Share Your Thinking section can assist students with sharing their solution paths and strategies.

Gather the class to view the selected work samples and lead a discussion. Invite the selected pairs to share their counting processes. The following dialogue models a sample discussion. Partitioning (Kayla and Leo’s Way)

• The Ask for Reasoning section can assist students with forming questions about the Leo .shared solutions.

For this counting collection, I am partners with

We are counting

. .

We think theywe cover a total area 15 square units . This is how organized andofcounted the collection: This is how we organized and counted the collection:

1 square 1 square 1 square 1 square 1 square 1 square 1 square 1 square 1 square 2 unit unit unit unit unit unit unit unit unit

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 = 82 Area: 8 1 square units 2

We counted a total area of

8__21 square units 8

altogether.

1 2 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + __ 21 = 8__21 . This is an equation that describes how we counted.

square units This is an equation describes We counted a totalthat area of 8 how we counted.altogether.

230

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 = 82

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 10

Examine Kayla and Leo’s work. How did they organize their tiles? They made long rows of tiles. They partitioned the row into groups of 4 tiles. Kayla and Leo, why did you decide to line up the tiles and then partition them into groups of 4? We lined up the tiles so we could see them all at once. We knew that each tile has an area of _ 1square unit, so we partitioned the row into groups of 4 because _ 1+ _ 1+ _1+ _1= 1. 4

It is easier for us to count by ones than it is to count by fourths.

1 square unit? How did you know that the 2 remaining tiles have a total area of _ 2 Because each tile has an area of _ 1square unit, we know that the 2 tiles have an area

of _ 1+ _ 1, or _1square unit. 4

How did you determine the total area that the collection covers? We added all our whole square units and the half square unit to get a total area of

8 _1square units. 2

Invite students to turn and talk about the similarities and differences between their own work and Kayla and Leo’s work.

231

2 Florida 2 MATH EUREKA MATH 5 ▸EUREKA M5 ▸ TB ▸ Lesson 10 Florida

5 ▸ M5 MATH ▸5 TB ▸ M5 ▸2 Florida Lesson ▸ TB 10 ▸ Lesson 10 EUREKA

Creating UnitFor Squares Unitcounting Squares collection, I am partners with this Creating Unit Squares (Adesh and Julie’s Way)Creating

Julie

1 work. 1 work. Count Count collection. the collection. Record Record your Countthe the collection. Recordyour your work. unit by unit tiles

We are counting

. . .

This is how This we is organized how we organized and counted and the counted collection: the collection:

1 square 1 square 1 square 1 square unit unit unit unit 1 square 1 1square square 1 1square square 1 1square square 1 square unit

unit unit

unit

1 square square 1 1square square1 square 1 1square square 1 square1 1square 1 square unit unit unit unit unit unit unit unit unitunit EUREKA MATH2 Florida

square 1 square 1 square 2 2 2 unit unit unit

square 11square unit unit

5 ▸ M5 ▸ TB ▸ Lesson 10

8 ×1= 8 8 ×1= 8 1

8 ×1 1 =1 8

8+ = 88 + =8 Creating Unit Squares 2 8 + 21 2 8 1 2

2 Florida 2 Florida EUREKA EUREKA MATHMATH

1 2 Area: 8 1 Area: square82units square units 2 2 Record your work. Count the collection. 1

5 ▸ M5 5 ▸▸ M5 TB ▸▸ TB Lesson ▸ Lesson 10 10

Area: 8 square units For this counting collection, 12 1I am partners with

Julie . square8units square units 8 area We counted We a total counted area a of total of altogether. altogether. 2 2 Creating Creating Unit Squares Unit Squares 1 by units unit tiles 81 1 unit square Weaare counting . 2 2 We counted total area of square units 8your This iscounted anCount equation This isa an that equation describes that how describes counted. how we counted. We total area ofRecord altogether. 2wework. Count the collection. the collection. Record your work. altogether. 1 1 1area of1 . square units We think they 8collection, × 1 + cover 8=I×aam +partners 8 1total 8 2with 7with . .. For this Forcounting this counting collection, I 2am= partners Julie. Julie 2 2 This is an equation that describes how we counted.

This is an equation that describes how we counted. 1 1 and 1 counted 1 This is how we1 + organized unit unitunit by1 tiles unit the tilescollection: 8× = 8 1by We areWe counting are counting . 2 2 2 2 .. 8×1+ =8 2 2 1 square 1 square 1 square 1 units squareunits . 7 square We think We they think cover they cover a total aarea totalofarea of7 square unit unit unit unit

This isThis howiswe how organized we organized and counted and counted the collection: the collection:

1 square 1 square 1 square 1 square 1 square 1 square 1 square 1 square unit unit unit unit unit unit unit unit 1 square 1 square 1 square 1 square unit unit unit unit

8 ×1= 8

1 square 2 unit

1 square 1 square

227

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 10

Examine Adesh and Julie’s work. What is similar between their work and Kayla and Leo’s work? What is different? Both pairs made groups of 4 tiles. Adesh and Julie didn’t line up all the tiles. They made groups of 4 tiles to create unit squares. Adesh and Julie, can you tell us why you grouped 4 tiles to make squares?

We knew that each tile has a side length of _ 1unit, so when we put 4 tiles together 2

to make a square, we knew that it was a unit square because the side lengths were all

1 unit. It is easier for us to count by ones than it is to count by fourths.

How did you know that the 2 remaining tiles have a total area of _ 1square unit? 2

We saw that 2 tiles make up half of a unit square. A unit square has an area of

1 square unit, so _1 of 1 square unit is _1square unit. 2

How did you determine the total area that the collection covers? We saw that we could make 8 unit squares. Each unit square has an area of 1 square unit, so we had 8 square units. Then we added the _ 1square unit to get 8 _1square units. 2

Invite students to turn and talk about the similarities and differences between their own work and Adesh and Julie’s work.

233

▸ M5 ▸55TB ▸▸ M5 ▸M5 Lesson ▸▸ TB ▸▸ Lesson TB10 Lesson10 10

For this counting collection, I am partners with Ryan .. For this counting collection, I am partners with Ryan 1 For this counting collection,11 Iunit am by partners with Ryan . 1 unit tiles Finding Area the of 1Multiply Area of 1to Tile and Multiplying Tile and Multiplying We arethe counting Find the Area ofFinding One Tilethe and Then Find Total (Sasha and 2Ryan’s Way) . unit by unit tiles 2 We are counting . 21 21 unit by unit tiles We are counting . 5 ▸ M5 ▸ TB ▸ Lesson 10 2 2 Count the collection. collection. Record your work. work. Count theCount collection. the Record your Record work. your . 7 square units We think they cover a total area of . square. units Wethis think they collection, cover a total area of with. 7Ryan For this counting For collection, counting I am partners I am with partners Ryan For this counting collection, I am partners 7with Ryan . . square units We think they cover a1 total area of Finding the Area of 1 Tile and Multiplying 1 1 1 This is how we organized and counted the collection: unit by unit unit tiles by unit tiles unit by unit tiles 5 ▸ M5 5 ▸ ▸M5 TB▸▸TB Lesson ▸ Lesson 10 10 We are counting We Weare are counting . . 22 This iscounting how we2 organized and2counted. the collection: thewe collection. Record your work. ThisCount is how organized and counted the collection: Wethink thinkthey cover a total area square units . . 7 square units 7 square7. units We think they We cover athey total cover areaa of total area of of Finding Finding the Area the Area of 1 Tile of 1 Tile andFor and Multiplying thisMultiplying counting collection, I am partners with Ryan .

10 we organized and counted the collection: This is how

1 10we organized This is how This we is organized how and counted and the counted collection: the1 collection: unit by unit tiles We collection. are 2 your 2 Count Count the the collection. Record Record your work. work. 10 counting

10 . . For this Forcounting this10 counting collection, collection, I am Ipartners am partners with with Ryan Ryan 7 square units We think they cover a total area of 10 10 1010 1 1 1 1 unit by unit unit by tiles unit tiles 10 We are Wecounting are counting . . 2 2 2 2 This is how we organized and counted the collection: 10 1010 4 . 7 square units units. 4they cover We We think they a total a total area area of of 7 square 10 think 10 cover 4 10

2 Florida 2 EUREKA MATH EUREKA MATH EUREKA MATH2Florida Florida

EUREKA MATH2 Florida

2 Florida 2 Florida EUREKA EUREKA MATH MATH

. .

This is how is4we howorganized we organized and counted and counted the collection: the collection: 1 unit 4 This

10 10

1 unit

1 unit 1 unit

101 unit10 4 1 unit1 unit 10

1 unit

Area of 1 1tile: 11 square unit 4 4 1 Area of 1 tile: unit 4 square Area of 1 tile: Area of square 1 tile:unit square 41 unit 4 4 Area of 1 tile: square unit 1 unit 34 total 4 tiles 1 unit unit 34 total tiles 34 total tiles 34 1total tiles 34 total 34 1 12 341 34 2tiles 2 34 × = 34=×834=× 1 = =8 34 = 8 2 44 × 441 = 44 = 8 4 1 4unitArea 1 4unitof34 1 tile:41 square 4 unit42 34 4 34 × =8 2 2 = Area of all Area tiles:of tiles: 8units 4 square 8 4allsquare 42 units 4 4 tiles 34 tiles: total Area of all 2 square units 8 Area of all tiles: 8 4 square units 42 2 1 1 34 Area Area ofArea 1 tile: ofof 1 1tile: square square unit all tiles: square units 8 =unit 2 4 34 ×4 2 = 48 square units

We counted a total area 8 of4 square altogether. 4 units 4 square units 8 area We counted We a counted total areaa of total of altogether. altogether. 4 4 2 34 total 34 total tiles tiles 2 2 square units This is an equation that howunits weunits counted.altogether. all describes tiles: We counted a Area totalof area of 88 square 4square 8 4 We counted a total area of altogether. This is an This equation is anthat equation describes that describes how 422 we counted. 1 how 1 we342counted. 34 square units 34 × 34=× = 8 =88 4 2area . We counted a 1total altogether. 4 1 of44 2 44 × = 8 34 that × =describes 8 42 This is an34equation how. we counted. . 4 4that 4 describes This is an equation how we counted. 2 of 28 square units We counted a of total altogether. Area Area of all tiles: allarea tiles: 4 units units 8 4 square 8 4 1square 2 This is an equation that 34 describes × 1 = 8 2how we counted. 228 234

34 × 4 = 8 4

This is an equation that describes how 41 42 we counted. 234 ×21 4 = 28 4 square units units 8 square 34 We counted We counted a total a total area area of 8 of altogether. altogether. 4 × 44 = 8 4

. . .

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 10

Examine Sasha and Ryan’s work. What do you notice about how they organized their tiles? They made rows of 10 tiles as many times as they could. They didn’t make groups of 4 like the other pairs made. Sasha and Ryan, can you tell us how you found the total area that the collection covers?

We knew that each tile has an area of _ 1square unit, so we knew that we could multiply 4

the total number of tiles by _ 1to find the total area. We organized the tiles into rows 4

of 10 because it was easy for us to count by tens to find the total number of tiles. How did you know you could multiply the total number of tiles by _ 1to get the 4 total area? Because each tile has the same area, it is like we have 34 groups of _ 1, or 3 4×_ 1. 4

Did Sasha and Ryan get the same answer as the other groups? How do you know? Yes. 8 _2is equivalent to 8 _1. 2

Invite students to turn and talk about the similarities and differences between their own work and Sasha and Ryan’s work. What if the collection had square tiles with side lengths of _ 1unit? Would the total 4

area be the same? How do you know?

No. The area of a square with side lengths of _ 1unit is less than the area of a square with 4

side lengths of _ 1unit, so the total area of the collection of squares with side lengths 2

of _ 1unit would be less than the total area of the collection of squares with side lengths 4

of _ 1 unit. 2

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Universal Design for Learning: Action & Expression Consider reserving time for the class to reflect after all work has been shared. First, have students review their self-reflection responses. Then ask partners if they heard a strategy from another group that they might try next time. Engage the class in discussion about reasons for trying a different approach. The sharing component of the counting collections activity serves as a formative feedback opportunity. After sharing multiple examples of annotated peer work for students to analyze, give them time to reflect on whether they would adjust their own approach in the future.

235

5 ▸ M5 ▸ TB ▸ Lesson 10

Land

EUREKA MATH2 Florida

Debrief 5 min Objective: Organize, count, and represent a collection of square tiles. Facilitate a class discussion about using square tiles to tile a rectangle and find its area by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Is a rectangle’s area in square units always equal to the number of square tiles that fit into the rectangle? Why? No, not always. The smaller the square tiles are, the more tiles it will take to cover the rectangle. If the tiles are unit squares, then yes, the rectangle’s area in square units is equal to the number of unit square tiles that fit in the rectangle. If the tiles are not unit squares, then the rectangle’s area in square units is not equal to the number of tiles that fit in the rectangle. How can you find the total area of a rectangle when you know the number of square tiles that it takes to cover the rectangle? You can multiply the number of tiles that it takes to cover the rectangle by the area of 1 of those tiles. What were you successful with when counting? What strategies helped you count your collection? Thinking about the number of tiles that make up 1 square unit helped me count the total area. I knew that I could find the total area by multiplying the area of 1 tile by the total number of tiles.

236

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 10

237

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 10

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET PRACTICE

I can use square tiles to find area.

Florida B.E.S.T. MA.5.GR.2.1

Name

3. What is the total area that can be covered with 100 squares like the ones shown in parts (a) and (b)? a.

Date

1. A rectangle with a length of 5 units and a width of 3 units is shown.

1 foot 4

1 foot 3

6 _1 square feet

11 _1 square feet

3 units

5 units a. Draw lines to partition the rectangle to show unit squares. b.

4. The rectangle shown represents the measurements of a laundry room.

squares with side lengths of 1 unit are needed to cover the rectangle.

1 unit are needed to cover the rectangle? c. How many square tiles with side lengths of _ 2

60 tiles

8 ft

1 unit to cover the rectangle shown. 2. It takes 48 square tiles with side lengths of _ 2

10 ft

6 units

a. How many square tiles with a side length of 1 foot are needed to cover the laundry room floor?

2 units

1 unit to cover a. Kayla says that it would take more than 48 square tiles with a side length of _ 3 the rectangle. Do you agree? Explain.

80 square tiles

1 unit take up less space than square tiles with side I agree. Square tiles with side lengths of _

1 foot are needed to cover the laundry b. How many square tiles with a side length of _ 2 room floor?

1 unit, so it takes more tiles with side lengths of _ 1 unit to cover the rectangle. lengths of _ 2

320 square tiles

b. How many tiles does it take?

108 tiles

238

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 10

EXIT TICKET

I can use square tiles to find area.

Florida B.E.S.T. MA.5.GR.2.1

Name

Date

1. What is the total area of 25 square tiles if each tile has side lengths of 3 unit?

The total area is 2 9 square units.

_1 × 1_ = 1 3 3 9 25 25 × _19 = __ 9

2. A rectangle has side lengths of 3 feet and 5 feet. How many square tiles with side lengths

of 6 foot would it take to cover the rectangle?

__1

The area of 1 tile is 36 square unit.

It takes 36 tiles to cover 1 square foot. The area of the rectangle is 15 square feet because 3 × 5 = 15. It takes 540 tiles to cover the rectangle because 36 × 15 = 540.

239

LESSON 11

Find the area of a rectangle with fraction side lengths by relating the rectangle to a unit square. I can use unit squares to find areas of rectangles that have fraction side lengths.

Lesson at a Glance Students find the part of a unit square that is covered by a rectangle with unit-fraction side lengths by thinking of the rectangle as a rectangular tile. They use the number of equal parts into which the unit square is partitioned to find the area of 1 rectangular tile, which is also the area of the rectangle. Students extend this understanding to finding areas of rectangles with one side length that is a unit fraction and one side length that is a fraction less than 1.

Key Questions • Does drawing a unit square help you find the area of rectangles with fraction side lengths? How?

Language Objective • In partners and as a class, orally describe how to find the area of a rectangle with fraction side lengths by using the relationship between the sizes of the unit square and the rectangular tile.

Mathematical Thinking and Reasoning Standard MTR.5.1

• How does knowing the area of a rectangle with unit-fraction side lengths help you find the area of a rectangle with one side length that is not a unit fraction?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 11

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 5 min Learn 35 min • Relate Rectangles with Unit-Fraction Side Lengths to a Unit Square • Draw Unit Squares to Find Areas of Rectangles • Problem Set

• None

Students • None

Land 10 min

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD8 Find and model the area

MA.5.GR.2.1 Find the perimeter and area of a

of rectangles with fraction side lengths by tiling. (MA.5.GR.2.1)

rectangle with fractional or decimal side lengths using visual models and formulas.

241

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 11

Fluency

Whiteboard Exchange: Subtract Decimals Students determine differences to develop procedural fluency with subtracting decimals from module 4. Display 0.824 − 0.808 =

Write and complete the equation. Show your method.

0.824 – 0.808 = 0.016

824 thousandths – 808 thousandths = 16 thousandths

Display the difference and sample student work. Repeat the process with the following sequence:

0.9 – 0.27 = 0.63

242

8.23 – 2.75 = 5.48

7.506 – 2.283 = 5.223

4 – 1.392 = 2.608

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 11

Choral Response: Multiply Fractions Students determine products to prepare for finding the area of rectangles with fraction side lengths by multiplying. Display _    × _    = 1 3

1 4

What is the product? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

1 ×1= 1 3 4 12

 __  1 12

Display the product. Repeat the process with the following sequence: Teacher Note

1 ×3= 3 3 4 12

5 ×1= 5 7 7 49

5 × 4 = 20 7 7 49

3 ×3= 9 6 4 24

4 × 6 = 24 6 4 24

4 × 7 = 28 5 10 50

8 × 8 = 64 5 11 55

In this activity, students are not expected to say the product by using the largest unit

__1

__3

(such as   for   ) or as a mixed number 4

__9

64 __

(such as 1   for   ). 55

Choral Response: Add Whole Numbers and Fractions

Students determine the sum of a whole number and a mixed number or fraction greater than one to prepare for adding partial products beginning in lesson 13. Display 1 + 1  _ = 1 3

What is the sum? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

2  _  1 3

1+ 11 = 21 3

Display the sum.

243

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 11

Repeat the process with the following sequence:

Teacher Note Beginning with the equation

4 3+2 = 6

4 5 6

3 1 +2= 4

3 3 4

5 2+ = 5

7 2+ = 5

2 +  5 =

2 3 5

, consider asking students

to rename the fraction as a whole or mixed number before determining the sum. For this equation students should think, “2 + 1 = 3.”

10 = 10

Launch

13 = 10

3 10

8 +3= 4

9 +1= 4

1 4

Students determine the area of a square that is part of a unit square. Display the picture of the unit square and use the Math Chat routine to engage students in mathematical discourse. Give students 1 minute of silent think time to determine the area of the shaded region. Have students give a silent signal to indicate they are finished.

244

1 unit

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 11

Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Purposefully choose work that allows for rich discussion about connections between strategies. Then facilitate a class discussion. Invite students to share their thinking with the whole group and record their reasoning. As students discuss, highlight thinking that shows strategies involving partitioning the unit square into smaller squares. I think the area is __    square unit. I could draw lines to partition the unit square into 1 16

1 16 squares that are the same size as the shaded square. Because __    of the unit square 16

is shaded, the shaded region has an area of __    square unit. 1 16

I could extend the vertical line in the middle to divide the unit square in half. The shaded square is _     of _    of the unit square, or __    of the unit square. So the area is __    square unit. 1 4

1 4

1 16

Ask questions that invite students to make connections and encourage them to ask questions of their own. Display the picture and invite students to turn and talk about how they can use what they know about the area of the shaded square to find the area of the shaded rectangle.

1 unit

Transition to the next segment by framing the work. Today, we will find areas of rectangles with fraction side lengths.

1 unit

245

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 11

Learn

Relate Rectangles with Unit-Fraction Side Lengths to a Unit Square Students create a unit square to find the area of a rectangle with unit-fraction side lengths. Direct students to problem 1 in their books. Have them read the problem and give them silent think time to consider how to approach it. Promoting Mathematical Thinking and Reasoning

Find the area of the rectangle.

Students use patterns and structure (MTR.5) as they find ways to build unit squares by using iterations of square or rectangular tiles. They find the area of rectangles with fraction side lengths by relating those areas to the area of the unit square.

Ask the following questions to promote MTR.5:

1 unit 4 1 unit 2

_ 1 square unit 8

What do you notice about this rectangle? The side lengths are fractions.

• How are the rectangular tile and the unit square related? How can that help you find the area of the rectangle? • How can what you know about the area of a unit square help you find the area of this rectangle? • How can you break the rectangle into smaller parts that would help you compare it to a unit square?

Its side lengths are both less than 1 unit. Do you think the area of this rectangle is more than 1 square unit or less than 1 square unit? Why? I think the area of the rectangle is less than 1 square unit because both side lengths are less than 1 unit. 246

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 11

I think the area of the rectangle is less than 1 square unit because when we had squares with side lengths that were fractions less than 1, it took several of the small squares to make 1 unit square. When the side lengths of a rectangle are both less than 1, the area of the rectangle is less than 1 square unit. Instead of thinking about how many square units it takes to cover the rectangle, we are thinking about how much of 1 unit square is covered by the rectangle. Let’s think about this rectangle as part of a unit square. What is the area of a unit square?

1 square unit How did knowing the area of a unit square help us find the area of square tiles with

fraction side lengths, such as     unit?

We knew 4 square tiles with side lengths of _    unit could fit in a unit square, so we knew 2 each tile’s area must be 1 square unit ÷ 4. 2

Invite students to Think–Pair–Share about how they think they could use what they know about the area of a unit square to find the area of a rectangle with side lengths of _     unit 1 4

and _     unit. 1 2

Maybe we could put the rectangles together to make a unit square.

Maybe we could add another rectangle with side lengths of _     unit and _    unit next to the 1 4

1 2

rectangle so that the side lengths of _    unit make a side length of 1 unit. 1 2

I think we could put 3 more rectangles with side lengths _     unit and _    unit above the 1 4

1 2

Differentiation: Support Consider using the Tiling digital interactive or showing a diagram to support students in determining the area of squares with unit-fraction side lengths by finding the number of squares with unit-fraction side lengths that fit in a unit square.

rectangle so that the side lengths of _    unit make a side length of 1 unit. 1 4

Teacher Note

Let’s draw to find how many rectangles with side lengths

1 1 of   unit and   unit fit in a unit square. 4 2

Model drawing to begin creating a unit square around the rectangle in problem 1 and direct students to do the same.

How many rectangles with side lengths of   unit does 4 it take to make 1 unit?

4 © Great Minds PBC

1 unit 4 1 unit 2

In module 3, students use area models to find the product of two fractions. They partition the area models and then shade the region to find the product. In this lesson, students are given a shaded rectangle and must create the area model to find the product.

247

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 11

Draw 3 additional rectangles on top of the given rectangle to extend the side length to 1 unit. 1 unit does it take to make 1 unit? How many rectangles with side lengths of _ 2

Draw 1 rectangle to the right of the given rectangle to extend the side length to 1 unit. Now we have 1 unit by 1 unit. Let’s finish drawing the sides of the square. Finish drawing the square and direct students to do the same. Did we create a unit square? How do you know? Yes. We drew a square that has side lengths of 1 unit. Does our unit square show equal parts? No, not yet. There is one bigger part that’s not partitioned into same-size parts.

1 unit 4 1 unit 2

Can we make equal parts? How? Yes. We can finish partitioning the rest of the unit square.

Partition the rest of the unit square into equal parts and direct students to do the same. What do you notice about the unit square now? It is partitioned into 8 equal parts.

Each part is a rectangular tile with side lengths of  _unit and  _unit. 1 4

Consider using the Tiling digital interactive to confirm the solution.

1 2

The unit square looks like it is tiled with rectangular tiles. Can we find the area of the shaded rectangle? Explain your thinking. I’m not sure. We’ve only tiled with square tiles. Now it looks like our unit square is tiled with rectangular tiles.

Universal Design for Learning: Representation

1 unit 4 1 unit 2

Yes. The unit square is partitioned into 8 rectangular tiles that are all the same size, so each rectangular tile takes up the same amount of area.

248

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 11

Because we have partitioned the unit square into equal parts, we can find the area of the parts. What is the area of the shaded rectangle? How do you know?

 _18 square unit. There is 1 shaded part out of 8 total parts. The unit square has an area of 1 square unit, and  _18 of 1 square unit is  _18 square unit.

 _18 square unit. I know because 1 square unit ÷ 8 =  _18 square unit.

You estimated that the area of the rectangle would be less than 1 square unit. Were you correct? Why? Yes. There are more equal parts in the unit square than in the rectangle, and the area of the unit square is 1, so the area of the rectangle had to be less than 1. Display the sample student work. Multilingual Learner Support

What do you notice about this student’s work compared to the work we just did? This work has more equal parts in the unit square than we had. This student partitioned the shaded rectangle into two 1 squares with side lengths of  _unit and created a

1 4 unit

Consider providing sentence starters to support students in discussing the two models and how they may be familiar:

unit square that is partitioned into 16 square tiles with

1 2 unit

Did this student get the same answer as we did? Why?

2 16 square units

side lengths of  _unit. 1 4

Yes, the student’s answer is equal to our answer 2 1 because  __ is equivalent to  _. 16

• The unit square that is partitioned into 8 equal-size rectangular tiles reminds me of . • The unit square that is partitioned into 16 equal-size square tiles reminds me of . • These models remind me of work we have done before in math class, such as .

Invite students to turn and talk about the two models and to discuss what other work from grade 5 this reminds them of. These models are similar to the models that students used in module 3 for multiplying fractions.

249

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 11

Draw Unit Squares to Find Areas of Rectangles Students create a unit square to find the area of a rectangle with one unit-fraction side length and one side length that is not a unit fraction. Display the problem and the sample student work.

Find the area of the rectangle.

2 units 3 1 unit 4

2 units 3

2 units 3 1 unit 4

1 unit 4

Area of 1 rectangle tile: 1 square unit 12 2 units 3

Area of shaded rectangle: 1

1 unit 4

250

2 x 12 square unit = 12 square units

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 11

Use the Five Framing Questions routine to invite students to analyze the work.

Notice and Wonder What do you notice about this work? I notice that one of the side lengths is not a unit fraction. I notice that the student created a unit square like we just did. I notice that the tiles are rectangles. I notice that two rectangular tiles in the unit square are shaded. From your observations, what do you wonder? I wonder why the student first partitioned the shaded rectangle into two equal parts. I wonder why the student did not make square tiles. I wonder if the student’s answer is correct.

Organize What steps did this student take? How do you know? In the first drawing, the student partitioned the shaded rectangle into two rectangular tiles with side lengths of  _unit and  _unit. 1 4

1 3

In the first drawing, the student added more rectangles with side lengths of  _unit and  _unit to create side lengths of 1 unit. 1 3

1 4

In the second drawing, the student completed the drawing to make a unit square. In the third drawing, the student partitioned the unit square into equal parts by drawing in the rest of the lines. Then the student found the area of one rectangular tile and wrote an equation to find the area of the shaded rectangle that shows multiplying the area of one rectangular tile by 2. Advance the discussion to focus on partitioning the shaded rectangle into rectangular tiles with unit-fraction side lengths and encourage student thinking that makes connections to tiling the unit square with rectangular tiles with unit-fraction side lengths.

251

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 11

Reveal Let’s focus on how the student tiled the unit square with rectangular tiles with unit-fraction side lengths. Where do you see that in this work?

The unit square is tiled with rectangular tiles that have side lengths of  _unit and  _unit. 1 4

1 3

Distill Why did the student first partition the shaded rectangle into two rectangular tiles? The side lengths of a unit square are 1 unit, so the student needed a rectangular tile that would go evenly into 1 unit. The student needed to draw more rectangles to extend the side length of  _units 2 3

to make 1 unit. The student partitioned the shaded rectangle into two rectangular tiles with unit-fraction side lengths to make it easier to create a unit square.

 _23 =  _13 +  _13 , so the student partitioned the shaded rectangle into two rectangular tiles with side lengths of  _unit to make it easier to create a unit square. 1 3

_ 1 and _ 1are both unit fractions. How is it helpful to create rectangular tiles with side 4

lengths that are unit fractions? Unit fractions help us because we can make 1 unit. If we had tiles with side lengths of  _units, we couldn’t make 1 unit evenly. 2 3

When the side lengths of the tile are unit fractions, it’s easy to use iterations of the tile to make side lengths of 1 unit to create a unit square. Why did the student multiply the area of one rectangular tile by 2 to find the area of the shaded rectangle? The shaded rectangle is made up of 2 rectangular tiles, so the student multiplied the area of one tile by 2 to find the area of the shaded rectangle.

252

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 11

Know How is drawing a unit square helpful in finding the area of a rectangle with fraction side lengths? If I can draw a unit square with equal-size tiles, I can find the area of each tile inside of the unit square. Once I know the area of each tile, I can find the area of the shaded rectangle by multiplying the number of shaded tiles by the area of each tile. Direct students to problems 2 through 4. We need to find the area of the rectangles in problems 2 through 4. Before we begin, what do you notice about the side lengths of the rectangle in problem 2? The rectangle has one side length that is a unit fraction and one side length that is not a unit fraction.

Because   is not a unit fraction, it is helpful to partition the shaded rectangle into 3

rectangular tiles with unit-fraction side lengths before drawing a unit square. Invite students to turn and talk with a partner about how they plan to partition the rectangle in problem 2. Once the shaded rectangle is partitioned into rectangular tiles with unit-fraction side lengths, we can use the denominators of each of the fractions to determine how many more rectangular tiles we need to draw to create a unit square. Keep this in mind as you draw unit squares to find the areas of the shaded rectangles in problems 2 through 4. Invite students to work with a partner to find the areas of the rectangles in problems 2 through 4.

253

5 ▸ M5 ▸ TB ▸ Lesson 11

EUREKA MATH2 Florida

1 unit 2 2 units 3

_ 2 square units 6

3 units 4

1 unit 5 3 __    square units 20

254

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 11

1 unit 6

3 units 5

3 __    square units 30

As students work, circulate and ask the following questions to advance their thinking: • Can you partition the shaded rectangle into rectangular tiles with unit-fraction side lengths? • Can you extend the side lengths of the shaded rectangle to create a unit square? How? • Can you partition the unit square into equal parts? • What is the area of one of the equal parts? • How many equal parts are shaded? • How can you use the area of one of the equal parts to find the area of the shaded rectangle? When students have finished, discuss the solutions. Revisit the problem from the Five Framing Questions routine. Invite students to turn and talk about how to find the area of a rectangle with one side length that is a unit fraction and one side length that is a fraction less than 1.

255

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 11

Display the sample student work. What do you notice about this student’s work compared to the work we just looked at?

Area of 1 square tile:

Area of shaded rectangle:

2 units 3

This student has a lot more equal parts in the unit square than the other student did.

1 square unit 144

24 ×

1 24 square unit = square units 144 144

1 unit 4

This student partitioned the shaded rectangle into square tiles with side lengths of  __ unit and created a unit square that is partitioned into 144 square tiles with side 1 12

lengths of  __ unit. 1 12

Did this student get the same answer as the other student? Why?

Yes, this student’s answer is equal to the other student’s answer because  ___ is equivalent to  __ . 2 12

24 144

Invite students to Think–Pair–Share about which method they prefer. I prefer the first method because I don’t have to draw as many tiles to tile the unit square. I prefer the first method because it is more efficient to draw rectangular tiles with fraction side lengths than it is to draw square tiles with fraction side lengths. I prefer the second method because I like working with square tiles better than rectangular tiles.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

256

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 11

Land

Debrief 5 min Objective: Find the area of a rectangle with fraction side lengths by relating the rectangle to a unit square. Gather the class with their Problem Sets. Facilitate a class discussion about finding the area of rectangles with fraction side lengths by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Does drawing a unit square help you find the area of rectangles with fraction side lengths? How? Yes. Drawing a unit square and then partitioning it into equal parts helps me to see how many shaded units out of the total number of units the rectangle takes up. I know that the area of the rectangle is the number of shaded parts out of the total number of equal parts. Yes. Drawing a unit square and then partitioning it into equal parts helps me to find the area of one of the parts. I know that the area of the rectangle is the number of shaded parts times the area of one of the parts. Direct students to problems 3 and 4 in the Problem Set. How does knowing the area of a rectangular tile with unit-fraction side lengths help you find the area of a rectangle with one side length that is not a unit fraction? In problem 3, I partitioned the shaded rectangle into rectangular tiles with side lengths of _     unit and _    unit to help me create a unit square and find the area of one tile. Each 1 6

1 4

rectangular tile has an area of  __ square unit. Because the shaded rectangle is made up 1 24

of 3 rectangular tiles, I can multiply the area of one tile by 3 to find the area of the shaded rectangle.

257

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 11

In problem 4, I partitioned the shaded rectangle into rectangular tiles with side lengths of  _unit and _    unit to help me create a unit square and find the area of one tile. Each 1 2

1 5

rectangular tile has an area of __    square unit. Because the shaded rectangle is made 1 10

up of 2 rectangular tiles, I can multiply the area of one tile by 2 to find the area of the shaded rectangle.

258

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 11

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can use unit squares to find areas of rectangles that have fraction side lengths.

PRACTICE

Florida B.E.S.T.

2. A rectangle with side lengths of _ unit and _ unit is shown. Use the rectangle to complete 5 1

MA.5.GR.2.1

1 3

parts (a)–(c).

Name

Date

1. Use this unit square, which is partitioned into equal-size rectangles, to answer parts (a) and (b).

1 unit 3

1 unit

1 unit 5

a. Create a unit square. Partition the unit square into equal parts. b. How many equal parts did you need to create a unit square?

a. The side lengths of the shaded rectangular tile are

_1 4

unit and

_1 6

c. What is the area of the rectangular tile with side lengths of _ unit and _ unit? How do you 5 1

know?

unit.

1 3

1 The area of the rectangular tile is __ square unit. There are 15 equal parts in the unit square, 15 1 __ so 1 part has an area of 15 square unit.

b. The area of the shaded rectangular tile is

__1

square unit because there are

equal-size rectangular tiles and 1 is shaded.

PROBLEM SET

259

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 11

4. A rectangle with side lengths of _5 units and _ unit is shown. Use the rectangle to complete

3. A rectangle with side lengths of _ units and _ unit is shown. Use the rectangle to complete 3 4

parts (a)–(c).

1 6

1 2

parts (a)–(c).

1 unit 2 2 units 5

1 unit 6

a. Create a unit square. Partition the unit square into equal parts.

3 units 4

b. How many equal parts do you need to create a unit square?

a. Create a unit square. Partition the unit square into equal parts.

c. What is the area of the rectangle with side lengths of _5 units and _ unit? 2

b. How many equal parts did you need to create a unit square?

2 __ square units 10

c. What is the area of the rectangle with side lengths of _ units and _ unit? 3 __ square units 24

3 4

1 6

1 2

5. What is the area of the rectangle shown?

3 units 5

1 unit 2

__3 square units 10

260

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 11

EXIT TICKET

I can use unit squares to find areas of rectangles that have fraction side lengths. 6. Sana creates a drawing to determine the area of a rectangle with a side length of _ units and 5 6

1 a side length of _ unit.

Florida B.E.S.T. MA.5.GR.2.1

Name

Date

_1 _7 Find the area of a rectangle with side lengths of unit and units. Sketch to show how you know. 2

1 unit 4

5 units 6

Sana says the area of the rectangle is _ square unit. 1 8

Is she correct? Why? No, Sana is not correct. The unit square is not partitioned into rectangular tiles of the correct size. The side length of the rectangular tile in her drawing is _ of the side length of the unit square, but 1 4

the other side length of the rectangular tile is not _ of the unit square. She cannot use her drawing to find the area of the shaded rectangle.

5 6

1 unit 2

7 units 8

__7

The area is 16 square units.

PROBLEM SET

261

LESSON 12

Find areas of rectangles with fraction side lengths by using multiplication. I can use multiplication to find areas of rectangles that have fraction side lengths.

Lesson at a Glance Students reason about and draw models to find the area of rectangles with side lengths that are fractions greater than 1. They discover that drawing a model is not an efficient method for finding the area of these rectangles. After analyzing lengths, widths, and areas, students conclude that they can find the area of any rectangle by multiplying its length by its width.

Key Questions • Why is the relationship between a rectangle’s side lengths and its area the same whether the side lengths are fractions or whole numbers?

Language Objective • Orally describe multiplicative relationships between the fraction side lengths and the area of a rectangle.

Mathematical Thinking and Reasoning Standard MTR.5.1

• How is finding the area of a rectangle with fraction side lengths the same as or different from finding the area of a rectangle with whole-number side lengths?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 12

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 5 min Learn 35 min • Rectangles with One Side Length That Is a Fraction Greater Than 1 • Rectangles with Both Side Lengths That Are Fractions Greater Than 1 • Find Area by Multiplying • Problem Set

• None

Students • None

Land 10 min

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD9 Find areas of rectangles with

MA.5.GR.2.1 Find the perimeter and area of a

fraction or decimal side lengths. (MA.5.GR.2.1)

rectangle with fractional or decimal side lengths using visual models and formulas.

263

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 12

Fluency

Whiteboard Exchange: Round Decimals Students round a number to the nearest tenth and hundredth to build fluency with rounding decimals from module 4. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display 0.826 ≈

Round 0.826 to the nearest tenth.

Display the rounded value and then display 0.826 ≈ Round 0.826 to the nearest hundredth.

0.826 ≈

0.8

0.826 ≈ 0.83

Display the rounded value. Repeat the process with the following sequence:

7.1

21.493 ≈ 21.5

90.604 ≈ 90.6

7.057 ≈ 7.06

21.493 ≈ 21.49

90.604 ≈ 90.60

7.057 ≈

264

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 12

Choral Response: Multiply Fractions Students determine products to develop fluency with finding the area of rectangular figures with fraction side lengths by multiplying. Display  _1 ×  _  = 3

1 5

What is the product? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

1 1 × = 3 5

1 15

1  __   15

Display the product. Repeat the process with the following sequence:

1 ×4= 4 3 5 15

7 ×1= 7 8 8 64

7 × 5 = 35 8 8 64

3 ×3= 9 7 4 28

4 × 7 = 28 7 4 28

5 × 7 = 35 6 10 60

9 × 9 = 81 6 11 66

Choral Response: Add Whole Numbers and Fractions Students determine the sum of a whole number and a mixed number or fraction greater than one to prepare for adding partial products beginning in lesson 13. 1 Display 1 + 1 _    = 4

What is the sum? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

1 4

1+1 =

1 4

2 1_   4 Display the sum.

265

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 12

Repeat the process with the following sequence:

3+2 =

3 8

6 6

4+ =

266

3 8

2 5

2+ =

3 3

2+ =

5 3

1 6

16 +5= 8

19 +1= 8

1 +2=

7 6

3+ =

2 3

3 8

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 12

Launch

Students reason about how to find the area of a rectangle with fraction side lengths greater than 1. Display the three rectangles and the sample student work.

1 unit 3

3 units 3

2 units 3 4 units 5

Area: 15 square units

4 units 5

Area: 15 square units

Invite students to Think–Pair–Share about what they notice and wonder. I notice that in each rectangle, one side length stays the same and one side length increases by  _1  unit. 3

I notice that when the side length increases by  _1 unit, the area increases by 3

4   square units.  __ 15

I notice that the product of each length and width is equal to the area of the rectangle. I wonder if we can multiply length and width to find the area of a rectangle with fraction side lengths instead of tiling. I wonder if this pattern continues.

I wonder how to draw a rectangle that is  _4  units by  _4  units. 3

Teacher Note Some students may notice a pattern: The area is the product of the side lengths, even if those side lengths are fractions. The work of this lesson is to notice this repeatedly until students can confidently answer the question, Can we find the area of a rectangle with fraction side lengths by multiplying the side lengths? Therefore, allow this noticing to emerge naturally during the lesson and to solidify by the end.

267

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 12

Point to the first rectangle.

How many rectangular tiles that are _  1 unit by _  1 unit make 1 square unit? 3

What is the area of 1 rectangular tile? 1  square unit  __ 15

Invite students to Think–Pair–Share about how they could find the area of a rectangle with side lengths of  _4  units and  _4  units. 5

We can continue the pattern. Because we are increasing one side length by  _1 unit, the 16 square units. 4  square units, so the area will be  __ area will increase by  __ 15

If we continue the pattern, we’ll need to add another unit square that is partitioned into rectangular tiles and there will be 16 shaded tiles. Because each tile has an area of 16 square units. 1  square unit, we can multiply 16 and  __ 1   to get  __  __ 15

Transition to the next segment by framing the work. Today, we will find areas of rectangles with fraction side lengths, including side lengths that are fractions greater than 1.

268

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 12

Learn

Rectangles with One Side Length That Is a Fraction Greater Than 1 Students find areas of rectangles with one side length that is a fraction greater than 1. Introduce the Critique a Flawed Response routine and present the student work for finding the area of a rectangle with side lengths of  _5  units and  _4  units. 3

Give students 2 minutes to identify the error or ambiguity. Invite students to share.

5 units 3

I see 20 shaded parts out of 30 total parts, but I don’t think the denominator should be 30 because each unit square shows 15 equal parts.

4 units 5

Each unit square shows 15 equal parts, so the answer should be in fifteenths.

Area: 30 square units

16 square units. The area of the rectangle with side lengths of  _4  units and  _4  units was  __ 5

16 square units because this rectangle The area of this rectangle should be greater than  __ 15

is larger than the rectangle with side lengths of  _4  units and  _4  units. 3

269

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 12

Give students 2 minutes to solve the problem based on their own understanding. Circulate and identify a few students to share their thinking. Purposefully choose work that allows for rich discussion about the number of total rectangles in the model compared to the number in the denominator of the solution. 20 square units. If we continue the pattern from earlier, each time The area is  __ 15

4  square units. we increase a side length by  _1 unit, the area increases by  __ 15

20 square units. Each shaded rectangular tile has an area of  __ 1  square unit. The area is  __ 15 15 1 20 square units. There are 20 shaded rectangular tiles, so the area is 2 0 × __     , which is  __ 15 15 20 square units. The whole is 1 unit square, and it is partitioned into The area is  __ 15

15 rectangular tiles. There are 20 rectangular tiles shaded out of the whole, so the area 20  square units. is  __ 15

Then facilitate a class discussion. Invite students to share their solutions with the whole group. Lead the class to a consensus about how best to correct the flawed response. 20 square units. The area of a rectangle with side lengths of  _5  units and  _4  units is  __ 3

Display the sample student work.

Area of 1 tile:

What do you 5 notice about this units 3 student’s work compared to the work we just looked at?

1 1 1 × = square unit 3 5 15

Area of rectangle: 20 ×

1 20 square unit = square units 15 15

4 Instead of drawing units 5 unit squares, this student tiled the rectangle with rectangular tiles with side lengths  _1  unit and  _1  unit. 5

It looks like the student used multiplication to find the area of 1 tile.

In the last student’s work, 15 tiles were used to cover a unit square, and that’s how they

1  square unit. I think this student multiplied  _1  and  _1  and knew the area of each tile was  __ 1  square unit. they also got  __

270

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 12

Invite students to Think–Pair–Share about why they think the student multiplied to find the area of 1 tile. Just like in the last student’s work, we would need 3 rows of 5 tiles to tile a unit square. 1   of 1 square unit, so it would have an area of 3 × 5 = 15 tiles, and each tile would be  __ 15

1  square unit. I think multiplying the denominators might always tell us the number  __ 15

of tiles and the area of each tile. I’ve noticed a pattern: The area of 1 rectangular tile is equal to the product of the length and width of the rectangular tile. Maybe that is always true. We know that to find the area of a rectangle with whole-number side lengths, we can multiply the length and width of the rectangle. Maybe that is true for fraction side lengths. The formula for finding the area of a rectangle with whole-number side lengths is A = l × w. Maybe the formula also works for finding the area of rectangles with fraction side lengths, and the student used the formula. When we corrected the previous student’s work, we found the area of 1 tile by using the rectangular tile to draw unit squares. What did we find as the area of 1 rectangular tile by drawing the unit square? 1  square unit  __ 15

Did both methods give us the same area for the rectangular tile? Yes. We can find the area of 1 rectangular tile with unit-fraction side lengths more than one way. We can use the rectangular tile to draw unit squares or multiply the unit-fraction side lengths of the rectangular tile. How did this student use what they knew about the area of each tile to find the area of the whole rectangle? 1  square unit. Because 20 tiles are needed They knew that each tile has an area of  __ 15

20 square units. 1   to get  __ to cover the rectangle, the student multiplied 20 and  __ 15

Invite students to turn and talk about how the two methods are similar and how they are different.

271

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 12

Rectangles with Both Side Lengths That Are Fractions Greater Than 1 Students use rectangular tiles to find areas of rectangles when both side lengths are fractions greater than 1. Direct students to problem 1 in their books. Find the area of the rectangle. 1. 6 units 5 3 units 2 18 square units  __ 10

What do you notice about this rectangle? Its side lengths are both fractions. The fractions are both greater than 1. What size tiles could we use to tile this rectangle? How do you know?

We could use rectangular tiles with side lengths of  _1  unit and  _1 unit. When we partition 5 2  _6  and  _3 into unit fractions, we get  _1  and  _1 . 5 5 2 2 1  because when we partition fifths and We could use square tiles with side lengths of  __ 10 1  . halves into squares, the squares will have side lengths of  __

Differentiation: Support Some students may need to create and tile a unit square to find the area of a rectangle with side lengths of  1 unit and  1 unit.

__ 5

__ 2

Partition the rectangle and direct students to do the same. What are the side lengths of each rectangular tile?  _1  unit and  _1  unit 5

1 unit 5

What is the area of each rectangular tile? How do you know?

1  square unit. The area of a rectangle with side lengths of  _1  unit and  _1 unit is  __ 10 1  _1 ×  _1 = __     square unit. 5 2 10

272

1 unit 2

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 12

How can we use what we know about the area of each tile to find the area of the rectangle? Explain your thinking. The rectangle is tiled with 18 rectangular tiles with side lengths of  _1  unit and  _1  unit, 1 so we can multiply 1 8 × __    to find the area of the entire rectangle.

What is the area of the rectangle?

Multilingual Learner Support

Provide sentence starters to support students in sharing what they notice and discussing how to find the area:

18 square units  __

Invite students to turn and talk about what they notice about the relationship between the side lengths of the rectangle and its area. Ask pairs to discuss how they can find the area of a rectangle with fraction side lengths without tiling.

• The rectangle’s area

Find Area by Multiplying Students conclude that they can find the area of a rectangle with any side lengths by multiplying its length and width. We can find the area of a rectangular tile with unit-fraction side lengths by multiplying the length and the width. I wonder whether this is true for any rectangle with fraction side lengths. Display the rectangles and their areas. Use the following questions to facilitate a discussion to guide students to conclude that they can multiply the side lengths when finding the area of any rectangle. The rectangles are from this lesson and previous lessons.

3 units 4 2 units 3

1 unit 3

6 units 5

2 9

3 units 2

Area:

6 square units 12

• To find the area of a rectangle with fraction side lengths without tiling, .

Promoting Mathematical Thinking and Reasoning Students use patterns and structure (MTR.5) as they notice that the multiplicative relationship between a rectangle’s fraction side lengths and its area is repeated for a variety of rectangles. They transition to using that relationship as a shortcut for finding the area of rectangles with fraction side lengths. Ask the following questions to promote MTR.5:

2 units 3

Area: square units

• The side lengths of the rectangle .

Area:

18 square units 10

• What patterns do you notice when you look at the side lengths and areas of these rectangles? • Will the area always equal the product of the side lengths? Explain.

273

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 12

We have previously found the areas of all these rectangles by tiling. Let’s record their lengths, widths, and areas in a table. Draw a table with the lengths, widths, and areas of the rectangles. Record the side lengths and the areas of the rectangles in the table.

Length (units)

Width (units)

Area (square units)

2 3

1 3

2 9

2 1 2 3×3=9

3 4

2 3

6 12

3 2 6 4 × 3 = 12

18 10

3 6 18 2 × 5 = 10

2 5 Invite students to turn and talk about the length, width, and area of each rectangle.

Universal Design for Learning: Representation

What do you notice about the length, width, and area of each rectangle? The area of each rectangle is the product of its length and width.

Consider annotating the table to highlight the factors and the product.

When the side lengths are both less than 1 unit, the area is less than 1 square unit. Write the multiplication equation for each row of the table. Ask questions such as the following to lead the class to the consensus that they can multiply the length and width of any rectangle to determine its area, even when the side lengths are fractions. What do you notice about the denominators in the length, width, and area? The denominator in the area is the product of the denominators in the length and width. What does this relationship tell us about the size of tiles we can use to tile the rectangle? Use one of the rectangles represented in the table to explain your thinking. We can tile the rectangle with tiles that have sides that are unit fractions with the same

Length (units)

Width (units)

Area (square units)

2 3

1 3

2 9

3 4

2 3

6 12

3 2

6 5

18 10

denominators as the length and width of the rectangle. We can tile a rectangle with

side lengths  _1  unit and  _2 units with rectangular tiles with side lengths of  _1  unit and  _1  unit. 3

274

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 12

We can multiply the denominators of the length and width of the rectangle to find the denominator of the area of the tile that we can use to tile the rectangle. We can use 1  square unit to tile a rectangle with side lengths  _2 units and a tile with an area of  __ 12

1  _3  units because  _1 ×  _1 = __    . 4

What do you notice about the numerators in the length, width, and area? The numerator in the area is the product of the numerators in the length and width. What does this relationship tell us about the number of tiles we need to tile the rectangle? Use one of the rectangles in the table to explain your thinking. When we tile the rectangle with tiles that have sides that are unit fractions with the same denominators as the length and width of the rectangle, the product of the numerators tells us how many tiles we need to cover the rectangle. If we tile the

rectangle that has side lengths of  _6  units and  _3 units with a rectangular tile with side 5

lengths  _1  unit and  _1 unit, we need 18 tiles because 6 × 3 = 18. 5

When we tile the rectangle with tiles with an area where the denominator is the product of the length and width of the rectangle, the product of the numerators tells us how many tiles we need to cover the rectangle. If we tile the rectangle that has side lengths

 _1  unit and  _2 units with a rectangle with an area of  _1 square unit, we need 2 tiles to cover 3

the rectangle because 1 × 2 = 2.

Can we find the area of a rectangle with fraction side lengths by multiplying the side lengths? How do you know? Use one of the rectangles in the table to explain your thinking. Yes. We know the area of a rectangle with fraction side lengths is the number of tiles multiplied by the area of each tile. The rectangle that has side lengths of  _6  units 5

1  square unit. So the area of the and  _3 units can be tiled with 18 tiles with an area of  __ 10

3 18 1 rectangle is 1 8 × __    =  __  , which is equal to multiplying  _6 ×  _ . 10

How is finding the area of a rectangle with fraction side lengths like finding the area of a rectangle with whole-number side lengths? Finding the area of a rectangle with fraction side lengths is just like finding the area of a rectangle with whole-number side lengths because you can multiply the side lengths. © Great Minds PBC

275

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 12

We know the formula for finding the area of a rectangle with whole-number side lengths is A = l × w. Do you think we can use the same formula to find the area of rectangles with fraction side lengths? Why? Yes. To find the area of a rectangle with fraction side lengths, we can multiply the length and width of the rectangle. We can use the formula A = l × w to find the area of rectangles with whole-number side lengths or fraction side lengths. Direct students to problems 2 through 5. Ask students to find the areas of the rectangles with the given side lengths. Have them check their work with a partner. Find the area of each rectangle with the given side lengths. Length (units)

Width (units)

Area (square units)

2_   3

5_   6

10 __   18

3_   2

5_   6

15 __   12

3_   2

7_   6

21 __   12

5_   6

3_   3

15 __   18

When students have finished, discuss the solutions. Why is the area in problem 4 greater than 1 square unit? We are multiplying a number greater than 1 by another number that is greater than 1, so the answer will also be a number greater than 1. 276

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 12

Why is the area in problem 2 less than _     square units? 2 3

We are multiplying  _2 by a fraction less than 1, so the area is less than  _2 square units. 3

Why is the area in problem 2 less than _  5  square units?

We are multiplying  _5 by a fraction less than 1, so the area is less than  _5 square units. 6

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Find areas of rectangles with fraction side lengths by using multiplication. Facilitate a class discussion about using multiplication to find the area of rectangles with fraction side lengths by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Why is the relationship between a rectangle’s side lengths and its area the same whether the side lengths are fractions or whole numbers? A rectangle takes up flat space no matter how long or short the sides are and no matter if the side lengths are parts of units instead of whole units. The area of a rectangle is the amount of flat space the rectangle takes up. We can find the area by multiplying its side lengths, whether the side lengths are whole numbers or fractions.

277

5 ▸ M5 ▸ TB ▸ Lesson 12

EUREKA MATH2 Florida

How is finding the area of a rectangle with fraction side lengths the same as or different from finding the area of a rectangle with whole-number side lengths? Finding the area of a rectangle with fraction side lengths is like finding the area of a rectangle with whole-number side lengths because for both, we can multiply the length and the width to find the area. With whole-number side lengths, we can always use unit squares to tile a rectangle and find the area. For fraction side lengths, we have to find the right size tile to use. Do you think finding the area of a rectangle with decimal side lengths is the same as or different from finding the area of a rectangle with fraction or whole-number side lengths? They are the same. Decimals are another way to name fractions, so the methods we use to find the area of a rectangle with fraction side lengths should also work to find the area of a rectangle with decimal side lengths. They are the same. Decimals, fractions, and whole numbers are all ways to name numbers. The relationships between the side lengths and the area are the same no matter how the numbers are written.

278

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 12

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can use multiplication to find areas of rectangles that have fraction side lengths.

PRACTICE

Florida B.E.S.T. MA.5.GR.2.1

Name

2. Use the partitioned unit squares to complete parts (a)–(d).

Date

1. Use the models to complete parts (a)–(d).

1 unit

2 units 3

a. How many equal parts is the unit square partitioned into?

6 units 5

a. Each unit square shown is partitioned into

1 unit

b. What is the area of each equal part? 1 __ square unit 12

equal parts.

b. What is the area of each equal part? 1 __ square unit 15

c. Use blue to color the equal parts of the unit squares to represent the blue rectangle shown. The rectangle represented by the green portion of the two unit squares has side lengths of

d. What is the area of the blue rectangle? 12 __ square units 15

_7 4

units and

_2 3

units.

d. What is the area of the rectangle represented in part (c)? 14 __ square units 12

103

104

PROBLEM SET

279

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 12

3. Use this model, which shows four unit squares partitioned into ninths, to complete parts (a)–(d).

Find the area of the rectangles shown. Show how you know. 4.

1 unit 2

7 units 3

3 units 2

6 units 5 21 _7 × _3 = __

21 __ square units 6

a. How many equal parts is each unit square partitioned into?

__6 square units

6 1_ 6_ __ × = 2 5 10

b. Use orange to color the equal parts of the unit squares to represent a rectangle with side

6. Find the areas of the rectangles with the given side lengths.

lengths of 3 units and 3 units.

c. What is the area of the rectangle? 20 __ square units 9

Length (units)

Width (units)

Area (square units)

9_ 8

6_ 5

54 __ 40

10 __ 9

11 __ 10

110 ___ 90

d. Show how to find the area by multiplying. 5 _ _ __ × 4 = 20 3 3 9

280

PROBLEM SET

105

106

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 12

EXIT TICKET

I can use multiplication to find areas of rectangles that have fraction side lengths.

Florida B.E.S.T. MA.5.GR.2.1

11 _3 __ Find the area of a rectangle with side lengths of 4 units and 9 units.

Name

33 The area is __ square units. 36

Date

_3 × 11 __ = 33 __ 4 9 36

107

281

LESSON 13

Multiply mixed numbers. I can multiply mixed numbers.

Language Objective

Lesson at a Glance Students determine areas of rectangles with mixed-number side lengths. By using an area model, students decompose mixed-number factors into whole-number parts and fractional parts and then represent and add the partial products. Students make connections between the area model and the distributive property before transitioning to multiplying mixed numbers without visual models, applying the distributive property, or converting mixed numbers into fractions greater than 1.

Key Questions • How are area models helpful when multiplying mixed numbers?

• Orally justify methods to multiply mixed numbers, including using an area model, writing mixed numbers as fractions greater than 1, and using the break apart and distribute strategy.

Mathematical Thinking and Reasoning Standard MTR.6.1

• How can we use the distributive property when multiplying mixed numbers? • How is multiplying mixed numbers similar to multiplying decimals? How is it different?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 13

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 5 min Learn 35 min • Explore Area Models • Multiply Two Mixed Numbers • Problem Set

• None

Students • None

Land 10 min

Achievement Descriptors

Florida B.E.S.T. Standard

FL.5.Mod5.AD1 Multiply mixed numbers and

MA.5.FR.2.2 Extend previous understanding

whole numbers, fractions, or mixed numbers. (MA.5.FR.2.2) FL.5.Mod5.AD2 Model products of fractions

of multiplication to multiply a fraction by a fraction, including mixed numbers and fractions greater than 1, with procedural reliability.

and mixed numbers by using rectangular areas. (MA.5.FR.2.2)

283

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 13

Fluency

Round 0.738 to the nearest tenth.

Display the rounded value and then display 0.738 ≈ Round 0.738 to the nearest hundredth.

0.738 ≈ 0.7 0.738 ≈ 0.74

Display the rounded value. Repeat the process with the following sequence:

284

8.056 ≈ 8.1

31.572 ≈ 31.6

80.504 ≈ 80.5

8.056 ≈ 8.06

31.572 ≈ 31.57

80.504 ≈ 80.50

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 13

Whiteboard Exchange: Add or Subtract Decimals Students determine sums and differences to develop procedural fluency with adding and subtracting decimals from module 4. Display 0.54 + 0.39 =

0.54 + 0.39 = 0.93 0.53 0.01

Display the sum and sample student work. Repeat the process with the following sequence:

3.6 + 8.82 = 12.42

0.8 − 0.26 = 0.54

5 − 1.47 = 3.53

9.04 − 3.86 = 5.18

285

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 13

Launch

Students relate the distributive property and the area model. Display the rectangle with side lengths of 2 units and 3.8 units. We know that we can find the area of rectangles with fraction side lengths by multiplying the side lengths. How can we find the area of this rectangle?

3.8 units 2 units

We can multiply the length by the width.

Universal Design for Learning: Representation Activate prior knowledge by reviewing prior understandings about area models and building connections to new information. Consider presenting the area model and asking the following questions:

We can multiply 2 by 3.8. Invite students to Think–Pair–Share about how to determine the product of 2 and 3.8.

180

600

I can record my work in vertical form. I can multiply 2 by 38 tenths. I can use the distributive property. Display the work that uses the distributive property to find 2 × 3.8. The work that uses the distributive property shows the partial products of 2 × 3.8. What are the partial products?

2 × 3.8 = 2 × (3 + 0.8) = (2 × 3) + (2 × 0.8) = 6 + 1.6 = 7.6

6 and 1.6 What is 2 × 3.8? How do you know?

7.6. I know because the sum of the partial products is 7.6.

• What are the partial products of 34 × 26? • What is the product of 34 and 26? How do you know?

7.6. I know because 6 + 1.6 = 7.6. 7.6. I know because 2 × 3 = 6 and 2 × 0.8 = 1.6 and 6 + 1.6 = 7.6.

286

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 13

Invite students to turn and talk about how they could use an area model to show the partial products. Why might we use an area model to record our work with the distributive property? The area model helps us keep factors and products organized. The area model clearly shows the partial products. The area model helps us keep track of the different place values of the factors and products.

Write 2 × 3  5  . Invite students to turn and talk about how they could use partial products to multiply mixed numbers. Transition to the next segment by framing the work. Today, we will multiply mixed numbers.

Learn

Explore Area Models Students use an area model to multiply a mixed number and a whole number.

Display the area model that represents 2 × 3  5  .

What multiplication expression does this area model represent? 4 2 × 3 _  5

4 5

Place students in groups of 2 or 3. Ask them to find 2 × 3  5  and to show their thinking on their whiteboards. Allow students to use any method to determine the product.

287

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 13

When students are finished, invite them to share their thinking, highlighting that there is more than one way to correctly determine the product. Anticipate any of the following responses. Addition:

2 × 34 = 34 + 34 5

5 4 4 = 3+ 3+ 5 + 5 = 6+8 5 = 6+ 1 3 5 = 73 5

Number bond:

(

3 54 3

) = (2 × 3) + (2 × 54)

2 × 3 54 = 2 × 3 + 54 4 5

= 6 + 58

= 6 + 1 35 = 7 35 Distributive property:

(

2 × 3 4 = (2 × 3) + 2 × 4 5

288

=6+8 5 = 6 + 13 5 3 =7 5

)

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 13

Area model:

2 × 34 = 6 + 5 5

= 6+ 1 3 = 73 5

4 5

8 5

Display the area model of 2 × 3  5  .

4 3 5

Universal Design for Learning: Representation

4 5

Consider color-coding to connect the area model and the distributive property.

One way to find 2 × 3   5 is to use an area model. Does the area model on the right represent the same expression as the area model on the left? How do you know?

Yes. Both area models have one side labeled with 2 and the other side labeled with 3  5  ,

which I know because 3 +  5 = 3  5  .

The area model on the right decomposes 3  5 into a whole number and a fraction.

It is decomposed into 3 and   5 so that we can multiply to get partial products. I can find 4

Display the partial products in the area model of 2 × 3  5  .

What is 2 × 3   5 ? How do you know?

7  5  . I know because 6 +  5 = 7  5 .

8 5

4 5

8 5

) 5) = (2× 3) + )2× 4 ) 5 8

2× 34 = 2× 3 + 4 = 6+

Why is 3   5 decomposed into a whole number and a fraction?

2 × 3 and 2 ×  _5 and add the products together.

4 5

How is the area model on the right different from the area model on the left?

4 5

8 5

5 = 6 + 13 5 = 73 5

289

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 13

How does the area model show the distributive property?

The area model shows decomposing the factor 3  5 into a whole unit and a fractional unit and then multiplying each unit by 2.

The area model shows decomposing 3  5 into two different parts and then multiplying each part by 2. Write the following work, pointing to the area model as you go and thinking aloud about how each step relates to the area model:

• 2 is multiplied by the lengths of both smaller rectangles to determine their areas.

3 6

8 5

2 3 6

4 5

5 = 6 + 13 5 3 =7 5

Display the four area models.

4 5

= 6+ 8

• The areas of the smaller rectangles, the partial products, are added to determine the total area.

4 5

( ) = (2 × 3) + (2 × )

2× 34 =2× 3+ 4

• The length of the rectangle in the area model is decomposed into 2 parts.

8 5

2 4 5

8 5

Can any of these area models be used to represent 2 × 3   5 ? How do you know?

Yes. Each area model shows 2 × 3 and 2 ×  5  .

Yes. The areas of the parts are the same in each area model.

290

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 13

When you draw an area model, can you draw it any of these ways? Why?

Yes. In every model 2 is being multiplied by 3  5  .

_3 _2 Display the area model for 2    × 1     . 4

Universal Design for Learning: Action & Expression

Invite students to Think–Pair–Share about how to use an area model to decompose the mixed numbers to determine the product.

I can decompose 1  3  into 1 and   3 . I can decompose 2  4  into 2 and   4  .

Display the area model with the decomposed mixed numbers.

Direct students to problem 1 in their books. Invite them to draw the area model with the decomposed mixed numbers. Have them work with a partner to complete problem 1.

Circulate as students work, asking the following questions:

2 3

Consider displaying a template that shows how to decompose mixed numbers. Some students may write the whole number of one of the mixed-number factors with the fraction from the other factor. Consider inviting students who make this mistake to color-code their whole numbers and fractions.

Whole number

• What partial product does this rectangle represent? How do you know? • How can you determine the product?

• What do you need to do before you add the fractions? • How can you write the fraction greater than 1 as a mixed number?

3 4

1. Use an area model to multiply.

2 3_ × 1 2_  3 4

Whole number

Fraction

2 3

6   2 3_ × 1 2_ = 2 + 3_ + 4_ +  __ 3 12 4 4 3

4 3

= 2 + 31 __  12

Fraction

3 4

6 12

Whole number Fraction

2 4 × 13

9  + 16 6   = 2 +  __ __ +  __ 12 12 12

7   = 2 + 2  __ 12 7   = 4  __ 12

Fraction

2 3

3 4 291

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 13

When students are finished, display the area models and the first two lines of work showing the multiplication with the distributive property. Whole number

Fraction

2 3

Whole number

4 3

Fraction

3 4

6 12

2 _4 × 1_3 = ( 2 + _4 )× ( 1 + _3 ) 3

Challenge students to predict and then

Differentiation: Challenge

_ _

3 3 2 2 = (2 × 1) + ( 4 × 1)+ ( 2 × 3 )+ ( 4 × 3 )

__3

__1

determine whether 2   × 3   has the same

__1

__3

value as 2   × 3   . 8

The work from the distributive property shows the sum of 4 partial products. Where do we see those 4 partial products in the area model? They are the areas of the 4 smaller rectangles. Display the remaining lines of work. 2

23 4

3 4

2 3

4 3 6 12

2 3_ × 1 2_ = (2 + 3_ )× (1 + 2_ ) 3 3 4 4 = (2 × 1)+ (3_ × 1)+ (2 × 2_ )+ (3_ × 2_ ) 3 4 4 3

6   = 2 + 3_ + 4_ +  __ 12 4 3

9  + 16 6   = 2 +  __ __ +  __ 12 12 12

= 2 + 31 __  12

7   = 2 + 2  __ 12 7 = 4  __ 12

292

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 13

What is 2     × 1    ? How do you know? 3

__7

4  12 . I know because that’s the sum of the 4 partial products.

Is it possible to multiply 2    by 1    without drawing an area model? Explain. 3

Yes. I can use the distributive property to multiply without drawing an area model.

5 11 _ __

2 13

5 3

Display the area model for 2  4 × 1  3 and the area model for   4 ×  3  .

23 4

11 4

What is similar about these two area models? What is different? The rectangles have the same lengths and widths. The length and width are written as mixed numbers for one rectangle and as fractions greater than 1 for the other rectangle. 11 _5 __

How can you find     ×    without decomposing the numbers? 3

I can multiply the fractions.

Multilingual Learner Support Mixed number and fraction greater than 1 are used throughout the lesson. Consider creating a chart that includes examples to support students as they hear and use these terms. Label the whole-number and fractional parts of the mixed numbers and annotate to show the numerator is greater than the denominator in fractions greater than 1. Mixed Number

Fraction Greater Than 1

2  3

1  2 3

__ 3

__ 4

11 _5 __

Write   3 ×   4 . Ask students to determine the product and to write the answer as a mixed number on their whiteboards. 11 _5 __

What is     ×    ? How do you know? 3

55 __    . I know because 5 × 11 = 55 and 3 × 4 = 12. 12

293

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 13

What is   7   4  __

55 __  written as a mixed number? 12

When we used an area model or the distributive property to multiply 1    and 2     ,

__7

we determined that the product is 4    . When we wrote the mixed numbers

as fractions greater than 1 and multiplied to determine the product, we also

__7

determined that the product is 4     . 12

What are some advantages of multiplying mixed numbers by writing them as fractions greater than 1? I only need to find the product of one multiplication expression. I do not need to use the distributive property. When might you not want to multiply mixed numbers by writing them as fractions greater than 1? I would not want to multiply fractions greater than 1 if the numbers are large or difficult to multiply. What are the advantages of using partial products to multiply mixed numbers? I don’t need to write the mixed numbers as fractions greater than 1. I can keep track of my work with an area model. I don’t need to write a fraction greater than 1 as a mixed number after multiplying.

Multiply Two Mixed Numbers Students multiply two mixed numbers by using two different methods.

Write 2  5 × 3  8  . Invite students to Think–Pair–Share to estimate the product and to use an equation to support their thinking. Write the estimates as students share and leave the estimates displayed while students complete problem 2.

Teacher Note When students multiply mixed numbers by writing the factors as fractions greater than 1, they may write their answers as fractions greater than 1 or as mixed numbers. Both are correct, but a common convention is to write the answer as a mixed number because the factors were given as mixed numbers.

I estimate 6 because 2 × 3 = 6. I estimate 9 because 3 × 3 = 9. 294

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 13

What methods can you use to determine the product?

I can draw an area model that represents ( 2 +  5 )× (3 +  8 ).

I can use the distributive property to multiply ( 2 +  5 ) by ( 3 +  8 ). I can write the mixed numbers as fractions greater than 1 and then multiply the fractions. Direct students to problem 2. Read the directions aloud. Invite students to complete the problem with a partner. Circulate as they work, asking the following questions: Promoting Mathematical Thinking and Reasoning

• Which methods did you choose? Why? • Which method do you prefer? Why?

When students multiply mixed numbers and use their estimate to explain whether their answer makes sense, they are assessing the reasonableness of solutions (MTR.6).

• How do you write the mixed number as a fraction greater than 1? • Is your answer correct? How do you know?

2. Use two different methods to evaluate 2  5 × 3  8  .

Ask the following questions to promote MTR.6:

Method 1: Sample:

• How can you estimate the product? Does your estimate sound reasonable?

2 3_ × 3 1_ = (2 + 3_ )× (3 + 1_ ) 5 5 8 8

= (2 × 3) + (3_ × 3)+ (2 × 1_ )+ (3_ × 1_ ) 5

3   = 6 + 9_ + 2_ +  __ 5 8 40

• Does your answer make sense mathematically?

3   = 6 + 72 __ + 10 __ +  __ 40 40 40

= 6 + 85 __  40

5   = 6 + 2  __ 40 5   = 8  __ 40

295

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 13

Method 2: Sample:

2 3_ × 3 1_ = 13 __ × 25 __  5 5 8 8 = 325 ___  40

5   = 8  __ 40

Invite students to share which two methods they used and why. Then ask the following questions.

__5

Why does an answer of 8    seem reasonable? 40

__5

Our estimates were 6 and 9, and 8  40 is close to those estimates. Sometimes, writing mixed numbers as fractions greater than 1 and multiplying those fractions is more efficient than using an area model, and sometimes it is not. In this problem, why might writing the mixed numbers as fractions greater than 1 and multiplying the fractions be less efficient than using an area model or the distributive property?

___

325

I have to determine what 13 times 25 is and then I have to write   40 as a mixed number. Invite students to turn and talk to discuss which method of multiplying mixed numbers they prefer and why.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

296

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 13

Land

Debrief 5 min Objective: Multiply mixed numbers. Facilitate a class discussion about multiplying mixed numbers by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Display the area model and work showing 2  5 × 3  8  .

1 8

2 8

3 5

9 5

3 40

( 5) ( 8) = (2 × 3) +(3 × 3) + (2 × 1)+(3 × 1 ) 5 8 5 8

2 3 × 3 1 = 2+ 3 × 3+ 1 5

=6+9+2+ 3

5 8 40 = 6 + 72 + 10 + 3 40 40 40

= 6 + 85

40 = 6+ 2 5 40 5 =8 40

How are area models helpful when we are multiplying mixed numbers? The area model helps us keep track of all the partial products to make sure we find them all. With an area model, we can do four smaller multiplication problems and then add the products together.

297

5 ▸ M5 ▸ TB ▸ Lesson 13

EUREKA MATH2 Florida

How can we use the distributive property when multiplying mixed numbers? We can decompose each mixed number into a whole-number part and a fractional part. Then we can multiply each part by each part of the other factor. How is multiplying mixed numbers similar to multiplying decimals? How is it different? I can use the distributive property or an area model to multiply decimals or mixed numbers. When I use the distributive property to multiply decimals, I decompose the decimal by place value. When I use the distributive property to multiply mixed numbers, I decompose a mixed number into a whole number and a fraction. I do not know how to use vertical form to record my work for multiplying mixed numbers, but I do know how to use it for multiplying decimals.

298

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 13

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET PRACTICE

I can multiply mixed numbers.

Florida B.E.S.T. MA.5.FR.2.2

Name

2. Write the multiplication expression that the area model represents.

Date

_2 1. Circle the area models that can be used to find 4 × 8 .

1 2

2 2_ 1

4 2 3 8

4 3

8 32

2 3

8 3

2 3

8 3

Multiply.

16 3

2 4. 5 5_ × 4 =

21 5_ 3

32 8

2 3

1 3. 3 _ × 8 = 4

2 3

8 3

111

112

PROBLEM SET

299

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 13

5. Consider the multiplication expression 4 2 × 2 5 .

a. Fill in the blanks in the area model.

4 5

b. Fill in the blanks to show the sum of the partial products.

4 __

16 __

+1+ 5 +

c. The product of 4 2 and 2 5 is

16 5

1 2

4 10

12 _5 .

Multiply by using a method of your choice. 1 1 4 13 __ 20 7. 2 _ × 5 5_ = 4

6. The area models shown both represent 3 3 × 6 3 . The side lengths of the first area model are labeled with mixed numbers. The side lengths of the second area model are labeled with

fractions greater than 1. Use both area models to determine 3 3 × 6 3 .

62 3

3 1 8. 6 _ × 3 _ = 2 4

20 3

24 8_ 3

10 3

= 22 _9 2

2 3

6 3

1 3

6 3

6 3 2 9

= 22 9_ 2

300

PROBLEM SET

113

114

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 13

EXIT TICKET

I can multiply mixed numbers.

Florida B.E.S.T. MA.5.FR.2.2

Name

Date

1 _3 Draw an area model to find 2 5 × 3 2 .

1 2

3 5

9 5

3 10

2 _5 × 3 _2 = 6 + 1 + _5 + __ 10 3

= 7 + __ + __ 10 10 18

= 7 + __ 10 21

1 = 7 + 2__ 10

= 9 __ 10 1

115

301

LESSON 14

Solve real-world problems involving areas of composite figures with mixed-number and decimal side lengths. I can solve real-world problems that involve area and mixed numbers.

Lesson at a Glance Students solve a variety of problems involving areas of a house and garden. Throughout the lesson, students multiply mixed numbers and decimals and apply a variety of methods for determining area of rectangles and composite figures. After working in pairs to complete each problem, students share their work and compare methods. This lesson formalizes the term composite figure.

Key Questions • How do you choose a method for determining areas of composite figures with mixed-number and decimal side lengths in realworld situations? • How can you keep your work organized when determining areas of composite figures with mixed-number and decimal side lengths?

Mathematical Thinking and Reasoning Standard MTR.1.1

Language Objectives • Read word problems involving areas of composite figures. For each word problem, write or draw to record a solution strategy, including decomposing into smaller rectangles to add and composing a larger rectangle to subtract. • Listen to and orally describe strategies to determine the area of composite figures, including decomposing into smaller rectangles to add and composing a larger rectangle to subtract, and compare and connect strategies.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 14

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Tear out and cut apart Rounding Cards, Set 1 from the student books. Place each set into an envelope. Prepare enough sets for 1 per student pair.

Launch 5 min Learn 35 min • The Garden • Share, Compare, and Connect • The House • Problem Set

• Envelopes (12) • Rounding Cards, Set 1 (1 per student pair, in the student book)

Students • Envelope of Rounding Cards (1 per student pair)

Land 10 min

Achievement Descriptors

Florida B.E.S.T. Standards

FL.5.Mod5.AD3 Solve real-world problems

MA.5.AR.1.2 Solve real-world problems

involving multiplication of mixed numbers. (MA.5.AR.1.2) FL.5.Mod5.AD9 Find areas of rectangles with

fraction or decimal side lengths. (MA.5.GR.2.1)

involving the addition, subtraction or multiplication of fractions, including mixed numbers and fractions greater than 1. MA.5.GR.2.1 Find the perimeter and area of a

rectangle with fractional or decimal side lengths using visual models and formulas.

303

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 14

Fluency

Flip: Round Materials—S: Rounding Cards, Set 1

Students round a number to the nearest one or tenth to build fluency with rounding decimals from module 4. Have students form pairs. Distribute an envelope of cards to each student group. Have them use the following procedure to play. Consider doing a practice round with students. • Place all the cards in a pile facedown. • Take turns flipping over a card. Both partners say the number aloud. • Partner A says the number when rounded to the nearest one. Partner B says the number when rounded to the nearest tenth.

0.53

Teacher Note Students should focus on accuracy in this activity and may not get through the entire deck of cards in ten minutes. Consider saving the envelopes of Rounding Cards and providing multiple opportunities for students to do this activity in the future.

Partners A and B: “0.53” Partner A: “0.53 rounded to the nearest one is 1.” Partner B: “0.53 rounded to the nearest tenth is 0.5.”

• Continue until all cards are discarded. Circulate during the activity and provide support as needed.

Launch

Students examine a floor plan for a house. Play part 1 of the Floor Plan video. If necessary, replay the video and ask students to note any details. Engage students in a brief conversation about the video. Guide the conversation to the garden. Display the site plan of the house. Tell students that this is the same diagram they viewed in earlier lessons.

304

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 14

20 ft

What do you notice about the garden? What do you wonder? I notice that the garden is in the shape of an L.

12 1 ft

I notice the garden does not have dimensions labeled. I wonder why the garden is in the shape of an L.

Island

I wonder how big the garden is. I wonder what will be planted in the garden. A composite figure is composed, or made, from other shapes. How is the garden a composite figure? What other shapes is it made from?

8 1 ft 2

6 1 ft

Bath

24 ft Garden

Driveway

Garage

Living Room and Kitchen

Bedroom

16 1 ft

12 ft

11 3 ft 4

The garden is made from two rectangles. The garden is made from two rectangles. How can we determine its area? Invite students to Think–Pair–Share about strategies they could use to find the area of the garden. Direct students to draw a diagram to represent their thinking.

A B

Subtract this area

Multilingual Learner Support Consider displaying tetrominoes or other composite figures and inviting students to point to the individual shapes that are composed to make the larger figure. Consider labeling the figures composite figures.

We could draw a vertical line to partition the garden into two rectangles and determine the area of each rectangle. Then we could add the areas together.

305

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 14

We could draw a horizontal line to partition the garden into two rectangles and determine the area of each rectangle. Then we could add the areas together. We could multiply the lengths of the long sides of the garden to get the area of a rectangle that includes the garden and then subtract the area of the region that is not part of the garden. Transition to the next segment by framing the work. Today, we will determine areas in real-world situations.

Learn

The Garden Students determine the area of the garden and the cost to cover it with compost.

Use the Read–Draw–Write process to solve the problem.

Differentiation: Support Consider providing students who need additional support with a drawing of the garden split into two separate rectangular gardens that have the same dimensions as the garden pictured in problem 1.

1. The drawing represents the vegetable garden behind the house. The owner of the house plans to cover the garden with a layer of compost. If compost costs $2 for each square foot, how much will it cost to cover the garden with compost?

8.5 ft Garden

9.5 ft

5 ft 6 ft 3.5 ft

306

3.5 ft

3.5 ft 9.5 ft

5 ft

6 ft

3.5 ft

Garden

Direct students to problem 1 in their books. Have students work with a partner to use the Read–Draw–Write process to solve the problem. Encourage students to self-select their approaches and materials.

3.5 ft

3.5 ft © Great Minds PBC

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 14

9.5 × 3.5 = (9 + 0.5) × (3 + 0.5) = (9 × 3) + (9 × 0.5) + (0.5 × 3) + (0.5 × 0.5) = 27 + 4.5 + 1.5 + 0.25 = 33.25 5 × 3.5 = 5 × (3 + 0.5) = (5 × 3) + (5 × 0.5) = 15 + 2.5 = 17.5 The total area of the garden is 33.25 + 17.5, or 50.75, square feet.

2 × 50.75 = 2 × (50 + 0.75) = (2 × 50) + (2 × 0.75) = 100 + 1.50 = 101.50 The total cost for compost is $101.50. Circulate and observe student methods. Select two or three students to share in the next segment. Look for work samples that use a variety of methods to multiply mixed numbers and to determine the area of the garden. The following student work samples demonstrate different methods for determining the area of the garden.

Universal Design for Learning: Engagement Consider providing mastery-oriented feedback that focuses attention on a student’s choice of strategy. Recognize students for choosing efficient strategies.

Teacher Note The sample student work shows common responses. Look for similar work from your students and encourage authentic classroom conversations about the key concepts. If your students do not produce similar work, choose one or two pieces of their work to share and highlight how it shows movement toward the goal of this lesson. Then select one work sample from the lesson that works best to advance student thinking. Consider presenting the work by saying, “This is how another student solved the problem. What do you think this student did?”

307

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 14

Decompose into Smaller Rectangles to Add

Compose a Larger Rectangle and Subtract 8.5 f t

3.5 ft 5 ft

9.5 ft

3.5 ft

9.5 × 3.5 = (9 + 0.5) × (3 + 0.5) = 9 × 3 + 9 × 0.5 + 0.5 × 3 + 0.5 × 0.5 = 27 + 4.5 + 1.5 + 0.25 = 27 + 6.25 = 33.25 5 × 3.5 = 5 × (3 + 0.5) = 15 + 2.5 = 17.5 The total area of the garden is 33.25 + 17.5, or 50.75, square feet. 50.75 × 2 = (50 + 0.75) × 2 = 100 + 1.50 = 101.50

The total cost for compost is $101.50.

9.5 f t

6 ft 5 ft

9.5 × 8.5 = (9 + 0.5) × (8 + 0.5) = 9 × 8 + 0.5 × 8 + 9 × 0.5 + 0.5 × 0.5 = 72 + 4 + 4.5 + 0.25 = 76 + 4.75 = 80.75 5 × 6 = 30 80.75 - 30 = 50.75 The total area of the garden is 50.75 square feet. 2 × 50.75 = 2 × (50 + 0.75) = 2 × 50 + 2 × 0.75 = 100 + 1.50 = 101.50 The total cost for compost is $101.50.

308

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 14

Share, Compare, and Connect Students compare methods for finding the area of the composite figure in problem 1. Invite two or three sets of partners to share their solutions. As each pair shares, ask questions to elicit their thinking and clarify the strategy they used. Ask the class questions that invite students to make connections between the different solutions strategies and their own work. Encourage students to ask questions of their own. Decompose into Smaller Rectangles to Add (Kayla and Adesh’s Way) Tell us how you solved the problem. We decomposed the garden into two rectangles. Then we found the area of each rectangle and added the areas together. We used the total area to find the cost of the compost. How did you decompose the rectangles? We drew a vertical line to decompose the rectangles into a left rectangle and a right rectangle. Why did you multiply 5 by 3  _  ? 1 2

3.5 ft 9.5 ft

After drawing a vertical line to decompose the garden into two rectangles, we saw that the rectangle on the right has a length of 5 feet and a width of 3  _  feet. 1 2

5 ft

The total cost for compost is $101.50.

What method did you use to multiply mixed numbers? Why? We used the distributive property because the numbers in the mixed numbers are large. Class, how did they determine the cost of the compost? They multiplied the number of square feet in the garden by the cost for each square foot of the compost.

309

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 14

How did planning ahead help you solve this problem? We knew when we looked at the diagram that we wanted to decompose the garden into two rectangles. Once we had a strategy, we were able to make the calculations. Invite students to turn and talk about the similarities and differences between Kayla and Adesh’s work and their work. Compose a Larger Rectangle and Subtract (Sana and Leo’s Way) How did you solve the problem? We composed a rectangle around the garden. Then we subtracted the area of the rectangle that is not part of the garden. We used the area of the garden to find the cost of the compost. Why did you choose that method? We noticed that the rectangle that is not part of the garden has whole-number side lengths, so we thought those numbers might be easier to work with. Why did you multiply 5 by 6? The length and width of the rectangle that is not part of the garden are 5 feet and 6 feet. How did planning ahead help you solve this problem?

8.5 f t

9.5 f t

6 ft 5 ft

We realized that if we drew a rectangle that included the garden, the part of the rectangle that is not in the garden would have whole-number side lengths. We thought it would be easier for us to determine that area and subtract it from the area of the large rectangle. Class, how is Sana and Leo’s method similar to Kayla and Adesh’s method? How is it different? Both pairs used methods that involved thinking about the garden in terms of rectangles. Kayla and Adesh decomposed the garden into two rectangles, and Sana and Leo composed a rectangle around the garden. Invite students to turn and talk about the similarities and differences between Sana and Leo’s work and their work.

310

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 14

The House Students determine the area of the house to be covered with flooring. Play part 2 of the Floor Plan video. If necessary, replay the video, and ask students to note any details. Give students 1 minute to turn and talk about what they noticed. Engage students in a brief conversation about the video. Discuss student observations and any relevant questions they have. Have students work with a partner to solve problem 2. Encourage them to self-select their strategies.

Multilingual Learner Support Consider directing students to use the Talking Tool as they complete problem 2. Invite them to use the Share Your Thinking section of the tool when communicating their ideas.

311

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 14

2. Given the following information, how many square feet of rolled flooring does the owner need? • The floor of the bedroom, the bathroom, and the garage will not be covered with rolled flooring. • The rest of the house will be covered with rolled flooring. • The island in the kitchen measures 3 feet by 7 feet. Flooring will not be put under the island.

20 ft 12 1 ft

Island

Living Room and Kitchen

Bedroom

16 1 ft

12 ft

312

8 1 ft 6 1 ft

Bath

24 ft Garden

Driveway

Garage

11 3 ft 4

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 14

16 1 + 12 = 28 1 2

= (24 × 28) + 24 × 1 2

61 × 81 = 6 + 1 × 8 + 1 2

Students actively participate in effortful learning (MTR.1) as they solve a word problem.

= 132 + 9 = 141

= 672 + 12 = 684 2

Promoting Mathematical Thinking and Reasoning

= (12 × 11) + 12 × 3 4 36 = 132 + 4

24 × 28 1 = 24 × )28 + 1) 2

12 × 11 4 = 12 × )11 + 4 )

Ask the following questions to promote MTR.1: • What is your plan for determining how much flooring is needed?

3 × 7 = 21

= (6 × 8) + 6 × 1 + 1 × 8 + 1 × 1

2 1 = 48 + 3 + 4 + 4 1 = 55 4

• Is your method working? Is there something else you could try?

684 − 141 − 55 4 − 21 = 466 4

The area of the house that needs rolled flooring is 466_ 43 square feet. Circulate and observe student strategies. Select two or three students to share their work. Look for work samples that use a variety of methods to multiply mixed numbers and to determine how much rolled flooring she needs. When students are finished, invite some pairs to show their work. Consider using the following prompts to help students make connections among the shared work: • Explain how you planned before you started making any calculations.

Universal Design for Learning: Representation Consider annotating students’ work to show what each calculation represents. For example: Length of house 1

• What method did you use for multiplying mixed numbers? Why? Invite students to turn and talk about the similarities and differences between other students’ work and their own work.

16 2 + 12 = 28 2

(

1 1 24 × 28 2 = 24 × 28 + 2

)

(

)(

) 3

)

62× 8 2= 6 + 2 × 8 + 2 1

= 12 × 11 + 12 × 4 36 = 132 + 4

= 672 + 12 = 684 1

(

12 × 11 4 = 12 × 11 + 4

1 = 24 × 28 + 24 × 2

• Why did you add or subtract those numbers? • How did you know to multiply those numbers?

Area of bedroom

= 132 + 9 = 141

3 × 7 = 21 1

= 6 × 8 + 6 × 2 + 2 × 8+ 2 × 2 1 = 48 + 3 + 4 + 4 1 = 55 4

Area of island

Area of bathroom

313

5 ▸ M5 ▸ TB ▸ Lesson 14

EUREKA MATH2 Florida

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Solve real-world problems involving areas of composite figures with mixed-number and decimal side lengths. Facilitate a class discussion about areas in the real world by using the following prompts. Encourage students to restate or add on to their classmates’ responses. How do you choose a method for determining areas of composite figures with mixed-number and decimal side lengths in real-world situations? I look at the composite figure and decide if I see two or more rectangles put together or if I see a large rectangle with pieces taken out of it. I consider the side lengths and think about whether it will be easier for me to find the areas of the smaller rectangles and add them or to find the area of the larger rectangle and the area of the rectangle that is not part of the garden and subtract. How can you keep your work organized when determining areas of composite figures with mixed-number and decimal side lengths? I can make a plan. I can draw on a given diagram. I can redraw a diagram or draw it as pieces, if that helps. I can find the area of one part at a time and then add or subtract as needed.

314

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 14

315

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 14

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can solve real-world problems that involve area and mixed numbers.

PRACTICE

Florida B.E.S.T. MA.5.AR.1.2, MA.5.GR.2.1

Name

2. Sana is building a dollhouse. The picture shows her plan for the living room floor. Find the area of the floor by using a method of your choice. Dollhouse Living Room

Date

3 in

6.5 in

1. Each expression represents the area of a figure. Draw a line to match the figure to the expression it is partitioned to represent.

1 in 4.5 in

3.5 in Figure

Expression

9.5 in

1 2

4 cm

36.25 square inches 5 cm

8 cm

4 cm 3 cm

1 1 (8 2 × 3) + (5 × 4 2 )

1 8 cm 2

1 2

4 cm 5 cm 8 cm

(8 × 8 2 ) − (5 × 4) 1

4 cm 3 cm 1 2

8 cm

1 2

4 cm 5 cm

(4 2 × 8) + (4 × 3) 1

8 cm 4 cm

3 cm

1 2

8 cm

316

123

124

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 14

EXIT TICKET

I can solve real-world problems that involve area and mixed numbers.

Florida B.E.S.T. MA.5.AR.1.2, MA.5.GR.2.1

3. The drawing shows the site plan of a backyard. Riley wants to lay stones for the patio, which is represented by the shaded area. Name

Date

How many square feet of tile are needed to cover the floor shown?

Yard

4 ft 15 ft 1

2 3 ft

9 ft

3 4 ft

1 2

4 ft

4 3 ft

1 2

11 ft

Patio 1 2

2 ft

10 ft 1

7 4 ft

House

Sample:

4 1_3 × 4 = (4 + 1_3 ) × 4

3 1_4 × 2 = (3 + 1_4 ) × 2

= 16 + 1 1_3

= 6 2_4

= 17 1_3

b. The patio stones are squares with a side length of 1 _1 feet. Find the area of one stone. 2

c. Riley purchases exactly 80 patio stones. Does Riley have enough stones to cover the patio? How do you know?

__ square feet 23 10 12

No, Riley does not have enough patio stones to cover the patio. The area of the patio is

182 _1 square feet, but the total area of the stones is only 180 square feet. 4

PROBLEM SET

= 6 + 2_4

= 16 + 34_

182 _1 square feet

2 _1 square feet

= (3 × 2) + (1_4 × 2)

= (4 × 4) + (1_3 × 4)

a. What is the area of the patio?

125

4 __ + 6 12 17 1_3 + 6 2_4 = 17 __ 12

__ = 23 10 12

127

317

LESSON 15

Solve word problems involving multiplication of mixed numbers. I can solve word problems with mixed numbers.

Lesson at a Glance Students use the Read–Draw–Write process to solve word problems involving multiplication of mixed numbers. A stations activity allows students to work in pairs through problems of various difficulty levels. Students use estimation to determine whether answers are reasonable and compare their methods for solving the problems.

Key Questions • Why is it helpful to estimate an answer before solving a word problem with mixed numbers? • How can drawing a diagram help you solve word problems with mixed numbers?

Mathematical Thinking and Reasoning Standard MTR.7.1

Language Objectives • Read word problems involving multiplication of mixed numbers. Listen to and orally describe the problems’ meanings and possible strategies for solving them to compare and connect solution strategies. • Write or draw to record a solution strategy and an equation to solve a problem involving multiplication of mixed numbers.

• How do you know which operations to use when you solve a word problem with mixed numbers?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 15

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

• Tear out and cut apart Rounding Cards, Set 2 from the student books. Place each set into an envelope. Prepare enough sets for 1 per student group of three. • Print or copy and cut out Station Cards. Prepare enough cards to set up four stations with a few copies of a single problem at each station.

Launch 5 min Learn 35 min • Problem Stations • Compare and Connect Methods • Problem Set

Land 10 min

• Envelopes (8) • Rounding Cards, Set 2 (1 per student group, in the student book) • Station Cards (in the teacher edition)

Students • Envelope of Rounding Cards (1 per student group)

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD3 Solve real-world problems

MA.5.AR.1.2 Solve real-world problems

involving multiplication of mixed numbers. (MA.5.AR.1.2)

involving the addition, subtraction or multiplication of fractions, including mixed numbers and fractions greater than 1.

319

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 15

Fluency

Flip: Round Materials—S: Rounding Cards, Set 2

Students round a number to the nearest one, tenth, or hundredth to build fluency with rounding decimals from module 4. Have students form groups of three. Distribute an envelope of cards to each student group. Have them use the following procedure to play. Consider doing a practice round with students. • Place all the cards in a pile facedown. • Take turns flipping over a card. All group members say the number aloud. • Partner A says the number when rounded to the nearest one. • Partner B says the number when rounded to the nearest tenth. Partner C says the number when rounded to the nearest hundredth. • Continue until all cards are discarded. Circulate during the activity and provide support as needed.

320

Teacher Note If it is not possible for all students to be in groups of three, consider having partners A and B take turns saying the number when rounded to the nearest hundredth or only focusing on two place values (e.g., rounding to the nearest one and tenth).

0.581 Partners A, B, and C: “0.581” Partner A: “0.581 rounded to the nearest one is 1.” Partner B: “0.581 rounded to the nearest tenth is 0.6.” Partner C: “0.581 rounded to the nearest hundredth is 0.58.”

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 15

Launch

Students interpret a tape diagram representing a real-world situation with mixed numbers. Display the tape diagram labeled Kayla and Sana. What information do we know from the tape diagram? How do we know it? We know that Kayla has 2 _1 of 2

Kayla

Sana

Teacher Note Use this lesson as a formative assessment to see how students approach word problems that involve multiplication of mixed numbers. The intent of this lesson is not to introduce new arithmetic skills. Highlight the similarities and differences in how students choose to make their calculations.

something because her tape has one unit labeled 2 _1. 2

We know that Sana has 4 times as much as Kayla because Sana’s tape has 4 units that are the same size as the unit in Kayla’s tape. Invite students to turn and talk to write an expression to represent Sana’s tape. Display the tape diagram that includes a tape for Tyler. What additional information do we know? How do we know it? We know Tyler has more than Kayla and Sana because Tyler’s tape is longer than Kayla’s tape and Sana’s tape. We know that Tyler has about

4 _1times as much as Kayla. Tyler’s

Kayla

1 22

Sana

Teacher Note Given a tape diagram without an accompanying written context, we cannot precisely determine that the partial unit is exactly _ 1of the whole unit. Consider 2 encouraging students to use estimation language such as about and looks like to describe the relationship.

Tyler

tape has 4 units that are the same size as the unit in Kayla’s tape and a unit that looks like it is _ 1as long as the unit in Kayla’s tape. 2

Sometimes we use mixed numbers to represent a times-as-much relationship. Invite students to turn and talk to write an expression to represent Tyler’s tape.

321

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 15

Display the completed tape diagram. What information is unknown? We do not know what the situation is or what they have. We do not know how much Sana and Tyler have. We do not know the exact length of Tyler’s tape diagram. Let’s make an estimate. Approximately how long is Tyler’s tape diagram? How do you know?

Tyler has approximately 10 units. I know because there are about 4 _1units in the tape 2

diagram and each one has a value of 2 _1 . 5 × 2 = 10. 2

Pair students and use the Co-construction routine to have partners create a context that could apply to the tape diagram. Give pairs 1 minute to compare the context they construct with other groups. Invite students to share their ideas and explain the relationship between the tape diagrams and the context. Sample student responses:

Kayla, Sana, and Tyler are making punch. Kayla uses 2 _1cups of juice. Sana uses 4 times 2

as much juice as Kayla. How much juice does Sana use? Tyler uses 4 _1times as much juice as Kayla. How much juice does Tyler use?

Kayla rides her bike 2 _1miles. Sana rides 4 times as far as Kayla. How far does Sana ride 2 her bike? Tyler rides 4 1 times as far as Kayla. How far does Tyler ride his bike? 2

322

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 15

Why might we choose to represent the situation with a tape diagram? A tape diagram shows what is known and unknown. A tape diagram shows how the parts of a problem are related. A tape diagram helps me make sure that all the information from the problem is represented in my plan for finding a solution. A tape diagram helps me see if there is a way to solve the problem efficiently. A tape diagram is a way to show that something is some number of times as much as something else. Transition to the next segment by framing the work. Today, we will solve word problems that involve multiplication of fractions and mixed numbers.

Learn

Problem Stations Materials—T: Station Cards

Students solve problems involving multiplication of mixed numbers. Pair students. Introduce the Problem Cards activity by giving the following instructions. There are four stations set up around the room. Each station has a set of cards with one problem, and there is a different problem at each station. Make an estimate and complete the problem by using the Read–Draw–Write process. Record all work in your book.

Multilingual Learner Support Consider showing pictures or videos of the contexts at the stations to support students with the word problems. For example, you might show a video of a ballet recital or a photo of a tray of doughnuts.

323

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 15

Station Cards:

Teacher Note

Station 1

Tara prepares for a ballet recital. Last week she rehearsed 2 _1hours each day for 4 days. 4 This week she rehearsed 1 _3hours each day for 3 days. How many hours did Tara rehearse 4

each week? Station 2

On Saturday, a bakery sells 2 _4trays of doughnuts. The bakery sells 2 _1times as many 5

doughnuts on Sunday as on Saturday. How many trays of doughnuts does the bakery

sell on Sunday? The bakery sells 1 _3times as many doughnuts on Monday as on Saturday. 4

How many trays of doughnuts does the bakery sell on Monday? Station 3

Scott runs 1 _1miles. Eddie runs 3 _1times as far as Scott. How far does Eddie run? Julie 2

runs _ 3as far as Eddie. How far does Julie run?

Stations are in order from least challenging, station 1, to most challenging, station 4. Decide whether to use a signal for when it is time for students to rotate through the room or to let students do so at their own pace. Similarly, decide whether they will follow a particular route and how many sets of partners are allowed at a station at one time. Consider making multiple copies of the cards and setting up more than one of each station, if needed, to reduce the number of students at each station at one time. If it is preferable for students to stay stationary, consider placing students in groups and giving each group all four station cards.

Station 4

A glass company makes 12 windows that measure 4 _3feet long by 3 _1feet wide and 4

7 windows that measure 2 _4feet long by 5 feet wide. How many square feet of glass 5

does the company need for the windows? Consider using the following prompts as you circulate while students work: • Tell me about your approach. • What do you know? • Can you draw something? What can you draw? How does your drawing connect to the story?

Differentiation: Support Consider breaking the problems into distinct parts. Invite students to draw tape diagrams to represent each part.

• What does this number represent? • Why did you use that operation? • Does your answer seem reasonable? Why? • How can you check your answer? 324

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 15

Use the Read–Draw–Write process to complete the problem at each station. Estimate before you solve the problem. Record your work for each station in the table.

Station 1

Students are not expected to write fractions

Station 2

Last week

1 2 4

This week

as mixed numbers or with the greatest unit.

4 2 5

Saturday

Teacher Note

Some students may realize, for example,

that 4 18 is equivalent to 4 9 , but recognize 20

all equivalent correct answers.

3 4

Sunday

Estimates:

Monday

2 × 4 = 8 2 × 3 = 6

Estimates:

4 × 2 _1 = 4 × _9

Solution:

=_ 36 4

2 _ 4 × 2 _ 1 = (2 + _ 4 )× (2 + _ 1 ) 5

3 × 1 _3 = 3 × _7 4

=_ 21 4

1 = 5_ 4

Tara rehearsed 9 hours last week and 1 _ 5 hours this week. 4

4 4 1 = (2 × 2) + _ × 2 + 2 × _ 1 + _ × _

)

(

=4+_ 8 + 1 + _ 4 5 10 =4+1+_ 16 + _ 4 = 5 + _ 20 10 10 10 = 5 + 2 = 7

= 9

2 _ 4 × 1 _ 3 = (2 + _ 4 )× (1 + _ 3 ) 5

When students determine the meaning of each real-world problem, estimate a solution, represent it with an expression, and recontextualize the solution while attending to units, they are applying mathematics to real-world contexts (MTR.7).

3 × 3 = 9 3 × 2 = 6

Solution: 4

Promoting Mathematical Thinking and Reasoning

Ask the following questions to promote MTR.7: • What does the expression represent in this situation? • What real-world situations are modeled by multiplication expressions with mixed numbers?

4 4 3 = (2 × 1) + _ × 1 + 2 × _ 3 + _ × _

(5 ) ( 4) 6 4 12 _ _ _ = 2 + + + 5 4 20 _ = 2 + 16 + _ 30 + _ 12 20 20 20 =2+_ 58 = 2 + 2 + _ 18 = 4 _ 9 10 20 20

The bakery sells 7trays of doughnuts on Sunday 9 trays of doughnuts on Monday. and 4 __ 10 © Great Minds PBC

325

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 15

Station 4

Station 3 Scott

1 3 feet 3

1 2

Eddie

12 windows 4 3 feet 4

Julie

4 2 feet 5

Estimates:

7 windows

2 × 4 = 8 1 × 8 = 8

Solution:

1 _ 1 × 3 _ 1 = (1 + _ 1 )× (3 + _ 1 ) 2

Estimates:

1 1 1 = ( 1 × 3)+ (1 × _ 1 )+ (_ × 3)+ (_ × _ 2 2 2) 2 =3+_ 1 + _ 3 + _ 1 2 2 4 =3+_ 2 + _ 6 + _ 1 4 4 4 =3+_ 9 4 =3+2+_ 1 4 = 5_ 1 4 _3 × 5 _1 = _3 × 5 + _1 4 4 4 ( 4) 3 3 1 _ = ( × 5)+ (_ × _ 4 4 4) =_ 15 + _ 3 4 16 =_ 60 + _ 3 16 16 =_ 63 16 = 3_ 15 16 1 Eddie ran 5 miles and Julie ran 3 15 miles. 2 16

326

5 feet

5 × 3 = 15 12 × 15 = 180 3 × 5 = 15 7 × 15 = 105

Solution:

A=l×w

A = (4 _ 3 × 3 _ 1 ) 4

A= _ 19 × _ 10 = _ 190 = 15 _ 5

(4 3 ) 12 6 190 5 _ _ 12 × 15 = 12 × (12 )= 190 6

A=l×w

A = (2 _ 4 × 5)

5 14 A = (_ × 5)= 14 5

7 × 14 = 98 98 + 190 = 288

The company needs 2 88square feet of glass for the windows. © Great Minds PBC

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 15

When most students are finished or time is up, bring the class back together to debrief the Problem Cards activity. Student methods will be discussed in the next segment.

Compare and Connect Methods Students share and compare solutions and reason about their connections. Choose one station for students to discuss. Invite students to share their solutions and methods. As each student shares, ask questions to elicit their thinking. Ask the class questions to help students make connections between the demonstrated solution methods and their own work. Encourage them to ask questions of their own. The following questions, for example, can be used when discussing station 1: • What numbers did you choose for your estimate? Why? • Explain the drawing that you used to make sense of the problem. • Why did you multiply 2 _1 by 4? 4

• Why did you multiply 1 _3 by 3? 4

• What method did you use to multiply mixed numbers? • Why did you use addition? After discussing one of the stations in depth, facilitate a class discussion by asking any or all the following questions: • How did you approach station 1 and station 3 differently? • What do the problems at station 2 and station 3 have in common? • Which stations were the most straightforward for you? Why? • How did drawing a diagram help you? Were some diagrams more helpful than others? • Which situations were easiest for you to understand? Why?

327

5 ▸ M5 ▸ TB ▸ Lesson 15

Invite students to Think–Pair–Share about which station was the most challenging and why. Station 4 was the most challenging for me because it involved so many calculations. It was challenging to keep track of all the calculations. Station 3 was the most challenging for me because of thinking about how the distances are related. Station 1 was the most challenging for me because it involved subtraction.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

EUREKA MATH2 Florida

Universal Design for Learning: Action & Expression Consider reserving time for students to reflect after all solutions have been shared and the discussion concludes. Ask students to consider if they might do anything differently next time. The sharing component of the activity provides a formative feedback opportunity as students reflect on their own work relative to the examples from their peers. Students may benefit from a think-aloud modeling how to determine if a classmate’s approach is more efficient than their own approach.

Debrief 5 min Objective: Solve word problems involving multiplication of mixed numbers. Facilitate a class discussion about solving word problems by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Why is it helpful to estimate an answer before solving a word problem with mixed numbers? Multiplying mixed numbers involves a lot of parts and steps. Comparing my answer to my estimate helps me know if I missed a part or a step. I cannot multiply mixed numbers in my head, but I can round the mixed numbers to whole numbers and get an estimate that I can use to determine if my answer is reasonable.

328

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 15

How can drawing a diagram help you solve word problems with mixed numbers? Drawing a diagram can help me make sure I read the problem carefully and think about everything that is known and unknown before I make a plan to solve it. When the problem is about shapes and areas, my drawing can help me see the shapes and think about how to find the area. A diagram is useful for showing relationships in the problem. Once I have a diagram, I can write numerical expressions to add, subtract, or multiply. How do you know which operations to use when you solve a word problem with mixed numbers? My diagram helps me see how the numbers are related, which tells me which operations to use to solve the problem.

329

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB ▸ Lesson 15

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET PRACTICE

I can solve word problems with mixed numbers.

Florida B.E.S.T.

23 _ 1 _ and runs a distance of 3 miles. Ryan participated in 4 sprint triathlons this year.

MA.5.AR.1.2

3. In a sprint triathlon, a participant swims a distance of 50 miles, bikes a distance of 12 5 miles, 10

Name

a. How many total miles did Ryan swim in the 4 sprint triathlons? 42 Ryan swam a total of 1 __ miles. 50

Date

Use the Read–Draw–Write process to solve each problem.

1 1. Leo walked 2 miles on Friday. He walked 3 _ times as far on Saturday as he did on Friday. 4

a. Estimate the distance Leo walked on Saturday.

2×3=6

b. How many total miles did Ryan run in all 4 triathlons? 2 Ryan ran a total of 12 _5 miles.

b. Determine the actual distance Leo walked on Saturday.

Leo walked 6 4 miles on Saturday.

2. Noah uses 36 tiles to cover a table. Each tile measures 4 4 inches by 4 4 inches. a. Estimate the area of the table.

4 × 4 = 16

c. How many total miles did Ryan swim in all 4 triathlons? 21 Ryan swam a total of 1 __ miles. 25

16 × 36 = 576

b. Find the actual area of the table. 4 The area of the table is 650 __ square inches. 16

330

135

136

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB ▸ Lesson 15

EXIT TICKET

I can solve word problems with mixed numbers.

Florida B.E.S.T. MA.5.AR.1.2

Name

Date

_3 _3 1 _ 4 batches of recipe A and 2 batches of recipe B. How much milk does Noah use in all for recipe A? Use the Read–Draw–Write process to solve the problem.

One batch of recipe A uses 1 4 cups of milk. One batch of recipe B uses 4 4 cups of milk. Noah makes 2

1 _4 × 4 1_2 = (1 + _4 ) × (4 + 1_2 )

How much milk does Noah use in all for recipe B? 3

= (1 × 4) + (1 × 1_2 ) + (_4 × 4) + (_4 × 1_2 ) = 4 + 1_2 + 3 + _8 3

= 7 + 7_8 = 7 7_8

4 _4 × 2 = (4 + _4 ) × 2 3

= (4 × 2) + (_4 × 2)

= 8 + 1 2_4

= 9 2_4

7 2 Noah uses 7 _ cups of milk in total for recipe A and 9 _ cups of milk in total for recipe B. 4 8

137

331

332 4

This page may be reproduced for classroom use only.

for the windows?

that measure 2 _4 feet long by 5 feet wide. How many square feet of glass does the company need

A glass company makes 12 windows that measure 4 _3 feet long by 3 _1 feet wide and 7 windows

Station 4

_3 as far as Eddie. How far does Julie run?

Scott runs 1 _1 miles. Eddie runs 3 _1 times as far as Scott. How far does Eddie run? Julie runs

Station 3

of doughnuts does the bakery sell on Monday?

The bakery sells 1 _3 times as many doughnuts on Monday as on Saturday. How many trays

on Sunday as on Saturday. How many trays of doughnuts does the bakery sell on Sunday?

On Saturday, a bakery sells 2 _4 trays of doughnuts. The bakery sells 2 _1 times as many doughnuts

Station 2

last week?

This week she rehearsed 1 _3 hours each day for 3 days. How many hours did Tara rehearse

Tara prepares for a ballet recital. Last week she rehearsed 2 _1 hours each day for 4 days.

Station 1

5 ▸ M5 ▸ TB ▸ Lesson 15 ▸ Station Cards EUREKA MATH2 Florida

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB

Topic B Quiz Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

A N SW E R K E Y ▸ TO PI C Q U I Z B -1

2. Consider the rectangle shown.

7 2 1. Consider the expression 5 _7 × 1 _ . 9

Part A Write numbers in the boxes to complete an area model to determine the product. Sample:

2 7

7 9

35 9

2 9

7 centimeters 8

2 centimeters 3

Part A What size rectangular tile can be used to tile the rectangle without any gaps or overlaps? Choose the area of the rectangular tile.

Part B

Multiply.

5 _27 × 1 7_9 =

1 _ square centimeter 64

1 B. _ square centimeter 24 1 C. _ square centimeter 11 1_ D. 9 square centimeter

25 9_ 63

Distractor rationale: A. Incorrect. The student may have used the larger given denominator to find the area of a rectangular tile. B. Correct. C. Incorrect. The student may have added the given denominators to find the denominator of a rectangular tile. D. Incorrect. The student may have used the smaller given denominator as the denominator to find the area of a rectangular tile.

Part B What is the area of the given rectangle? The area of the rectangle is © Great Minds PBC

231

232

14 _ 24

square centimeters.

A N S W E R K E Y ▸ TO P I C Q U I Z B -1

333

EUREKA MATH2 Florida

5 ▸ M5 ▸ TB

3. Multiply.

6 4_5 × 27 =

183 _5

2 1_2 × 5_7 =

_ 1 11 14

5 2_3 × 4 1_6 =

_ 23 11 18

4 4_7 × 3 1_2 =

3 5. Use the model shown to find the area of the shaded rectangle with side lengths of _ units 4 3 _ and units. 8

The area of the shaded rectangle is

9 _

square units.

1 1 6. Mr. Evans builds a patio. The patio is 5 _ yards long and 2 _ yards wide. What is the area of 3 6 the patio?

2 2 4. A rectangle has side lengths of _5 inches and _7 inches. What is the area of the rectangle?

4 _ square inches

35 16 B. _ square inches 35 24 C. _ square inches 35 48 D. _ square inches 35

334

Distractor rationale: A. Correct. B. Incorrect. The student may have doubled the given side lengths before multiplying. C. Incorrect. The student may have added the given side lengths.

The area of the patio is

D. Incorrect. The student may have found the perimeter.

A N S W E R K E Y ▸ TO P I C Q U I Z B -1

233

234

11 _9 5

square yards.

A N S W E R K E Y ▸ TO P I C Q U I Z B -1

EUREKA MATH2 Florida 5 ▸ M5 ▸ TB

Observational Assessment Recording Sheet Grade 5 Module 5 Topic B

Student Name

Areas of Rectangular Figures with Fraction and Decimal Side Lengths Standards and Achievement Descriptors

MA.5.FR.2.2

FL.5.Mod5.AD1 Multiply mixed numbers and whole numbers, fractions, or mixed numbers.

MA.5.FR.2.2

FL.5.Mod5.AD2 Model products of fractions and mixed numbers by using rectangular areas.

MA.5.AR.1.2

FL.5.Mod5.AD3 Solve real-world problems involving multiplication of mixed numbers.

MA.5.GR.2.1

FL.5.Mod5.AD8 Find and model the area of rectangles with fraction side lengths by tiling.

MA.5.GR.2.1

FL.5.Mod5.AD9 Find areas of rectangles with fraction or decimal side lengths.

Dates and Details of Observations

PP Partially Proficient P Proficient HP Highly Proficient

This page may be reproduced for classroom use only.

335

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC

Topic C Volume Concepts In topic C, students develop foundational understanding of volume concepts and relate volume to both addition and multiplication. In topic A, students are introduced to three-dimensional figures. Topic C begins with students comparing two-dimensional and three-dimensional figures. After identifying and comparing attributes and properties of three-dimensional figures, students build on their knowledge of three-dimensional figures, with a focus on right rectangular prisms. They explain how changing the length, width, and height of a right rectangular prism affects it. Students learn that because all three-dimensional objects take up space, they have volume. To find the volume of a right rectangular prism, students count the number of unit cubes that pack a model of a prism. They determine that prisms with different dimensions can have the same volume. After packing a model of a prism with cubes, students build prisms with improvised units, realizing that they can find volume by using something other than unit cubes. Building on their experience with packing, students transition to composing and decomposing prisms. They find volumes of prisms by decomposing the prisms into layers, finding the volume of each layer, and multiplying the number of layers by the volume of each layer. With the understanding that a prism is composed of layers, that a layer is composed of rows, and that each row is composed of individual cubes, students explain why prisms can be decomposed into layers in multiple ways and why one group of cubes can be arranged in multiple ways to compose a prism. The topic concludes with students exploring conceptual ideas about volume and capacity. Students differentiate between packing and filling, and they identify situations in which filling is more appropriate than packing. They relate volumes of solids to volumes of liquids, knowing that volume is the same regardless of whether a space is packed or filled. After establishing that 1 cubic centimeter has the same volume as 1 milliliter, students use the relationship between volume of a solid and liquid volume to solve real-world problems. In topic D, students extend their understanding of volume to apply the formulas V = l × w × h and V = B × h to solve mathematical and real-world problems involving right rectangular prisms and solids composed of right rectangular prisms. © Great Minds PBC

337

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC

Progression of Lessons Lesson 16

Lesson 17

Lesson 18

Identify attributes and properties of right rectangular prisms.

Find the volume of right rectangular prisms by packing with unit cubes and counting.

Find the volume of right rectangular prisms by packing with improvised units.

Height

Length

Width

Height Width

Length

I know that a three-dimensional figure has a height, is not flat, and does not lie in a plane. Right rectangular prisms have 6 rectangular faces, 12 edges, and 8 vertices. I can describe a right rectangular prism’s size by identifying its length, width, and height.

338

I can find the volume of a right rectangular prism by packing it with unit cubes and counting the number of cubic units. I know that two prisms with different dimensions can have the same volume.

I can use right rectangular prisms to build a larger right rectangular prism. I can find the volume of the larger prism by multiplying the volume of a group of unit cubes by the number of groups.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC

Lesson 19

Lesson 20

Lesson 21

Compose and decompose right rectangular prisms to find their volume by using layers.

Interpret volume as filling.

Relate volumes of solids and liquid volume.

1 cubic centimeter = 1 milliliter 10 mL 9 8 7 6 5 4

I can decompose a right rectangular prism into layers. Then I can find the volume of the prism by multiplying the volume of each layer by the number of layers. There is more than one way to decompose the prism to make the layers.

I know that every three-dimensional object has volume. I can find volume by packing a container with individual solids or filling it with liquid. The volume is the same whether the container is packed or filled.

3 2 1

I know that volumes of solids and liquid volumes both indicate how much space the solid or liquid takes up. I know that 1 centimeter cube has a volume of 1 milliliter.

339

LESSON 16

Identify attributes and properties of right rectangular prisms. I can identify properties of right rectangular prisms.

Lesson at a Glance Students identify similarities and differences between a square and a cube and learn that a cube is one of the five Platonic solids. They determine that three-dimensional figures do not lie in a plane because they have height. Through a digital interactive, students compare attributes and properties of a rectangle with those of a right rectangular prism. They explain how changing the length, width, and height of a prism affects its attributes and properties. This lesson introduces the term right rectangular prism.

Key Questions • What makes a figure three-dimensional? • How can we describe a right rectangular prism’s size?

Mathematical Thinking and Reasoning Standard MTR.5.1

Language Objectives • Read statements involving attributes and properties of three-dimensional figures to determine whether the statements are true or false. Rewrite false statements so they use the correct attribute or property. • Orally compare a rectangle to a right rectangular prism by describing relationships between the side lengths of a rectangle and the length and width of the base of a right rectangular prism.

• What does changing the length, width, or height of a right rectangular prism change about the prism’s faces, edges, and vertices?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 16

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

• Tear out and cut apart Decimal Number Cards from the student books. Place each set in an envelope. Prepare enough sets for each student pair to have one. Consider whether your students should only use cards with tenths, hundredths, thousandths, and/or ones. • Tear out Hidden Addends Mat from the student books. Consider whether to prepare this material in advance or during the lesson. Consider saving for reuse in lesson 17. • Consider tearing out Right Rectangular Prism for students in advance of the lesson. • Prepare four signs on paper: Rectangle, Right Rectangular Prism, Both, and Neither. Hang the signs in different locations in the classroom. Leave one sheet of paper blank for use in a demonstration. • Review the Math Past resource to support delivery of Launch.

Launch 10 min Learn 30 min • Length, Width, and Height • Faces, Edges, and Vertices • Rectangle or Right Rectangular Prism? • Problem Set

Land 10 min

• Decimal Number Cards (1 per student pair, in the student book) • Envelopes (12) • Centimeter cube • Right Rectangular Prism (in the teacher edition) • Scissors • Tape • Paper (5 sheets)

Students • Envelope of Decimal Number Cards (1 per student pair) • Hidden Addends Mat (1 per student pair, in the student book) • Centimeter cube • Right Rectangular Prism (in the student book) • Scissors • Tape • Highlighter

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD6 Classify three-dimensional

MA.5.GR.1.2 Identify and classify

figures into categories based on attributes. (MA.5.GR.1.2)

three-dimensional figures into categories based on their defining attributes. Figures are limited to right pyramids, right prisms, right circular cylinders, right circular cones and spheres.

341

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 16

Fluency

Whiteboard Exchange: Evaluate by Using the Order of Operations Students evaluate expressions to build fluency with evaluating expressions by using the order of operations. Display 42.6 − 5 × 5.3 =

Write and complete the equation. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to evaluate their corrections.

42.6 − 5 × 5.3 = 16.1

Display the answer. Repeat the process with the following sequence:

96 + 12.6 × 10 =

222

39.45 ÷ 0.1 − 3 ÷ 0.1 = 364.5

342

15.3 × 4.2 + 0.8 = 65.06

14.6 × (3 + 5) ÷ 2 =

114.4 ÷ (4 × 2) =

14.3

58.4

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 16

Hidden Addends Materials—S: Decimal Number Cards, Hidden Addends Mat

Students determine the sum then write and say an addition equation or related subtraction equation to develop procedural fluency with adding and subtracting decimals from module 4. Have students form pairs. Distribute a deck of Decimal Number Cards to each pair. Have them use the following procedure to play. Consider doing a practice round with students.

0.5

0.3

• Partners place the deck of cards facedown next to the Hidden Addends Mat. • Partners flip over a card from the top of the pile and place it on a blue square. • Both students say the sum.

Partners A and B: “0.8” Partner A: “0.5 + 0.3 = 0.8” Partner B: “0.8 − 0.5 = 0.3”

Teacher Note Consider whether to limit the game to using cards with only one place value or using cards with other place values as well. For example, students may play to add hundredths and hundredths, tenths and hundredths, or whole numbers and thousandths.

• Students discard the cards into a separate pile. Circulate during the activity to ensure that students are recording and saying accurate equations. If students run out of cards before the time ends, they can switch roles, shuffle the cards, and continue playing.

343

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 16

Launch

Materials—T/S: Centimeter cube, Right Rectangular Prism, scissors, tape

Students identify similarities and differences between a square and a cube. Display the square. Invite students to Think–Pair–Share about what they know about a square, including properties of squares and other names for squares.

Square

A square has four sides that are the same length. A square has four right angles. A square has four lines of symmetry. A square is a rectangle, rhombus, parallelogram, trapezoid, and kite. A square is a quadrilateral. A square is a polygon. A square is a two-dimensional figure. We can tile a square, and we can use squares to tile other figures. A square has area. Distribute a centimeter cube to each student. Invite students to Think–Pair–Share about similarities and differences between the cube and the square. A square is flat, but a cube is not. A cube has squares on all sides. Both a cube and a square have right angles. A cube is a three-dimensional figure, and a square is a two-dimensional figure. A square has four corners, but a cube has more than four corners. A square is a polygon, but a cube is not.

344

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 16

A square is a two-dimensional figure and lies in a plane. A cube is a three-dimensional figure, or a solid, and does not lie in a plane. Hold up a cube. Point to one of the cube’s faces. Have students do the same. What shape are the faces of a cube? A cube’s faces are squares.

Multilingual Learner Support Consider providing a visual to support the term plane. Ask students to recall that a plane is a flat surface that extends forever in all directions.

How many faces does a cube have? A cube has six faces. Display the Platonic solids. Tetrahedron

Octahedron

Cube

Dodecahedron

Icosahedron

4 equilateral

8 equilateral

6 squares

12 regular pentagons

20 equilateral

triangles

Plane

triangles

The ancient Greeks studied solids, including a group of solids known as the Platonic solids. All the faces of Platonic solids are identical regular polygons, or identical polygons that have sides that are the same length and angles that are the same measure. Are all sides of a square the same length? Do all angles of a square have the same measure?

Teacher Note Students learn about regular polygons in earlier grades and revisit them in later grades. Students are not expected to know the names of the Platonic solids, other than the cube. Instead, focus on the shapes of the faces of the Platonic solids.

Yes. A square’s sides are the same length, and its angles have the same measure. A cube is one of the Platonic solids because its faces are squares that are identical regular polygons.

345

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 16

Invite students to turn and talk about what they notice and wonder about the shape and number of faces of the Platonic solids. Then display the Platonic solids and the picture of the right rectangular prism. Math Past Tetrahedron

Octahedron

Cube

Dodecahedron

Right Rectangular Prism

Icosahedron

Direct students to remove Right Rectangular Prism from their books. Cut out the figure, fold it into a prism on the lines, and tape the prism together. Have students do the same.

This lesson provides an ideal time to incorporate the material in the Math Past resource. The Math Past resource features more information about the history of the study of solids and a more detailed explanation of the Platonic solids, including the vertices, edges, and faces of solids. If students are interested in further information on the history of the study of solids, consider incorporating other activities provided in the Math Past resource.

You have made a right rectangular prism. What shape are the faces of a right rectangular prism? Its faces are rectangles. A right rectangular prism is a solid where all the faces are rectangles. Ask students to Think–Pair–Share about the following prompt. Is the right rectangular prism a Platonic solid? Why? No. The right rectangular prism’s faces are rectangles, and not all rectangles are regular polygons. No. All angles of the rectangle have the same measure, but the rectangle’s sides are not all the same length. A right rectangular prism is a solid, and all its faces are rectangles. However, this right rectangular prism is not a Platonic solid because all its faces are not identical regular polygons. In fact, there are only five Platonic solids.

346

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 16

Invite students to turn and talk about which Platonic solid is also a right rectangular prism. Have students keep the constructed right rectangular prism to use later in the lesson. Transition to the next segment by framing the work. Today, we will identify characteristics of right rectangular prisms.

Learn

Length, Width, and Height Materials—T: Constructed right rectangular prism

Students predict and explain how changing length, width, or height affects the size of a rectangle and of a right rectangular prism. Display the Right Rectangular Prism digital interactive. Create a rectangle with a length of 5 units and a width of 3 units. Invite students to Think–Pair–Share about what they know about the rectangle. The rectangle’s length and width are 5 units and 3 units. The rectangle’s area is 15 square units. The rectangle has four right angles. The rectangle has two lines of symmetry through the midpoints of its sides. The rectangle is a polygon, quadrilateral, parallelogram, and trapezoid. The rectangle is a two-dimensional figure.

Universal Design for Learning: Representation The digital interactive Right Rectangular Prism allows students to observe the effects of changing the length and the width of a rectangle and the length, width, and height of a right rectangular prism in real time. The digital interactive also allows students to observe the number of faces, edges, and vertices of right rectangular prisms of different sizes. Consider allowing students to experiment with the tool individually.

We can tile the rectangle, and we can use the rectangle to tile other rectangles.

347

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 16

Then create a right rectangular prism with a length of 5 units, a width of 3 units, and a height of 1 unit. Invite students to Think–Pair–Share about any similarities and differences they notice between the right rectangular prism and the rectangle. Both the rectangle and the right rectangular prism have a length of 5 units and a width of 3 units. The right rectangular prism has a height of 1 unit, but the rectangle has no height. The rectangle is flat, but the right rectangular prism is not.

Teacher Note In the digital interactive, perspective may make it look like the right rectangular prism is on a rectangular grid. Show other views so students see that the bottom face of the right rectangular prism lies on a square grid.

The right rectangular prism has rectangles on all sides. Both the right rectangular prism and the rectangle have right angles. The right rectangular prism is a three-dimensional figure, or solid, but the rectangle is a two-dimensional figure. The rectangle has four corners, but the right rectangular prism has more than four corners. The rectangle is a polygon, but the right rectangular prism is not. The base of a right rectangular prism is one of the faces of the prism. We can choose the face on which the prism rests to be the base. The base of this right rectangular prism has a length of 5 units and a width of 3 units. When we identify a face as the base of a right rectangular prism, the length and width of the base are the length and width of the prism. Invite students to turn and talk about what they think what will happen to the rectangle and to the right rectangular prism if you change the length and width. Then direct students to watch the rectangle and right rectangular prism as you adjust the length and then adjust the width. Slowly increase the length and then decrease the length to 1 unit. Repeat several times before returning to a length of 5 units. Slowly increase the width and then decrease the width to 1 unit. Repeat several times before returning to a width of 3 units.

348

Promoting Mathematical Thinking and Reasoning When students apply their understanding of the length and width of a rectangle to identify the length, width, and height of a right rectangular prism, they are using patterns and structure (MTR.5). Ask the following questions to promote MTR.5: • How are rectangles and right rectangular prisms related? How could that help you identify the length, width, and height of a right rectangular prism? • How can what you know about the length and width of rectangles help you identify the length, width, and height of right rectangular prisms?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 16

What do you notice about the right rectangular prism as the length and width change? As the length changes, the right rectangular prism’s length gets longer and shorter, but its height and width stay the same. As the width changes, the right rectangular prism’s width gets longer and shorter, but its height and length stay the same. What do you notice about the rectangle as the length and width change? As the length changes, the rectangle’s length gets longer and shorter, but its width stays the same. As the width changes, the rectangle’s width gets longer and shorter, but its length stays the same. Invite students to turn and talk about what they think will happen to the rectangle and right rectangular prism if the height changes. Then direct students to watch the rectangle and right rectangular prism as you change the height. Slowly increase the height and then decrease it back to 1 unit. Repeat several times before returning to a height of 1 unit. What do you notice about the right rectangular prism as the height changes? The right rectangular prism gets taller and shorter, but its length and width stay the same.

Teacher Note Later in the lesson, students determine that any face of a right rectangular prism can be called the base. In later grades, students learn that all right prisms have two bases that are parallel faces. For example, a right triangular prism has two bases that are triangles. Any face of a right rectangular prism can be a base, but not any face of a right triangular prism can be a base.

What do you notice about the rectangle as the height changes? The rectangle does not change at all. It stays the same size. Why doesn’t the size of the rectangle change when we change the height? A rectangle is flat. It has no height. A rectangle is a two-dimensional figure. It has a length and width, but it doesn’t have a height. A rectangle is flat because it lies in a plane. Figures that lie in a plane are two-dimensional. Is a right rectangular prism two-dimensional? Why? No. A right rectangular prism is not two-dimensional because it isn’t flat and it has a height. A right rectangular prism is three-dimensional. No. A right rectangular prism does not lie in a plane, so it is not two-dimensional.

Teacher Note As students describe the size of the right rectangular prism and how the prism changes as an edge length changes, they may use language that describes volume (e.g., they may say that there are more layers or there are more small boxes). The language for describing volume is introduced in lesson 17, and the language in this lesson is foundational to understanding volume. Focus the discussion in this lesson on length, width, and height.

349

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 16

A right rectangular prism has length, width, and height. A right rectangular prism is three-dimensional and does not lie in a plane. Direct students to problem 1 in their books. Have them label the right rectangular prism with the terms length, width, and height. 1. Label the length, width, and height of the right rectangular prism.

Height

Length

Universal Design for Learning: Representation Consider posting pictures that show different ways to turn and label the same right rectangular prism. Highlight that the length, width, and height can be labeled in many ways.

Height

Width Length

Width

Height Width

Length

Hold up a constructed right rectangular prism. Point at the bottom of the prism and slide your finger to the top. Is this the length, width, or height of the right rectangular prism? The height Turn the prism so that what students identified as the height is now along the bottom of the prism. Is the prism still the same size even though what you identified as the height has moved?

Teacher Note Students may have the misconception that the figure shown in problem 1 must be transparent and that it has an inside and an outside. Tell them that the dashed lines show the edges that would not be visible when viewing a solid from the front.

Yes. Can we label the length, width, and height of the prism differently? Why? Yes. We can turn a prism in different ways. The prism stays the same size no matter which way we turn it.

350

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 16

Invite students to turn and talk about how they can label the length, width, and height of the right rectangular prism in problem 1 a different way.

Faces, Edges, and Vertices Materials—T: Constructed right rectangular prism; S: Constructed right rectangular prism, highlighter

Students compare a rectangle to a right rectangular prism and identify faces, edges, and vertices of a right rectangular prism. Continue to display the Right Rectangular Prism digital interactive with a 5 unit by 3 unit rectangle and a 5 unit by 3 unit by 1 unit right rectangular prism. We know all rectangles have four sides. The points where two sides meet are vertices. Do all rectangles have the same number of vertices? If so, how many? Yes. All rectangles have four vertices. Do right rectangular prisms have four sides and four vertices like rectangles? No. A right rectangular prism has faces, edges, and vertices.

Differentiation: Support Consider preparing several right rectangular prisms of various sizes, each with one edge highlighted as the height. Invite students to turn each prism in various ways to see how the height changes with different orientations. This concrete experience helps students confirm that the height of the prism can change depending on how it is turned. Invite students to notice how the length and width change as the height changes.

Hold up the constructed right rectangular prism. Point to two adjacent faces. Then slide your hand along the edge where the two faces meet. The line segment where two faces meet is called an edge. Point to one of the right rectangular prism’s vertices. The point where three edges meet is called a vertex. Direct pairs to count the number of faces of their constructed right rectangular prism. How many faces does a right rectangular prism have?

6 Have students keep count of the faces as you point to and identify each face with the appropriate description top, bottom, front, back, left, or right. Turn the prism so that a new side is on the bottom and ask students to identify each face in the new orientation.

351

5 ▸ M5 ▸ TC ▸ Lesson 16

EUREKA MATH2 Florida

Often, the bottom face of a right rectangular prism is called the prism’s base. Can any face be called the base? Why? Yes. We can turn a prism in different ways, so any face can be called the base. Invite students to turn and talk and identify each face of their constructed right rectangular prism. Then invite students to Think–Pair–Share about which faces are the same size. The top and bottom faces are the same size. The front and back faces are the same size.

Multilingual Learner Support If an anchor chart was created for the terms face, edge, and vertex in topic A, consider referring to the chart as needed.

The left and right faces are the same size. Use the digital interactive to create right rectangular prisms of different sizes (e.g., tall and narrow and then short and wide). Count the number of faces of each prism. Next, point to an edge of the right rectangular prism. Direct pairs to count the number of edges that their right rectangular prism has. How many edges does your right rectangular prism have?

12 Have students count as you point to each of the constructed right rectangular prism’s twelve edges: • the four edges where the bottom face meets the front, back, left, and right faces; • the four edges where the top face meets the front, back, left, and right faces; and • the four edges where the front and back faces meet the left and right faces. Use the digital interactive to create right rectangular prisms of different sizes. Count to confirm the number of edges of each prism. Next, point to a vertex of the right rectangular prism. Direct pairs to count the number of vertices that their constructed right rectangular prism has.

352

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 16

How many vertices does your right rectangular prism have?

8 Have students count as you point to each of the eight vertices on the constructed right rectangular prism: • the four vertices at the corners of the bottom face, and • the four vertices at the corners of the top face. Use the digital interactive to create right rectangular prisms of different sizes. Count to confirm the number of vertices of each prism. Invite students to turn and talk about what they think happens to the right rectangular prism’s faces, edges, and vertices when the length, width, and height changes. Then direct students to watch the right rectangular prism in the digital interactive as you change the length, width, and height. Slowly increase the length and then decrease the length back to 5 units. Then slowly increase the width and then decrease the width back to 3 units. Slowly increase the height and then decrease the height back to 1 unit. Repeat several times as necessary. Do the number of faces, edges, and vertices of a right rectangular prism change when the length, width, or height changes? Why? No. A right rectangular prism always has 6 faces, 12 edges, and 8 vertices. A property of right rectangular prisms is that they have 6 faces, 12 edges, and 8 vertices. All angles of a rectangle are right angles. Are the angles of a right rectangular prism right angles? Why? Yes. The faces are all rectangles, so all the angles are right angles. A property of right rectangular prisms is that all its angles are right angles. Direct students to highlight the edges and label the vertices of the right rectangular prism in problem 2.

353

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 16

2. Highlight the edges and label the vertices of the right rectangular prism with a V.

V Vertex

Face

Edge

V V

Invite students to work with a partner to point to and count each of the faces of the right rectangular prism in problem 2. Invite students to Think–Pair–Share about the following prompts. Use the digital interactive to adjust the length, width, and height of the right rectangular prism to help students answer the questions. Which faces of the right rectangular prism change in size when the length changes? Which faces stay the same size? When the length changes, the top, bottom, front, and back faces get larger and smaller. When the length changes, the left and right faces stay the same size.

Teacher Note Consider modeling how to point to each of the faces of the right rectangular prism before inviting students to do so with their partners. Use your finger to trace the edges of one face and then swipe across and down the face. Repeat for each of the faces. Invite students to name the faces (i.e., front face, back face, left face, right face, top face, and bottom face) as you point.

Which faces of the right rectangular prism change in size when the width changes? Which faces stay the same size? When the width changes, the top, bottom, right, and left faces get larger and smaller. When the width changes, the front and back faces stay the same size. Which faces of the right rectangular prism change in size when the height changes? Which faces stay the same size? When the height changes, the left, right, front, and back faces get larger and smaller. When the height changes, the top and bottom faces stay the same size.

354

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 16

Rectangle or Right Rectangular Prism? Materials—T: Signs, paper

Students reason about whether a sheet of paper is a rectangle or a right rectangular prism. Invite students to Think–Pair–Share about the following question. How can you tell whether a figure is a rectangle or a right rectangular prism? A rectangle is flat, and a right rectangular prism is not flat. A rectangle lies in a plane, but a right rectangular prism does not lie in a plane. A right rectangular prism has a height, but a rectangle does not have height. A right rectangular prism is three-dimensional, but a rectangle is two-dimensional.

Universal Design for Learning: Representation To support the idea of a sheet of paper’s thickness being its height, show a stack of paper and have students reason about it before they think about a single sheet of paper.

Introduce the Take a Stand routine to the class. Draw students’ attention to the signs hanging in the classroom: Rectangle, Right Rectangular Prism, Both, and Neither. Hold up a sheet of paper. Present the statements and invite students to stand beside the sign that best describes their thinking. • The sheet of paper is a rectangle. • The sheet of paper is a right rectangular prism. • The sheet of paper is both a rectangle and a right rectangular prism. • The sheet of paper is neither a rectangle nor a right rectangular prism. When all students are standing near a sign, allow 1 minute for groups to discuss the reasons why they chose that sign. Then call on each group to share reasons for their selection. Invite students who change their minds during the discussion to join a different group. Have students return to their seats. As a class, reflect on the reason that a sheet of paper is actually a right rectangular prism: It has a height, or thickness, and can be stacked to create a taller right rectangular prism.

Teacher Note A rectangle has two dimensions: length and width. A right rectangular prism has three dimensions: length, width, and height. Consider comparing a piece of patty paper to a piece of cardstock. Show students that both have thickness, or height, which makes them both right rectangular prisms.

355

5 ▸ M5 ▸ TC ▸ Lesson 16

EUREKA MATH2 Florida

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Identify attributes and properties of right rectangular prisms. Facilitate a class discussion about properties and attributes of right rectangular prisms by using the following prompts. Encourage students to restate or add on to their classmates’ responses. What makes a figure three-dimensional? A three-dimensional figure has a height. A three-dimensional figure is not flat. A three-dimensional figure does not lie in a plane. How can we describe a right rectangular prism’s size? We can identify its length, width, and height. Does changing the length, width, or height of a right rectangular prism affect how many faces, edges, and vertices it has? Explain your thinking. No, the numbers of faces, edges, and vertices of a right rectangular prism do not change when we change the prism’s length, width, or height. No, all right rectangular prisms have 6 faces, 12 edges, and 8 vertices. Changing the length, width, or height does not change the number of faces, edges, or vertices.

356

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 16

What about the faces, edges, and vertices of a right rectangular prism changes when we change the prism’s length, width, or height? What does not change? When a right rectangular prism’s length, width, or height changes, some faces change size. When a right rectangular prism’s length, width, or height changes, some edges in the prism change length. The vertices do not change when a right rectangular prism’s length, width, or height changes.

357

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 16

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET PRACTICE

I can identify properties of right rectangular prisms.

Florida B.E.S.T. MA.5.GR.1.2

Name

2. Circle the two right rectangular prisms that are identical.

4 in

Date

3 in

1. Consider the figures.

3 in

12 in 12 in

3 in

4 in

2 in 3 in

12 in 10 in

4 in

a. Use red to color all two-dimensional figures. b. Circle all three-dimensional figures.

358

155

156

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 16

3. Consider the rectangle and the right rectangular prism.

Consider how the length, width, and height are labeled on the right rectangular prism.

5 in

Height

4 in

Length

3 in

Width

Find the length, width, and height of the prisms in problems 4–7.

Fill in the blanks.

a. A rectangle has

sides.

b. A rectangle has

vertices.

5 in

c. A right rectangular prism has

faces.

d. A right rectangular prism has

edges.

e. A right rectangular prism has

vertices.

5 in 4 in

2 in

6 in

Length:

4 in

Length:

6 in

Width:

2 in

Width:

3 in

Height:

5 in

Height:

5 in

3 in

7 in 7 in 6 in

8 in

3 in

6 in

PROBLEM SET

157

158

Length:

6 in

Length:

6 in

Width:

8 in

Width:

3 in

Height:

7 in

Height:

7 in

PROBLEM SET

359

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 16

EXIT TICKET

I can identify properties of right rectangular prisms.

Florida B.E.S.T. MA.5.GR.1.2

Name

Date

Indicate whether each statement is true or false. If the statement is false, correct the statement. Statement A three-dimensional figure is flat.

True

False

A three-dimensional figure has length, width, and height.

True

False

A three-dimensional figure does not lie in a plane.

True

False

All right rectangular prisms have 4 faces.

True

False

All right rectangular prisms have 12 edges.

True

False

All right rectangular prisms have 6 vertices.

True

False

360

True or False

Correct Statement A three-dimensional figure is not flat.

All right rectangular prisms have 6 faces.

All right rectangular prisms have 8 vertices.

159

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 16 ▸ Right Rectangular Prism

This page may be reproduced for classroom use only.

361

LESSON 17

Find the volume of right rectangular prisms by packing with unit cubes and counting. I can use unit cubes to find the volume of a right rectangular prism.

Lesson at a Glance Students find the volume of a right rectangular prism by counting the number of centimeter cubes it takes to pack the prism. When they find it takes the same number of cubes to pack prisms with different lengths, widths, and heights, they realize that volumes can be the same even when prisms have different dimensions. When using cubes of different sizes, students determine that prisms can have different volumes even if the prisms are made with the same number of cubes. This lesson introduces the terms unit cube, cubic unit, volume, cubic centimeter, and cubic inch.

Key Questions • How can we measure the amount of space a right rectangular prism takes up?

Language Objectives • Orally reason about the volume of a right rectangular prism by estimating how many unit cubes pack the prism. • Orally and in writing, explain how two prisms made from the same number of cubes could have different volumes.

Mathematical Thinking and Reasoning Standard MTR.2.1

• Do two right rectangular prisms that have different dimensions always have different volumes? Why? • How can right rectangular prisms made from the same number of cubes have different volumes?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 17

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

• Tear out and cut apart Decimal Number Cards from the student books. Place each set in an envelope. Prepare enough sets for each student pair to have one. Consider whether your students should only use cards with tenths, hundredths, thousandths, and/or ones. • Tear out Hidden Addends Mat from the student books. Consider whether to prepare this material in advance or to have students prepare it during the lesson. • Consider labeling the prisms in the rectangular prisms set as A, B, C, and D with a permanent marker to help students identify each prism during the lesson.

Launch 5 min Learn 35 min • Pack Prisms • Compare Volumes • Compare Cubic Units • Problem Set

Land 10 min

• Decimal Number Cards (1 set per student pair, in the student book) • Envelopes (12)

Students • Envelope of Decimal Number Cards (1 per student pair) • Hidden Addends Mat (1 per student pair, in the student book) • Rectangular prisms set (1 set per student pair) • Centimeter cubes (40 per student pair) • Ruler

▶ A: 4 cm × 3 cm × 2 cm ▶ B: 3 cm × 3 cm × 3 cm ▶ C: 5 cm × 4 cm × 1 cm ▶ D: 2 cm × 2 cm × 5 cm

Achievement Descriptors

Florida B.E.S.T. Standard

FL.5.Mod5.AD10 Recognize that volume can

MA.5.GR.3.1 Explore volume as an attribute

be measured by using unit cubes and that a solid packed without gaps or overlaps by n unit cubes has a volume of n cubic units (MA.5.GR.3.1)

of three-dimensional figures by packing them with unit cubes without gaps. Find the volume of a right rectangular prism with whole-number side lengths by counting unit cubes.

FL.5.Mod5.AD11 Measure volumes by counting

unit cubes. (MA.5.GR.3.1)

363

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 17

Fluency

Whiteboard Exchange: True or False Number Sentences Students decide whether a number sentence is true or false and make false number sentences true to build fluency with reasoning about products without evaluating. Display 18.902 × 1 < 18.902. Is the number sentence true or false? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond. False.

18.902 × 1 < 18.902 False

18.902 × 1 = 18.902

Write the number sentence by using the correct comparison symbol. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the corrected number sentence. Repeat the process with the following sequence:

0.29 × 2.1 > 0.29 True

364

0.45 × 1.05 < 0.45

12.5 × 0.73 > 12.5

0.45 × 1.05 > 0.45

12.5 × 0.73 < 12.5

False

6.04 × 5.04 > 6.04 True

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 17

Hidden Addends Materials—S: Decimal Number Cards, Hidden Addends Mat

Students determine the sum and then write and say an addition equation or related subtraction equation to develop procedural fluency with adding and subtracting decimals from module 4. Have students form pairs. Distribute a deck of Decimal Number Cards to each pair. Have them use the following procedure to play. Consider doing a practice round with students. • Partners place the deck of cards facedown next to the Hidden Addends Mat. • Partners flip over a card from the top of the pile and place it on a blue square. • Both partners say the sum.

0.5

0.3

Teacher Note Consider whether to limit the game to using only one place value or using cards with other place values as well. For example, students may play to add hundredths and hundredths, tenths and hundredths, or whole numbers and thousandths.

Partners A and B: “0.8” Partner A: “0.5 + 0.3 = 0.8” Partner B: “0.8 − 0.5 = 0.3”

• Partner A records an addition equation on their whiteboard and then reads it. Partner B records a related subtraction equation on their whiteboard and then reads it. See the sample partner dialogue under the sample mat. • Students discard the cards into a separate pile. Circulate during the activity to ensure that students are recording and saying accurate equations. If students run out of cards before the time ends, they can switch roles, shuffle the cards, and continue playing.

365

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 17

Launch

Students reason about which container takes up more space. Present the question and two containers shaped like right rectangular prisms. Use the Math Chat routine to engage students in mathematical discourse. Teacher Note

Which prism takes up more space?

Right Rectangular Prism A

A right rectangular prism is a solid. Because it is physically impossible to pack a solid, we pack containers shaped like right rectangular prisms and use the capacity of the containers to explore the volume of right rectangular prisms with the same dimensions. Because these containers are a model for right rectangular prisms, the plastic is treated as having no thickness.

Right Rectangular Prism B

Give students 1 minute of silent think time to consider whether prism A takes up more space, prism B takes up more space, or prisms A and B take up the same amount of space. Have students give a silent signal to indicate they are finished.

366

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 17

Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Purposefully choose students whose ideas initiate a rich discussion about how to measure the amount of space a prism takes up. Then facilitate a class discussion. Invite students to share their thinking with the whole group and record their reasoning. As students discuss, highlight thinking about measurement relationships and strategies. I am not sure which prism takes up more space. Prism B is taller, but prism A is wider. So either prism could take up more space or they could take up the same amount of space, depending on the measurements. We could fill the prisms with water and see which one holds more water. We could measure each prism’s length, width, and height with a ruler and compare them. As a class, reflect on ideas about how the amount of space that a three-dimensional figure takes up can be measured. Ask questions that invite students to make connections and encourage them to ask questions of their own. • How can you measure how much flat space a two-dimensional figure takes up? • How do you reason about how much space a three-dimensional figure takes up? • One prism looks taller, but one prism looks wider. How do the height and width affect how much space a prism takes up? Transition to the next segment by framing the work. Today, we will measure the amount of space that right rectangular prisms take up.

367

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 17

Learn

Pack Prisms Materials—S: 4 cm × 3 cm × 2 cm prism, centimeter cubes, ruler, 3 cm × 3 cm × 3 cm prism

Students pack containers shaped like right rectangular prisms with centimeter cubes to find volumes of solids with the same dimensions. Pair students and give each pair a 4 cm × 3 cm × 2 cm prism and 40 centimeter cubes. Invite students to turn and talk about how they might use the materials to determine the amount of space that the right rectangular prism takes up. Allow students a couple of minutes to work with the materials. Then invite students to share anything they noticed. I put cubes in the right rectangular prism and found that I can fit 24 cubes in the prism. I put 24 cubes together to create a right rectangular prism that is the same size. I was able to fit 12 cubes along the bottom, and then I stacked another 12 cubes on top of those. We can use unit squares to tile two-dimensional figures and measure the figure’s area in square units. One way to measure the amount of space a three-dimensional object takes up is to pack it with cubes. What shape do a cube’s faces have? A cube’s faces are squares. Because a cube’s faces are squares and the sides of a square are the same length, the edges of a cube are the same length. When the edges of a cube each measure 1 unit, we call the cube a unit cube. Hold up one of the cubes.

Teacher Note In earlier grades, students use centimeter cubes in a weight context because each cube weighs 1 gram; in this topic, they use the cubes to measure the amount of space a right rectangular prism takes up. Point out that a different type of cube that is the same size could be made of something heavier or lighter, and the prism could still be packed with the same number of cubes.

Teacher Note If necessary, invite students to confirm that the edges of each cube are the same length by placing a cube on a piece of paper and drawing tick marks to indicate the length of an edge. Direct them to rotate the cube and confirm that all the edges are the same length. Later in the lesson, students measure the edges and determine that the cube is a centimeter cube.

Is this a unit cube? How do you know? Yes. The faces are squares, and the edges are the same length.

368

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 17

Direct students who have not already done so to pack the prism completely, without gaps or overlaps, and count the number of cubes it takes. Be sure the cubes do not have gaps between them and do not go above the height of the prism. We don’t have to worry about the cubes overlapping because they are hard and hold their shape. Have students complete problem 1 in their books. 1. Sketch to show the number of unit cubes visible on the faces of the right rectangular prism. In the blank, write the total number of unit cubes it takes to pack the prism.

Universal Design for Learning: Action & Expression Consider offering a straightedge to support drawing the cubes. If sketching a right rectangular prism packed with unit cubes presents a challenge to students, consider offering them the option of writing a description of what the prism looks like instead.

Multilingual Learner Support

Number of unit cubes:

When students are finished, use the following prompts. We can use unit cubes to measure the amount of space a three-dimensional figure takes up. We can count the number of unit cubes it takes to pack a figure completely without gaps or overlaps. The cubes are the units, so we say cubic units.

Consider providing a visual to accompany the introduction of unit cube and cubic unit. A unit cube is a cube with edges that each measure 1 unit. A cubic unit is a unit of measurement that describes the threedimensional space a unit cube takes up.

Unit Cube 1 unit

1 unit

Volume is 1 Cubic Unit. 1 unit

1 unit 1 unit

How many cubic units of space does the right rectangular prism in problem 1 take up?

24 cubic units Because there are 24 unit cubes packed in the prism, we can say the prism’s volume is 24 cubic units, or that the prism takes up 24 cubic units of space. The volume of a three-dimensional figure is the amount of space it takes up.

Consider adding visuals for centimeter cube, cubic centimeter, inch cube, and cubic inch as the terms are introduced later in the lesson.

369

5 ▸ M5 ▸ TC ▸ Lesson 17

Have students write volume and 24 cubic units by the prism in problem 1. If someone packed only 20 cubes in the prism, can we say the volume of the prism is 20 cubic units? Why? No. If someone packed only 20 cubes in the prism, then they did not pack the prism all the way. To pack a prism, there must be no gaps or overlaps. If someone packed 24 cubes and then put 4 more on top, can we say the volume of the prism is 28 cubic units? Why? No. If someone used 28 cubes, then they must have gone over the top of the prism. That would be too many cubes. Give each student a ruler. Invite students to measure a centimeter cube’s length, width, and height in centimeters. Then ask the following question. We call this a centimeter cube. Why do you think we call this a centimeter cube? Its length, width, and height are each 1 centimeter. This cube is called a centimeter cube because its edges each measure 1 centimeter. A centimeter cube takes up 1 cubic centimeter of space. How many cubic centimeters of space does the right rectangular prism in problem 1 take up? Why?

EUREKA MATH2 Florida

Multilingual Learner Support The word volume has multiple meanings in mathematics and in everyday life. Consider highlighting some different meanings of the term and activating prior knowledge from previous grades. • In grade 3, students learn about liquid volume. We know that gallon, quart, pint, and cup are the customary units of liquid volume. • Volume can also refer to how loud something is. The volume of the television was turned up to the highest setting. • Volume can refer to a book in series. I am reading Volume 2 of my favorite book series. There are 3 volumes left.

24 cubic centimeters. Each centimeter cube takes up 1 cubic centimeter of space, and there are 24 centimeter cubes. Because there are 24 centimeter cubes packed in the prism, we can say the prism’s volume is 24 cubic centimeters. Have students write volume and 24 cubic centimeters by the prism in problem 1. Then direct them to count the cubes to see how long, how wide, and how tall the prism is in centimeters. How long is the prism? How wide is the prism? How tall is the prism? The prism is 4 centimeters long, 3 centimeters wide, and 2 centimeters tall. I see the length as 3 centimeters, the width as 4 centimeters, and the height as 2 centimeters. Students’ perspective on length, width, and height may differ. Accept all correct responses. 370

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 17

Have students label the right rectangular prism in problem 1 with its length, width, and height in centimeters. Does a paper clip take up space? Yes. So, although we can’t pack a paper clip, a paper clip does have volume. Do you think a paper clip’s volume is greater than or less than 24 cubic centimeters? Why? A paper clip’s volume is less than 24 cubic centimeters because a paper clip takes up less than 24 cubic centimeters of space. Does a person take up space? Yes. So, although we can’t pack a person, a person does have volume. Do you think a person’s volume is greater than or less than 24 cubic centimeters? Why? A person’s volume is greater than 24 cubic centimeters because a person takes up more than 24 cubic centimeters of space. Next, hold up a 3 cm × 3 cm × 3 cm prism. What do you notice about this prism? It is taller than the other prism, but it is not as long. Invite students to turn and talk about whether the volume of this prism is greater than or less than the volume of the prism in problem 1. How can we measure the amount of three-dimensional space this prism takes up, its volume, in cubic centimeters?

Promoting Mathematical Thinking and Reasoning When students find volumes of right rectangular prisms by packing containers shaped like the right rectangular prisms with centimeter cubes, they are representing problems in multiple ways (MTR2).

We can pack it with centimeter cubes and count the number of cubic centimeters.

Ask the following questions to promote MTR.2:

Estimate this prism’s volume in cubic centimeters, or the number of centimeter cubes you can pack in the prism. Write your estimate on your whiteboard.

• What can you draw to help you understand this problem? • What does the number of centimeter cubes packed in a right rectangular prism tell you about the volume of the prism?

371

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 17

Invite several students to share the reasoning for their estimates. Then give each pair a 3 cm × 3 cm × 3 cm prism. Direct students to pack the prism and count the number of cubes they need to pack it completely without gaps or overlaps. Have students complete problem 2. 2. Sketch to show the number of centimeter cubes visible on the faces of the right rectangular prism. Then complete the table.

Length (centimeters)

Width (centimeters)

Height (centimeters)

Volume (cubic centimeters)

Teacher Note Some students may start to realize they can multiply the edge lengths of right rectangular prisms to find volume. Acknowledge the pattern they notice. However, this is not the intent of this lesson. Rather, guide students to use their reasoning to estimate and check the number of cubes it takes to pack right rectangular prisms completely without gaps or overlaps to reinforce a conceptual understanding of volume. Finding volume by multiplying the length, width, and height of a right rectangular prism is formally introduced in topic D.

Ask students to compare the actual volume of the prism to their estimates for the volume of the prism written on their whiteboards.

372

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 17

What did you notice about the right rectangular prism in problem 2? Its volume is 27 cubic centimeters. Its length, width, and height all measure 3 centimeters. I know it is a cube because its length, width, and height are the same. Invite students to turn and talk to compare the lengths, widths, heights, and volumes of the prisms in problems 1 and 2.

Compare Volumes Materials—S: 5 cm × 4 cm × 1 cm prism, 2 cm × 2 cm × 5 cm prism, centimeter cubes

Students compare volumes of right rectangular prisms. Give each student pair a 5 cm × 4 cm × 1 cm prism and a 2 cm × 2 cm × 5 cm prism. Before having students pack the prisms, ask them to write one of the following options on their whiteboards: • Shorter if they think the shorter prism has a greater volume • Taller if they think the taller prism has the greater volume • Same if they think the prisms have the same volume Invite a student who wrote shorter to share their reasoning. Repeat with students who wrote taller and same. Then ask each student in the pair to pack one of the two prisms and have pairs complete problem 3.

373

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 17

3. Complete the table for each right rectangular prism. Right Rectangular Prism

Length Width Height Volume (centimeters) (centimeters) (centimeters) (cubic centimeters)

When most students have completed the table, ask them to sketch to show the number of centimeter cubes visible on the faces of each right rectangular prism. Then ask the following questions. What do you notice about the volume of the two prisms? The volumes are the same.

374

Differentiation: Challenge Challenge students by giving them the volume of a right rectangular prism and only two of its three dimensions. Ask them to find the third dimension. For example, find the length of a right rectangular prism with a width of 6 centimeters, a height of 2 centimeters, and a volume of 36 cubic centimeters.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 17

Are the dimensions—or lengths, widths, and heights—of the two prisms the same? No. Which prism takes up more space? How do you know? They take up the same amount of space. I know because they have the same volume. Then invite students to Think–Pair–Share about the following prompt. If we double only one dimension of a right rectangular prism, what do you think happens to the volume? The volume will be greater. The volume will be twice as much. Join pairs to form groups of four. Direct pairs to stack their two 5 cm × 4 cm × 1 cm prisms to create a new prism. What are the dimensions of the new prism you created? The length is 5 centimeters, the width is 4 centimeters, and the height is 2 centimeters. What is the volume of this new prism? How do you know? The volume is 40 cubic centimeters. I know because 20 × 2 = 40. The volume is 40 cubic centimeters. I know because 20 + 20 = 40. What happened to the volume when you doubled the height? The volume doubled. What would happen if we doubled the length instead of the height? How do you know? The volume would double. If we lay the prisms side by side, there are still 40 cubic centimeters. Each time we double a dimension of a right rectangular prism, the volume is doubled. Invite students to turn and talk about why the volume is doubled when one dimension is doubled.

375

5 ▸ M5 ▸ TC ▸ Lesson 17

EUREKA MATH2 Florida

Compare Cubic Units Materials—S: Centimeter cubes

Students compare volumes of right rectangular prisms. Hold up 1 centimeter cube. What is the length, width, and height of 1 centimeter cube? Its length is 1 centimeter, its width is 1 centimeter, and its height is 1 centimeter. How much space does 1 centimeter cube take up? Why? It takes up 1 cubic centimeter of space because that is its volume. Invite students to think about what a cube would look like if the length, width, and height were doubled so that each measures 2 centimeters rather than 1 centimeter. Have them Think–Pair–Share about the volume of a cube with edge lengths that measure 2 centimeters rather than 1 centimeter. The volume would be 8 cubic centimeters because it takes 8 centimeter cubes to build a cube with edge lengths that measure 2 centimeters. If we double the length, that doubles the volume of 1 cubic centimeter to 2 cubic centimeters because 1 × 2 = 2. Then if we double the width, that doubles the volume to 4 cubic centimeters because 2 × 2 = 4. Then if we double the height, that doubles the volume to 8 cubic centimeters because 4 × 2 = 8. Direct students to build a right rectangular prism with a length, width, and height that each measure 2 centimeters to confirm that its volume is 8 cubic centimeters. Have students use their hands to show the approximate size of 1 centimeter and then 1 inch.

376

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 17

If we build two right rectangular prisms with the same number of cubes, one with 8 centimeter cubes and the other with 8 inch cubes, which prism will take up more space? Why? The prism made of 8 inch cubes would take up more space because inches are longer than centimeters. A prism built with inch cubes takes up more space and has a greater volume than a prism built with the same number of centimeter cubes. A cubic inch is the volume of an inch cube and is greater than a cubic centimeter. Have students use their hands to show you the size of 1 foot. We can measure volume in other cubic units, such as cubic feet, cubic yards, or even cubic miles. Which do you think takes up more space: 1 cubic foot or 8 cubic centimeters?

1 cubic foot Although the number of cubes is less, 1 cubic foot takes up more space than 8 cubic centimeters. So, 1 cubic foot has a greater volume than 8 cubic centimeters. Invite students to turn and talk about how the volume of two prisms can be different even if the number of cubes used to build the prisms is the same.

Differentiation: Support To help students visualize different cubic units, show scaled drawings of different cubic units, such as a cubic centimeter, a cubic inch, a cubic foot, and a cubic yard. 1 cubic centimeter 1 cubic inch

1 cubic foot

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex. 1 cubic yard

377

5 ▸ M5 ▸ TC ▸ Lesson 17

Land

EUREKA MATH2 Florida

Debrief 5 min Objective: Find the volume of right rectangular prisms by packing with unit cubes and counting. Facilitate a class discussion about volume of right rectangular prisms by using the following prompts. Encourage students to restate or add on to their classmates’ responses. How can we measure the amount of space a right rectangular prism takes up? We can find the volume of the prism by packing it with cubes and counting the number of cubic units. What size cube can we use to pack a prism to find its volume? We can use any size cube if it packs the prism completely with no gaps or overlaps. We can use centimeter cubes, inch cubes, or even foot cubes if the prism is big enough. Do two right rectangular prisms that have different dimensions always have different volumes? Why? No. Two prisms with different dimensions can have the same volume if both prisms are packed with the same number of cubes that are the same size. How can two right rectangular prisms be made from the same number of cubes but have different volumes? Give an example. If each right rectangular prism is made with a different size of cube, then the volumes will be different. For example, 1 cubic centimeter takes up less space than 1 cubic foot. So, even though both prisms are made of 1 cube, the volumes are different.

378

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 17

379

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 17

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can use unit cubes to find the volume of a right rectangular prism.

PRACTICE

Florida B.E.S.T. MA.5.GR.3.1

Name

9. The picture represents a right rectangular prism packed with centimeter cubes. Use the prism to complete parts (a) and (b).

Date

For problems 1–3, fill in the blanks. 1. A unit cube takes up 2. A

centimeter

cubic unit of space.

cube takes up 1 cubic centimeter of space.

a. How many centimeter cubes are packed in the right rectangular prism?

30 centimeter cubes 3. An inch cube takes up 1 cubic

inch

of space.

b. Complete the table. Sample:

For problems 4–7, circle the measurement with the greater volume. 4. 1 cubic foot or 1 cubic inch

Length (centimeters)

Width (centimeters)

Height (centimeters)

Volume (cubic centimeters)

5. 1 cubic inch or 1 cubic centimeter 10. A right rectangular prism is 4 units long, 3 units wide, and 3 units tall. It is packed with unit cubes.

6. 1 cubic centimeter or 1 cubic foot

a. Sketch to show the number of unit cubes visible on the faces of the right rectangular prism.

7. 1 cubic foot or 13 cubic inches

8. Sana says a right rectangular prism with a volume of 10 cubic inches takes up the same amount of space as a right rectangular prism with a volume of 10 cubic centimeters because both volumes are 10 cubic units. Do you agree or disagree with Sana? Why?

b. How many unit cubes does it take to pack the prism?

I disagree. 1 cubic inch takes up more space than 1 cubic centimeter, so the volume of 10 cubic inches takes up more space than the volume of 10 cubic centimeters.

380

36 unit cubes

175

176

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 17

EXIT TICKET

I can use unit cubes to find the volume of a right rectangular prism.

Florida B.E.S.T. MA.5.GR.3.1

11. Use right rectangular prism A and right rectangular prism B to complete parts (a)–(c). Right Rectangular Prism A

Right Rectangular Prism B Name

Date

Tara draws to represent a right rectangular prism packed with centimeter cubes. Use the prism to complete parts (a)–(c).

3 in 4 in

a. How many centimeter cubes are packed in the right rectangular prism?

2 in

6 in

24 centimeter cubes

2 in

b. Complete the table.

a. What is the volume of right rectangular prism A?

Sample:

24 cubic inches b. What is the volume of right rectangular prism B ?

24 cubic inches c. Which prism takes up more space? How do you know ?

Length (centimeters)

Width (centimeters)

Height (centimeters)

Volume (cubic centimeters)

c. Blake’s right rectangular prism has the dimensions shown.

The prisms take up the same amount of space because they have the same volume.

12. Use the right rectangular prism shown to determine whether the statements are true or false. Statement

True

Doubling the length of the prism doubles the prism’s volume.

Doubling the width of the prism halves the prism’s volume. Tripling the height of the prism triples the prism’s volume.

4 ft

False

2 ft

Blake says his right rectangular prism has the same volume as Tara’s right rectangular prism. Is Blake correct? Explain your thinking.

Blake is not correct. The volume of Blake’s prism is measured in cubic feet. The volume of Tara’s right rectangular prism is measured in cubic centimeters. Cubic feet take up more space than cubic centimeters, so Blake’s right rectangular prism has a greater volume than Tara’s right rectangular prism.

PROBLEM SET

3 ft

177

179

381

LESSON 18

Find the volume of right rectangular prisms by packing with improvised units. I can use smaller prisms to find the volume of a right rectangular prism.

Lesson at a Glance Students reason about how to determine the number of packs of gum and sticks of gum there are in a box. Then they use interlocking cubes to make right rectangular prisms. They use the prisms to build larger right rectangular prisms that are the same size as given boxes and find the boxes’ volumes. Students realize that they can arrange the smaller prisms in different ways to build the larger prisms and that they can use multiple strategies to find volume.

Key Questions • Do right rectangular prisms of the same size have the same volume if they are built by using different arrangements of smaller prisms? Why?

Language Objectives • Write or draw to record how to arrange right rectangular prisms to find the volume of a box. • Listen to and orally describe various strategies for finding the volume of a right rectangular prism and compare and connect strategies.

Mathematical Thinking and Reasoning Standard MTR.4.1

• How do you find the volume of a right rectangular prism when it is made with something other than cubes?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 18

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 5 min Learn 35 min • Build a Prism with Prisms • Build a Larger Prism • Problem Set

Land 10 min

• None

Students • Interlocking cubes, 1 cm (80 per student pair) • Box 1 (1 per student pair, in student book) • Scissors (1 per student pair) • Tape (1 per student pair) • Box 2 (1 per student pair, in student book)

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD11 Measure volumes by counting

MA.5.GR.3.1 Explore volume as an attribute

unit cubes. (MA.5.GR.3.1)

of three-dimensional figures by packing them with unit cubes without gaps. Find the volume of a right rectangular prism with whole-number side lengths by counting unit cubes.

383

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 18

Fluency

Whiteboard Exchange: Divide Whole Numbers Students determine the quotient and express it as a mixed number to build procedural fluency with dividing two-digit divisors and two- and three-digit dividends from module 1. 213 ÷ 40 =

Display 213 ÷ 40. Find the quotient and express it as a mixed number if needed. Show your method.

5 13 40

5 40 213 –200 13

91 ÷ 22 =

384

422

425 ÷ 17 =

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 18

Whiteboard Exchange: Write and Evaluate Expressions Students write and evaluate an expression to build fluency with two-step calculations involving decimals from module 4. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the statement: The sum of 1.2 and 1.8, doubled. Write an expression to represent the statement. Display the sample expression. Write the value of the expression.

The sum of 1.2 and 1.8, doubled

(1.2 + 1.8) × 2 6

Display the answer. Repeat the process with the following sequence:

The difference between 1 and 0.1, divided by 3

Add 5 to the product of 0.2 and 4

(1 − 0.1) ÷ 3

(0.2 × 4) + 5

0.3

5.8

385

5 ▸ M5 ▸ TC ▸ Lesson 18

Launch

EUREKA MATH2 Florida

Students reason about finding the number of pieces of gum in a pack and in a box. Display the pack of gum.

How many sticks of gum are in the pack?

5 Display both the pack of gum and the box of packs of gum.

386

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 18

Invite students to Think–Pair–Share about the following prompts. How many sticks of gum are in the box? I think there are 50 sticks of gum in the box because I can see 10 packs of gum and 10 × 5 = 50. I think there are 100 sticks of gum in the box. I can see 10 packs of gum, and it looks like there might be 10 more packs under those. 20 × 5 = 100. I think there are 150 sticks of gum in the box. I can see 10 packs of gum, and it looks like there might be 2 more groups of 10 packs under those. 30 × 5 = 150. If you know the volume of 1 stick of gum and you know the number of sticks of gum in a pack, how could you find the total volume of the gum in a pack? We could add the volumes of each stick of gum in the pack. We could multiply the volume of 1 stick of gum by the number of sticks of gum in the pack. If you know the volume of 1 pack of gum and you know the number of packs of gum in a box, how could you find the total volume of the packs of gum in a box? We could add the volumes of each pack of gum in the box. We could multiply the volume of 1 pack of gum by the number of packs of gum in the box. Transition to the next segment by framing the work. Today, we will find the volume of right rectangular prisms by using smaller right rectangular prisms.

387

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 18

Learn

Build a Prism with Prisms Materials—S: Cubes, Box 1, scissors, tape

Students build 3 unit × 1 unit × 1 unit right rectangular prisms and use them to compose a larger right rectangular prism and find its volume. Pair students and direct them to complete problems 1(a) and 1(b) in their books. For problem 1, use box 1 to complete parts (c)–(f). 1. A right rectangular prism is 3 units long, 1 unit wide, and 1 unit tall. a. Draw the prism. Sample:

b. What is the volume of the prism?

3 cubic units How much space does the right rectangular prism from problem 1 take up? How do you know?

3 cubic units. I know because its volume is 3 cubic units. 3 cubic units. I know because it would take 3 unit cubes to make the prism. Give each student pair 80 interlocking cubes and have them make the prism from problem 1 by connecting 3 cubes that are the same color.

Teacher Note Consider showing a centimeter cube and an interlocking cube side by side. Explain that one cube is better for finding volume than the other: Although the interlocking cubes are not the best cubes for packing, they are a good tool for building. When interlocking cubes are used to represent cubic units, it is assumed that they are complete cubes. It is not necessary to show the connectors or holes when sketching a right rectangular prism made from interlocking cubes.

Teacher Note In lesson 17, the outline of the prism is provided, and students draw line segments to indicate individual cubes. This is students’ first experience drawing a cube on isometric dot paper. Consider modeling how to connect the dots to represent a threedimensional figure by drawing at least the first cube in the prism. Provide students with straightedges if they have difficulty drawing straight lines between the dots.

Then direct them to remove Box 1 from one of their books. Tell them to cut out box 1, fold the sides up, and tape the box together. 388

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 18

How can we determine the amount of space the box takes up? We can find its volume. We can pack it with centimeter cubes and count the number of centimeter cubes it takes to pack it. We can build a right rectangular prism that is the same size and count the number of cubes. One way to find the area of a rectangle is to tile it with smaller rectangles. One way to find the volume of the box is to use smaller prisms to build and find the volume of a right rectangular prism that is the same size as the box. Direct students to erase the interior lines that show individual cubes, leaving only the edges of the prism in part (a). Then read problem 1(c) aloud. Have students complete problem 1(c) with their partners. c. Estimate how many prisms like the ones from part (a) it takes to build a right rectangular prism that is the same size as box 1.

10 prisms Invite several students to share their estimates and their reasoning. Direct students to use interlocking cubes to make more right rectangular prisms that are 3 units long, 1 unit wide, and 1 unit tall. Invite them to make as many prisms as necessary to build a right rectangular prism that is the same size as box 1. Each small prism can be a different color, but have students use cubes of the same color to make each small prism if possible.

Universal Design for Learning: Engagement As students work, offer feedback that focuses on their effort and strategy use. For example, call out good examples of the following: • Arranging prisms in a unique way • Drawing an accurate picture or writing a good explanation for how the prisms were arranged • Finding the volume of the box by using the volume of one prism or groups of prisms and multiplying If students are not applying strategies, ask questions to prompt their use: • How is finding volume by using something other than unit cubes similar to finding area by tiling rectangles with something other than unit squares?

389

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 18

Allow pairs to select their own strategies for completing the remaining parts of problem 1. Circulate and observe student strategies. Look for work samples that use a variety of methods for building a right rectangular prism that is the same size as box 1 and for finding the box’s volume. Purposely choose work that allows for rich discussion about different ways to determine the box’s volume. d. Use right rectangular prisms that are 3 units long, 1 unit wide, and 1 unit tall to build a right rectangular prism that is the same size as box 1. Explain or draw lines on the right rectangular prism to show how you arranged the prisms.

Teacher Note In this lesson, it is not important that students use terms such as stacks, towers, and layers. Instead, emphasize the reasoning around using groups of units to compose a right rectangular prism the same size as the box, such as 12 groups of 3 cubes or 4 groups of 9 cubes.

Sample:

Differentiation: Support If partners need support in determining how to arrange their prisms, ask questions to help them make connections between the right rectangular prisms and the box.

e. How many prisms did you use to build a right rectangular prism that is the same size as box 1?

12 prisms f. What is the volume of box 1? Explain.

• What dimensions do the right rectangular prism and the box have in common? • What have you tried that didn’t work? Why didn’t it work? What else could you try?

36 cubic units We used 12 prisms to build a right rectangular prism that is the same size as box 1. Each prism has a volume of 3 cubic units. The volume of the box is 36 cubic units because 12 × 3 = 36.

390

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 18

The student work samples demonstrate possible strategies for building a right rectangular prism the same size as box 1 and for finding the box’s volume. Standing Prisms Up and Counting Towers

Arranging in Groups and Counting Groups

There are 12 prisms standing up.

There are 2 groups of 6 prisms, which is 12 prisms.

Each prism is 3 cubic units.

Each group has 6 prisms, and each prism is 3 cubic units.

12 × 3 = 36 The volume of the box is 36 cubic units.

Ask the following questions to promote MTR.4:

Each group is 18 cubic units. There are 2 groups of 18 cubic units.

• Why does your strategy work? Convince your classmate.

18 + 18 = 36

• When does this strategy work? • What questions can you ask your classmate to make sure you understand how they found the volume of the box?

Making Stacks of Prisms and Counting Stacks

There are 3 layers of 4 prisms, which is 12 prisms.

There are 4 stacks of 3 prisms, which is 12 prisms.

Each layer has 4 prisms, and each prism is 3 cubic units.

Each stack has 3 prisms, and each prism is 3 cubic units.

4 × 3 = 12

3×3=9

Each layer is 12 cubic units. There are 3 layers of 12 cubic units.

Each stack is 9 cubic units. There are 4 stacks of 9 cubic units.

3 × 12 = 36

4 × 9 = 36

The volume of the box is 36 cubic units.

Students engage in discussions that reflect on mathematical thinking (MTR.4) when they determine the best way to use right rectangular prisms to build a prism that is the same size as a box, justify how they find the volume of the box, and listen to their classmates’ arguments.

6 × 3 = 18

The volume of the box is 36 cubic units. Arranging in Layers and Counting Layers

Promoting Mathematical Thinking and Reasoning

391

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 18

Invite the students you identified to share their solutions one at a time. As each student shares, ask questions to elicit their thinking. Ask the class questions to make connections between the different solutions and their own work. Encourage students to ask questions of their own. Standing Prisms Up and Counting Towers (Scott and Kayla’s Way) How did Scott and Kayla arrange the prisms to build a right rectangular prism that is the same size as the box? They stood the prisms up and arranged them in 4 rows of 3 towers. Scott, why did you stand the prisms up? The prisms are 3 units long, which is the same size as the height of the box.

There are 12 prisms standing up. Each prism is 3 cubic units. 12 × 3 = 36 The volume of the box is 36 cubic units.

Multilingual Learner Support Consider providing sentence frames to support the explanations of how students found the volume of the box in part (f): • There are

stacks.

• There are

prisms.

• Each stack has • Each group has • The volume of the box is

. . .

Kayla, how did you use multiplication to find the volume of the box? We used 12 prisms to build a right rectangular prism the same size as the box, and each prism is 3 cubic units. So we multiplied 12 by 3 to find that the volume of the box is 36 cubic units. Invite students to turn and talk about the similarities and differences between Scott and Kayla’s work and their own work. Arranging in Groups and Counting Groups (Tyler and Tara’s Way)

There are 2 groups of 6 prisms, which is 12 prisms.

Did Tyler and Tara use the same number of prisms as Scott and Kayla? Did they arrange the prisms the same way?

Each group has 6 prisms, and each prism is 3 cubic units.

They used 12 prisms, but they had 2 groups of 6. They stood 1 group up and laid the other group flat.

6 × 3 = 18 Each group is 18 cubic units. There are 2 groups of 18 cubic units. 18 + 18 = 36 The volume of the box is 36 cubic units.

392

Teacher Note The sample student work shows several ways to build a right rectangular prism that is the same size as box 1 and to find the box’s volume. Look for similar work from students and encourage authentic classroom conversations about key concepts. If students do not produce similar work, choose one or two pieces of their work to share, and highlight how it shows movement toward the goal of this lesson. Then select a sample in the lesson that works best to advance student thinking. Consider presenting the work by saying, “This is how another student built a prism the same size as the box. What do you think about what this student did to find the volume of the box?”

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 18

Tyler, why did you make groups of 6 prisms? We each made the same number of prisms, which turned out to be 6. I arranged the prisms standing up and Tara arranged the prisms laying down so that we would know who had arranged which prisms. Tara, did you use addition, multiplication, or both to find the volume of the box? We used both addition and multiplication. We each arranged 6 prisms, and each prism is 3 cubic units. Each group of 6 prisms is 18 cubic units because 6 × 3 = 18. Because we each built half of the prism, we added 18 and 18 to get 36. So the volume of the box is 36 cubic units. Invite students to turn and talk about the similarities and differences between Tyler and Tara’s work and their own work. Arranging in Layers and Counting Layers (Leo and Sasha’s Way) How did Leo and Sasha arrange the prisms? They laid the prisms along the bottom to make a layer of 4 prisms. Then they stacked 3 layers of 4 prisms to build a prism the same size as the box. Leo, how did you know how many prisms to put in each layer? The box is 4 units long and 3 units wide. Because the prisms are 3 units long, we needed to have 4 prisms in each layer so the layer would be as wide as the bottom of the box. Sasha, how did you find the volume of the box?

There are 3 layers of 4 prisms, which is 12 prisms. Each layer has 4 prisms, and each prism is 3 cubic units.

Differentiation: Challenge Consider having students experiment or mentally determine what other right rectangular prisms besides those that are 3 units long, 1 unit wide, and 1 unit tall can be used to build a right rectangular prism the same size as box 1.

4 × 3 = 12 Each layer is 12 cubic units. There are 3 layers of 12 cubic units. 3 × 12 = 36 The volume of the box is 36 cubic units.

Each layer has 4 prisms, and each prism is made of 3 cubic units, so each layer is 12 cubic units. It took 3 layers to be as tall as the box, and 3 × 12 = 36. So the volume of the box is 36 cubic units. Invite students to turn and talk about the similarities and differences between Leo and Sasha’s work and their own work.

393

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 18

Making Stacks of Prisms and Counting Stacks (Adesh and Riley’s Way) How did Adesh and Riley arrange the prisms? They made stacks of 3 prisms and arranged 4 of the stacks of 3 prisms side by side. Adesh, how did you know how many prisms to put in each stack? The box is 3 units tall, so we needed stacks 3 prisms high to have a height of 3 units. Riley, how did you find the volume of the box? Each prism is 3 cubic units, so each stack of 3 prisms is 9 cubic units. It took 4 stacks, and 4 × 9 = 36. So the volume of the box is 36 cubic units.

There are 4 stacks of 3 prisms, which is 12 prisms. Each stack has 3 prisms, and each prism is 3 cubic units. 3×3=9 Each stack is 9 cubic units. There are 4 stacks of 9 cubic units. 4 × 9 = 36 The volume of the box is 36 cubic units.

Invite students to turn and talk about the similarities and differences between Adesh and Riley’s work and their own work. Ask the class the following questions. How is the way Leo and Sasha arranged the prisms and found the box’s volume similar to Adesh and Riley’s way? How is it different? Both pairs laid the prisms down. Both grouped several prisms rather than counting each prism separately. Leo and Sasha counted 3 layers of 4 prisms, or 3 groups of 12 cubic units. Adesh and Riley counted 4 stacks of 3 prisms, or 4 groups of 9 cubic units. What is the length, width, and height of the box? The length is 4 units, the width is 3 units, and the height is 3 units. Invite students to Think–Pair–Share about the following question. Does the volume of the box change depending on how the prisms are arranged? Why? No. The volume is 36 cubic units no matter how the prisms are arranged because it takes 12 prisms, each with a volume of 3 cubic units, to build a right rectangular prism that is the same size as the box. 394

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 18

Build a Larger Prism Materials—S: Box 2, scissors, tape, cubes

Students build 3 unit × 2 unit × 2 unit right rectangular prisms and use them to compose a larger right rectangular prism and find its volume. Direct pairs to remove Box 2 from their books. Tell them to cut out box 2, fold the sides up, and tape the box together. Direct students to use interlocking cubes to make right rectangular prisms that are 3 units long, 2 units wide, and 2 units tall. Invite them to make as many prisms as necessary to build a right rectangular prism that is the same size as box 2. Each small prism can be a different color, but have students use cubes of the same color to make each small prism if possible. Direct pairs to complete problem 2. Encourage students to self-select their strategies. Circulate and observe student strategies. Look for work samples that use a variety of methods for building a right rectangular prism that is the same size as box 2 and for finding the box’s volume. Purposely choose work that allows for rich discussion about different ways to determine the box’s volume. For problem 2, use box 2 to complete parts (c)–(f). 2. A right rectangular prism is 3 units long, 2 units wide, and 2 units tall. a. Draw the prism. Sample:

b. What is the volume of the prism?

12 cubic units © Great Minds PBC

395

5 ▸ M5 ▸ TC ▸ Lesson 18

EUREKA MATH2 Florida

c. Erase the interior lines that show individual cubes, leaving only the edges of the prism in part (a). Estimate how many prisms it takes to build a right rectangular prism that is the same size as box 2.

5 prisms d. Use right rectangular prisms that are 3 units long, 2 units wide, and 2 units tall to build a right rectangular prism that is the same size as box 2. Explain or draw lines on the right rectangular prism to show how you arranged the prisms. Sample:

e. How many prisms did you use to build a right rectangular prism that is the same size as box 2?

6 prisms f. What is the volume of box 2? Explain.

72 cubic units We used 6 prisms to build a right rectangular prism that is the same size as box 2. Each prism has a volume of 12 cubic units. The volume of the box is 72 cubic units because 6 × 12 = 72.

396

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 18

When students are finished, invite several pairs to show their work. Consider using the following prompts to help students make connections between other students’ work for problem 2 and their own work: • What is the volume of each small prism that you used to build a right rectangular prism the same size as box 2? • How did you arrange the small prisms? • How many small prisms did it take to build a right rectangular prism the same size as box 2? • What is the volume of box 2? How did you find the volume? Ask the following after students share. Each small prism for problem 1 was 3 units long, 1 unit wide, and 1 unit tall. Each small prism for problem 2 was twice as wide and twice as tall. How does the volume of each small prism for problem 2 compare to the volume of each small prism for problem 1? The volume of each prism for problem 2 is 12 cubic units. The volume of each prism for problem 1 is 3 cubic units. The volume of each prism for problem 2 is 4 times as much as the volume of each prism for problem 1. Invite students to turn and talk about the similarities and differences between other students’ work for problem 2 and their own work.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

397

5 ▸ M5 ▸ TC ▸ Lesson 18

Land

EUREKA MATH2 Florida

Debrief 5 min Objective: Find the volume of right rectangular prisms by packing with improvised units. Facilitate a class discussion about finding the volume of right rectangular prisms by using the following prompts. Encourage students to restate or add on to their classmates’ responses. If we build right rectangular prisms in different ways, but the prisms are the same size, do they have the same volume? Explain your thinking. Yes. We can build a prism in different ways, but it still takes the same number of cubes to build a prism that is the same size. Yes. The volume stays the same no matter how we build the prism. How do you find the volume of a right rectangular prism when it is made with something other than cubes, such as smaller right rectangular prisms? We count the number of prisms used to make the larger right rectangular prism and multiply that number by the volume of one of the smaller prisms. If you know the volume of 1 stick of gum, that there are 5 sticks of gum in a pack, and that there are 30 packs of gum in a box, how can you find the volume of the gum in the box? You can multiply the volume of 1 stick of gum by 5 to find the volume of the gum in a pack. Then you can multiply the volume of the gum in a pack by 30 to find the volume of gum in the box.

398

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 18

399

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 18

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can use smaller prisms to find the volume of a right rectangular prism.

PRACTICE

Florida B.E.S.T. MA.5.GR.3.1

e. What is the volume of the larger right rectangular prism?

140 cubic units Name

Date

1. Use the right rectangular prism to complete parts (a)–(f).

The prisms like the one from part (a) are arranged in a different way to build another right rectangular prism. There are two groups. One group has 5 stacks of 4 prisms and the other group has 2 stacks of 4 prisms.

a. Sketch lines on the prism to show that it is 5 units long, 1 unit wide, and 1 unit tall. b. What is the volume of the prism?

5 cubic units

c. Several prisms like the one from part (a) are used to build a larger right rectangular prism. There are 7 stacks of 4 prisms as shown. Fill in the blanks for the unknown measurements. Toby says that because this right rectangular prism is built differently, the volume is different from the volume found in part (e). Is Toby correct? Justify your answer. Toby is not correct. The volume is the same. The length of the larger right rectangular prism is still 7 units, the width is still 5 units, and the height is still 4 units. The number and volume of small prisms used to build the larger right rectangular prism are the same.

4 units

7 units

5 units

d. How many prisms are used to build the larger right rectangular prism?

28 prisms

400

189

190

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 18

EXIT TICKET

I can use smaller prisms to find the volume of a right rectangular prism.

Florida B.E.S.T. MA.5.GR.3.1

2. A right rectangular prism is 5 units long, 3 units wide, and 2 units tall. a. Sketch the prism. Name

Date

A right rectangular prism is 4 units long, 1 unit wide, and 1 unit tall.

b. Fill in the blanks. The prism has the same length as the prism from problem 1(a), but the width is as wide and the height is

times

a. What is the volume of the prism?

4 cubic units

times as high. b. Prisms like the one from part (a) are packed in a box that is 4 units long, 2 units wide, and 3 units tall. How many prisms are packed in the box?

c. What is the volume of the prism?

30 cubic units

3. 12 prisms that are 5 units by 3 units by 2 units are used to build a large right rectangular prism. What is the volume of the large right rectangular prism?

360 cubic units

6 prisms c. What is the volume of the box?

6 × 4 = 24 24 cubic units

PROBLEM SET

191

193

401

LESSON 19

Compose and decompose right rectangular prisms to find their volume by using layers. I can use layers to find the volume of right rectangular prisms.

Lesson at a Glance Students determine the volume of a right rectangular prism made of a single layer of cubes. Then they add layers and find the volume of the prism they create by multiplying the number of layers by the volume of each layer. Students find the volumes of prisms by decomposing them into layers, calculating the volume of each layer, and multiplying the number of layers by the volume of each layer. Students explain why prisms can be decomposed into layers in multiple ways.

Key Questions • Why do we decompose right rectangular prisms into layers?

Language Objectives • Orally and in writing, describe the layers of a right rectangular prism and make connections between the number of layers, the number of cubes in each layer, and the volume of the prism. • Write to explain how decomposing a prism into layers helps find the volume.

Mathematical Thinking and Reasoning Standard MTR.5.1

• Why is there more than one way to decompose a right rectangular prism into layers?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 19

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 5 min Learn 35 min • Add Layers to Compose Right Rectangular Prisms • Decompose Right Rectangular Prisms into Layers • Problem Set

• Interlocking cubes, 1 cm (30)

Students • Interlocking cubes, 1 cm (30)

Land 10 min

Achievement Descriptors

Florida B.E.S.T. Standard

FL.5.Mod5.AD10 Recognize that volume

MA.5.GR.3.1 Explore volume as an attribute

can be measured by using unit cubes and that a solid packed without gaps or overlaps by n unit cubes has a volume of n cubic units. (MA.5.GR.3.1)

of three-dimensional figures by packing them with unit cubes without gaps. Find the volume of a right rectangular prism with whole-number side lengths by counting unit cubes.

FL.5.Mod5.AD11 Measure volumes by counting

unit cubes. (MA.5.GR.3.1)

403

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 19

Fluency

Whiteboard Exchange: Divide Whole Numbers Students determine the quotient and express it as a mixed number to build procedural fluency with dividing three- and four-digit dividends by two-digit divisors from module 1. Display 318 ÷ 50. Find the quotient and express it as a mixed number if needed. Show your method.

18 318 ÷ 50 = 6 50

6 50 318 –300 18

882 ÷ 21 =

404

2,493 ÷ 18 = 13818 9

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 19

Whiteboard Exchange: Write and Evaluate Expressions Students write and evaluate an expression to build fluency with two-step calculations involving fractions from module 3. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the statement: The difference between 1 half and 1 fourth, multiplied by 3. Write an expression to represent the statement. Display the sample expression.

The difference between 1 half and 1 fourth, multiplied by 3 1 2

1 4

( − )× 3

Write the value of the expression.

3 4

Display the answer. Repeat the process with the following sequence: 2 times as much as the sum of 3 tenths and 1 fifth 2×(

3 1 + ) 10 5

The difference between 2 thirds and 5 ninths, divided by 3 2 3

5 9

( − )÷ 3 1 27

405

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 19

Launch

Students examine layers of figures. Introduce the Which One Doesn’t Belong? routine. Display the picture of the four three-dimensional figures. Figure A

Figure B

Figure C

Figure D

Invite students to study the figures. Give students 1 minute to find a category in which three of the items belong, but a fourth item does not. When time is up, invite students to explain their chosen categories and to defend why one item does not fit. Highlight responses that emphasize reasoning about length, width, height, and volume. Ask questions that invite students to use precise language, make connections, and ask questions of their own. Which one doesn’t belong? Figure A doesn’t belong because it’s not a right rectangular prism. Figure B doesn’t belong because its height is not 5 units. Figure C doesn’t belong because its length is not 3 units. Figure D doesn’t belong because its bottom layer, or base, doesn’t have 6 cubes.

406

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 19

How many cubes are in the base in figure B? How do you know? There are 6 cubes in the base. I know because the base is 2 units wide and 3 units long, and 2 × 3 = 6. We can think of the base of the prism as 1 layer in the prism. Every layer in a prism has 1 edge that has a length of 1 unit. The height of the base in figure B is 1 unit, so we can call it a layer of the prism. We can say that there are 6 cubes in the bottom layer in figure B. How many cubes are in figure B? How do you know? There are 36 cubes because there are 6 horizontal layers and each layer has 6 cubes. Transition to the next segment by framing the work. Today, we will find volume of right rectangular prisms by using layers.

Learn

Add Layers to Compose Right Rectangular Prisms

Multilingual Learner Support The term layer has multiple meanings in mathematics and everyday life (e.g., layers of a cake, layers of clothing). In this lesson, layer refers to a right rectangular prism with one dimension that is one unit. Consider activating prior knowledge by asking when students may have used the word layer.

Materials—T/S: Cubes

Students compose right rectangular prisms from layers. Display the picture of a right rectangular prism made from 6 cubes. Tell students that each cube represents 1 cubic centimeter. Build this figure with your cubes.

Universal Design for Learning: Action & Expression Consider partnering students to build the figures together if the fine motor demands of the task present a barrier.

What is the volume of this figure? How do you know? The volume is 6 cubic centimeters. I know because the figure is made of 6 cubes.

407

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 19

This figure shows 1 layer because it has a height of 1 centimeter. Let’s see what happens to the volume when we add more layers to the top. Add another layer on top of the first layer. Display the figure with the layer added on top. Describe the layers in this prism. There are 2 layers. In each layer, there are 2 rows of 3 cubes, so each layer has 6 cubes. What is the volume of this figure now? How do you know? The volume is 12 cubic centimeters. I know because the figure is made of 2 layers of 6 cubes each, and 2 × 6 = 12. Create the following table. Complete the first two rows. Add 2 more layers on top of the first 2 layers. Describe the layers in this prism. There are 4 layers. In each layer, there are 2 rows of 3 cubes, so each layer has 6 cubes.

Number of Layers 1 2 4

Number of Cubes Volume in Each Layer (cubic centimeters) 6 6 6 12 6 24

Teacher Note Interlocking cubes are used in this activity, but they are not a perfect representation of a cubic centimeter. Centimeter cubes can also be used, but because they do not stick together, students may find them harder to stack. Choose the material that works best for your students. If the cubes used are not 1 cm × 1 cm × 1 cm, rephrase the sample dialogue to units and cubic units. Ask students to recall that in either case, the prisms formed by the cubes serve as models and are not solids themselves.

What is the volume of the figure now? How do you know? The volume is 24 cubic centimeters. I know because there are 4 layers and each layer has a volume of 6 cubic centimeters. The volume is 24 cubic centimeters. I know because I added 12 cubic centimeters to the 12 cubic centimeters I already had.

408

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 19

Complete the table. Then display the prism with 4 layers. Describe this prism. The prism is 2 centimeters wide, 3 centimeters long, and 4 centimeters tall. The prism has a length of 3 centimeters, a width of 2 centimeters, and a height of 4 centimeters. The prism has 4 layers, and each layer is 2 centimeters wide and 3 centimeters long. The base of the prism is 2 centimeters wide and 3 centimeters long. There are 4 layers. Direct students to the completed table. Ask students to turn and talk about what the volume of the figure would be if it had 30 layers, each with a volume of 6 cubic centimeters. We created this 2 cm × 3 cm × 4 cm prism by making four 2 cm × 3 cm layers from the bottom up. We can create the same prism by starting with a layer that is 2 centimeters wide and 4 centimeters tall and adding layers to the side. Demonstrate, starting with a layer that is 2 centimeters wide and 4 centimeters tall and adding 2 additional layers to the side. What is the volume of this prism? How do you know? The volume is 24 cubic centimeters. I know because there are 3 layers and each layer has a volume of 8 cubic centimeters. Direct students to problem 1 in their books. Ask students to complete the problem with a partner. 1. Use 24 cubes to create a right rectangular prism. Create a prism that is different from the one created in class. a. Describe the layers in the right rectangular prism you created. Sample: My prism has 6 layers that are 2 centimeters wide and 2 centimeters long. b. What is the volume of the prism? How do you know? The volume of the prism is 24 cubic centimeters. I know because it has 6 layers that each have a volume of 4 cubic centimeters. © Great Minds PBC

409

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 19

When students are finished, bring the class together. Invite students to share their descriptions and show the class their prisms. As they share, create a new table to represent their prisms. Do not repeat rows in the table for prisms that Volume Number of Cubes have the same number Number of Layers in Each Layer (cubic centimeters) of layers and same number of cubes in each 4 6 24 layer but look different 6 4 24 because the length and width are interchanged 2 12 24 (e.g., 2 layers with 3 8 24 a length of 4 centimeters 1 24 24 and a width of 3 centimeters and 2 layers with a length of 3 centimeters and a width of 4 centimeters). Discuss why the prism is already represented in the table. Invite students to turn and talk about the patterns they notice in the table. Notice that the number of cubes in each layer is not the same. How do you determine the number of cubes in each layer of a prism? I count the number of cubes in each row and multiply that number by the number of rows. How can you find the volume of a right rectangular prism if you know the number of layers and the volume of each layer? I can multiply the number of layers by the volume of each layer.

Teacher Note The digital interactive Building Prisms helps show that layers are composed of rows and that rows are composed of individual cubes. Consider allowing students to experiment with the tool individually or demonstrating the activity for the whole class.

Gesture to several different rows of the table when making the following statement. When I look at this row of the table, I can picture a prism that has 2 layers with 12 cubes each. When I look at this row, I can picture a prism that has 1 layer with 24 cubes.

410

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 19

Invite students to turn and talk about prisms they can picture from the table. Encourage them to use their hands to help describe the prism. Several prisms that look different from one another have a volume of 24 cubic centimeters. How is that possible? The prisms have different numbers of layers of different sizes, but the number of layers times the volume of each layer is still 24 cubic centimeters. The prisms are all made from 24 cubes with a volume of 1 cubic centimeter. A cube is a special type of right rectangular prism. What do we know about a cube? A cube has the same length, width, and height. Invite students to use their cubes to create a larger cube. When they are finished, invite students to share their larger cubes, describing the cubes’ volumes and what they look like. My cube is 1 centimeter wide, 1 centimeter long, and 1 centimeter tall. It has a volume of 1 cubic centimeter. My cube is 2 centimeters wide, 2 centimeters long, and 2 centimeters tall. It has a volume of 8 cubic centimeters. My cube has 3 layers, and each layer is 3 centimeters wide and 3 centimeters long. It has a volume of 27 cubic centimeters. Consider asking students whether they have enough cubes to make a 4 cm × 4 cm × 4 cm cube. Kelly says that her cube has 5 layers and that each layer is 4 centimeters wide and 4 centimeters long. Is Kelly’s prism a cube? How do you know? No. I know because the number of layers of a cube must be equal to the number of centimeters of width and number of centimeters of length.

Promoting Mathematical Thinking and Reasoning When students repeatedly create prisms with 24 cubes to notice the pattern that the volume of a right rectangular prism can be found by multiplying the volume of each layer by the number of layers, they are using patterns and structure (MTR.5). Ask the following questions to promote MTR.5: • What is the same about each prism built with 24 cubes? • What patterns do you notice when you build a right rectangular prism with layers? How can that help you determine the volume more efficiently?

Multilingual Learner Support Consider supporting student responses with the Talking Tool. Invite students to use the Share Your Thinking section to explain how to create a prism with a volume of 20 cubic centimeters.

Invite students to turn and talk about how to create a prism with a volume of 20 cubic centimeters.

411

5 ▸ M5 ▸ TC ▸ Lesson 19

EUREKA MATH2 Florida

Decompose Right Rectangular Prisms into Layers Students decompose right rectangular prisms into layers. Build or display the 3 cm × 4 cm × 5 cm right rectangular prism. Invite students to turn and talk about what the base of the prism looks like. Then ask students to work with a partner to build the prism with their cubes. Earlier in the lesson, we created right rectangular prisms by composing them from layers. We determined that we could multiply the volume of each layer by the number of layers to find the volume of the prism. We can also find the volume of a right rectangular prism by decomposing it into layers and finding the volume of each layer. Display the hand-drawn prism from problem 2. This picture shows a drawing of what you just constructed. Let’s slice the prism horizontally to decompose this prism into layers. What does it mean to decompose something? To break something apart To break something into pieces To separate something Invite students to turn and talk about how many horizontal layers the prism has. We can draw lines on the drawing of the prism to show how to slice it to create horizontal layers. Display the drawing of the prism with the horizontal layers. Point to each layer and count to show the 5 layers. How many cubes are in each layer? How do you know? There are 12 cubes in each layer. I know because each layer is 3 centimeters wide and 4 centimeters long. 412

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 19

What is the volume of each layer? How do you know? The volume of each layer is 12 cubic centimeters. I know because each layer is made of 12 centimeter cubes. Direct students to problem 2. For the first prism in part (a), ask students to show horizontal layers by drawing the decomposition shown. Teacher Note

2. The right rectangular prism shown is composed of centimeter cubes. a. Draw lines on the prisms to show how to decompose the prism into layers in three different ways.

The digital interactive Volume Model helps show how to decompose a prism into layers. Consider allowing students to experiment with the tool individually or demonstrating the activity for the whole class. For example, consider building the prism from this segment of the lesson and decreasing the height one unit at a time to show the layers.

b. Use your work from part (a) to complete the table. Number of Layers

Number of Cubes in Each Layer

Volume (cubic centimeters)

In addition, consider showing the same prism so that students can see layers but not every cube.

413

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 19

Then ask students to work with a partner to complete problem 2. Encourage students to use the 3 cm × 4 cm × 5 cm prisms they created from cubes to complete the problem. Circulate as students work, asking the following questions: • How can you represent slicing to make layers on the drawing of the prism? • How can you use your model of the prism with cubes to determine how to decompose the prism into layers? • What is the volume of that layer? How do you know? • How can you use the layers to find the volume of the prism? Differentiation: Challenge

When students are finished, bring the class together and ask the following questions.

For students who need additional challenge, consider asking the following questions:

Describe the three different ways you decomposed the prism. I decomposed into horizontal layers and two different types of vertical layers. I decomposed from bottom to top, from left to right, and from front to back. Do you think every right rectangular prism can be decomposed into layers in three different ways? Why? Yes. Every right rectangular prism has a width, length, and height. Yes. Every right rectangular prism can be decomposed from bottom to top, left to right, and front to back.

• How does the volume of the prism in problem 2 change if it is built from 2-centimeter cubes instead of 1-centimeter cubes? • How does the volume of a prism change if the dimensions of the cubes used to compose it are tripled?

Does it matter how you slice a right rectangular prism to find its volume? Why? No. The total number of cubes is the same regardless of how they are partitioned. No. The sum of the number of cubes in the layers should be the same regardless of how the layers are created.

3 cm

Display the prism from problem 3. How is the drawing of this prism different from other prisms you’ve seen in this lesson? The drawing doesn’t show the individual cubes in the prism. 414

4 cm

6 cm

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 19

Invite students to turn and talk about how to think of the prism as being composed of centimeter cubes even if students cannot see the cubes. Describe this right rectangular prism. It is longer than it is tall. It is twice as long as it is tall. It is 4 centimeters wide, 6 centimeters long, and 3 centimeters tall. It is 4 cm × 6 cm × 3 cm. Invite students to turn and talk about ways they can slice the prism to make layers. Ask students to complete problem 3 with a partner. Circulate as students work, asking the following questions: • What three ways can you slice the prism to make layers? • How do you want to slice the prism? Why? • What is the volume of one layer? How do you know? • How can you use the volume of one of the layers you created to find the volume of the prism?

415

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 19

3. The right rectangular prism shown is 4 centimeters wide, 6 centimeters long, and 3 centimeters tall. Sample:

3 cm

6 cm

4 cm

a. Draw lines to show how to decompose the prism into layers. b. Use the layers you created in part (a) to complete the following sentences. Sample:

For students who need support with problem 3, consider any of the following modifications: • Ask students to create the prism with cubes before starting problem 3. • Provide a version of the prism that shows it composed of individual cubes.

The prism has

layers.

Each layer has

cubic centimeters.

The volume of this prism is

Differentiation: Support

cubic centimeters.

When students are finished, invite them to share their strategies for decomposing the prism into layers.

• Provide a hand-drawn version of the prism with layers shown.

Universal Design for Learning: Representation Consider drawing to show the individual cubes in one layer for each decomposition if students need support visualizing the volume of each layer.

416

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 19

Display three different ways to decompose the prism.

3 cm

4 cm

6 cm

Differentiation: Challenge

3 cm

4 cm

6 cm

4 cm

Consider presenting the following problem for students who need an additional challenge: The layers for problem 3 are 1 unit thick. Do the layers need to be 1 unit thick? Why? How else could layers be made from this prism?

6 cm

6 layers

4 layers

3 layers

The volume of each layer is 12 cubic centimeters.

The volume of each layer is 18 cubic centimeters.

The volume of each layer is 24 cubic centimeters.

Once you decide which way you want to create layers, such as left to right or bottom to top, how do you determine how many layers there are? There is one layer for each unit of length. For example, if a side is 4 centimeters long, I can slice it into 4 layers. How do you find the volume of each layer? I determine the number of cubes that would be in each layer by multiplying the length and width of the layer. How do you find the volume of the prism once you know the volume of each layer? I multiply the volume of the layer by the number of layers.

417

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 19

Display the prism with dimensions shown in feet. The volumes we calculated so far in this lesson were measured in cubic centimeters. How could we use the strategies from today’s lesson to find the volume of this right rectangular prism? We could decompose the prism into layers, find the volume of each layer, and multiply the volume of each layer by the number of layers.

3 ft

7 ft 2 ft

We could think of what this prism would look like if it was made of cubes. Then we could think of the volume of each cube as a cubic foot instead of as a cubic centimeter. Invite students to turn and talk to summarize the process they use to calculate volume when given a right rectangular prism.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

418

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 19

Land

Debrief 5 min Objective: Compose and decompose right rectangular prisms to find their volume by using layers. Gather the class with their Problem Sets. Facilitate a class discussion about decomposing right rectangular prisms by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Why do we decompose a right rectangular prism into layers? Give an example. By decomposing a prism into layers, we can find the volume of 1 layer. Then we can multiply the volume of 1 layer by the number of layers to find the volume of the prism. For example, in problem 4, I decomposed the prism into 3 layers and determined that each layer had 5 rows of 2 cubes, or 10 cubes. Because 3 × 10 = 30, the volume of the prism is 30 cubic centimeters. Why is there more than one way to decompose a right rectangular prism into layers? No matter how the prism is turned, the layers are always along the width, the length, or the height. There is more than one way because the layers can go front to back, left to right, or bottom to top.

419

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 19

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can use layers to find the volume of right rectangular prisms.

PRACTICE

Florida B.E.S.T. MA.5.GR.3.1

Name

2. Use the picture of the horizontal layer and the right rectangular prism to complete the table. Each cube represents 1 cubic unit.

Date

1. Draw a line to match each right rectangular prism with the picture of a horizontal layer of that prism. Right Rectangular Prism

420

Horizontal Layer

197

198

Number of Layers

Number of Cubes in Each Layer

Volume (cubic units)

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 19

4. The right rectangular prism shown is 2 centimeters wide, 5 centimeters long, and 3 centimeters tall. Use the prism to complete parts (a) and (b).

3. Three pictures of the same right rectangular prism are shown. Each cube represents 1 cubic centimeter.

Sample:

a. Draw lines to decompose the prism into layers in three different ways.

3 cm 2 cm 5 cm a. Draw lines to decompose the prism into layers. b. Use the different ways you decomposed the right rectangular prism in part (a) to complete the table.

Number of Layers

Number of Cubes in Each Layer

Volume (cubic centimeters)

140

PROBLEM SET

b. Use the layers you created in part (a) to complete the following sentences. Sample: The prism has

layers.

Each layer has

cubic centimeters.

The volume of this prism is

199

200

PROBLEM SET

cubic centimeters.

421

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 19

EXIT TICKET

I can use layers to find the volume of right rectangular prisms.

Florida B.E.S.T. MA.5.GR.3.1

Name

Date

The right rectangular prism shown is composed of centimeter cubes. Sample:

a. Draw lines to decompose the prism into layers. b. Use the layers you created in part (a) to complete the following sentences. Sample: The prism has There are

layers.

centimeter cubes in each layer.

The volume of the prism is

cubic centimeters.

c. How does decomposing a prism into layers help you find the volume? I can find the volume of 1 layer. Then I multiply the volume of 1 layer by the number of layers to find the volume of the prism.

422

201

LESSON 20

Interpret volume as filling. I can measure and explain volume by filling a container.

Lesson at a Glance Students explore conceptual ideas about volume. After realizing that packing with cubes to find volume is not always possible, students learn that they can find volume by filling with a liquid. Students learn that every three-dimensional object has volume and that volume is the same regardless of whether a space is packed or filled. There is no Problem Set for this lesson. The Exit Ticket for this lesson serves as an opportunity for students to reflect on the lesson.

Key Questions • When we measure the volume of a liquid or a solid, what are we measuring?

Mathematical Thinking and Reasoning Standard

Language Objectives • Orally justify why statements about volume and capacity are true or false, and listen to the reasoning of others to support or challenge their thinking. • Write to explain concepts related to volume and capacity by using knowledge about packing and filling a three-dimensional space.

MTR.2.1

• How are packing and filling similar? How are they different?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 20

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Tear out Equivalent Expression Cards, Set 1 from the student books. Cut the cards apart and place each set in an envelope. Prepare enough sets for 1 set per student pair.

Launch 10 min Learn 30 min • Pack Prisms and Cylinders • From Packing to Filling

Land 10 min

• Envelopes (12) • Equivalent Expression Cards, Set 1 (in the student book)

Students • Envelope of Equivalent Expression Cards, Set 1 (1 per student pair)

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD10 Recognize that volume can

MA.5.GR.3.1 Explore volume as an attribute

be measured by using unit cubes and that a solid packed without gaps or overlaps by n unit cubes has a volume of n cubic units. (MA.5.GR.3.1)

of three-dimensional figures by packing them with unit cubes without gaps. Find the volume of a right rectangular prism with whole-number side lengths by counting unit cubes.

425

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 20

Fluency

(63.12 − 56.34) × 8 = 54.24

Display the answer. Repeat the process with the following sequence:

9.6 + 18 × 4.3 =

4.5 × 13.4 − 4 =

56.3

12 ÷ 2 × 4.5 − 24 =

8.75 ÷ 2.5 × 5 =

17.5

426

621.6 ÷ (8.64 + 6.16) =

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 20

Match: Equivalent Expressions Materials—S: Equivalent Expression Cards, Set 1

Students identify equivalent expressions and create equations to develop procedural reliability with adding and subtracting fractions with unlike units from module 2. Have students form pairs. Distribute a set of cards to each student pair and have them complete the activity by using the following procedure: • Lay out all cards faceup. • Match cards that show two equivalent expressions. Note that two cards do not have matches. • Lay each set of matched cards side by side. Form an equation by placing an equal sign card between equal expressions.

1 1 − 3 6

2 1 − 6 6

4 1 − 6 3

−

• Continue until all but two cards are matched. • Find common units for the two expressions without matches and use them to write two expressions that are equal on the blank cards and form two additional equations.

Circulate as students work and provide support as needed. Encourage students to explain their thinking to their partners.

427

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 20

Launch

Students compare volumes of various objects. Teacher Note

Invite students to Think–Pair–Share about the following question. Does every three-dimensional object, including liquid, have volume? How do you know? Yes. Every three-dimensional object takes up space, and volume measures how much space an object takes up. I think so, but I don’t know how to figure out the volume of most objects. All three-dimensional objects, including liquids, have volume because every three-dimensional object takes up space. Display the visual with the objects, quantities, and the question about volume.

This lesson is designed to be an exploratory lesson in which students explore concepts about volume. Encourage students to visualize conceptual aspects of the context, to talk about what they notice and wonder, and to ask questions. Because there is no Problem Set, allow students the time needed to grapple with tough concepts. The video problem presented in this lesson is intentionally unresolved until the next lesson.

Which quantity has the greatest volume?

1 cubic foot of popcorn

428

1 cubic foot of juice boxes

1 cubic foot of marbles

1 cubic foot of water

Multilingual Learner Support Support the contexts in the visual by discussing each object and, if possible, showing an example of it. For example, show a container of popcorn, a juice box, and a jar of marbles.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 20

Read the question aloud and invite students to Think–Pair–Share about the answer. Do not indicate whether any answers are correct or incorrect. Allow students to give any answer they can justify. Encourage students to respectfully debate with one another. They all have the same volume because they’re all 1 cubic foot.

Multilingual Learner Support

The juice boxes have the greatest volume because I can pack them in a box.

As students discuss which quantity has the greatest volume, encourage them to use the Agree or Disagree section of the Talking Tool to support their discussions.

The marbles have the greatest volume because they have space between them that can’t be filled, so they will take up more space. The water has the greatest volume because I can fill 1 cubic foot without any gaps or holes. The popcorn has the greatest volume because I can keep squishing the popcorn into 1 cubic foot until there aren’t any gaps or holes left. The popcorn doesn’t have the greatest volume because it’s lighter than the other materials. Consider asking any of the following questions to advance student thinking: • About how big is 1 cubic foot? Show me with your hands or show me an object in the room with a volume of about 1 cubic foot. • What does it mean to pack a container in the shape of a solid? • What is the difference between trying to pack a box with popcorn and trying to pack a box with marbles? • Do you know for sure that juice boxes can pack 1 cubic foot? Why? • Can you pack a space with a liquid, such as water?

Teacher Note The question about which quantity has the greatest volume is purposely open ended, allowing students to consider aspects of a unit measure that has volume on its own and that can pack or fill a container to give the volume of the container. Accept all student responses and encourage students to be flexible in their thinking. Before starting the lesson, consider reading the discussion near the end of the lesson when Launch is revisited.

Transition to the next segment by framing the work. Today, we will learn the difference between packing and filling, and we will revisit this question about which quantity has the greatest volume.

429

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 20

Learn

Pack Prisms and Cylinders Students determine why packing with cubes is not always useful for finding volume. Play part 1 of the Cubes in a Cylinder video. If necessary, replay the video and ask students to note any details. Give students 1 minute to turn and talk about what they noticed. Engage students in a conversation about the video. Discuss student observations and any relevant questions they have. Consider the following possible sequence. What do you notice? I notice that cubes fit nicely into a prism but not in a cylinder.

Teacher Note In lesson 17, containers shaped like right rectangular prisms served as models for solids. In that lesson, students packed the containers with unit cubes to determine the volume of a right rectangular prism. Similarly, containers shaped like right rectangular prisms and cylinders serve as models in this lesson.

The cubes fill all the space in the prism but not in the cylinder. I see someone trying to stick parts of cubes into the cylinder to see whether they can fill more of the empty space in the cylinder. What do you wonder? I wonder how you find the volume of a cylinder if you can’t pack it with unit cubes. I wonder why you would try to pack something with pieces of centimeter cubes. I wonder whether we can use smaller cubes so there won’t be any spaces between them.

Universal Design for Learning: Representation If the materials from the video are readily available, consider acting out the scenario in the video instead of or in addition to viewing it. Invite students to participate.

I wonder whether we can somehow squish the cubes to pack it without any gaps.

430

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 20

What is the volume of the prism? How do you know? The volume is 20 cubic centimeters. I know because that is the number of centimeter cubes that packed into the prism. What is the volume of the cylinder? How do you know? I don’t know. It’s not possible to pack the cylinder with centimeter cubes. I think the volume is more than 20 cubic centimeters. The cubes seem to fit in the cylinder with space left over. Give students several minutes to turn and talk with a partner about how they would try to find the volume of a cylinder. Invite pairs to share their ideas with the class. Play part 2 of the Cubes in a Cylinder video. Give students 1 minute to turn and talk about what they noticed. Engage students in a brief conversation about the video. Discuss student observations and any relevant questions they have. Consider the following possible sequence. What do you notice? I notice that the person tried filling the cylinder with rice instead of cubes. I notice that rice doesn’t create as many gaps as the cubes did, so it appears to work better for filling a cylinder. What do you wonder? I wonder how you find the volume of the rice. I wonder whether the rice actually packs the cylinder. I wonder what else we could use to fill the cylinder with no gaps. For the rice to have the same volume as the cylinder, the rice must take up all the space in the cylinder. Do you think the rice takes up all the space in the cylinder? Why?

Promoting Mathematical Thinking and Reasoning Students represent problems in multiple ways (MTR.2) when they look at models of a right rectangular prism and cylinder being packed with centimeter cubes and discuss how to determine the volume of objects that are not shaped like a right rectangular prism. Ask the following questions to promote MTR.2: • How does thinking about packing with centimeter cubes help you think about volume? • What does trying to pack a cylinder with centimeter cubes tell you about volume? • What does volume mean in this situation?

No. I think there are tiny amounts of empty space between the grains of rice.

431

5 ▸ M5 ▸ TC ▸ Lesson 20

EUREKA MATH2 Florida

Play part 3 of the Cubes in a Cylinder video. Give students 1 minute to turn and talk about what they noticed. Engage students in a brief conversation about the video. Discuss student observations and any relevant questions they have. Consider the following possible sequence. What do you notice? I notice that the person dumped out the rice and filled the cylinder with water instead. I notice that when the water fills the cylinder, there are no gaps or holes. What do you wonder? I wonder whether we can pour the water into a graduated cylinder to find the volume. I wonder why water fills the cylinder, but the cubes and rice don’t. I wonder why we don’t use water to fill prisms instead of packing them with cubes.

Multilingual Learner Support If students do not suggest pouring water into a graduated cylinder, consider activating prior knowledge about a graduated cylinder. Consider showing real graduated cylinders of different sizes or pictures of them and discussing the units with which graduated cylinders are labeled. In addition, display a picture of a graduated cylinder with the written words Graduated cylinder.

Why does water fill the cylinder when the cubes and rice do not? There is empty space around the cubes and the pieces of rice. The water fills all the space in the cylinder. What is the volume of the cylinder? How do you know? I still don’t know the volume of the cylinder. I think it’s the volume of water that was poured into the cylinder, but I don’t know the volume of the water. Give students several minutes to turn and talk about ways they could find the volume of the water that was poured into the cylinder. Invite students to share their thinking with the class. Do not yet provide feedback to students about their ideas. Tell students that they resolve this problem of how to find the volume of a figure that is not a prism in the next lesson.

432

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 20

From Packing to Filling Students make connections between packing and filling as methods for finding volume. Pause part 3 of the Cubes in a Cylinder video to display the picture of the cylinder filled with cubes, the cylinder filled with rice, and the cylinder filled with water.

Teacher Note

We need to find the volume of something that can’t be packed with centimeter cubes.

This lesson differentiates between packing and filling. While any container can be filled with water, only certain containers, such as those shaped like right rectangular prisms, can be packed with cubes.

When we packed a right rectangular prism with cubes, there were no gaps or overlaps. The cubes filled all the space inside the prism. Can we also find the volume of a container shaped like a right rectangular prism by filling it with water? Why?

Both packing and filling can be used to find volume. In either case, the substance that packs a container or the substance that fills a container takes up all the space in the container.

What problem are we having with finding volume?

Yes. We can fill the space with water and find the volume of the water. As we saw with the cylinder, not all containers can be packed with cubes to find their volumes.

As you discuss various containers and substances in this lesson, consider asking students to differentiate between packing and filling.

Display the picture of the different-shape containers of water. Highlight how containers come in all shapes and sizes, not just prisms and cylinders.

433

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 20

What strategy can we use to measure the volume of a container when we cannot pack the container with cubes? We can fill the container with water or another liquid. Why is filling a container with water or any liquid an effective method for finding the volume of a container? Water fills a container with no gaps and no overlaps, just as cubes do when packing a prism. Ask students to Think–Pair–Share about the following question. What are some differences between measuring volume with centimeter cubes and measuring volume with water? We can count centimeter cubes, but we can’t count water. Centimeter cubes are solid, and water is a liquid. We measure the volume of centimeter cubes in cubic centimeters and the volume of water in milliliters or liters. Centimeter cubes pack into a container while water fills a container. Water takes the shape of its container, but cubes don’t change shape. We know that we can fill a container with water to find the volume of the container. What we do not know yet is how the volume of the water in the container relates to the volume of the centimeter cubes. We will explore that problem in the next lesson. You learned previously that capacity is the maximum amount that a container can hold and that liquid volume is the amount of space a liquid takes up.

Teacher Note Students study liquid volume and capacity in previous grades. Students learn that capacity is the maximum amount that a container can hold and that liquid volume is the amount of space a liquid takes up. If time permits, consider asking students to discuss and record everything they know about volume and capacity before beginning this segment of the lesson. Use the discussion and the one in the lesson to address the following common misconceptions: • Solid objects do not have volume because they cannot be filled with something. • Volume and capacity are the same. • If an object does not have capacity, it cannot have volume. • Thin objects such as pieces of paper do not have volume. • Volume is only about liquids or only about solids.

434

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 20

Display the two boxes.

These two boxes are identical except one box has extra padding to protect what goes inside it. These two boxes have the same dimensions when closed. Do they have the same volume when closed? Why? Yes. If the boxes are the same size, they take up the same amount of space and therefore have the same volume. Invite students to Think–Pair–Share about whether the boxes have the same capacity and how they know. No. The boxes do not have the same capacity. I know because the one on the left has more padding, so it can’t hold as much. No. The boxes do not have the same capacity. I know because the one on the right has less padding, so it can hold more.

435

5 ▸ M5 ▸ TC ▸ Lesson 20

EUREKA MATH2 Florida

To engage students in a discussion about their conceptual understanding of volume and capacity, display and read any or all the following statements one at a time. Invite students to determine whether the statement is true or false, write True or False on their whiteboards, and hold the whiteboards up. After each statement, encourage students to respectfully debate with one another if they disagree and to provide explanations for why the statement is true or false. • Liquids and solids have volume. True. Every substance and every three-dimensional object has volume. • Solids take up more space than liquids. False. Solids and liquids can take up the same amount of space. • When we measure a liquid’s volume, we are measuring how much space the liquid takes up.

Differentiation: Challenge For students who need an additional challenge, encourage them to think about whether the volume of an uninflated balloon is equal to the volume of a balloon once it has been inflated and why.

True. That’s the definition of volume. • When we measure a solid’s volume, we are measuring how much space the solid takes up. True. That’s the definition of volume. • Liquids, such as water, can change shape, but solids, such as cubes and rice, cannot change shape. True. Liquids fill a container, and solids pack a container. • A ruler does not have volume because it is not packed or filled with anything. False. A ruler has volume because it takes up space. • The volume of a three-dimensional figure is the same whether the figure is a solid or not. True. The figure takes up the same amount of space whether it is solid or not. • The volume of a prism is the same regardless of whether we think of it being packed with cubes or filled with water. True. The amount of space taken up is the same whether the space is packed or filled. • A prism with a capacity of 20 cubic centimeters holds the same amount of water as a cylinder with a capacity of 20 cubic centimeters. True. The prism and the cylinder hold the same amount of water. • Every three-dimensional object has volume, even Earth. True. Every three-dimensional object takes up space. 436

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 20

Display the visual with the objects, quantities, and the question about volume from Launch. Teacher Note Which quantity has the greatest volume?

1 cubic foot of popcorn

1 cubic foot of juice boxes

1 cubic foot of marbles

1 cubic foot

Consider creating a cubic foot container from cardboard to use with this discussion.

of water

The following series of questions is challenging and is designed to advance student thinking about volume. Add to or omit questions to meet the class’s needs. Let’s think some more about this question from earlier in the lesson. This question is more complicated than it might appear at first. Let’s begin by thinking about the water. Suppose we have a container with a capacity of 1 cubic foot. Could we fill it with exactly 1 cubic foot of water? Explain. Yes. There would not be any gaps or holes in the water, so we could fill the container with exactly 1 cubic foot of water. Yes. The water would take up all the space in the container, so it would have a volume of exactly 1 cubic foot of water. Now, let’s think about the juice boxes. Suppose we have a container with a capacity of 1 cubic foot. Could we pack it with exactly 1 cubic foot of juice boxes? Explain. Maybe. It depends on how the juice boxes fit into the container. Maybe. If the juices boxes take up all the space in the container, then we could pack it with the juice boxes. Maybe. The volumes of multiple boxes may never add to exactly 1 cubic foot. © Great Minds PBC

437

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 20

Now, let’s think about the marbles. Suppose we have a container with a capacity of 1 cubic foot. Could we pack it with exactly 1 cubic foot of marbles? Explain. No. There will always be empty space around the marbles. No. There will always be gaps between the marbles. Is it possible to have exactly 1 cubic foot of marbles? Explain. It depends on the marble. If we knew the volume of one marble, we could multiply that marble by the number of marbles needed to get to a total volume of 1 cubic foot.

1 of a cubic foot, then 8 of those marbles would For example, if a marble has a volume of  _

have a volume of exactly 1 cubic foot.

Now, let’s think about the popcorn. Suppose we had a container with a capacity of 1 cubic foot. Could we pack it with exactly 1 cubic foot of popcorn? Explain. No. There will always be space between the popcorn, no matter how much we squish it in. The popcorn container has a capacity. When we put popcorn into a container, the volume of the popcorn is not the capacity of the container because there is empty space between the pieces of popcorn. But when we fill the container with water, the volume of the water is the capacity of the container because there is no empty space in the container. Invite students to turn and talk about what substances besides water they could use to fill a container.

438

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 20

Land

Debrief 5 min Objective: Interpret volume as filling. Facilitate a class discussion about filling and packing by using the following prompts. Encourage students to restate or add on to their classmates’ responses. When we measure the volume of a liquid or a solid, what are we measuring? We are measuring how much space the liquid or solid takes up. Why can we use water but not marbles to find the volume of a right rectangular prism? Water takes up all the space in a prism, but there are gaps between the marbles. So the marbles do not take up all the space in the prism. How are packing and filling similar? How are they different? With both packing and filling, we are putting something into a container to find the volume of the container, or how much space it takes up. When we pack a container, we use individual solids, such as cubes. We have to be able to pack the container without any gaps or overlaps. When we fill a container, we use a liquid, such as water. Packing or filling can help us to find volume as long as the substance we pack with or the substance we fill with takes up all the space in a container.

439

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 20

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EXIT TICKET

I can measure and explain volume by filling a container.

1 ___ 1 ___ 3–6 2 – ___ 1 ___ 3 6

2 – ___ 1 ___ 6 6

4 – ___ 1 ___ 6 6

Florida B.E.S.T. MA.5.GR.3.1

Name

Date

1. Blake measures the volume of a right rectangular prism. a. What does it mean to measure the volume of an object? To measure the volume of an object means to measure how much space that object takes up.

2 – ___ 1 ___ 4 8

This page may be reproduced for classroom use only .

440

4 + ___ 1 ___ 8 8

3 + ___ 1 ___ 4 8

1 + ___ 1 ___ 5 2

5 2 _____ _____ 10 + 10

2 + ___ 1 ___ 5 2

5 4 _____ _____ 10 + 10

6 1 + ___ ___ 4 8

Sample: __14 + _34

4 – ___ 1 ___ 6 3

Sample: __46 – _26

4 + ___ 1 ___ 8 8

b. Blake has marbles, rice, and water. Which should Blake use to fill the right rectangular prism completely? Explain. Blake should use water. The marbles and rice will not fill the right rectangular prism completely and will leave gaps.

2. Kayla measures the volume of the pretzels shown.

Is the volume of the jar the same as the volume of the pretzels? Explain. No. The pretzels leave gaps in the jar. There is air between the pretzels. The volume of the jar is greater than the volume of the pretzels.

211

LESSON 21

Relate volumes of solids and liquid volume. I can solve problems that involve the volumes of solids and liquids.

Lesson at a Glance Through watching a video and class discussion, students learn that 1 cubic centimeter has the same volume as 1 milliliter. Students know that they can determine the volume of any container in cubic centimeters by filling the container with water, measuring the volume of water in a graduated cylinder, and converting the volume in milliliters to cubic centimeters. Students use the relationships between cubic centimeters and milliliters to solve real-world problems.

Key Questions • How are volumes of solids and liquids related?

Mathematical Thinking and Reasoning Standard

Language Objectives • Read real-world problems involving liquid volume and solve the problems by applying strategies for finding the volumes of solids and by using the relationship between cubic centimeters and milliliters. • Orally and in writing, describe how cubic centimeters and milliliters are related.

MTR.2.1

• How are cubic centimeters and milliliters related?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 21

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Tear out Equivalent Expression Cards, Set 2 from the student books. Cut the cards apart and place each set in an envelope. Prepare enough sets for 1 set per student pair.

Launch 5 min Learn 35 min • Relate Cubic Centimeters and Milliliters • Solve Problems Involving Cubic Centimeters and Milliliters • Problem Set

• Envelopes (12) • Equivalent Expression Cards, Set 2 (in the student book)

Students • Envelope of Equivalent Expression Cards, Set 2 (1 per student pair)

Land 10 min

Achievement Descriptors

Florida B.E.S.T. Standard

FL.5.Mod5.AD10 Recognize that volume

MA.5.GR.3.1 Explore volume as an attribute

can be measured by using unit cubes and that a solid packed without gaps or overlaps by n unit cubes has a volume of n cubic units. (MA.5.GR.3.1)

of three-dimensional figures by packing them with unit cubes without gaps. Find the volume of a right rectangular prism with whole-number side lengths by counting unit cubes.

FL.5.Mod5.AD11 Measure volumes by counting

unit cubes. (MA.5.GR.3.1)

443

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 21

Fluency

Whiteboard Exchange: True or False Number Sentences Students decide whether a number sentence is true or false and make false number sentences true to build fluency with reasoning about products without evaluating. Display 23.5 × 1 > 23.5. Is the number sentence true or false? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond. False.

23.5 × 1 > 23.5 False

23.5 × 1 = 23.5

1.77 × 183.2 < 1.77 False

1.77 × 183.2 > 1.77

444

1.77 × 0.03 < 1.77 True

37.4 × 2.3 > 37.4 True

0.59 × 2.3 < 0.59 False

0.59 × 2.3 > 0.59

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 21

Match: Equivalent Expressions Materials—S: Equivalent Expression Cards, Set 2

Students identify equivalent expressions and create equations to develop procedural reliability with adding and subtracting mixed numbers with unlike units from module 2. Have students form pairs. Distribute a set of cards to each student pair and have them complete the activity by using the following procedure: • Lay out all cards faceup. • Match cards that show two equivalent expressions. Note that two cards do not have matches.

1 3

1 +3

2 9

3 9

1 +3

−

2 9

• Lay each set of matched cards side by side. Form an equation by placing an equal sign card between equal expressions. • Continue until all but two cards are matched. • Find common units for the two expressions without matches and use them to write two expressions that are equal on the blank cards and form two additional equations. Circulate as students work and provide support as needed. Encourage students to explain their thinking to their partners.

Launch

4 8

2 −1

1 2

Students explore a method for finding the volume of a space that cannot be packed with cubes. Play parts 1–4 of the Cubes in a Cylinder video. If necessary, replay the video and ask students to note any details.

445

5 ▸ M5 ▸ TC ▸ Lesson 21

EUREKA MATH2 Florida

Engage students in a brief discussion about the video, with a focus on part 4. Discuss student observations and any relevant questions they have. Consider the following possible sequence. What do you notice? I notice that the same prism is packed with cubes first and then filled with water. I notice that the person is using a graduated cylinder to measure the volume of water that was in the prism. What do you wonder? I wonder what the volume of the water is. I wonder how many milliliters of water fit in a centimeter cube. What unit did the person use when they packed the prism with cubes? Cubic centimeters Invite students to Think–Pair–Share about why the person poured the water from the prism into the graduated cylinder. The person tried to measure liquid volume with a graduated cylinder. It’s possible to use the graduated cylinder to find the volume of the water that was in the prism. What unit do we use when measuring volume with a graduated cylinder? Milliliters We want to know the volume of a cylinder in cubic centimeters, but we cannot pack a cylinder with centimeter cubes. Have students turn and talk with a partner about the following question. How can pouring the water from the prism into a graduated cylinder help us solve this problem?

Universal Design for Learning: Engagement If the materials are readily available, consider modeling the activity from the video in person. Invite students to help with the activity, including pouring the water, measuring the water, and dropping the cube in the graduated cylinder. In addition, consider activating prior knowledge about how many milliliters are in 1 liter.

Transition to the next segment by framing the work. Today, we will relate volumes of solids to volumes of liquids.

446

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 21

Learn

Relate Cubic Centimeters and Milliliters Students determine that 1 cubic centimeter has the same volume as 1 milliliter. Play part 5 of the Cubes in a Cylinder video, pausing when the person holds the centimeter cube above the graduated cylinder. What is the volume of the prism in milliliters? How do you know? The volume of the prism is 20 milliliters. I know because that is how much water was poured into the graduated cylinder. What do you notice about the volume of the prism in cubic centimeters and the volume of the prism in milliliters? They are equal. A volume of 20 cubic centimeters is equal to a volume of 20 milliliters. Invite students to turn and talk about what they think will happen when the cube is dropped into the graduated cylinder. Show the rest of part 5 of the Cubes in a Cylinder video. Engage students in a brief discussion about the video. What do you notice? The centimeter cube sank. When the centimeter cube was dropped into the water, the volume reading on the graduated cylinder changed from 1 milliliter to 2 milliliters.

Teacher Note When students watch the centimeter cube drop into the graduated cylinder of water, they are witnessing Archimedes’ principle. Archimedes was a mathematician and inventor in ancient Greece. Archimedes’ principle states, in part, that when any object is submerged in a fluid, the volume of displaced fluid is equivalent to the volume of the object.

The volume of the water in the graduated cylinder was originally 1 milliliter. We did not put more water in, and yet the cube caused the volume reading on the graduated cylinder to increase from 1 milliliter to 2 milliliters.

447

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 21

What is the volume of 1 centimeter cube in milliliters? How do you know?

1 milliliter. I know because the volume of the water and the centimeter cube together is 2 milliliters. There was already 1 milliliter of water, and 2 − 1 = 1.

Universal Design for Learning: Representation

1 milliliter. I know because the volume of the prism was 20 cubic centimeters, and the volume of the prism is also 20 milliliters, so 1 centimeter cube has the volume of 1 milliliter.

Consider creating an anchor chart to show that 1 cubic centimeter and 1 milliliter have the same volume.

We know the volume of a centimeter cube is 1 cubic centimeter. What is another way to describe the volume of a centimeter cube?

1 cubic centimeter = 1 milliliter

1 milliliter Display the picture of 1 cubic centimeter and 1 milliliter of water. How can knowing that 1 centimeter cube has the same volume as 1 milliliter of water help us solve problems?

10 mL 9 8 7 6 5 4

It can help us when we’re dealing with both units of measurement.

It can help us when we’re trying to fill containers that are not right rectangular prisms.

1 cubic centimeter = 1 milliliter 10 mL 9 8 7 6 5 4 3 2 1

Display a picture of the cylinder from the Cubes in a Cylinder video. This is the cylinder that was packed with centimeter cubes in the previous lesson. Invite students to Think–Pair–Share about how to find the volume of this cylinder in cubic centimeters. Accept all student thinking and responses. We can fill the cylinder with water and then pour the water into a graduated cylinder. We can use the graduated cylinder to find the volume of water in milliliters. The number of milliliters is equal to the number of cubic centimeters. We can fill the graduated cylinder with some water and then pour it into the cylinder. We can continue doing that and adding the amounts of milliliters we pour in together until the cylinder is filled up. The number of cubic centimeters will be the same number as the total milliliters. 448

Differentiation: Challenge Consider inviting students to reason about whether similar relationships exist between other units of volume. For example, ask whether they think an object with a volume of 8 cubic inches has a volume of 8 liquid ounces (or 1 cup), or whether they think an object with a volume of 1 cubic foot has a volume of 1 gallon or 1 liter. Invite students to design an experiment that would allow them to confirm their thinking.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 21

Solve Problems Involving Cubic Centimeters and Milliliters Students use the fact that an object with a volume of 1 cubic centimeter also has a volume of 1 milliliter to solve real-world problems. Let’s use what we have learned about filling to determine volume to solve real-world problems. How can you find the volume of a right rectangular prism? I can decompose it into layers and find the volume of each layer. Then I multiply the volume of each layer by the number of layers. Direct students to problem 1 in their books. Ask students to complete the problem with a partner. Circulate as students work, asking the following questions: • How can you decompose the prism into layers? • How do you find the volume of each layer? • How do you find the volume of the prism when you know the volume of each layer? • How many milliliters of water are in the graduated cylinder? • How many cubic centimeters are in 10 milliliters?

Multilingual Learner Support

1. Tyler plans to pour the water from the graduated cylinder into the container shaped like a right rectangular prism.

10 mL

3 cm 3 cm

To support students with communicating their ideas to their partner, consider inviting them to use the Share Your Thinking section of the Talking Tool to complete problem 1.

Differentiation: Support Encourage students who need additional support with problem 1 to use their centimeter cubes to create the prism that packs the container.

1 cm

449

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 21

a. Draw lines on the container to decompose it into layers. b. Determine the volume of the container. The volume of one layer is 3 cubic centimeters. There are 3 layers.

3 × 3 cubic centimeters = 9 cubic centimeters The volume of the container is 9 cubic centimeters. c. Can Tyler pour all the water from the graduated cylinder into the container? Explain. No. The water from the graduated cylinder does not fit in the container. When students are finished, bring the class together. Invite students to share how they decomposed the prism into layers to find its volume. Highlight different layers that they create. Then ask the following question. How do you know that all the water in the graduated cylinder does not fit in the prism? The volume of water in the graduated cylinder is 10 milliliters, which is equal to 10 cubic centimeters. The volume of the prism is only 9 cubic centimeters. Ask students to complete problem 2 with a partner. Circulate as students work, asking the following questions: • How can you decompose the prism into layers? • How do you find the volume of each layer? • How do you find the volume of the prism when you know the volume of each layer?

Promoting Mathematical Thinking and Reasoning Students represent problems in multiple ways (MTR.2) when they use layers to determine the volume of water that an object shaped like a right rectangular prism can hold, use the relationship between cubic centimeters and milliliters, and interpret their result in context. Ask the following questions to promote MTR.2:

• What is the relationship between cubic centimeters and milliliters?

• What does the problem ask you to do?

• How many milliliters are in a liter?

• How do the units involved in the length, width, and height of the prism help you think about this problem?

450

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 21

2. A company advertises that its glass vase, which is shaped like a right rectangular prism, holds 2 liters of water. The base of the inside of the vase is a square. A side of the base measures 8 centimeters. a. Decompose the vase into layers to find its volume. Volume of 1 layer:

8 × 8 = 64

30 cm

64 cubic centimeters Volume of the vase:

Teacher Note Real-world objects, such as the vase, have thickness, so to measure the amount of water the vase can hold, it is important to know the volume of the space inside the vase. Discuss with students why measuring the length, width, and height of the outside of the vase would not accurately tell them how much water the vase can hold.

64 × 30 = 1,920 1,920 cubic centimeters b. What is the volume of the vase in milliliters?

8 cm

1,920 milliliters c. What is the volume of the vase in liters?

1,920 mL = 1,920 × 1 mL = 1,920 × 0.001 L

Differentiation: Support For students who need additional support with problem 2, provide a drawing of the vase that is already decomposed into layers or into individual cubes.

= 1.92 L 1.92 liters d. Is the company’s advertisement correct? Explain. No. The company said that the vase can hold 2 L of water, but it can only hold 1.92 L, so the advertisement is not correct. When most students are finished, bring the class together. Invite students to share how they decomposed the prism into layers to find its volume. Highlight different layers that students create. Then ask the following questions. How did you find the volume of the vase in milliliters? I knew that an object with a volume of 1 cubic centimeter also has a volume of 1 milliliter. So 1,920 cubic centimeters is equal to 1,920 milliliters.

451

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 21

How did you find the volume of the vase in liters?

1 as much as 1 liter. I multiplied the number of milliliters by 0.001 because 1 milliliter is  ____ 1,000

Invite students to Think–Pair–Share about another way they could have determined if the company’s advertisement is correct. I could have decomposed the prism into layers in a different way. I could have determined the number of cubic centimeters in 2 liters and then compared that to the volume of the vase in cubic centimeters.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Relate volumes of solids and liquid volume. Facilitate a class discussion about volumes of solids and liquid volume by using the following prompts. Encourage students to restate or add on to their classmates’ responses. How can you find the volume of a solid object, such as a cube? I can decompose the cube into layers and multiply the volume of each layer by the number of layers. I can place the object in a graduated cylinder with water and determine how much the water level rises.

452

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 21

We often use cubic centimeters to measure volumes of solids and milliliters to measure volumes of liquids. How are volumes of solids and liquid volume related? Volumes of solids and liquid volume both indicate how much space the solid or the liquid is taking up. Cubic centimeters can be used for either solid or liquid volume. A solid that is put into a liquid will increase the liquid volume, so you can add the volume of a liquid to the volume of a solid to find the total volume if the units are both cubic centimeters. How are cubic centimeters and milliliters related? The volume of an object in cubic centimeters is equal to the volume of the object in milliliters.

453

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 21

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

This page may be reproduced for classroom use only .

429

454

1_13 + 3_29

1_39 + 3_29

1_23 + 3_29

1_69 + 3_29

2_12 − 1_18

2_48 − 1_18

2_12 − 1_38

2_48 − 1_38

7_23 + 5_34

9 8 __ 7__ 12 + 5 12

7_34 + 5_16

9 2 __ 7__ 12 + 5 12

2_48 − 1_12

Sample: 2_21 + 1_12

9 7_34 + 5__ 12

9 9 _ Sample: 7_ 12 + 5 12

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC ▸ Lesson 21

PROBLEM SET

I can solve problems that involve the volumes of solids and liquids.

PRACTICE

Florida B.E.S.T. MA.5.GR.3.1

Name

3. The right rectangular prisms shown are made of centimeter cubes. Draw a line to match each right rectangular prism with the amount of water in milliliters a container shaped like the prism can hold. Right Rectangular Prism

Date

Amount of Water

Use the right rectangular prisms to complete the statements.

8 mL

1. The right rectangular prism is made

centimeter cubes.

A container shaped like this right rectangular prism can hold exactly

mL of water.

2. The right rectangular prism is made of

centimeter cubes.

24 mL

A container shaped like this right rectangular prism can hold exactly

mL of water.

27 mL

221

222

PROBLEM SET

455

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC ▸ Lesson 21

EXIT TICKET

I can solve problems that involve the volumes of solids and liquids.

Florida B.E.S.T. MA.5.GR.3.1

4. How many milliliters of juice fit in the juice box?

Name

Date

Eddie’s juice box is shaped like a right rectangular prism as shown.

8 cm

4 cm

6 cm

192 milliliters of juice fit in the juice box.

11 cm

5. Jada has a fish tank that is the shape of a right rectangular prism. The length, width, and height of the fish tank are shown.

6 cm

4 cm

a. Decompose the prism into layers to find its volume in cubic centimeters.

30 cm

Volume of one layer: 6 × 4 = 24

20 cm

24 cubic centimeters

40 cm Jada’s fish tank can hold

24,000

cubic centimeters of water.

Jada’s fish tank can hold

24,000

milliliters of water.

Jada’s fish tank can hold

Volume of the juice box: 24 × 11 = 264

264 cubic centimeters

b. What is the volume of Eddie’s juice box in milliliters?

liters of water.

The volume of Eddie’s juice box is 264 milliliters.

c. How are cubic centimeters and milliliters related?

1 cubic centimeter has the same volume as 1 milliliter.

456

PROBLEM SET

223

225

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC

Topic C Quiz Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

A N SW E R K E Y ▸ TO PI C Q U I Z C-1

2. The right rectangular prisms shown are made of centimeter cubes. Indicate whether each prism represents a container that can hold exactly 24 milliliters. Right Rectangular Prism

1. Consider the right rectangular prism shown. Each cube represents 1 cubic unit.

Yes

What is the volume of the prism? A. 30 cubic units B. 60 cubic units C. 140 cubic units D. 300 cubic units

Distractor rationale:

A. Incorrect. The student may have found the area of the front face. B. Incorrect. The student may have found the area of the top face. C. Incorrect. The student may have found the area of the three visible faces. D. Correct.

239

240

A N S W E R K E Y ▸ TO P I C Q U I Z C-1

457

EUREKA MATH2 Florida

5 ▸ M5 ▸ TC

3. Which attribute describes how much space a fish tank takes up? A. Area B. Length

Distractor rationale: A. Incorrect. The student may not have considered that a fish tank also has height.

C. Volume

B. Incorrect. The student may not have considered that a fish tank also has height and width.

D. Width

C. Correct. D. Incorrect. The student may not have considered that a fish tank also has length and height.

4. Determine the volume of the right rectangular prism shown. Each

represents 1 cubic centimeter.

The volume of the prism is

458

126

cubic centimeters.

A N S W E R K E Y ▸ TO P I C Q U I Z C-1

241

EUREKA MATH2 Florida 5 ▸ M5 ▸ TC

Observational Assessment Recording Sheet Grade 5 Module 5 Topic C

Student Name

Volume Concepts Standards and Achievement Descriptors

MA.5.GR.1.2

FL.5.Mod5.AD6 Classify three-dimensional figures into categories based on attributes.

MA.5.GR.3.1

FL.5.Mod5.AD10 Recognize that volume can be measured by using unit cubes and that a solid packed without gaps or overlaps by n unit cubes has a volume of n cubic units.

MA.5.GR.3.1

FL.5.Mod5.AD11 Measure volumes by counting unit cubes.

Dates and Details of Observations

PP Partially Proficient P Proficient HP Highly Proficient

This page may be reproduced for classroom use only.

459

5 ▸ M5 ▸ TD

EUREKA MATH2 Florida

Topic D Volume and the Operations of Multiplication and Addition In topic D, students extend their understanding of volume concepts from topic C. They learn the volume formulas V = B × h and V = l × w × h for right rectangular prisms and use them to solve mathematical and real-world problems. The topic begins with students building on their experience decomposing right rectangular prisms into layers. Students determine that when the height of a layer is 1 unit, the number of square units in the base is equal to the number of cubic units in a layer, and that the number of units in the height is equal to the number of layers. They find that the volume of the prism can be determined by multiplying the area of the prism’s base by its height, and they learn to represent this thinking with the formula V = B × h. In applying the formula, students come to recognize that any face of a right rectangular prism can be the base. After determining that the area of the base of a prism is the prism’s length multiplied by its width, students write the formula V = B × h as V = l × w × h, aware that any of the edge lengths of a prism can be defined as the length, width, or height. To find volumes or unknown dimensions of right rectangular prisms in both mathematical and real-world situations, students use the volume formulas. When given the volume of a prism, students determine possible dimensions of the prism by decomposing numbers into factors and by using the commutative and associative properties of multiplication.

460

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD

To conclude their study of volume, students explore solid figures composed of right rectangular prisms. They find volumes of right rectangular prisms that compose a figure and add the volumes together. They solve real-world problems involving not only volume, but also perimeter and area, including perimeter of figures with fraction and decimal side lengths. In an optional, open-ended activity, students design and create a sculpture by using right rectangular prisms. By following guidelines to create their sculpture, students choose heights of prisms with given bases, multiply edge lengths to find volumes of specific prisms, and calculate to find the total volume of the sculpture. In grade 6, students find the volumes of right rectangular prisms with fraction edge lengths by packing with unit cubes with fraction edge lengths and by applying the formulas V = B × h and V = l × w × h.

461

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD

Progression of Lessons Lesson 22

Lesson 23

Lesson 24

Find the volumes of right rectangular prisms by using the area of the base.

Find the volumes of right rectangular prisms by multiplying the edge lengths.

Solve word problems involving volumes of right rectangular prisms.

Area of the top face: 6 square centimeters

Volume of the top layer: 6 cubic centimeters

4 layers

Height: 4 centimeters

V=6×4

I can use the formula V = B × h to determine the volume of a right rectangular prism. I know that any face of a right rectangular prism can be its base.

462

32 cm (2 × 3) × 4

(3 × 4) × 2

(2 × 4) × 3

I can multiply a prism’s length, width, and height to use the formula V = l × w × h and determine the prism’s volume. I know that formulas are always true and that I can use a formula to solve problems about the volume of any right rectangular prism if I have enough information.

25 cm l cm The volume of the aquarium is 20,000 cubic centimeters. Volume formulas help me find volumes or unknown dimensions of right rectangular prisms in real-world situations. When I know the volume of a right rectangular prism, I can determine possible dimensions for the prism.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD

Lesson 25

Lesson 26

Lesson 27

Find the volumes of solid figures composed of right rectangular prisms.

Solve word problems involving perimeters of rectangles.

Solve word problems involving perimeter, area, and volume.

w in

2m 4m 2m 6m

I can find the volume of a figure composed of right rectangular prisms by finding the volumes of right rectangular prisms within the solid figure and adding them.

12 ft

12.25 in

The perimeter is 73.5 inches. I can use the Read–Draw–Write process to help me make sense of word problems involving perimeter of rectangles with dimensions that are fractions or decimals. Once I understand the problem, I can use a perimeter formula to help me solve the problem.

2 ft 4 ft

12 ft

8 ft 8 ft

2 ft 4 ft

I know that the distance around something is its perimeter. The amount it takes to cover a two-dimensional space is its area. The amount of three-dimensional space something takes up is its volume. Sometimes I need to use all three measurements to solve a problem.

463

LESSON 22

Find the volumes of right rectangular prisms by using the area of the base. I can use the area of the base of a right rectangular prism to find its volume.

Lesson at a Glance Students determine that they can calculate the volume of a right rectangular prism by multiplying the area of the prism’s base by the prism’s height. They write, define, and use the formula V = B × h. They explain why any face of a right rectangular prism can be its base.

Key Questions • Why can we think of any face of a right rectangular prism as its base?

Mathematical Thinking and Reasoning Standard MTR.5.1

Language Objectives • Orally describe how using layers to find the volume of a right rectangular prism is related to using the formula V = B × h. • Read a description of a right rectangular prism and analyze its measurements to orally determine the unknown volume, height, or area of the base by using the formula V = B × h.

• Why can we calculate the volume of a right rectangular prism by multiplying the area of its base by its height?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 22

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

• Tear out and cut apart Decimal Number Cards from the student book. Place each set in an envelope. Prepare enough sets for each student pair to have one. Consider saving them for reuse in lesson 23. • Tear out Hidden Factors Mat from the student book. Consider whether to prepare this material in advance or during the lesson. Consider saving it for reuse in lesson 23.

Launch 5 min Learn 35 min • Write a Formula • Use the Area of a Base to Find Volume • Find Unknown Height or Area of a Base • Problem Set

Land 10 min

• Decimal Number Cards (1 set per student pair, in the student book) • Envelopes (12) • Centimeter cube

Students • Envelope of Decimal Number Cards (1 per student pair) • Hidden Factors Mat (1 per student pair, in the student book)

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD12 Find volumes of right

MA.5.GR.3.2 Find the volume of a right

rectangular prisms with whole-number edge lengths by using a formula. (MA.5.GR.3.2)

rectangular prism with whole-number side lengths using a visual model and a formula.

465

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 22

Fluency

Hidden Factors Materials—S: Decimal Number Cards, Hidden Factors Mat

Students determine the product and then write and say a multiplication equation or related division equation to build fluency with multiplying and dividing decimals from module 4. Have students form pairs. Distribute a deck of Decimal Number Cards to each pair. Have them use the following procedure to play. Consider doing a practice round with students. • Partners place the deck of cards facedown next to the Hidden Factors Mat. • Partner A and partner B flip over a card from the top of the pile and place it on a blue square.

0.5 × 0.3

• Both students say the product. • Partner A records a multiplication equation on their personal whiteboard and then reads it. Partner B records a related division equation on their whiteboard and then reads it. See the sample partner dialogue under the sample mat.

Partners A and B: “0.15” Partner A: “0.5 × 0.3 = 0.15” Partner B: “0.15 ÷ 0.5 = 0.3”

466

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 22

Counting with Centimeter Cubes Students determine how many centimeter cubes are in one layer and then count to find the volume of a right rectangular prism. Display the right rectangular prism and the layer of centimeter cubes. What is the volume of the layer of centimeter cubes next to the prism? Raise your hand when you know. = 1 cubic centimeter

3 cubic centimeters Display the layer of cubes inside the prism. The 3 centimeter cubes represent one layer of the prism. Let’s count by 3 cubic centimeters to pack the prism and find the volume. The first measurement you say is 3 cubic centimeters. Ready?

Display each layer in the prism one at a time as students count to 15 cubic centimeters.

3 cubic centimeters, 6 cubic centimeters, 9 cubic centimeters, 12 cubic centimeters, 15 cubic centimeters What is the volume of the prism? Raise your hand when you know.

15 cubic centimeters Display the volume.

The volume is

15 cubic centimeters.

Repeat the process with the following sequence:

The volume is 16 cubic centimeters.

The volume is 18 cubic centimeters.

467

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 22

Launch

Students compare methods for finding the volume of a right rectangular prism. Display the three decomposed right rectangular prisms. Tell students the layers are composed of centimeter cubes.

Yuna

Ryan

Jada

Yuna, Ryan, and Jada each use layers composed of centimeter cubes to build a right rectangular prism. Invite students to Think–Pair–Share about whether the three prisms will have the same volume and why. Yes. They will have the same volume because 4 × 6 = 24, 2 × 12 = 24, and 3 × 8 = 24. Yes. They will have the same volume because there are 24 cubes in each prism. Yes. They will have the same volume because 4 layers of 6 cubic centimeters, 2 layers of 12 cubic centimeters, and 3 layers of 8 cubic centimeters each have a volume of 24 cubic centimeters. Will the prisms have the same dimensions? Yes. What are the dimensions of the prisms?

2 centimeters by 3 centimeters by 4 centimeters The prisms that Yuna, Ryan, and Jada composed use different layers, but the composed prisms have the same dimensions. 468

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 22

Display the three prisms and the table representing their volumes.

Yuna

Student

Ryan

Number of Cubes in Each Layer

Jada

Number of Layers

Volume (cubic centimeters)

Yuna Ryan Jada Let’s record some information about the compositions in the table. Invite the class to help complete the columns for the number of cubes in each layer and for the number of layers.

Student

Number of Cubes in Each Layer

Number of Layers

Yuna

Ryan

Jada

Volume (cubic centimeters)

469

5 ▸ M5 ▸ TD ▸ Lesson 22

EUREKA MATH2 Florida

Invite students to turn and talk to summarize how they could use the information in the table to determine the volume of each prism. Transition to the next segment by framing the work. Today, we will determine another way to find the volume of a right rectangular prism.

Learn

Write a Formula Materials—T: Centimeter cube

Students determine that they can find the volume of a right rectangular prism by multiplying the area of its base by its height. Continue to display the prisms and the table from the launch. Use the following prompts to discuss Yuna’s composition further. What is the volume of Yuna’s prism? How do you know?

Universal Design for Learning: Representation Consider creating Yuna’s prism with interlocking cubes, as shown, to serve as a concrete model throughout the discussion. The prism may be helpful for discussing the meaning of the term base and how the base could be any face of the prism. Consider leaving the cubes in 1 by 3 prisms that are not locked together so the prism can be easily decomposed and recomposed.

The volume is 24 cubic centimeters. I know because we can multiply the volume of each layer, 6 cubic centimeters, by the number of layers, 4. There is another way we can use Yuna’s strategy to think about volume. Hold up a centimeter cube. We know that this cube has a volume of 1 cubic centimeter. What is the area of one face of this cube? How do you know? The area of one face is 1 square centimeter. I know because the length is 1 centimeter, the width is 1 centimeter, and 1 × 1 = 1.

470

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 22

Point to the top face of Yuna’s top layer.

Teacher Note

What is the area of this face? How do you know? The area of the face is 6 square centimeters. I know because it has 6 squares that each have an area of 1 square centimeter. What do you notice about the number of square centimeters in this top face and the number of cubic centimeters in this top layer?

In lesson 19, students start to decompose right rectangular prisms by using layers. These layers always have a thickness of 1 unit, which helps students transition to the formula V = B × h in this lesson.

They are equal. The number of square centimeters in this face is equal to the number of cubic centimeters in this layer. How many layers are there?

4 What do you notice about the number of layers and the number of centimeters in the height of the prism? They are equal. Display the color-coded diagram of Yuna’s decomposition.

Volume of the top layer: 6 cubic centimeters

Area of the top face: 6 square centimeters

4 layers

Height: 4 centimeters

V=6×4

471

5 ▸ M5 ▸ TD ▸ Lesson 22

Point to the decomposed prism as you make the following statement. We can calculate the volume of a prism by multiplying the number of cubic centimeters in one layer, 6, by the number of layers, 4. Point to the composed prism as you make the following statement. We can also calculate the volume by multiplying the number of square centimeters in the top face, 6, by the number of centimeters in the height of the prism, 4. In either case, we are multiplying 6 by 4. The top face of the prism has an area of 6 square centimeters. What is the area of the base, the bottom face of the prism? How do you know? The area of the base is 6 square centimeters. I know because opposite faces have the same length and width, which means they have the same area. Because we can multiply the area of the top face by the height of the prism to determine the prism’s volume, we can also determine volume by multiplying the area of the base by the height of the prism. Write V = B × h. This formula is a way of representing how we can find the volume of any right rectangular prism. It tells us that we can always find the volume of a right rectangular prism by multiplying the area of its base by its height. Invite students to Think–Pair–Share about what each letter in the equation V = B × h stands for. The V stands for volume. The B stands for the area of the base.

EUREKA MATH2 Florida

Multilingual Learner Support Support the term calculate by emphasizing that to find the volume, students are using the operation of multiplication. Ask students to consider other times that they have made calculations and what operations they needed to use in each circumstance. Brainstorming circumstances for operations first may be helpful. • We multiply length and width to determine the area of a rectangle. • We add decimals to determine the total cost of several items.

Multilingual Learner Support This lesson uses the multiple-meaning term formula. Students are familiar with the area formula for a rectangle and perimeter formulas for a rectangle. Consider facilitating a class discussion focusing on students’ prior experience with the term in mathematics and with other meanings in everyday life. For example, a student might say that a chef has a special formula for the seasoning in a recipe.

The h stands for height.

472

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 22

Display the prism composed of two layers. Kelly composes the same prism as Yuna, also by using horizontal layers, but he uses two layers instead of four. What is the volume of each of Kelly’s layers?

12 cubic centimeters What is the area of the base of Kelly’s prism?

6 square centimeters Can Kelly calculate the total volume of his prism by multiplying the area of the base by the number of layers? Why? No. Multiplying the area of the base by the number of layers only gives half of the volume. No. Kelly needs to multiply the area of the base by the height of the prism, which is 4, not 2. When we calculate volume, we multiply the area of the base by the height of the prism, not by the number of layers because the number of layers is not always equal to the height.

Use the Area of a Base to Find Volume

Universal Design for Learning: Representation Consider creating an anchor chart to show the ways students know to determine the volume of a right rectangular prism.

4 layers Each layer has 6 cubes. 6 x 4 = 24 The volume is 24 cubic units. The area of the base is 6 square units. The height is 4 units. V=Bxh 24 = 6 x 4 The volume is 24 cubic units.

Students use the area of a base to determine the volume of a right rectangular prism. Display the two prisms. Imagine that we laid the prism we have been working with on a different side. Do the two prisms have the same volume? How do you know? Yes. They are the same prism, but one is laying on its side. Yes. Both prisms have the same dimensions. Yes. Both prisms have the same number of cubes. Yes. Both prisms take up the same amount of space.

473

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 22

What is the area of the base of the prism on the right? How do you know? The area of the base is 8 square centimeters. I know because the length of the base is 4 centimeters, the width of the base is 2 centimeters, and 4 × 2 = 8. What is the height of the prism on the right?

3 centimeters What is the formula for the volume of a right rectangular prism?

V=B×h Guide students in using the formula, with 8 as the number for the area of the base and 3 as the number for the height. Display the calculations for the volume of the prism and the volume of the prism when it is turned on its side.

V=B×h =8×3 = 24 The volume is 24 cubic centimeters.

V=B×h

=6×4

=8×3

= 24

The volume is 24 cubic centimeters.

Point to the taller prism as you make the following statement. When we first calculated the volume of this prism, we used the face with side lengths of 2 centimeters and 3 centimeters as the base. We used the edge that measures 4 centimeters as the height.

474

Promoting Mathematical Thinking and Reasoning When students apply their understanding of using layers to determine the volume of a right rectangular prism to make sense of and use the formula V = B × h, they are using patterns and structure (MTR.5). Ask the following questions to promote MTR.5: • How is multiplying the volume of each layer by the number of layers related to the formula V = B × h? How can that help you calculate the volume? • How can what you know about layers help you with using the formula?

Teacher Note These equations are intentionally written without units (i.e., V = 8 × 3, not V = 8 square inches × 3 inches). The product of the height and the area of the base, without units, is the total number of unit cubes in the rectangle. The volume, or the product with the unit, is written in the solution statement.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 22

Point to the wider prism as you make the following statement. Then we used the face with side lengths of 4 centimeters and 2 centimeters as the base. We used the edge that measures 3 centimeters as the height. We determined that, no matter which face is on the bottom of the prism, its volume is 24 cubic centimeters. Can you choose any face of the prism to be the base? Why? Yes. A right rectangular prism can be turned so that any face is the base.

Teacher Note It is true that any face of a right rectangular prism may serve as its base because all faces of a right rectangular prism are rectangles. However, it is not true for other prisms, pyramids, or cylinders. For example, a right triangular prism has two triangular bases, but the remaining rectangular faces are not bases.

Once you choose the base, how do you know which edge is the height? If the base is on the bottom, the height is the edge that goes up. The height is an edge that is not on the base or the face opposite the base. Direct students to problem 1 in their books. Invite students to turn and talk about whether they need to determine the edge lengths of the top face before determining the prism’s volume. Ask students to complete problem 1 with a partner. Circulate as students work, asking the following questions: • Can you use a formula to calculate the volume of the prism? • How do you use the formula? • Does your answer make sense? Why?

475

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 22

1. The area of the base of a right rectangular prism is 28 square inches and the height is 6 inches. What is the volume of the right rectangular prism?

V=B×h = 28 × 6 = 168 The volume is 168 cubic inches. When students are finished, ask the following questions. What are possible lengths and widths for the base? Explain your thinking. The length and width of the base could be 7 inches and 4 inches, 14 inches and 2 inches, or 28 inches and 1 inch because 7 × 4 = 28, 14 × 2 = 28, and 28 × 1 = 28. 14 inches and 2 inches is the most reasonable pair of side lengths based on the drawing.

Differentiation: Support For students who need additional support with problem 1, consider providing the formula and helping them determine which numbers to use for B and h.

Did we need to use a formula to calculate the volume of the prism? If not, what other strategy could we have used? No. We could have decomposed the prism into 6 layers. If the area of the base is 28 square inches, then the volume of each layer is 28 cubic inches. We could have multiplied the volume of each layer by 6, the number of layers. Why might we use a formula instead of decomposition to find the volume of the prism? The formula is more efficient. It takes more time to decompose the prism into layers.

476

Teacher Note Thinking about the possible corresponding lengths and widths for the base with a given area can help students connect the formula V = B × h to the formula V = l × w × h in lesson 23. It is not necessary for students to determine every pair of possible side lengths.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 22

Ask students to consider a right rectangular prism where the area of the base is 28 square inches and the height is 12 inches. Invite students to Think–Pair–Share about the following question. How would doubling the height of the right rectangular prism from 6 inches to 12 inches affect the volume of the prism? How do you know? The volume would double because we would multiply the area of the base by twice as much. The volume would double because we would find B × (2 × h) instead of B × h. Ask students to consider a right rectangular prism where the area of the base is 28 square inches and the height is 3 inches. How would the volume of the right rectangular prism be different if the height of the prism were 3 inches instead of 6 inches? Why? The volume would be half because we would multiply the area of the base by half as much.

The volume would be half because we would find B × ( _ 1 × h) instead of B × h. 2

477

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 22

Find Unknown Height or Area of a Base Students determine an unknown height or face area of a right rectangular prism. Display the three prisms.

Prism A

Prism B

Prism C

The area of the shaded face is 6 square centimeters.

Universal Design for Learning: Action & Expression Consider providing students who need additional support with a copy of the prisms. Ask students to label the base and the height of each prism.

2 cm The area of the shaded face is 12 square centimeters.

V = 30 cubic centimeters

V = 24 cubic centimeters

V = 36 cubic centimeters

Invite students to turn and talk with a partner about the following question. The volumes of the right rectangular prisms are shown. What other information can you determine about each prism? After a couple of minutes, invite students to share their thinking. Encourage students to justify their conclusions. For prism A, the shaded face is the base, and its area is 6 square centimeters. The height is not an edge of the shaded face. We know V = B × h, so the height of the prism is 5 centimeters because 6 × 5 = 30.

478

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 22

For prism B, the height is 2 centimeters, and the height is not an edge of the base. We know V = B × h, so the area of the base of the prism is 12 square centimeters because 12 × 2 = 24. I can think about turning prism C so that the shaded face is on the bottom to help me think of it as the base. The area of the base is 12 square centimeters. The height is not an edge of the shaded face, and V = B × h. So the height of prism C is 3 centimeters because 12 × 3 = 36. As students discuss their ideas, consider asking the following questions: • Do you know the length and width of the shaded face? • What are possible values for the length and width of the shaded face? • What is the height of the prism? How do you know? • What face can you determine the area of? Why? What formula did we use today? What does the formula mean?

V = B × h. We can find the volume of a right rectangular prism by multiplying the area of its base by its height. Write V = B × h. We can use this formula to solve many types of problems. If we know the height and the area of the base of a right rectangular prism, we can find the volume. If we know the area of the base and the volume of a right rectangular prism, we can find the height. If we know the height and volume of a right rectangular prism, we can find the area of the base.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

479

5 ▸ M5 ▸ TD ▸ Lesson 22

Land

EUREKA MATH2 Florida

Debrief 5 min Objective: Find the volumes of right rectangular prisms by using the area of the base. Facilitate a class discussion about using multiplication to find the volume of right rectangular prisms by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Why can we think of any face of a right rectangular prism as its base? A right rectangular prism can be turned so that any face is the base. How do we find the volume of a right rectangular prism if we know the prism’s height and the area of the base? We multiply the area of the base of the prism by the height. Why can we calculate the volume of a right rectangular prism by multiplying the area of its base by its height? The number of square units in the base is equal to the number of cubic units in a layer with a height of 1 unit. The number of units in the height is equal to the number of layers. So the volume can be found by multiplying the area of the base of the prism by its height.

480

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 22

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can use the area of the base of a right rectangular prism to find its volume.

PRACTICE

Florida B.E.S.T. MA.5.GR.3.2

Name

2. The right rectangular prism shown is made of centimeter cubes. Use the prism to complete parts (a)–(d).

Date

1. The right rectangular prisms shown are made of centimeter cubes. Circle the two right rectangular prisms that have the same volume.

a. The top layer is made of b. The area of the top face is c. The area of the base is

235

236

cubes.

square centimeters.

d. The height of the right rectangular prism is

e. The volume of the right rectangular prism is

140 cubic centimeters.

PROBLEM SET

centimeters.

481

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 22

Calculate the volume of the right rectangular prism. 3. The area of the base is 24 square inches and the height is 10 inches.

240 cubic inches

Consider each right rectangular prism. 4. The area of the base is 48 square inches and the height is 5 inches.

7. What is the area of the base if the volume is 126 cubic centimeters?

240 cubic inches

8. What is the height if the volume is 180 cubic inches and the area of the base is 45 square inches?

4 inches 7 centimeters

18 square centimeters

5. The area of the base is 24 square centimeters 6. The area of the base is 24 square centimeters and the height is 4 centimeters. and the height is 8 centimeters.

96 cubic centimeters

482

192 cubic centimeters

PROBLEM SET

237

238

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 22

EXIT TICKET

I can use the area of the base of a right rectangular prism to find its volume.

Florida B.E.S.T. MA.5.GR.3.2

Name

Date

Calculate the volume of each right rectangular prism. 1. The area of the base is 18 square centimeters 2. The area of the base is 18 square centimeters and the height is 4 centimeters. and the height is 2 centimeters.

V= B× h

= 18 × 2

= 18 × 4

= 36 36 cubic centimeters

= 72 72 cubic centimeters

239

483

LESSON 23

Find the volumes of right rectangular prisms by multiplying the edge lengths. I can use the edge lengths of a right rectangular prism to find its volume.

Lesson at a Glance Students determine that they can multiply a right rectangular prism’s length, width, and height to determine its volume. They write, define, and use the formula V = l × w × h, recognizing that any of the edges of a prism can be defined as the length, width, or height. Students determine an unknown edge length, such as height, in a right rectangular prism when given the volume and two other dimensions.

Key Questions • How are the formulas for finding the volume of a right rectangular prism similar?

Language Objective • Orally explain why the formulas V = B × h and V = l × w × h can be used to determine the volume of a right rectangular prism by making connections between the area of the prism’s base and the length and width of the prism.

Mathematical Thinking and Reasoning Standard MTR.5.1

• Why is having two formulas for the volume of a right rectangular prism helpful?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 23

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

• Tear out and cut apart Decimal Number Cards (in the student book). Place each set in an envelope. Prepare enough sets for each student pair to have one. • Tear out Hidden Factors Mat (in the student book). Consider whether to prepare this material in advance or during the lesson.

Launch 5 min Learn 35 min • Write Another Formula • Use Edge Lengths to Find Volume • Find an Unknown Edge Length • Problem Set

Land 10 min

• Decimal Number Cards (1 set per student pair, in the student book) • Envelopes (12)

Students • Envelope of Decimal Number Cards (1 per student pair) • Hidden Factors Mat (1 per student pair, in the student book)

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD12 Find volumes of right

MA.5.GR.3.2 Find the volume of a right

rectangular prisms with whole-number edge lengths by using a formula. (MA.5.GR.3.2)

rectangular prism with whole-number side lengths using a visual model and a formula.

485

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 23

Fluency

Hidden Factors Materials—S: Decimal Number Cards, Hidden Factors Mat

Students determine the product and then write and say a multiplication equation or related division equation to build fluency with multiplying and dividing decimals from module 4. Have students form pairs. Distribute a deck of Decimal Number Cards to each pair. Have them use the following procedure to play. Consider doing a practice round with students. • Partners place the deck of cards face down next to the Hidden Factors Mat. • Partner A and partner B flip over a card from the top of the pile and place it on a blue square.

0.5 × 0.3

Partners A and B: “0.15” Partner A: “0.5 × 0.3 = 0.15” Partner B: “0.15 ÷ 0.5 = 0.3”

486

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 23

Counting with Centimeter Cubes Students determine how many centimeter cubes are in one layer and then count to find the volume of a right rectangular prism. = 1 cubic centimeter

Display the right rectangular prism and the layer of centimeter cubes. What is the volume of the layer of centimeter cubes next to the prism? Raise your hand when you know.

5 cubic centimeters Display the layer of cubes inside the prism. The 5 centimeter cubes represent one layer of the prism. Let’s count by 5 cubic centimeters to pack the prism and find the volume. The first measurement you say is 5 cubic centimeters. Ready? Display each layer in the prism one at a time as students count to 25 cubic centimeters.

5 cubic centimeters, 10 cubic centimeters, 15 cubic centimeters, 20 cubic centimeters, 25 cubic centimeters What is the volume of the prism? Raise your hand when you know.

25 cubic centimeters Display the volume. Repeat the process with the following sequence:

The volume is 12 cubic centimeters.

The volume is

25 cubic centimeters.

The volume is 24 cubic centimeters.

487

5 ▸ M5 ▸ TD ▸ Lesson 23

Launch

EUREKA MATH2 Florida

Students determine whether they have enough information to find the volume of a right rectangular prism. Display the prism with a row and a column of cubes. Present the following question and use the Math Chat routine to engage students in mathematical discourse. Do you have enough information to determine the volume of this right rectangular prism? Why? Give students 1 minute of silent think time to think about the question, sketching ideas on their whiteboards if necessary. Have students give a silent signal to indicate they are finished. Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Purposefully choose reasoning that allows for rich discussion about finding volume. Then facilitate a class discussion. Invite students to share their thinking with the whole group. Ask questions that invite students to make connections and encourage them to ask questions of their own. What do you notice about the picture? What do you wonder? I notice part of one layer of cubes in a right rectangular prism. I wonder how tall the prism is.

Multilingual Learner Support To support student discourse, consider inviting students to use the Agree or Disagree section of the Talking Tool when discussing whether there is enough information given to determine the volume of the right rectangular prism.

I wonder what the volume of the prism is.

488

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 23

What do you know about the prism from the picture? I know that it has a width of 3 units and a length of 4 units. I know that its bottom layer has a volume of 12 cubic units. I know that the area of the bottom face is 12 square units. What changes if we turn the prism so it lays on a different face? Do we know the same information? We still know that the prism has one edge length that is 3 units and another that is 4 units. We still know that a layer has a volume of 12 cubic units, but it might not be the bottom layer. We still know that the area of a face is 12 square units. Do you have enough information to determine the volume of this right rectangular prism? Why? No. I don’t know the height of the prism. No. I don’t know how many layers of cubes there are. We can estimate the prism’s volume. What is an underestimate of the volume? Why? An underestimate is 24 cubic units. It looks like more than 2 layers of cubes fit in the prism. What is an overestimate of the volume? Why? An overestimate is 120 cubic units. It does not look like 10 layers of cubes fit in the prism. Transition to the next segment by framing the work. Today, we will determine another way to find the volume of a right rectangular prism.

489

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 23

Learn

Write Another Formula Students determine that they can calculate the volume of a right rectangular prism by multiplying its length, width, and height.

Universal Design for Learning: Representation Consider creating the prism out of interlocking cubes, as shown, to support students throughout the discussion.

Display the prism packed with cubes. Imagine that we packed the prism with unit cubes as shown. What is the volume of the prism? How do you know? The volume is 60 cubic units. I know because we determined that the volume of the bottom layer is 12 cubic units, we have 5 layers, and 12 × 5 = 60. The volume is 60 cubic units. I know because the area of the base is 12 square units, the height is 5 units, and 12 × 5 = 60. Write the formula V = B × h and use it to find the volume: We know that we can calculate the volume of a right rectangular prism by multiplying the area of its base by its height. How do you know that the area of the base of this prism is 12 square units?

V=B×h = 12 × 5 = 60 The volume is 60 cubic units.

The base is a rectangle, so I can multiply its length by its width, or 3 by 4, to get 12, the area. To determine the area of the base, you multiply the length of the base, which is also the length of the prism, by the width of the base, which is also the width of the prism.

490

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 23

Display the prism with the length, width, and height labeled. Use the formula V = B × h and write the area of the base as (3 × 4) next to the calculation for volume that was already written.

V=B×h = 12 × 5 = 60

V = (3 × 4) × 5 = 12 × 5 = 60

Height

Length

Width

The volume is 60 cubic units. The volume is 60 cubic units. How did we calculate volume differently than we have before? We determined the area of the base instead of just writing the number that represents the area of the base. We calculated by multiplying three numbers instead of two. Write the formulas V = B × h and V = l × w × h next to one another. Invite students to turn and talk about how the formulas are similar and how they are different. How is it possible that both formulas can be used to determine the volume of a right rectangular prism? We know that we can determine the area of the base of the prism by multiplying the length and the width of the base, so we can also write the formula V = B × h as V = l × w × h. Invite students to Think–Pair–Share about what each letter in the formula stands for. The V stands for volume. The l stands for length. The w stands for width. The h stands for height. How do we know that multiplying the length and width by the height will always give us the volume of a right rectangular prism? Multiplying the length and the width gives us the area of the base, and we know that we can always multiply the area of the base by the height to find the volume. © Great Minds PBC

Universal Design for Learning: Representation If you created an anchor chart in a previous lesson to summarize ways to find volume, add the V = l × w × h formula to that chart. In addition, consider color-coding the chart to show that l × w has the same value as B.

The area of the base is 6 square units. The height is 4 units. V=Bxh 24 = 6 x 4 The volume is 24 cubic units. The length is 3 units. The width is 2 units. The height is 4 units. V=lxwxh 24 = 3 x 2 x 4 The volume is 24 cubic units.

491

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 23

Display the compositions of the prism from lesson 22. Yuna’s Composition

Ryan’s Composition

Jada’s Composition

(2 × 3) × 4

(3 × 4) × 2

(2 × 4) × 3

Invite students to turn and talk about how Yuna, Ryan, and Jada thought about the layers differently. We can connect how the different students thought about the layers to the formula V = B × h. Point to the corresponding prisms as you make the following statements. Yuna’s composition uses the bottom face as the base and 4 as the height. Ryan’s composition uses the front face as the base and 2 as the height. Jada’s composition uses the right face as the base and 3 as the height. Look at the three multiplication expressions that represent the compositions. What patterns do you notice? Each expression represents the same volume. The same three numbers are multiplied in each expression, but in different orders. The three numbers that are multiplied are the length, width, and height of the prism. Multiplying the lengths of the sides gives the same volume as counting the number of cubes in each layer.

492

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 23

What is the volume of this prism?

24 cubic units What are some multiplication equations we can use to represent the volume?

(2 × 3) × 4 = 24 2 × (3 × 4) = 24 (2 × 4) × 3 = 24 2 × (4 × 3) = 24 (3 × 4) × 2 = 24 Record the equations. Point to the factors in the equations and ask the following question. What do the factors represent? The edge lengths of the prism The width, length, and height of the prism Point to the products in the equations and ask the following questions. What does the product in each equation represent? The volume of the prism Does the order in which the factors, or the edge lengths, are multiplied change the volume? Why? No. The commutative property of multiplication says that we can multiply in any order and get the same product. No. The associative property of multiplication says that we can group factors in any way and get the same product. Invite students to turn and talk about the different strategies they can use to find the volume of a right rectangular prism.

Promoting Mathematical Thinking and Reasoning Students use patterns and structure (MTR.5) when they apply their understanding of the formula V = B × h to make sense of the formula V = l × w × h and use it to determine the volume of a right rectangular prism. Ask the following questions to promote MTR.5: • How are the formulas V = B × h and V = l × w × h related? How can that help you determine the volume of a right rectangular prism? • What is another way you can identify the length, width, and height of the prism that will help you determine the volume?

493

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 23

Use Edge Lengths to Find Volume Students use V = l × w × h to determine the volumes of right rectangular prisms. Display prism A from problem 1. Why might using a formula be a good method for finding the volume of this right rectangular prism? We know the measures of the edge lengths, so using a formula may be more efficient than decomposing the prism into layers.

14 in

3 in 6 in

Direct students to problem 1 in their books. Write V = l × w × h. Invite students to write the formula in their books. Then guide students to identify the length, width, and height of the prism by labeling one edge as l, one edge as w, and one edge as h. Pair students and direct them to complete problem 1 with their partner. Circulate as students work, asking the following questions: • What strategies can you use to determine the volume of the prism? • Can you use a formula to determine the volume of the prism? Which formula can you use? Why? • How do you use the formula?

Differentiation: Challenge Consider challenging students to identify three edge lengths that would create a prism with a volume that is • 4 times the volume of prism A from problem 1 or • half the volume of prism A from problem 1.

• Does your answer make sense? Why?

494

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 23

1. Which right rectangular prism has the greater volume? Prism A

Prism B

The area of the base is 20 square inches and the height is 12 inches.

3 in

14 in

6 in

V = 14 × 6 × 3

V = 20 × 12

= 84 × 3

= 240

= 252

The volume of prism B is 240 cubic inches.

The volume of prism A is 252 cubic inches. Prism A has the greater volume. When students are finished, continue to display prism A.

What methods can we use to find the volume of prism A? We can use the formula V = l × w × h. We can multiply the length and width to determine the area of the base. Then we can multiply the area of the base by the height. We can decompose the prism into layers, determine the volume of each layer, and then multiply the volume of each layer by the number of layers. Do you need to use 14 for the length, 6 for the width, and 3 for the height of prism A? Explain. No. We can multiply the three edge lengths in any order and find the same volume. © Great Minds PBC

495

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 23

No. If we turn the prism on its side, we might have a different way of looking at length, width, and height. What method did you use to find the volume of prism B? I multiplied the area of the base by the height. Write the equations showing various ways to find the volume of prism A. When looking at prism A from problem 1, different students calculated their volume in these different ways. Will all students calculate the same volume? Why? Yes. Each way shows the same three numbers being multiplied but in different orders or with different groupings. Write (14 × 6) × 3 = (3 × 6) × 14. Encourage students to point to the prism to support their reasoning.

V = 14 × 6 × 3 V = (14 × 6) × 3 V = 14 × (6 × 3) V = 6 × 14 × 3 V = 6 × 3 × 14 Differentiation: Challenge

How can we use prism A to explain why this equation is true? If we think of the base of the prism as the face with dimensions of 14 inches and 6 inches, then the area of the base of the prism is 14 × 6 square inches and the volume of the prism is (14 × 6) × 3 cubic inches.

Consider presenting either of the following problems for students who need an additional challenge.

If we think of the base of the prism as the face with dimensions of 3 inches and 6 inches, then the area of the base of the prism is 3 × 6 square inches and the volume of the prism is (3 × 6) × 14 cubic inches.

• The length of prism A is 3 times as long

Display the 5-inch cube.

of prism A

is   1 as tall as the height of prism B. The 2

width of prism A is equal to the width of

This right rectangular prism is a cube. Do you think we can use the volume formula for a right rectangular prism to find the volume of this cube? Why?

prism B. Complete the following sentence: The volume of prism A is

times

as much as the volume of prism B.

Yes. A cube is a right rectangular prism, so the formula should work. How do you know which edge is the length, which edge is the width, and which edge is the height?

as the length of prism B. The height

5 in

• A face on a cube has an area of 49 square centimeters. What is the volume of the cube?

All edges are the same length, so it doesn’t matter which one is which. The multiplication equation would be the same no matter how you order the factors.

496

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 23

Direct students to complete problem 2 with their partner. 2. Find the volume of the cube.

V=5×5×5 = 125 The volume is 125 cubic inches.

5 in What is the volume of the cube?

125 cubic inches Invite students to turn and talk about how it can be helpful to use one of the formulas to find the volume of a right rectangular prism.

Find an Unknown Edge Length

Differentiation: Support

Students write an equation for finding the unknown edge length of a right rectangular prism. Display the right rectangular prism from problem 3 that has dimensions of 5 feet by 4 feet by h feet. Invite students to turn and talk about what is different about this prism.

h ft

What information is given about this right rectangular prism?

V=l×w×h 100 = 5 × 4 × h

The volume is 100 cubic feet. One edge measures 4 feet. Another edge measures 5 feet. Write V = l × w × h and direct students to do the same in their books.

5 ft

For students who need additional support, write the formula and the equation below it, aligning the numbers that are known and those that are not. Then make the connection to layers to cement understanding of the meaning of the formula.

4 ft

The volume is 100 cubic feet.

Then invite students to Think–Pair–Share about how to use the given information to write an equation with the volume formula.

• What does the 5 × 4 in this equation mean? • What does the h mean? How does it relate to the 100? • Tell me, using the word layers, how the 100 relates to the 5 and 4.

497

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 23

Write 100 = 5 × 4 × h and invite students to do the same. Direct students to work with a partner to find the unknown height. 3. What is the height of the right rectangular prism?

V=l×w×h 100 = 5 × 4 × h 100 = 20 × h

h ft

h=5 The height is 5 feet.

5 ft

4 ft

The volume is 100 cubic feet. What is the value of h?

5 What is the height of the prism?

5 feet What method did you use to find the height? I thought about it as an unknown factor problem. I know 4 × 5 = 20 and 20 × 5 = 100. So 100 must be equal to 5 × 4 × 5. Invite students to turn and talk about how to use one of the volume formulas to find unknown edge lengths.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

498

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 23

Land

Debrief 5 min Objective: Find the volumes of right rectangular prisms by multiplying the edge lengths. Facilitate a class discussion about using multiplication to find the volume of prisms by using the following prompts. Encourage students to restate or add on to their classmates’ responses. What is a formula for the volume of a right rectangular prism?

V=B×h V=l×w×h What does the formula V = l × w × h mean? The volume of a right rectangular prism is equal to the product of its length, width, and height. How are the formulas similar? Both formulas represent the area of the base times the height. In the V = l × w × h formula, the area of the base is written out as length times width. Why is having two formulas for the volume of a right rectangular prism helpful? I do not have to develop a new strategy for new problems. I can use one of the formulas to solve problems about the volume of any right rectangular prism if I have enough information. I can choose which formula to use based on the information I have about the right rectangular prism.

499

5 ▸ M5 ▸ TD ▸ Lesson 23

EUREKA MATH2 Florida

500

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 23

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can use the edge lengths of a right rectangular prism to find its volume.

PRACTICE

Florida B.E.S.T. MA.5.GR.3.2

Find the volume of each right rectangular prism. 2.

Name

Date

3 cm

1. A right rectangular prism with a length of 5 centimeters, a width of 3 centimeters, and a height of 6 centimeters and its composition are shown. Complete each multiplication expression to match the composition. a.

3 cm

7 in

27 cubic centimeters 4 in (5 × 3) ×

2 in 56 cubic inches

(

5 ft

)×3

4m 6m 10 ft

150 cubic feet

3 ft

5m 120 cubic meters

(

)×

249

250

PROBLEM SET

501

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 23

6. Use the right rectangular prism shown to complete parts (a) and (b).

Find the unknown edge length of each right rectangular prism. Show your work. 8.

5 cm

5 cm w cm 5 cm l cm

6 in

V=l×w×h

a. Fill in the blanks to complete the equation that represents the volume of the right rectangular prism.

200 =

h in

4 cm The volume is 140 cubic centimeters.

140 = 4 × w × 5

140 = 4 × 5 × w

3 in

The volume is 90 cubic inches.

140 = 20 × w

b. What is the value of l?

V=l×w×h

w=7

90 = 3 × 6 × h

7 centimeters

90 = 18 × h h=5 5 inches

502

PROBLEM SET

251

252

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 23

EXIT TICKET

I can use the edge lengths of a right rectangular prism to find its volume.

Florida B.E.S.T. MA.5.GR.3.2

Name

Date

Find the volume of the right rectangular prism.

10 units

4 units 8 units V=l×w×h = 8 × 4 × 10 = 32 × 10 = 320 The volume is 320 cubic units.

253

503

LESSON 24

Solve word problems involving volumes of right rectangular prisms. I can find the volumes of right rectangular prisms to solve word problems.

Lesson at a Glance Students work in pairs to solve word problems involving volumes of right rectangular prisms. Students use the volume formulas to find either volumes or unknown dimensions of prisms. When given the volume of a prism, students determine possible dimensions of the prism by decomposing numbers into factors and by using the commutative and associative properties of multiplication.

Key Questions • How can we use the formulas for the volume of a right rectangular prism to solve real-world problems? • If you know a right rectangular prism’s volume, how can you determine possible dimensions of the prism?

Mathematical Thinking and Reasoning Standard MTR.7.1

Language Objectives • Read word problems involving volumes of right rectangular prisms. Listen to and orally describe the problems’ meanings and possible strategies for solving them to compare solution strategies. • Write or draw to record a solution strategy and an equation to solve a problem involving the volume of a right rectangular prism. • Orally describe how to find the dimensions of a right rectangular prism by decomposing the prism’s volume into factors.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 24

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Consider tearing out the Sprint pages in advance of the lesson.

Launch 5 min Learn 35 min • Find an Unknown Edge Length of an Aquarium • Use Volume to Solve Problems

• None

Students • Multiply with Decimals Sprint (in the student book)

• Find Dimensions of an Aquarium • Problem Set

Land 10 min

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD13 Solve real-world problems

MA.5.GR.3.3 Solve real-world problems

involving volumes of right rectangular prisms. (MA.5.GR.3.3)

involving the volume of right rectangular prisms, including problems with an unknown edge length, with whole-number edge lengths using a visual model or a formula. Write an equation with a variable for the unknown to represent the problem.

505

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 24

Fluency

Sprint: Multiply with Decimals Materials—S: Multiply with Decimals Sprint

Students write the product to build fluency multiplying with decimals from module 4. Have students read the instructions and complete the sample problems. Teacher Note

Write the product. 1. 3 × 8 = 2. 0.3 × 0.8 = 3. 0.3 × 0.08 =

24 0.24

Validate all correct responses that may not be shown. For example, students may choose to write 0.04 instead of 0.040 when multiplying 0.5 × 0.08 on Sprint A. Either representation is correct.

0.024

Direct students to Sprint A. Frame the task. I do not expect you to finish. Do as many problems as you can, your personal best. Take your mark. Get set. Think! Time students for 1 minute on Sprint A. Stop! Underline the last problem you did. I’m going to read the answers. As I read the answers, call out “Yes!” if you got it correct. If you made a mistake, circle the answer. Read the answers to Sprint A quickly and energetically. Count the number you got correct and write the number at the top of the page. This is your personal goal for Sprint B.

506

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 24

Celebrate students’ effort and success. Provide about 2 minutes to allow students to complete more problems or to analyze and discuss patterns in Sprint A. If students are provided time to complete more problems on Sprint A, reread the answers but do not have them alter their personal goals. Lead students in one fast-paced and one slow-paced counting activity, each with a stretch or physical movement. Point to the number you got correct on Sprint A. Remember this is your personal goal for Sprint B.

Teacher Note Consider asking the following questions to discuss the patterns in Sprint A: • How can you use problem 1 to help solve problem 2? • How do problems 1–12 compare to problems 13–21?

Direct students to Sprint B. Take your mark. Get set. Improve! Time students for 1 minute on Sprint B. Stop! Underline the last problem you did. I’m going to read the answers. As I read the answers, call out “Yes!” if you got it correct. If you made a mistake, circle the answer. Read the answers to Sprint B quickly and energetically. Count the number you got correct and write the number at the top of the page.

Teacher Note Count forward by 3 tenths in decimal form from 0 to 3 for the fast-paced counting activity. Count backward by 2 tenths in decimal form from 2 to 0 for the slow-paced counting activity.

Calculate your improvement score and write the number at the top of the page. Celebrate students’ improvement.

507

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 24

Launch

Students compare aquariums with different dimensions. Display the two aquariums.

Aquarium A

Aquarium B

Multilingual Learner Support Consider activating prior knowledge by discussing student experiences with aquariums. Show real-world pictures as students share what they know about home or public aquariums.

What do you notice about the two aquariums? What do you wonder? I notice that they have different widths and lengths, but it looks like they might have the same height. I notice that aquarium B is wider than aquarium A. I wonder if they have the same volume. I wonder which one holds more water. I wonder what the length, width, and height is of each aquarium. Is it possible that the two aquariums have the same volume? Why? Yes. They could have different lengths, widths, and heights that multiply to the same number.

508

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 24

Display the two aquariums with the widths and heights labeled.

Aquarium A

Aquarium B

32 cm

40 cm 25 cm 25 cm

What do you notice now? Both aquariums have a height of 25 cm. The aquariums have different widths. Point to an unlabeled edge representing the length of each aquarium. We still do not know the lengths of the aquariums. We said it is possible that the aquariums have the same volume. If the aquariums have the same volume, what else do we know about the aquariums? The areas of the top faces of the aquariums must be equal. The areas of the bases of the aquariums must be equal. Have students turn and talk about what they need to know to determine the length of each aquarium. Transition to the next segment by framing the work. Today, we will solve real-world problems involving volumes of right rectangular prisms.

509

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 24

Learn

Find an Unknown Edge Length of an Aquarium Students use a formula for the volume of a right rectangular prism to find an unknown edge length. Display the two aquariums with the widths, heights, and volumes labeled.

Aquarium A

Aquarium B

32 cm

40 cm 25 cm 25 cm

The volume of aquarium A is 20,000 cubic centimeters.

The volume of aquarium B is 20,000 cubic centimeters.

What new information do we have now? The volumes of the aquariums are the same. Each aquarium has a volume of 20,000 cubic centimeters. Direct students to problem 1 in their books and have students label the widths, heights, and volumes of the aquariums.

510

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 24

1. Consider the two aquariums.

Aquarium A

Aquarium B

32 cm

40 cm 25 cm

25 cm

l cm

V = 20,000 cubic centimeters

a. Find the length of aquarium A.

b. Find the length of aquarium B.

V=l×w×h

20,000 = l × 32 × 25

20,000 = l × 40 × 25

20,000 = l × 800

20,000 = l × 1,000

25 = l The length of aquarium A is 25 cm.

20 = l The length of aquarium B is 20 cm.

Let’s label the unknown length of each aquarium l cm. Invite students to Think–Pair–Share about how to use the given information to write an equation by using the volume formula for each aquarium. For aquarium A, we could write 20,000 = l × 32 × 25. For aquarium B, we could write 20,000 = l × 40 × 25. Direct students to work with a partner to find the unknown length of each aquarium. When students are finished, invite pairs to share their work. For aquarium A, we multiplied 32 and 25 and found a product of 800. We found the unknown factor by dividing 20,000 by 800. The length of aquarium A is 25 cm.

511

5 ▸ M5 ▸ TD ▸ Lesson 24

EUREKA MATH2 Florida

For aquarium B, we multiplied 25 and 40 and found a product of 1,000. We know 20 × 1,000 = 20,000. So the length of aquarium B is 20 cm. Invite students to turn and talk about how they found the unknown edge lengths of the aquariums.

Use Volume to Solve Problems Students use a formula for the volume of a right rectangular prism to find an unknown edge length without a diagram of the prism. To prepare students for finding an unknown edge length of a prism without a diagram of the prism, ask the following question. Do you need a diagram of a right rectangular prism to determine its volume? Why? No. If I know its length, width, and height, I can determine its volume by using the formula V = l × w × h. No. If I know the area of its base and its height, I can determine its volume by using the formula V = B × h. Do you need a diagram of a right rectangular prism to find an unknown edge length? Why? No. If I know the volume and two of the three dimensions, I can find the other dimension by thinking about it as an unknown factor problem. No. If I know the volume and the area of the base, I can find the height by dividing. Pair students and direct them to complete problems 2 and 3. Circulate as students work and support their understanding by asking questions such as the following: • What information do you know? • How can you use the information you know to write an equation by using the volume formula? • How can you find the unknown edge length?

512

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 24

2. A pet store has the following aquariums for sale. Write an equation with a letter to represent the unknown for each aquarium. Then find each unknown. Aquarium Length (cm)

Width (cm)

Height (cm)

Volume (cubic cm)

Equation

9,000

V = 30 × 15 × 20

12,500

12,500 = 20 × w × 25

30,000

30,000 = l × 20 × 30

32,000

32,000 = 40 × 20 × h

54,000

54,000 = 60 × w × 30

3. Riley buys an aquarium that is shaped like a right rectangular prism. The aquarium has a volume of 15,000 cubic centimeters. It is 20 centimeters long and 25 centimeters wide. How tall is the aquarium?

V=l×w×h 15,000 = 20 × 25 × h 15,000 = 500 × h 30 = h The aquarium is 30 centimeters tall. When most students are finished, invite pairs to share their work for each problem and confirm the answers. Ask questions that encourage students to make connections and encourage students to ask questions of their own. Have students turn and talk about how they can determine an unknown edge length for any right rectangular prism.

513

5 ▸ M5 ▸ TD ▸ Lesson 24

EUREKA MATH2 Florida

Find Dimensions of an Aquarium Students determine possible dimensions of a right rectangular prism when given its volume. Use the following prompts to facilitate a discussion about how to determine the dimensions of an aquarium when the volume is known. Fish are different sizes. Some require less space in an aquarium, and others require more space. For the type of fish Kelly wants, he needs an aquarium with a volume of 30,000 cubic centimeters. Invite students to Think–Pair–Share about the following prompts. If we know that the volume of an aquarium shaped like a right rectangular prism is 30,000 cubic centimeters, can we determine a possible length, width, and height of the aquarium? How? Yes. We can find three numbers that we can multiply to get 30,000. I don’t know. I think we need to know two of the dimensions so we can divide to find the third. I don’t know. We could try multiplying some numbers, but it could take a long time to figure it out. We can find factors of 30,000 to help us determine the length, width, and height of an aquarium with a volume of 30,000 cubic centimeters. One way to get 30,000 is to multiply 1 by 1 by 30,000, but does it make sense to have an aquarium that is 1 centimeter by 1 centimeter by 30,000 centimeters? Why? No. The aquarium would be too long and narrow. There would not be enough room for fish to swim around. Let’s start by finding two factors of 30,000. How many times as much as 100 is 30,000?

300 We need three factors. Can we find two factors of 100 or two factors of 300? How? Yes. We can find two numbers that multiply to get 100, or we can find two numbers that multiply to get 300. 514

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 24

Let’s decompose 300 into two factors. What are some factor pairs for 300?

5 and 60 30 and 10 15 and 20 Let’s think about 5 and 60. We know 100 × 5 × 60 = 30,000, which means a right rectangular prism that is 100 centimeters long, 5 centimeters wide, and 60 centimeters tall has a volume of 30,000 cubic centimeters. Does it make sense to have an aquarium that size? Why? No. The aquarium would be not be wide enough for fish to swim around. Display the picture of Leo’s work. Invite students to Think–Pair–Share about what steps Leo took to write 30,000 as a product of three factors. Leo wrote 30,000 by using two factors, 250 and 120. He broke apart 250 and 120 and wrote each of them as a multiplication expression.

30,000 = 250 × 120 = (25 × 10 10)) × (30 (30 × 4) = 25 × 10 × 30 × 4 = 25 × 30 × 10 × 4 = 25 × 30 × (10 (10 × 4) = 25 × 30 × 40

He switched the order of the factors and grouped them in another way to get 25 × 30 × 40. Could Leo have written 250 as a product of factors other than 25 and 10? What about 120?

Promoting Mathematical Thinking and Reasoning When students find factors of a number to help them determine possible dimensions of an aquarium in the shape of a right rectangular prism with a given volume, they are applying mathematics to real-world contexts (MTR.7). Ask the following questions to promote MTR.7:

Yes. There are other pairs of numbers that we can multiply to get 250, and there are other pairs of numbers we can multiply to get 120.

• What do the factors of a number mean in a situation about volume of a right rectangular prism?

Where did Leo use the commutative property of multiplication in his work?

• What else do you wish you knew to help find possible dimensions of the aquarium? Can you make a reasonable assumption about that?

Leo changed the order of the factors 10 and 30. How did Leo use the associative property of multiplication to find that 40 is a factor of 30,000? He grouped the factors 10 and 4 together to get 40. So a right rectangular prism that is 25 centimeters long, 30 centimeters wide, and 40 centimeters tall has a volume of 30,000 cubic centimeters. Does it make sense to have an aquarium with those dimensions?

515

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 24

Do you think there are other prisms that have a volume of 30,000 cubic centimeters? Why? Yes. There are other prisms that have a volume of 30,000 cubic centimeters because 30,000 has many factors. Direct students to problem 4. Invite them to work with their partner to find the length, width, and height of at least two more right rectangular prisms with a volume of 30,000 cubic centimeters. Differentiation: Support 4. Find the length, width, and height of at least two different right rectangular prisms that each have a volume of 30,000 cubic centimeters. Sample:

30,000 = 100 × 300 = (20 × 5) × (50 × 6) = 20 × 5 × 50 × 6 = 20 × 50 × 5 × 6 = 20 × 50 × (5 × 6)

Consider providing students who need additional support with problem 4 with one of the following problems, which allows them to find factors of smaller numbers. • Find the length, width, and height of two right rectangular prisms with a volume of 36 cubic centimeters. • Find the length, width, and height of two right rectangular prisms with a volume of 200 cubic centimeters.

= 20 × 50 × 30 The length is 20 centimeters, the width is 50 centimeters, and the height is 30 centimeters.

30,000 = 200 × 150 = (40 × 5) × (50 × 3) = 40 × 5 × 50 × 3 = 40 × 50 × 5 × 3 = 40 × 50 × (5 × 3) = 40 × 50 × 15 The length is 40 centimeters, the width is 50 centimeters, and the height is 15 centimeters.

516

Universal Design for Learning: Action & Expression To support metacognition, encourage students to monitor their progress by having them take turns thinking aloud with a partner as they complete problem 4. For example, partner A thinks aloud and explains how they decompose 30,000, first into two factors and then into three factors, while partner B listens and asks questions. Then partners switch roles and repeat.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 24

When most students are finished, create a table to record their responses. Include 25 centimeters, 30 centimeters, and 40 centimeters from Leo’s work in the top row of the table. Invite students to share possible dimensions of right rectangular prisms with a volume of 30,000 cubic centimeters. As students share, record the dimensions in the table for the class to see. Consider including some of the sample answers shown if students do not share these dimensions.

Length (cm) 25 20 40 20 75

Width (cm) 30 50 50 60 25

Height (cm) 40 30 15 25 16

Kelly cannot have an aquarium taller than 25 centimeters because it will not fit on his shelf. How does that affect his choices? That means he can’t have an aquarium with a length of 25 centimeters, a width of 30 centimeters, and a height of 40 centimeters because it is too tall. He can’t have an aquarium with a length of 20 centimeters, a width of 50 centimeters, and a height of 30 centimeters because it is too tall. Invite students to turn and talk about how to determine possible lengths, widths, and heights of right rectangular prisms when given the volume.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

517

5 ▸ M5 ▸ TD ▸ Lesson 24

Land

EUREKA MATH2 Florida

Debrief 5 min Objective: Solve word problems involving volumes of right rectangular prisms. Facilitate a class discussion about solving word problems involving volumes of right rectangular prisms by using the following prompts. Encourage students to restate or add on to their classmates’ responses. How can we use the formulas for the volume of a right rectangular prism to solve real-world problems? We can use the formula V = l × w × h to find the volume of a real-world object if it is shaped like a right rectangular prism and we know its length, width, and height. We can use the formula V = l × w × h to find one dimension when the object is shaped like a right rectangular prism and we know its volume and its other two dimensions. We can use the formula V = B × h to find the volume of a real-world object if it is shaped like a right rectangular prism and we know its height and the area of its base. We can use the formula V = B × h to find the area of an object’s base when it is shaped like a right rectangular prism and we know its volume and height. Or we can use it to find the object’s height when we know its volume and the area of its base. Can there be right rectangular prisms that have different lengths, widths, and heights but equal volume? How? Yes. Right rectangular prisms can have different lengths, widths, and heights but still take up the same amount of space if the products of the lengths, widths, and heights are equal. Yes. We can multiply different sets of numbers and get the same product. So we can multiply different lengths, widths, and heights and get the same volume. If you know a right rectangular prism’s volume, can you determine possible dimensions of the prism? How? Yes. We can find three numbers that multiply to get the number of cubic units. 518

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 24

Yes. We can break apart the number of cubic units into two factors. Then we can break apart one of those factors into two factors. Yes. We can break apart the number of cubic units into two factors. Then we can break apart both of those factors and use the commutative property of multiplication and associative property of multiplication to rearrange and regroup the factors as necessary to get three factors.

519

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 24

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Number Correct:

Write the product.

23. 6 × 7

2. 0.2 × 0.3 =

0.06

3. 0.2 × 0.4 =

0.08

4. 3 × 4

26. 7 × 7

5. 0.3 × 0.4 =

0.12

6. 0.3 × 0.5 =

0.15

7. 4 × 7

29. 8 × 7

8. 0.4 × 0.7 =

0.28

9. 0.4 × 0.8 =

0.32

10. 5 × 6

32. 9 × 7

11. 0.5 × 0.6 =

0.30

12. 0.5 × 0.7 =

0.35

13. 3 × 6

35. 2 × 8

14. 0.3 × 0.06 =

0.018

36. 2 × 9

15. 0.3 × 0.07 =

0.021

17. 0.4 × 0.08 =

0.032

18. 0.4 × 0.09 =

0.036

19. 5 × 8

20. 0.5 × 0.08 =

0.040

21. 0.5 × 0.09 =

0.045

22. 0.09 × 0.5 =

0.045

16. 4 × 8

256

520

Improvement:

Write the product.

1. 2 × 3

Number Correct:

1. 2 × 2

24. 0.6 × 0.8 =

0.48

2. 0.2 × 0.2 =

0.04

25. 0.6 × 0.09 =

0.054

3. 0.2 × 0.3 =

0.06

23. 6 × 6

24. 0.6 × 0.7 =

0.42

25. 0.6 × 0.08 =

0.048

26. 7 × 6

4. 3 × 3

27. 0.7 × 0.8 =

0.56

5. 0.3 × 0.3 =

0.09

27. 0.7 × 0.7 =

0.49

28. 0.7 × 0.09 =

0.063

6. 0.3 × 0.4 =

0.12

28. 0.7 × 0.08 =

0.056

7. 4 × 6

29. 8 × 6

30. 0.8 × 0.8 =

0.64

8. 0.4 × 0.6 =

0.24

30. 0.8 × 0.7 =

0.56

31. 0.8 × 0.09 =

0.072

9. 0.4 × 0.7 =

0.28

31. 0.8 × 0.08 =

0.064

10. 5 × 5

32. 9 × 6

33. 0.9 × 0.8 =

0.72

11. 0.5 × 0.5 =

0.25

33. 0.9 × 0.7 =

0.63

34. 0.9 × 0.09 =

0.081

12. 0.5 × 0.6 =

0.30

34. 0.9 × 0.08 =

0.072

13. 3 × 5

35. 2 × 7

14. 0.3 × 0.05 =

0.015

36. 2 × 8

37. 0.6 × 0.11 =

0.066

15. 0.3 × 0.06 =

0.018

37. 0.5 × 0.11 =

0.055

38. 0.12 × 0.6 =

0.072

16. 4 × 7

38. 0.12 × 0.5 =

0.060

39. 0.7 × 0.11 =

0.077

17. 0.4 × 0.07 =

0.028

39. 0.6 × 0.11 =

0.066

40. 0.12 × 0.7 =

0.084

18. 0.4 × 0.08 =

0.032

40. 0.12 × 0.6 =

0.072

41. 0.8 × 0.11 =

0.088

41. 0.7 × 0.11 =

0.077

42. 0.12 × 0.8 =

0.096

20. 0.5 × 0.07 =

0.035

42. 0.12 × 0.7 =

0.084

43. 0.9 × 0.11 =

0.099

21. 0.5 × 0.08 =

0.040

43. 0.8 × 0.11 =

0.088

44. 0.12 × 0.9 =

0.108

22. 0.08 × 0.5 =

0.040

44. 0.12 × 0.8 =

0.096

19. 5 × 7

258

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 24

PROBLEM SET

I can find the volumes of right rectangular prisms to solve word problems.

PRACTICE

Florida B.E.S.T. MA.5.GR.3.3

Name

2. The right rectangular prism shown represents a fish tank. The volume of the fish tank is 24,000 cubic centimeters. What is the width of the fish tank?

Date

Use a volume formula to write an equation with a letter to represent the unknown. Then solve the problem. 1. The juice box shown is in the shape of a right rectangular prism. The volume of the juice box is 192 cubic centimeters. What is the height of the juice box?

30 cm

w cm 40 cm V=l×w×h

h cm

24,000 = 40 × w × 30 24,000 = 1,200 × w 4 cm

20 = w

6 cm

The width of the fish tank is 20 cm.

V=l×w×h 192 = 6 × 4 × h 192 = 24 × h 8=h

3. A store sells different-size gift boxes that are each shaped like a right rectangular prism. Write an equation with a letter to represent the unknown for each box. Then find each unknown.

The height of the juice box is 8 cm.

261

262

Box

Length (cm)

Width (cm)

Height (cm)

Volume (cu cm)

Equation

108

V=6×6×3

144

144 = 8 × w × 3

400

400 = l × 10 × 5

1,000

1,000 = 20 × 10 × h

1,440

1,440 = l× 12 × 10

PROBLEM SET

521

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 24

4. A teacher has a filing cabinet in the shape of a right rectangular prism with a base of 300 square inches. The volume of the filing cabinet is 9,000 cubic inches. How tall is the filing cabinet?

6. Mr. Perez asks his students to find the unknown height of the right rectangular prism.

9,000 = 300 × h 30 = h

h cm

The filing cabinet is 30 inches tall.

8 cm 12 cm Does Mr. Perez give the students enough information to find the unknown height? Explain. No. Mr. Perez does not give the students enough information. They need to know the volume of the right rectangular prism to find the unknown height.

5. A baseball card company puts packages of cards into larger boxes for shipping. Each shipping box is a right rectangular prism with a volume of 720 cubic inches. The base measures 72 square inches. a. What is the height of each shipping box?

720 = 72 × h

7. Adesh has a fish tank that is a right rectangular prism with a volume of 25,000 cubic centimeters. Write one possible length, width, and height for Adesh’s fish tank.

10 = h Each shipping box is 10 inches tall.

Sample:

100 centimeters, 10 centimeters, and 25 centimeters are possible dimensions for Adesh’s fish tank. b. Find at least three possible pairs of dimensions of the bases of the shipping boxes. Sample:

8 × 9 = 72 12 × 6 = 72 18 × 4 = 72 The bases of the boxes could be 8 inches by 9 inches, 12 inches by 6 inches, or 18 inches by 4 inches.

522

PROBLEM SET

263

264

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 24

EXIT TICKET

I can find the volumes of right rectangular prisms to solve word problems.

Florida B.E.S.T. MA.5.GR.3.3

8. Find the length, width, and height of at least two different right rectangular prisms that each have a volume of 20,000 cubic centimeters. Sample: One prism has a length of 50 centimeters, a width of 20 centimeters, and a height of 20 centimeters.

Name

Date

A fish tank is shaped like a right rectangular prism with a length of 20 centimeters and a width of

A second prism has a length of 40 centimeters, a width of 25 centimeters, and a height of 20 centimeters.

20 centimeters. The volume of the fish tank is 10,000 cubic centimeters. How tall is the fish tank? V=l×w×h 10,000 = 20 × 20 × h 10,000 = 400 × h 25 = h The fish tank is 25 centimeters tall.

PROBLEM SET

265

267

523

LESSON 25

Find the volumes of solid figures composed of right rectangular prisms. I can find the volumes of figures composed of right rectangular prisms.

Lesson at a Glance Students find the volumes of solid figures composed of right rectangular prisms. They decompose the figures into right rectangular prisms, find the volumes of the prisms, and add the volumes together. Students compare different ways that a figure can be decomposed.

Key Question • How can we use the volumes of right rectangular prisms to find the volume of a solid figure composed of those prisms?

Mathematical Thinking and Reasoning Standard MTR.1.1

Language Objectives • Orally decide how to decompose a figure into right rectangular prisms and how to use the decompositions to find the figure’s volume. Orally justify the decisions. • Orally compare different decompositions of a solid figure to conclude that there are multiple ways to find the volume of a solid figure composed of right rectangular prisms.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 25

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Gather 20 cubes of one color for each student. Provide a different color to each student in every group of three.

Launch 5 min Learn 35 min • Build a Solid • Find the Volume of a Solid Composed of Right Rectangular Prisms • Problem Set

• None

Students • Interlocking cubes, 1 cm (20)

Land 10 min

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD13 Solve real-world problems

MA.5.GR.3.3 Solve real-world problems

involving volumes of right rectangular prisms. (MA.5.GR.3.3)

525

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 25

Fluency

Whiteboard Exchange: Decimals Written Two Ways Students write a number in standard and expanded forms when given the word form to build fluency with reading and writing decimals to the thousandths place from module 4. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display three hundred seventy-five thousandths in word form. Write the number in standard form. Display the answer. Write the number in expanded form by using whole numbers and fractions. Display the answer.

Three hundred seventy-five thousandths

0.375 3×

1 1 1 +7× +5× 10 100 1,000

Teacher Note Consider having students use parentheses as a support to separate each digit. For example:

___

_____

(3 ×  1 )+ ( 7 ×   1 )+ ( 5 ×   1 ) 10 100 1,000 Students practice expanded form with decimals in lesson 27.

526

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 25

Repeat the process with the following sequence:

Five and two hundred eighty-nine thousandths

Seventy-four and thirty-six thousandths

5.289

74.036

5×1+2×

1 1 1 +8× +9× 10 100 1,000

7 × 10 + 4 × 1 + 3 ×

1 1 +6× 100 1,000

Sixteen and eight hundred two thousandths

Ninety and fifty-eight thousandths

16.802

90.058

1 × 10 + 6 × 1 + 8 ×

1 1 +2× 10 1,000

9 × 10 + 5 ×

1 1 +8× 100 1,000

527

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 25

Whiteboard Exchange: Multiply with Decimals Students determine the product to build fluency multiplying with decimals from module 4. Display 2 × 0.47 =

2 × 0.47 = 0.94

47 × 2

hundredths

Display the product and sample method. Teacher Note

Repeat the process with the following sequence:

6 × 3.52 = 21.12

8 × 10.4 = 83.2

14 × 0.32 = 4.48

Validate all correct methods of multiplication that may not be displayed. For example, students may choose to include the decimal in vertical form rather than writing the name of the units, or students may choose to use an area model.

0.47 2

4 tenths

7 hundredths

8 tenths

14 hundredths

0.94

528

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 25

Launch

Students discuss methods they can use to determine the volume of a solid figure composed of right rectangular prisms. Display the picture of the L-shaped fishpond. In the previous lesson, we made calculations about aquariums. Now let’s look at a fishpond. What do you notice about this pond? What do you wonder? I notice that the top of the water is shaped like two rectangles put together. I notice that it has lily pads in it. I wonder how deep it is. I wonder how you find the volume of something that’s not shaped like a right rectangular prism. I wonder how many fish live in that pond. Can we use the formula V = l × w × h to determine the volume of this pond? Why? No. It is not shaped like a right rectangular prism. Transition to the next segment by framing the work. Today, we will find the volumes of solid figures composed of right rectangular prisms.

Multilingual Learner Support Consider asking multilingual learners to expand on their wonders about the fishpond. Recognize and elevate their ideas while making explicit their mathematical thinking, especially as it pertains to the volume of the fishpond. Provide time for additional exploration independently or in interest groups. As students answer their own questions using their selected methods, invite them to present their findings to the class.

529

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 25

Learn

Build a Solid Materials—S: Cubes

Students build solid figures out of prisms to determine that volume is additive. Display the five solids each composed of six cubes.

Teacher Note The terms solid, figure, and solid figure are all used to describe three-dimensional figures.

What is similar about these five solids? What is different? They each have six cubes. Some of the solids are right rectangular prisms and some are not. Even though these solids look different, each is composed of the same number of cubes and takes up the same amount of space. What can we say about solids that take up the same amount of space? They have the same volume. Place students in groups of three such that each student has a different color of interlocking cubes. Invite students to use their cubes to build a right rectangular prism of any size. They do not need to use all their cubes. Direct students to record the volumes of their prisms on their whiteboards.

Teacher Note It is not necessary for students in a group to build right rectangular prisms with different dimensions or volumes as shown. Students may continue with the activity if two or three of the prisms built by the group are the same size.

When students are finished, ask them to combine their prisms to create a solid figure and then determine the volume of the figure they created. 530

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 25

Your group created a solid figure composed of right rectangular prisms. How did you determine the volume of your group’s solid? We added the volumes of each of the individual prisms. We added the number of cubes that each of us used. Ask groups to build a different figure by using the three prisms and find the volume. How did you determine the volume of your group’s new figure? We added the volumes of each of the individual prisms. We added the number of cubes that each of us used. When you built the second figure, did the volume change? Why? No. We used the same number of cubes both times. No. It doesn’t matter how we put the three prisms together. The volume of each prism is the same every time, so the total volume is the same every time. No. The total volume is always going to be the sum of the volumes of the three prisms, no matter how we put them together. Invite students to Think–Pair–Share about whether they can always find the volume of a figure composed of right rectangular prisms by adding the volumes of the prisms that compose the figure. Yes. The sum of the volumes of the pieces is the volume of the whole solid. Yes. The number of unit cubes stays the same no matter how they are arranged.

531

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 25

Find the Volume of a Solid Composed of Right Rectangular Prisms Students determine the volume of figures composed of right rectangular prisms. Display the figure composed of cubes.

Universal Design for Learning: Representation

How can you describe this figure? The figure is composed of cubes.

Consider providing a concrete experience by asking students to build the figure shown out of interlocking cubes, which will help them make connections between the figure and the prisms that compose it. Students can make the bottom prism out of one color and the top prism out of a different color and then use the colors to color the solid in problem 1.

There is a smaller prism on top of a larger prism. The figure is composed of a short prism attached to a tall prism. Invite students to Think–Pair–Share about how they can find the volume of this figure. I can count the cubes. I can decompose the figure into layers, find the volume of each layer, and add the volumes together. I can decompose the figure into two prisms, find the volume of each prism, and add the volumes together. Display the figure with the same shape that doesn’t show the cubes.

This solid figure has the same shape but doesn’t show the individual cubes. How can you find the volume of this figure when the cubes are not shown?

I can find the volume of the prism on the bottom and the volume of the prism on the top. Then I can add those volumes together. What are the length, width, and height of the prism on the bottom?

2m 6m

6 meters, 3 meters, and 2 meters What are the length, width, and height of the prism on the top?

2 meters, 3 meters, and 2 meters

532

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 25

How do you know that the unmarked edge of the prism on the top measures 3 meters? That edge is 3 meters because it has the same length as the edge directly below in the bottom prism. The prisms match up exactly on that edge. That means the unmarked edge of the top prism has the same length as the edge on the bottom prism. Direct students to problem 1 in their books. Invite them to label the unmarked edge of the top prism. Then pair students and direct them to complete the problem with their partner. Differentiation: Support

1. The figure is composed of right rectangular prisms. Calculate its volume.

L-sidebar-bodyConsider asking students to draw and label the dimensions of each prism individually so they can fully see each prism.

6 × 3 × 2 = 36 2 × 3 × 2 = 12

36 + 12 = 48 The volume is 48 cubic meters.

When students are finished, ask the following question.

Did you use a volume formula when finding the volume of the figure? If so, which one did you use and why did you use it? I used the V = l × w × h formula to determine the volume of each prism because I knew the length, width, and height of each prism. Display the prism from problem 1 without the decomposition.

How is this solid figure different from the one in problem 1? It doesn’t show how the figure is decomposed. It doesn’t show that one prism is sitting on top of another. It looks like one solid instead of two.

4m 2m 6m

Differentiation: Challenge Consider providing the following problem for students who need an additional challenge. Two right rectangular prisms have a combined volume of 135 cubic meters. Prism A has double the volume of prism B. If the base of prism A has an area of 10 square meters, what is its height?

533

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 25

What are the different ways that you decompose this figure into two right rectangular prisms? I could slice it vertically to create a left prism and a right prism. I could slice it horizontally to create a lower prism and an upper prism. Display the different ways to decompose the figure.

4m 2m 6m

Differentiation: Support

2m 6m

Consider providing printed copies of the prisms sliced both ways that students can draw on and label. If color copies are not available, invite students to color the individual prisms to match the displayed figures.

Which of these figures is decomposed the same way as the figure in problem 1? The figure on the right Point to the figure on the left. This is another way to decompose the figure in problem 1 to find its volume. What are the length, width, and height of the red prism?

2 meters, 3 meters, and 4 meters Invite students to Think–Pair–Share about the length, width, and height of the blue prism. Consider asking students to point to the displayed decomposition of the solid as they explain their thinking. The height of the blue prism is 2 meters. The width is 3 meters. It has the same width as the red prism. The entire edge length on the bottom is 6 meters, and the length of the red prism is 2 meters, so the length of the blue prism must be 4 meters.

534

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 25

Direct students to complete problem 2 with their partner. Circulate as they work, asking the following questions: • How can you decompose the figure? • What edge lengths can you label? • Why did you choose to decompose the figure that way? • What are the dimensions of that prism? How do you know? • How can you determine the volume of that prism? 2. The figure shown is composed of right rectangular prisms. a. Draw on the figure to decompose it into right rectangular prisms.

Promoting Mathematical Thinking and Reasoning

4m 3m

b. Find the volume of the figure. Sample:

V=7×4×3

= 28 × 3

When students find the volume of a solid figure composed of right rectangular prisms by determining how to decompose it, finding unlabeled edge lengths, finding the volume of each prism, and then deciding whether their answer makes sense, they are actively participating in effortful learning (MTR.1). Ask the following questions to promote MTR.1:

= 84 V=2×2×3

• What steps can you take to start finding the volume?

=4×3

• What is your plan for decomposing the figure?

= 12

• What information or facts do you need to find the volume?

84 + 12 = 96 The volume is 96 cubic meters. When students are finished, display the figure from problem 2.

Invite students to share how they decomposed the solid and how they determined the unknown dimensions that they needed for determining the volume of the prisms.

2m 535

5 ▸ M5 ▸ TD ▸ Lesson 25

EUREKA MATH2 Florida

Consider using the following prompts to guide the discussion: • Point to the solid to show how you decomposed the figure. • Why did you choose to decompose the figure that way? • What information did you need for finding the volume of that prism that you did not have? How did you find that information? • How did you know which edges to label? How did you determine an edge length that was not given? • What method did you use for finding the volume of the prism? Why? Invite students to turn and talk about methods they can use to determine the volume of a solid figure composed of right rectangular prisms.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

536

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 25

Land

Debrief 5 min Objective: Find the volumes of solid figures composed of right rectangular prisms. Gather the class with their Problem Sets. Facilitate a discussion about finding the volume of solid figures composed of right rectangular prisms by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Display the L-shaped solid figure. This figure is composed of right rectangular prisms. Can we use a volume formula to determine the volume of the figure? Why? No. We only have a volume formula for a right rectangular prism, and this figure is not a right rectangular prism.

4m 2m 6m

How can we use the volumes of right rectangular prisms to find the volume of this figure? We can decompose this figure into two right rectangular prisms. Then we can determine the volume of each prism and add the volumes together. Invite students to share how they decomposed the solids in problems 4–7 in the Problem Set and why they decomposed each solid in that way.

537

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 25

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can find the volumes of figures composed of right rectangular prisms.

PRACTICE

Florida B.E.S.T. MA.5.GR.3.3

Name

2. Two right rectangular prisms are shown. Find the volume of each right rectangular prism.

Date

1. Each solid figure shown is composed of centimeter cubes. Circle the figures with a volume of 8 cubic centimeters.

5 cm

4 cm

2 cm

3 cm

4 cm

40 cubic centimeters

24 cubic centimeters

3. The two right rectangular prisms from problem 2 are combined to make one solid figure. What is the volume of the figure?

4 cm 3 cm

5 cm 4 cm

538

271

272

PROBLEM SET

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 25

The solid figures shown are composed of right rectangular prisms. Calculate the volume of each figure.

10 in

7 cm

5 in

12 cm

440 cubic inches

6 ft

6 cm 2 in

4 cm

3 cm

2 ft

4 ft

15 in

2 cm

4 cm

8 in

11 cm

5 ft

8 cm

10 in 18 in

3 ft

10 ft

9 cm

300 cubic feet

552 cubic centimeters

PROBLEM SET

273

274

PROBLEM SET

3 cm

6 cm

8 cm 5 cm

592 cubic centimeters

539

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 25

EXIT TICKET

I can find the volumes of figures composed of right rectangular prisms.

Florida B.E.S.T. MA.5.GR.3.3

Name

Date

The solid figure shown is composed of right rectangular prisms. What is the volume of the figure?

V = 16 × 4 × 8 = 64 × 8

4 in 6 in

= 512 V=6×6×8

8 in

= 36 × 8 = 288 512 + 288 = 800

16 in

10 in

The volume is 800 cubic inches.

540

275

LESSON 26

Solve word problems involving perimeters of rectangles. I can solve word problems involving perimeters of rectangles.

Lesson at a Glance Students work in groups to complete a stations activity where they solve word problems involving perimeters with fractions and decimals. Students make sense of the problems by using the Read–Draw–Write process and solve the problems by using the formulas for perimeter of a rectangle.

Key Questions • How does the Read–Draw–Write process help you solve word problems involving perimeter with fractions and decimals?

Language Objective • Read word problems involving perimeters of rectangles. In groups and as a class, orally discuss solution strategies, and compare formulas chosen for each problem.

Mathematical Thinking and Reasoning Standard MTR.7.1

• How can you use a formula for perimeter to help you solve word problems involving perimeter with fractions and decimals?

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 26

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Launch 10 min

• Perimeter Stations (in the teacher edition)

Learn 30 min

Students

Print or copy Perimeter Stations and cut each page in half. Post each station at a location around the classroom. Additional sets of stations can be used if needed.

• Perimeter: Stations • Problem Set

• None

Land 10 min

Achievement Descriptor

Florida B.E.S.T. Standard

FL.5.Mod5.AD7 Find perimeters of rectangles

MA.5.GR.2.1 Find the perimeter and area of a

with fraction or decimal side lengths. (MA.5.GR.2.1)

rectangle with fractional or decimal side lengths using visual models and formulas.

543

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 26

Fluency

Whiteboard Exchange: Evaluate by Using the Order of Operations Students evaluate expressions to build fluency with evaluating expressions by using the order of operations. Display (145.23 − 45.23) × 5.36 = . Write and complete the equation. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to evaluate their corrections.

(145.23 ‒ 45.23) × 5.36 = 536

Display the answer. Repeat the process with the following sequence:

5.6 ÷ 0.8 × 9 − 16.04 = 46.96

7.5 + 2.5 × 0.2 =

5 × (23.91 − 8.42) = 77.45

62.923 + 0.324 − 27.113 = 36.134 (8.82 − 4.32) × 0.2 ÷ 15 = 0.06

544

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 26

Whiteboard Exchange: True or False Number Sentences Students decide whether a number sentence is true or false and make false number sentences true to build fluency with reasoning about products without evaluating. Display 1 × 4.65 < 4.65. Is the number sentence true or false? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

1 × 4.65 < 4.65 False 1 × 4.65 = 4.65

False. Write the number sentence by using the correct comparison symbol. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the corrected number sentence. Repeat the process with the following sequence: 182.45 × 0.99 > 182.45

2.74 × 292.3 > 292.3

14.8 < 14.8 × 0.8

8.13 > 8.13 × 0.21

False

True

False

True

182.45 × 0.99 < 182.45

14.8 > 14.8 × 0.8

545

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 26

Launch

Students determine the perimeter of the base of a wooden chest. Display the prompt and the picture of the wooden chest. Lisa builds a wooden toy chest. What do you notice? What do you wonder? I notice the toy chest is 18 inches tall. I notice the toy chest is 31.5 inches long.

Multilingual Learner Support

18 in

16.25 in

I notice the toy chest is 16.25 inches wide.

31.5 in

Consider activating prior knowledge by discussing students’ familiarity with wooden chests or toy boxes. Show real-world pictures as students share what they know about wooden chests and toy boxes.

I wonder what kind of toys Lisa will put in the toy chest. I wonder what the volume of the toy chest is. I wonder how heavy the toy chest is. What math questions could we answer about this toy chest? What is the volume of the toy chest? What is the area of the base of the toy chest? What is the difference between the length and the width of the toy chest? Lisa wants to glue a decorative ribbon around the base of the toy chest. Invite students to Think–Pair–Share about how they could determine the number of inches of ribbon Lisa needs. We could add the lengths of each of the four edges of the base. We could add two lengths and two widths. We could add the length and the width and then multiply the sum by 2.

546

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 26

To determine the number of inches of ribbon Lisa needs, we need to find the length around the base of the chest. What do we call the length of the boundary of a shape? Perimeter Do we need to know the height of the chest to determine the perimeter of the base? No. We only need to know the length and the width of the base to determine its perimeter. Direct students to work with a partner to determine how much ribbon Lisa needs. Circulate and observe student work. Select two or three pairs to share. Look for work samples that show different ways to find the perimeter. The student work samples demonstrate using different formulas to find the perimeter of the base.

P = l + w + l +w = 31 .5 + 16.25 + 31 .5 + 16.25 = 95.5

P = 2 l + 2w = 2 × 31.5 + 2 × 16.25 = 63 + 32.5 = 95.5

P = 2 × (l + w ) = 2 × (31.5 + 16.25) = 2 × 47.75 = 95.5

Lisa needs 95.5 inches of ribbon.

Universal Design for Learning: Representation To support recall of the formulas for area and perimeter of a rectangle, consider creating an anchor chart for the formulas.

w l Formula for Area of a Rectangle A =l xw Formulas for Perimeter of a Rectangle P = l + w +l + w P = 2l + 2w P = 2 x (l +w )

When students are finished, invite the pairs you selected to share. As students share, ask questions that elicit their thinking and encourage students to make connections between the different methods and their work. Encourage students to ask questions of their own. How are perimeter and volume different?

Teacher Note

Perimeter is a measurement of the length around a shape. Volume is a measurement of how much space a solid takes up.

Consider making connections during the discussion about how the order of operations applies to each method (e.g., in the first work sample shown, we evaluate the expression from left to right).

When we find the perimeter of a rectangle, we use two dimensions: length and width. When we find the volume of a right rectangular prism, we use three dimensions: length, width, and height. Transition to the next segment by framing the work. Today, we will solve word problems involving perimeters of rectangles.

547

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 26

Learn

Perimeter: Stations Materials—T: Perimeter Stations

Students use the Read–Draw–Write process to solve word problems involving perimeter with fractions and decimals. Organize students into groups of 3 or 4. Direct each group to one of the posted stations to begin the activity. Allow groups a few minutes to solve that station’s problem in their books. Then direct groups to rotate to a new station. Continue until all groups have solved the problems at all stations. Sample student work and answers can be found in Sample Solutions. Record your work for each station. Use the Read–Draw–Write process to solve each problem. Station 1

Teacher Note Six different stations in this activity accommodate six groups of four students each. If needed, print and post additional copies of each set of stations to reduce crowding at any one station or to complete the activity in smaller groups or pairs. As an alternative to having students rotate around the classroom to different posted stations, students could remain seated in their groups and pass around the printed station pages. Students may not complete all six stations in the time allowed. Consider differentiating the stations provided to each group.

Station 2 Universal Design for Learning: Engagement

Station 3

548

To foster collaboration during the stations activity, consider assigning group roles such as reader, notetaker, and timekeeper. Define responsibilities for each role. Review the activity directions and group norms before groups begin. Suggest a target completion time, such as 3 minutes per station, and project a visual timer.

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 26

Station 4

Station 5

Promoting Mathematical Thinking and Reasoning Students apply mathematics to real-world contexts (MTR.7) as they solve problems in groups during the stations activity.

Station 6

Ask the following questions to promote MTR.7: • What math can you draw to help you understand the problem? • What does perimeter mean in this situation?

Circulate as students work, and advance student understanding of solving word problems involving perimeter by asking questions such as the following: • How did using the Read–Draw–Write process help you make sense of the problem? • Which perimeter formula did you use? Why did you choose that formula? • What does the unknown in your equation represent? • How do you see a perimeter formula in your work? • How did you use a perimeter formula to find the unknown side length in station 2? • How is the problem in station 4 different from the other problems?

Differentiation: Support If students struggle to make sense of a problem involving fractions or decimals, consider replacing the fractions or decimals with whole numbers. Once students understand the underlying structure of the problem, reinsert the fractions or decimals and have students continue to solve the problem.

• How did you determine whether Kayla has enough ribbon in station 5?

549

5 ▸ M5 ▸ TD ▸ Lesson 26

Have students return to their seats. Call on a different group to share their solution for each station. Ask questions that encourage students to make connections and ask questions of their own. Invite students to turn and talk about how they used perimeter formulas to help them solve the problems.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

EUREKA MATH2 Florida

Differentiation: Challenge Consider challenging students to provide their answers in a different unit than what is specified in the problem. For example, in station 1, students could provide the length of lights Mr. Evans needs in inches rather than in feet. In station 4, students could give the perimeter of the sign in centimeters rather than in meters.

Debrief 5 min Objective: Solve word problems involving perimeters of rectangles. Gather the class with their Problem Sets. Facilitate a discussion about solving word problems involving perimeter. Encourage students to restate or add on to their classmates’ responses. How does the Read–Draw–Write process help you solve word problems involving perimeter with fractions and decimals? Use an example from the Problem Set to support your answer. The Read–Draw–Write process helps me solve problems involving perimeter by helping me make sense of the problem. In problem 3, I read the problem one sentence at a time. When I drew a rectangle and labeled it with the information in each sentence, I knew I had to find the length of the rectangle before I could find the perimeter.

550

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 26

How can you use a formula for perimeter to help you solve word problems involving perimeter? Use an example from the Problem Set to support your answer. For problem 1, I already knew the length of the bulletin board. The height of the bulletin board is the width of the rectangle. So I used the formula P = 2l + 2w. I multiplied the length by 2 and the width by 2 and then added the products together. For problem 2, I thought of it as an unknown factor problem. I used the formula P = 2 × (l + w). I knew the perimeter is equal to 2 times another factor, which is the sum of the length and the width. I divided the perimeter by 2 and then subtracted the length from the quotient to find the width.

551

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 26

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

556

Mr. Evans wants to put a strand of lights around the entire border of his rectangular window. The window is 6 1_ feet wide and 3 1_ feet tall. How many feet of lights does Mr. Evans need?

This page may be reproduced for classroom use only.

Station 1

P = 2 × (l + w) 3

7 P = 2 × 9 __ 12 2 P = 19 __

There are 305.6 meters of fence around a rectangular playground. The length of the playground is 65.9 meters. What is the width of the playground?

6 14 ft

P = 2 × (3 1_ + 6 1_) 4

3 13 ft

Station 3 Julie tries to determine the unknown side length in the following problem, but she makes a mistake.

a. What is Julie’s mistake?

2 × 65.9 = 131 .8 305.6 - 131 .8 = 173.8

w = 173.8

Julie found the length of two widths. She did not divide 173.8 by 2.

2 feet of lights. Mr. Evans needs 19 __ 12

P = 2l + 2w 305.6 = 2 × 65.9 + 2 × w

The width of the playground is 173.8 meters.

b. What is the width of the playground? 173.8 ÷ 2 = 86.9 The width of the playground is 86.9 meters.

Station 2 Tara has a poster with a perimeter of 73.5 inches. The poster is 12.25 inches long. How many inches wide is the poster? P = 2 × (l + w) 73.5 = 2 × (12.25 + w)

w = 24.5 The poster is 24.5 inches wide.

12.25 in

The perimeter is 73.5 inches.

This page may be reproduced for classroom use only.

73.5 ÷ 2 = 36.75 36.75 – 12.25 = 24.5

Station 4

w in

A rectangular sign on a billboard covers 33.6 square meters. The height of the sign is 3 meters. What is the perimeter of the sign?

wm 3m The area is 33.6 square meters. A=l×w 33.6 = 3 × w 33.6 ÷ 3 = 11.2 w = 11.2

P = 2 × (l + w) = 2 × (3 + 11.2) = 2 × 14.2 = 28.4

The perimeter of the sign is 28.4 meters.

557

552

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 26

558 This page may be reproduced for classroom use only.

Kayla has a piece of ribbon that is 150 centimeters long. She wants to use it to decorate the border of a rectangular place mat. The length of the place mat is 14.7 centimeters less than the width of the place mat. The place mat is 29.5 centimeters long. Is the ribbon long enough to decorate the entire border of the place mat? Explain.

14.7 cm 44.2 cm

length 29.5 cm

PRACTICE

Florida B.E.S.T. MA.5.GR.2.1

Name

29.5 cm

Date

Use the Read–Draw–Write process to solvse each problem. 1. Mrs. Chan wants to put a border around a rectangular bulletin board in her classroom. _3 The bulletin board is 48 inches long and 33 4 inches tall. How many inches of border does Mrs. Chan need? 1 3 2 × 48 + 2 × 33 _4 = 163 _2 _1 Mrs. Chan needs 163 2 inches of border.

width w

w = 29.5 + 14.7 w = 44.2

PROBLEM SET

I can solve word problems involving perimeters of rectangles.

Station 5

P = 2 × (l + w) = 2 × (29.5 + 44.2) = 2 × 73.7 = 147.4

Yes, the ribbon is long enough. The perimeter of the place mat is 147.4 centimeters, which is less than the length of the ribbon.

2. Some football players run laps around a rectangular football field during their practice. One lap is 316.98 meters. The length of the football field is 109.73 meters. How many meters wide is the football field?

Station 6 Some students paint a mural on a wall at school. First, they use tape to mark a rectangular space on the wall. The space is 3 times as long as it is tall. The height of the space is 8 1_ feet. How many feet of tape do the students use to mark 6 the space on the wall?

The football field is 48.76 meters wide.

158.49 − 109.73 = 48.76

3 × 8 1_ = 24 1_

24 12 ft

The students use 65 1_ feet of tape to mark the space on the wall.

316.98 ÷ 2 = 158.49

P = 2 × (l + w)

= 2 × (24 1_ + 8 1_) = 2 × 32 2_ = 65 1_

279

553

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 26

3. The deck around a rectangular pool is 27.8 meters wide. The deck is twice as long as it is wide. What is the perimeter of the deck?

2 × 27.8 = 55.6 2 × (55.6 + 27.8) = 166.8

5. Ryan and Sana want to rope off a rectangular space to play beach volleyball. They measure the length and width of the space as shown. Ryan and Sana each try to find the perimeter of the space to determine how much rope they need, but one of them makes a mistake. Who made a mistake? How do you know?

18 m

The perimeter of the deck is 166.8 meters.

9.02 m

Ryan’s Way

P = l ×w = 18 × 9.02 = 18 × 9 + 18 × 0.02 = 162 + 0.36 = 162.36 We need 162.36 meters of rope.

4. A city creates a new park. a. A rectangular space is roped off to create a small soccer field. The perimeter of the field is 76.2 meters. The length of the field is 23.2 meters. What is the width of the soccer field?

76.2 ÷ 2 = 38.1

Sana’s Way

P = 2 × (l + w ) = 2 × (18 + 9.02) = 2 × 27.02 = 54.04 We need 54.04 meters of rope.

Ryan made a mistake. He multiplied the length and the width of the rectangle, which means he found the area of the space for the volleyball court instead of the perimeter. He should have added the length and width and then multiplied the sum by 2.

38.1 − 23.2 = 14.9 The width of the soccer field is 14.9 meters.

b. The rectangular basketball court has an area of 384 square meters. The court is 15 meters wide. What is the perimeter of the basketball court?

384 ÷ 15 = 25.6 2 × (25.6 + 15) = 81.2 The perimeter of the basketball court is 81.2 meters.

280

554

PROBLEM SET

281

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 26

EXIT TICKET

I can solve word problems involving perimeters of rectangles.

Florida B.E.S.T. MA.5.GR.2.1

Name

Date

Use the Read–Draw–Write process to solve each problem. 1. Mr. Evans uses 38.5 meters of fence to make a rectangular pigpen. The pigpen is 7.65 meters wide. How many meters long is the pigpen?

7.65 m

The perimeter is 38.5 meters.

P = 2 × (l + w) 38.5 = 2 × (l + 7.65) 38.5 ÷ 2 = 19.25 The pigpen is 11.6 meters long.

19.25 − 7.65 = 11.6

2. A rectangular game board is 17.3 centimeters wide. The board is 3 times as long as it is wide. What is the perimeter of the game board?

l cm 17.3 cm 3 × 17.3 = 51.9 P = 2 × (l+ w) = 2 × (51.9 + 17.3) = 2 × 69.2 = 138.4 The perimeter of the game board is 138.4 centimeters. © Great Minds PBC

283

555

556 4

This page may be reproduced for classroom use only.

Tara has a poster with a perimeter of 73.5 inches. The poster is 12.25 inches long. How many inches wide is the poster?

Station 2

Mr. Evans wants to put a strand of lights around the entire border of his rectangular window. The window is 6 1_ feet wide and 3 1_ feet tall. How many feet of lights does Mr. Evans need?

Station 1

5 ▸ M5 ▸ TD ▸ Lesson 26 ▸ Perimeter Stations EUREKA MATH2 Florida

A rectangular sign on a billboard covers 33.6 square meters. The height of the sign is 3 meters. What is the perimeter of the sign?

Station 4

b. What is the width of the playground?

a. What is Julie’s mistake?

There are 305.6 meters of fence around a rectangular playground. The length of the playground is 65.9 meters. What is the width of the playground?

The width of the playground is 173.8 meters.

w = 173.8

305.6 - 131 .8 = 173.8

2 × 65.9 = 131 .8

P = 2l + 2w 305.6 = 2 × 65.9 + 2 × w

Julie tries to determine the unknown side length in the following problem, but she makes a mistake.

Station 3

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 26 ▸ Perimeter Stations

This page may be reproduced for classroom use only.

557

558 This page may be reproduced for classroom use only.

Some students paint a mural on a wall at school. First, they use tape to mark a rectangular space on the wall. The space is 3 times as long as it is tall. The height of the space is 8 1_ feet. How many feet of tape do the students use to mark 6 the space on the wall?

Station 6

Station 5

5 ▸ M5 ▸ TD ▸ Lesson 26 ▸ Perimeter Stations EUREKA MATH2 Florida

LESSON 27

Solve word problems involving perimeter, area, and volume. I can solve word problems that involve perimeter, area, and volume.

Lesson at a Glance Students work in pairs to sort real-world situations into categories based on whether the situations require finding perimeter, area, or volume. When students consider the length of wood that encloses a garden box, the amount of the lawn a garden box covers, and the amount of soil needed to fill a garden box, they recognize that building a garden box requires finding all three measurements: perimeter, area, and volume. Students choose real-world problems to solve: one involving perimeter, one involving area, and one involving volume.

Key Question • How do you know whether a situation involves perimeter, area, or volume?

Mathematical Thinking and Reasoning Standard

Language Objectives • Read word problems involving perimeter, area, and volume, and orally justify a solution strategy by using language from the problem that indicates which measurement to find. • Write or draw to record a solution strategy and an equation to solve a problem involving perimeter, area, and volume.

MTR.1.1

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 27

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Consider whether to remove Perimeter, Area, or Volume Cards from the student books and cut out the cards in advance or to have students prepare them during the lesson.

Launch 10 min Learn 30 min • Build a Garden Box • Perimeter, Area, and Volume • Problem Set

• None

Students • Perimeter, Area, or Volume Cards (in the student book) • Scissors

Land 10 min

Achievement Descriptors

Florida B.E.S.T. Standards

FL.5.Mod5.AD7 Find perimeters of rectangles

MA.5.GR.2.1 Find the perimeter and area of a

with fraction or decimal side lengths. (MA.5.GR.2.1)

rectangle with fractional or decimal side lengths using visual models and formulas.

FL.5.Mod5.AD9 Find areas of rectangles with

MA.5.GR.3.3 Solve real-world problems

fraction or decimal side lengths. (MA.5.GR.2.1) FL.5.Mod5.AD13 Solve real-world problems

involving volumes of right rectangular prisms. (MA.5.GR.3.3)

561

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 27

Fluency

Whiteboard Exchange: Decimals Written Two Ways Students write a number in standard and expanded forms when given the word form to build fluency with reading and writing decimals to the thousandths place from module 4. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display four hundred eighty-six thousandths in word form. Write the number in standard form. Display the answer. Write the number in expanded form by using whole numbers and decimals.

Four hundred eighty-six thousandths

0.486 4 × 0.1 + 8 × 0.01 + 6 × 0.001

Display the answer.

562

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 27

Repeat the process with the following sequence:

Six and three hundred ninety-eight thousandths

Eighty-five and forty-seven thousandths

6.398

85.047

6 × 1 + 3 × 0.1 + 9 × 0.01 + 8 × 0.001

8 × 10 + 5 × 1 + 4 × 0.01 + 7 × 0.001

Twelve and six hundred three thousandths

Fifty and fifty-two thousandths

12.603

50.052

1 × 10 + 2 × 1 + 6 × 0.1 + 3 × 0.001

5 × 10 + 5 × 0.01 + 2 × 0.001

Whiteboard Exchange: Multiply with Decimals Students determine the product to build fluency multiplying with decimals from module 4. Display 0.56 × 3 =

. 0.56 × 3 = 1.68

5 6 hundredths 3 1

1 6 8 hundredths

Display the product and sample method.

563

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 27

Repeat the process with the following sequence:

4.27 × 7 = 29.89

Launch

10.5 × 9 = 94.5

5.08 × 32 = 162.56

Materials—S: Perimeter, Area, or Volume Cards; scissors

Students sort situations into categories based on whether the situation requires finding perimeter, area, or volume. Have students remove Perimeter, Area, or Volume Cards and cut the cards apart. Pair students and have them work with their partner to sort the cards into three categories based on whether the situation requires finding perimeter, area, or volume. Though students may sort the cards in a number of ways and provide explanations to support their organization, expect them to sort the cards as follows:

Perimeter

Area

Volume

Fencing a playground

Painting a wall

Filling a pool

Decorating with rope lights

Tiling a shower

Baking a cake

Building a garden box*

Carpeting a floor

Packing a suitcase

Multilingual Learner Support

Building a garden box*

Presenting the situations in a picture format helps students by removing barriers associated with written and spoken language. Use these pictures to build background knowledge by inviting students to share their past experiences with each situation.

The asterisk (*) indicates that students may place the card in any of the three categories.

564

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 27

When students are finished sorting their cards, facilitate a discussion by using the following prompts. Give an example of a situation in which you would need to know perimeter. Explain your thinking. Fencing a playground. The length of the fence around a playground is the perimeter of the playground. Decorating with rope lights. If the rope lights go all the way around a room, then the length of the rope lights is the perimeter of the room. Building a garden box. The length of the wood around a garden box is the perimeter of the garden box. Give an example of a situation in which you would need to know area. Explain your thinking. Painting a wall. Paint covers the surface of a wall, so we need to find the area of the wall being painted. Tiling a shower. Tiles cover a shower wall, so we need to find the area of the shower wall being tiled. Carpeting a floor. Carpet covers a floor, so we need to find the area of the floor being carpeted. Building a garden box. A garden box covers part of a patio, so we need to find the area of the patio that the garden box sits on. Give an example of a situation in which you would need to know volume. Explain your thinking. Filling a pool. A pool has space to fill with water, so we need to know the volume of the pool. Baking a cake. A pan has space to pour cake batter into, so we need to know the volume of the pan. Packing a suitcase. A suitcase has space to pack clothes, so we need to know the volume of the suitcase. Building a garden box. A garden box has space to fill with soil, so we need to know the volume of the garden box.

565

5 ▸ M5 ▸ TD ▸ Lesson 27

EUREKA MATH2 Florida

If students do not mention that building a garden box requires knowing perimeter, area, and volume, ask probing questions to help them recognize the need for each calculation: • Do you need to know perimeter, area, or volume when building a garden box? • Why might we need to know perimeter when building a garden box? • Why might we need to know area when building a garden box? • Why might we need to know volume when building a garden box? • Does every garden box have enough room for any type of plant to grow? • Could a palm tree fit in a garden box? Invite students to correct any mistakes in their sort and save the cards, in their categories, for later in the lesson. Transition to the next segment by framing the work. Today, we will solve word problems involving perimeter, area, and volume.

Learn

Build a Garden Box Students solve real-world problems by finding perimeter, area, and volume. Direct pairs to problem 1 in their books, which correlates to the card about building a garden box from Launch. Have students read each part of the problem silently as you read aloud. Display the unlabeled garden box.

566

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 27

1. Consider building a garden box. The 6 sides of the garden box are built by using 6 wooden boards that are each 2 feet high. The two longest boards are each 12 feet long. The two shortest boards are each 4 feet long.

12 ft

2 ft 4 ft

12 ft

8 ft

2 ft 4 ft

a. What is the total length of wooden boards used to build the garden box?

12 + 12 + 4 + 4 + 8 + 8 = 48 The total length of wooden boards used to build the garden box is 48 feet. b. How much of the patio does the garden box cover?

12 × 4 = 48 8 × 4 = 32 48 + 32 = 80 The garden box covers 80 square feet of the patio. c. How much soil does it take to fill the box?

48 × 2 = 96 32 × 2 = 64 96 + 64 = 160 It takes 160 cubic feet of soil to fill the box. Invite students to turn and talk about what they know about the garden box. Then ask the following question. What can we label?

Differentiation: Challenge Consider challenging students by having them find the following: • Perimeter of the garden box in inches • Perimeter of the garden box in yards • Area the garden box covers in square inches

We can label the longest two sides as 12 feet.

• Area the garden box covers in square yards

We can label the shortest two sides as 4 feet.

• Volume of the garden box in cubic inches

We can label the height as 2 feet.

• Volume of the garden box in cubic yards

567

5 ▸ M5 ▸ TD ▸ Lesson 27

EUREKA MATH2 Florida

Label the two longest sides, the two shortest sides, and the height. Have students do the same. Then point to the other two sides of the box. Invite students to Think–Pair–Share about how to find the length of these boards. We can subtract 4 from 12 to find that the length of each of the unlabeled boards is 8 feet. Label the two remaining sides. Have students do the same. In a real garden box, the boards have thickness, which means the measurements inside the box and outside the box will be slightly different. As we find the perimeter, area, and volume of the box today, we will simplify the problem by using the measurements given instead of finding and using the inside and outside measurements. Then direct pairs to complete problem 1. Circulate as students work. Look for students who add the numbers in a different order for part (a) and for students who decompose the figure in different ways for parts (b) and (c). Ask the following questions to support student thinking: • What is the question asking? • What do you need to know to answer the question? • Do you need to find perimeter, area, or volume? How do you know? • Does the question ask for something that can be measured in feet, square feet, or cubic feet? • Can you decompose the figure? How? When most students are finished, ask which part of the problem required them to find perimeter, which part required them to find area, and which part required them to find volume. Select students to share their answers and reasoning for each part of the problem. Then ask the following questions.

Differentiation: Support To support students with parts (b) and (c), sketch the figure as two separate labeled rectangles and then as two separate labeled right rectangular prisms. Alternatively, consider building the prisms with cubes.

568

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 27

How did you know to find the perimeter of the box in part (a)? The question asked for the total length of wooden boards used to build the box. The box was built by using 6 boards placed end to end. So, we had to add the lengths of the sides of the box to find the total length of the boards. The boards make up the sides of the box. Because we needed to know the total length of the sides, we found the perimeter of the box to find the total length of the boards. How did you know to find the area of the box in part (b)? The question asked for how much space the box takes up on the patio. The garden box covers part of the patio, which means we need to know the area it covers. Why didn’t you use the height of the box to find the total length of the boards? Why didn’t you use the height to find the amount of patio covered by the garden box? We didn’t need to use the height because the height does not affect the length of the boards or the amount of the patio covered by the garden box. How did you know to find the volume of the box in part (c)? The question asked how much dirt it takes to fill the box. We needed to know how much dirt fits inside the box, which means we need to know the volume. Why is the number of cubic feet in the volume twice the number of square feet in the area? The area of the base of the garden box is 80 square feet. The height of the garden box is 2 feet. We can find volume by multiplying the area of the base by the height, and 80 × 2 = 160. Invite students to turn and talk about how they can tell whether they need to know perimeter, area, or volume to solve real-world problems.

Perimeter, Area, and Volume Materials—S: Perimeter, Area, or Volume Cards

Students solve real-world problems involving perimeter, area, and volume. Direct pairs to look back at the cards they sorted. Building a garden box required us to find perimeter, area, and volume. © Great Minds PBC

569

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 27

Point out that each situation in their book correlates to one of the Perimeter, Volume, or Area Cards. With your partner, choose three situations: one situation involving perimeter, one situation involving area, and one situation involving volume. In your books, find the problems for the situations you choose and complete the three problems. Post an answer key so students can check their answers as they complete each problem. Circulate as students work. As needed, ask the following questions to support students’ thinking: • What do you know? What is the question asking? • Do you have enough information to answer the question? • What can you draw? What can you label? • Can you decompose the figure? How? Invite students who finish early to choose additional problems to complete.

Universal Design for Learning: Action & Expression Including a variety of real-world problem contexts provides an opportunity for student choice. Allowing students to select problems they find interesting and challenging puts them in charge of their learning and promotes relevance.

2. Consider decorating with rope lights. Tara decorates her room with rope lights by lining the ceiling and the edges of her mirror.

Her room is rectangular and measures 12  2 feet by 10  4 feet. Her mirror is rectangular

and measures 4  2 feet by 1 foot. How many feet of the rope lights does she use?

12 1 + 12 1 + 10 1 + 10 1 = 45 1 2 2 2 4 4 4 1_ + 4 1_ + 1 + 1 = 11 2

_1 Tara uses 56    feet of rope lights. 2

45 1_ + 11 = 56 1_ 2

Universal Design for Learning: Action & Expression Providing students with an answer key helps support them in monitoring their own progress. Pose the following questions to prompt students to reflect after they have checked their solution: • What worked well? • What will you do the same next time? What will you do differently next time? • Do you still have a question about anything? Consider modeling possible responses or asking for volunteers to share reflections. Provide time to address questions.

570

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 27

3. Consider fencing a playground. An elementary school needs new fencing to surround the playground. How many yards of fencing does the school need?

9 yd 6 yd

1 2

8 yd

1 4

6 yd

Promoting Mathematical Thinking and Reasoning Students actively participate in effortful learning (MTR.1) when they work with partners to solve real-world problems involving perimeter, area, and volume by figuring out what information is given and what is being asked when solving the problem, looking for entry points, and monitoring their progress. Ask the following questions to promote MTR.1:

2 yd

1 2

8 yd

6 yd

• How can you explain this context in your own words? • What is your plan for solving the problem? • How can you simplify the problem?

3 yd 4

8 _1 + 9 + 6 + 6 + 8 1_ + 12 3_ + 6 + 2 1_ = 59 2

The school needs 59 yards of fencing. 4. Consider painting a room.

Ryan paints two walls in his room. One wall measures 10  2 feet by 8 feet. The other wall

measures 12  4 feet by 8 feet. How many square feet does Ryan paint?

10 1_ × 8 = 84 2

12 3_ × 8 = 102 4

84 + 102 = 186 Ryan paints 186 square feet.

571

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 27

5. Consider carpeting a floor. A family installs carpet in their living room. How many square yards of carpet do they use?

6 yd

1 yd 1 yd

1 7 yd 2

Differentiation: Challenge To challenge students, ask them to sketch and label the dimensions of a right rectangular prism with the following attributes: • The perimeter of at least one face is 18 units.

1 2

6 yd

• The area of at least one face is 18 square units. • The volume of the right rectangular prism is 18 cubic units.

7 yd

7 × 6 1_ = 45 1_ 2

6×1=6

45 1_ + 6 = 51 1_ 2

The family uses 51  2 square yards of carpet. 6. Consider tiling a shower.

Two of the walls of a shower need to be tiled. One wall of the shower is 5  2 feet by 8 feet.

The other wall of the shower is 6  4 feet by 8 feet. The cost to tile the shower walls is $7.99 per square foot. What is the total cost to tile the shower walls?

5 1_ × 8 = 44

2 6 1 × 8 = 50 4

44 + 50 = 94 94 × 7.99 = 751.06 The total cost to tile the shower walls is $751.06. 572

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 27

7. Consider packing a suitcase. A suitcase is shaped like a right rectangular prism. The suitcase measures 76 centimeters by 48 centimeters by 29 centimeters. What is the volume of the suitcase?

76 × 48 × 29 = 105,792 The volume of the suitcase is 105,792 cubic centimeters. 8. Consider filling a pool. Riley fills up her swimming pool. When the pool is full, the water is 6 feet deep. What is the volume of water in the pool?

16 ft 6 ft 25 ft 41 ft 8 ft

6 ft

16 ft 24 ft 25 × 16 × 6 = 2,400 24 × 16 × 6 = 2,304 2,400 + 2,304 = 4,704 The volume of water in the pool is 4,704 cubic feet.

573

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 27

9. Consider baking a cake. Adesh bakes a cake with three layers that are each shaped like a right rectangular prism. The bottom layer measures 9 inches by 13 inches by 2 inches. The other two layers each measure 8 inches by 8 inches by 2 inches. What is the total volume of the cake?

9 × 13 × 2 = 234 8 × 8 × 2 = 128 234 + 128 + 128 = 490 The total volume of the cake is 490 cubic inches. When most students are finished, invite them to share their strategies as time permits. Ask any of the following questions as students share: • How many sides does the figure have? How did that affect your calculations? • What are the side lengths of the figure? • What are the edge lengths of the solid? • How did you decompose the figure? Could you have decomposed it a different way? • How did you decompose the solid? Could you have decomposed it a different way? • Why did you use addition in this problem? • What was challenging about the problem? How did you figure it out?

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

574

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 27

Land

Debrief 5 min Objective: Solve word problems involving perimeter, area, and volume. Facilitate a class discussion about solving problems involving perimeter, area, and volume by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Give an example of a time you used addition in today’s lesson. When we found perimeter, we added the side lengths. When we wanted to know how many square feet were painted, we found the area of each wall and added them. When we wanted to know the total volume of Adesh’s cake, we found the volume of each layer and added them. How do you know whether a situation involves perimeter, area, or volume? When we need to know a length or distance around something, that is perimeter. When we need to know the amount of two-dimensional space something covers, that is area. When we need to know the amount of three-dimensional space something takes up, that is volume. Are there situations that involve all three measurements: perimeter, area, and volume? Yes. To build a garden box, we needed to know perimeter, or the length of the border. We needed to know the area so we would know how much space the box takes up on the patio. We needed to know the volume so we would know the amount of soil that it would take to fill the box. Invite students to turn and talk about whether they can think of another situation that requires all three measurements: perimeter, area, and volume.

575

5 ▸ M5 ▸ TD ▸ Lesson 27

EUREKA MATH2 Florida

576

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD ▸ Lesson 27

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

PROBLEM SET

I can solve word problems that involve perimeter, area, and volume.

PRACTICE

Florida B.E.S.T. MA.5.GR.2.1, MA.5.GR.3.3

2. A city is building a dog park. The measurements are shown.

8 ft Name

Date

1 2

7 ft

1. Miss Song is building a sandbox in her backyard. The measurements of the sandbox are shown.

1 10 ft 2

3 ft 2 ft 4 ft

3 ft

2 ft

6 ft

4 ft

a. How much space will the dog park take up? The dog park will take up 78 square feet of space.

6 ft

b. How many feet of fence are needed to surround the dog park?

37 feet of fence are needed to surround the dog park.

7 ft a. Miss Song uses one board per side. What is the total length of boards Miss Song uses to build the sides of the sandbox? Miss Song uses 26 feet of boards to build the sides of the sandbox. b. How much of the lawn does the sandbox cover? The sandbox covers 34 square feet of lawn. c. How much sand does it take to fill the sandbox to a height of 1 _1 feet? 2

It takes 51 cubic feet of sand to fill the sandbox to a height of 1 _1 feet. 2

293

294

PROBLEM SET

577

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD ▸ Lesson 27

EXIT TICKET

I can solve word problems that involve perimeter, area, and volume.

Florida B.E.S.T. MA.5.GR.2.1, MA.5.GR.3.3

3. A fishpond is being built around a corner of a building. The measurements are shown. What is the volume of the water in the pond when it is filled to the top? 1 2

2 ft Name

Date

A pool is shaped like an L as shown.

5 ft

4 ft

10 ft

3 ft

1 2

8 ft

2 ft 12 ft

2 ft

8 ft

a. A dog walks around the border of the pool. How far does the dog walk? The dog walks 32 feet.

1 The volume of the water in the pond is 97 _ cubic feet. 2

4. Some of the rain gutters on Eddie’s home are being replaced. The measurements of Eddie’s home where the gutters are being replaced are shown. How many feet of gutters are being replaced on Eddie’s home? 6 ft

8 + 8 + 5 + 4 + 3 + 4 = 32

b. The bottom of the pool is covered with tiles. How much space do the tiles cover?

4 × 8 = 32

4 ft 1 2

24 ft

The tiles cover 52 square feet.

30 ft

5 × 4 = 20

32 + 20 = 52

c. Julie fills the pool with water. When the pool is full, the height of the water is 3 feet. How much water does it take to fill the pool?

4 × 8 × 3 = 96 5 × 4 × 3 = 60 96 + 60 = 156

1 2

20 ft

It takes 156 cubic feet of water to fill the pool.

578

PROBLEM SET

295

297

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD

Topic D Quiz Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

A N SW E R K E Y ▸ TO PI C Q U I Z D -1

2. Tara makes a candle in the shape of a right rectangular prism. The candle has a volume of 480 cubic centimeters. Its length is 8 centimeters, and its height is 12 centimeters. Part A

1. Consider the right rectangular prism shown.

Write an equation to find the width of the candle. Use the letter w to represent the unknown. Write one number or symbol from the given answer choices in each blank. Answers may be used more than once.

𝑤

480

Answer Choices

8 feet 8

20 feet

480

−

12 feet What is the volume of the prism?

Part B

A. 40 cubic feet

What is the width of the candle?

B. 496 cubic feet C. 992 cubic feet D. 1,920 cubic feet

Distractor rationale: A. Incorrect. The student may have added the three given numbers. C. Incorrect. The student may have found the area of all six faces.

The width of the candle is

B. Incorrect. The student may have found the area of the three faces shown.

centimeters.

3. A rectangle has a length of 8.9 meters and a width of 7.5 meters. Find the area and perimeter of the rectangle. Write one number or unit from the given answer choices in each blank.

D. Correct.

The area of the rectangle is

66.75

square meters

32.8

The perimeter of the rectangle is

meters

Answer Choices

16.4

247

248

32.8

35.6

65.6

66.75

A N S W E R K E Y ▸ TO P I C Q U I Z D -1

133.5

meters

square meters

579

EUREKA MATH2 Florida

5 ▸ M5 ▸ TD

5. Mrs. Chan has a sandbox in the shape of a right rectangular prism. The inside of the sandbox is 48 inches wide, 90 inches long, and 3 inches deep. How many cubic inches of sand does Mrs. Chan need to completely fill the sandbox?

4. Yuna carves a piece of wood into the shape shown.

2 inches 2 inches 10 inches

8 inches 6 inches

12 inches

5 inches Mrs. Chan needs

12,960

cubic inches of sand to completely fill the sandbox.

What is the volume of the shape?

The volume of the shape is

580

380

cubic inches.

A N S W E R K E Y ▸ TO P I C Q U I Z D -1

249

250

A N S W E R K E Y ▸ TO P I C Q U I Z D -1

EUREKA MATH2 Florida 5 ▸ M5 ▸ TD

Observational Assessment Recording Sheet Grade 5 Module 5 Topic D

Student Name

Volume and the Operations of Multiplication and Addition Standards and Achievement Descriptors

MA.5.GR.2.1

FL.5.Mod5.AD7 Find perimeters of rectangles with fraction or decimal side lengths.

MA.5.GR.2.1

FL.5.Mod5.AD9 Find areas of rectangles with fraction or decimal side lengths.

MA.5.GR.3.2

FL.5.Mod5.AD12 Find volumes of right rectangular prisms with whole-number edge lengths by using a formula.

MA.5.GR.3.3

FL.5.Mod5.AD13 Solve real-world problems involving volumes of right rectangular prisms.

Dates and Details of Observations

PP Partially Proficient P Proficient HP Highly Proficient

This page may be reproduced for classroom use only.

581

EUREKA MATH2 Florida

5 ▸ M5

Spiral Review Sample Solutions Each module contains two Spiral Review sets to reinforce skills and concepts through distributed practice. The sets do not contain content from the current module so they can be given to students at any time during the module. Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Spiral Review

Add or subtract.

Name:

6. 15.93 + 12.207 =

28.137

1. Draw an acute angle XYZ. Explain how you know the angle is acute. Sample:

7. 28.4 − 9.271 =

19.129

X Use the Read–Draw–Write process to solve each problem.

8. David spends 1_1 hours studying history and math. He spends _5 hours studying math. About how 2 6 long does David spend studying history?

5 ≈ 1 _ 6

I know this angle is acute because it is smaller than a right angle.

1 _12 − 1 = _12

2. Draw an obtuse angle PRS. Explain how you know the angle is obtuse.

David spends about _1 hour studying history.

Sample:

10 inches tall. Last 9. Liz measures the height of her plant every Sunday. This Sunday it measures _ 12 5 inches tall. About how much has her plant grown since last Sunday? Sunday Liz’s plant was _

10 ≈ 1 _ 12

1 5 ≈ _ _ 2 12

1 − _12 = _12 Liz’s plant has grown about _1 inch.

I know this angle is obtuse because it is larger than a right angle.

Convert each measurement. 3. 5.3 yards =

190.8

inches

4. 4.36 cups =

2.18

pints

5. 8.48 ounces =

582

0.53

pounds

299

300

EUREKA MATH2 Florida 5 ▸ M5

Spiral Review

Name

7. Draw a model to divide.

Date

_1 ÷ 2 =

_1 12

Sample:

Multiply.

1. 0.8 × 9.3 =

7.44

2. 3.4 × 4.6 =

1 6

15.64 ?

8. Draw a model to find the quotient. Then complete the statement to check your work.

6 ÷ _1 =

Divide. 3. 1.5 ÷ 0.6 =

2.5

4. 0.42 ÷ 0.03 =

Sample:

1 2

Write an expression to represent the statement.

12 5. 5 times as much as the difference of 1.07 and 0.88

groups of _1 make 6. 2

5 × (1.07 − 0.88)

6. Write one pair of parentheses to make the equation true.

9.5 × (4.3 + 7.8 + 5.9) = 171

301

302

583

5 ▸ M5

EUREKA MATH2 Florida

Standards B.E.S.T. Standards Perform operations with fractions. MA.5.FR.2.2 Extend previous understanding of multiplication to multiply a fraction by a fraction, including mixed numbers and fractions greater than 1, with procedural reliability. Solve problems involving the four operations with whole numbers and fractions. MA.5.AR.1.2 Solve real-world problems involving the addition, subtraction or multiplication of fractions, including mixed numbers and fractions greater than 1. Classify two-dimensional figures and three-dimensional figures based on defining attributes. MA.5.GR.1.1 Classify triangles or quadrilaterals into different categories based on shared defining attributes. Explain why a triangle or quadrilateral would or would not belong to a category. MA.5.GR.1.2 Identify and classify three-dimensional figures into categories based on their defining attributes. Figures are limited to right pyramids, right prisms, right circular cylinders, right circular cones and spheres. Find the perimeter and area of rectangles with fractional or decimal side lengths. MA.5.GR.2.1 Find the perimeter and area of a rectangle with fractional or decimal side lengths using visual models and formulas. Solve problems involving the volume of right rectangular prisms. MA.5.GR.3.1 Explore volume as an attribute of three-dimensional figures by packing them with unit cubes without gaps. Find the volume of a right rectangular prism with whole-number side lengths by counting unit cubes. MA.5.GR.3.2 Find the volume of a right rectangular prism with whole-number side lengths using a visual model and a formula. MA.5.GR.3.3 Solve real-world problems involving the volume of right rectangular prisms, including problems with an unknown edge length, with whole-number edge lengths using a visual model or a formula. Write an equation with a variable for the unknown to represent the problem.

584

EUREKA MATH2 Florida 5 ▸ M5

Mathematical Thinking and Reasoning Standards MA.K12.MTR.1.1

Actively participate in effortful learning both individually and collectively.

MA.K12.MTR.2.1

Demonstrate understanding by representing problems in multiple ways.

MA.K12.MTR.3.1

Complete tasks with mathematical fluency.

MA.K12.MTR.4.1

Engage in discussions that reflect on the mathematical thinking of self and others.

MA.K12.MTR.5.1

Use patterns and structure to help understand and connect mathematical concepts.

MA.K12.MTR.6.1

Assess the reasonableness of solutions.

MA.K12.MTR.7.1

Apply mathematics to real-world contexts.

ELA Expectations ELA.K12.EE.1.1

Cite evidence to explain and justify reasoning.

ELA.K12.EE.2.1

Read and comprehend grade-level complex texts proficiently.

ELA.K12.EE.3.1

Make inferences to support comprehension.

ELA.K12.EE.4.1

se appropriate collaborative techniques and active listening skills when engaging in discussions in a variety U of situations.

ELA.K12.EE.5.1

Use the accepted rules governing a specific format to create quality work.

ELA.K12.EE.6.1

Use appropriate voice and tone when speaking or writing.

585

EUREKA MATH2 Florida

5 ▸ M5

Module Achievement Descriptors and B.E.S.T. Standards by Lesson Focus content A Topic 1 Lesson Achievement Descriptor

Aligned Florida B.E.S.T. Standards

FL.5.Mod5.AD1

MA.5.FR.2.2

FL.5.Mod5.AD2

MA.5.FR.2.2

FL.5.Mod5.AD3

MA.5.AR.1.2

FL.5.Mod5.AD4

MA.5.GR.1.1

FL.5.Mod5.AD5

MA.5.GR.1.1

FL.5.Mod5.AD6

MA.5.GR.1.2

FL.5.Mod5.AD7

MA.5.GR.2.1

FL.5.Mod5.AD8

MA.5.GR.2.1

FL.5.Mod5.AD9

MA.5.GR.2.1

FL.5.Mod5.AD10

MA.5.GR.3.1

FL.5.Mod5.AD11

MA.5.GR.3.1

FL.5.Mod5.AD12

MA.5.GR.3.2

FL.5.Mod5.AD13

MA.5.GR.3.3

586

A 1

A 2

A 3

A 4

A 5

A 6

A 7

A 8

B 9

B 10

B 11

B 12

B 13

B 14

B 15

C 16

C 17

C 18

C 19

C 20

C 21

D 22

D 23

D 24

D 25

D 26

D 27

EUREKA MATH2 Florida 5 ▸ M5

Mathematical Thinking and Reasoning Standards by Lesson Focus content A Topic 1 Lesson MTR Standard

A 1

A 2

A 3

A 4

A 5

A 6

A 7

A 8

B 9

B 10

B 11

B 12

B 13

B 14

B 15

C 16

C 17

C 18

C 19

C 20

C 21

D 22

D 23

D 24

D 25

D 26

D 27

MA.K12.MTR.1.1 MA.K12.MTR.2.1 MA.K12.MTR.3.1 MA.K12.MTR.4.1 MA.K12.MTR.5.1 MA.K12.MTR.6.1 MA.K12.MTR.7.1

587

EUREKA MATH2 Florida

5 ▸ M5

Assessment to Standards Alignment Assessment

Aligned Florida B.E.S.T. Standards

Achievement Descriptor

Module 5 Topic A Quiz

MA.5.GR.1.1 MA.5.GR.1.2

FL.5.Mod5.AD4 FL.5.Mod5.AD5 FL.5.Mod5.AD6

Module 5 Topic B Quiz

MA.5.FR.2.2 MA.5.AR.1.2 MA.5.GR.2.1

FL.5.Mod5.AD1 FL.5.Mod5.AD2 FL.5.Mod5.AD3 FL.5.Mod5.AD8 FL.5.Mod5.AD9

Module 5 Topic C Quiz

MA.5.GR.3.1

FL.5.Mod5.AD10 FL.5.Mod5.AD11

Module 5 Topic D Quiz

MA.5.GR.2.1 MA.5.GR.3.2 MA.5.GR.3.3

FL.5.Mod5.AD7 FL.5.Mod5.AD9 FL.5.Mod5.AD12 FL.5.Mod5.AD13

Module 5 Module Assessment

MA.5.FR.2.2 MA.5.AR.1.2 MA.5.GR.1.1 MA.5.GR.2.1 MA.5.GR.3.1 MA.5.GR.3.3

FL.5.Mod5.AD1 FL.5.Mod5.AD3 FL.5.Mod5.AD4 FL.5.Mod5.AD5 FL.5.Mod5.AD7 FL.5.Mod5.AD8 FL.5.Mod5.AD9 FL.5.Mod5.AD10 FL.5.Mod5.AD11 FL.5.Mod5.AD13

588

EUREKA MATH2 Florida 5 ▸ M5

Achievement Descriptors: Overview Addition and Multiplication with Area and Volume Achievement Descriptors (ADs) are standards-aligned descriptions that detail what students should know and be able to do based on the instruction. ADs are written by using portions of various standards to form a clear, concise description of the work covered in each module. Each module has its own set of ADs, and the number of ADs varies by module. Taken together, the sets of module-level ADs describe what students should accomplish by the end of the year. ADs and their proficiency indicators support teachers with interpreting student work on • informal classroom observations, • data from other lesson-embedded formative assessments, • Exit Tickets, • Topic Quizzes, and • Module Assessments. This module contains the 13 ADs listed.

FL.5.Mod5.AD1

FL.5.Mod5.AD2

FL.5.Mod5.AD3

FL.5.Mod5.AD4

Multiply mixed numbers and whole numbers, fractions, or mixed numbers.

Model products of fractions and mixed numbers by using rectangular areas.

Solve real-world problems involving multiplication of mixed numbers.

Classify triangles into categories based on attributes and explain the classification.

MA.5.AR.1.2

MA.5.GR.1.1

MA.5.FR.2.2

589

EUREKA MATH2 Florida

5 ▸ M5

FL.5.Mod5.AD5

FL.5.Mod5.AD6

FL.5.Mod5.AD7

FL.5.Mod5.AD8

Classify quadrilaterals into categories based on attributes and explain the classification.

Classify three-dimensional figures into categories based on attributes.

Find perimeters of rectangles with fraction or decimal side lengths.

Find and model the area of rectangles with fraction side lengths by tiling.

MA.5.GR.1.1

MA.5.GR.1.2

MA.5.GR.2.1

FL.5.Mod5.AD9

FL.5.Mod5.AD10

FL.5.Mod5.AD11

FL.5.Mod5.AD12

Find areas of rectangles with fraction or decimal side lengths.

Recognize that volume can be measured by using unit cubes and that a solid packed without gaps or overlaps by n unit cubes has a volume of n cubic units.

Measure volumes by counting unit cubes.

Find volumes of right rectangular prisms with whole-number edge lengths by using a formula.

MA.5.GR.2.1

MA.5.GR.3.1

MA.5.GR.3.2

FL.5.Mod5.AD13 Solve real-world problems involving volumes of right rectangular prisms.

MA.5.GR.3.3

590

EUREKA MATH2 Florida 5 ▸ M5

The first page of each lesson identifies the ADs aligned with that lesson. Each AD may have up to three indicators, each aligned to a proficiency category (i.e., Partially Proficient, Proficient, Highly Proficient). While every AD has an indicator to describe Proficient performance, only select ADs have an indicator for Partially Proficient and/or Highly Proficient performance. An example of one of these ADs, along with its proficiency indicators, is shown here for reference. The complete set of this module’s ADs with proficiency indicators can be found in the Achievement Descriptors: Proficiency Indicators resource. ADs have the following parts: • AD Code: The code indicates the grade level and the module number and then lists the ADs in no particular order. For example, the first AD for grade 5 module 1 is coded as FL.5.Mod1.AD1. • AD Language: The language is crafted from standards and concisely describes what will be assessed. • AD Indicators: The indicators describe the precise expectations of the AD for the given proficiency category. • Related Standard: This identifies the standard or parts of standards from the Florida Benchmarks for Excellent Student Thinking (B.E.S.T.) that the AD addresses. AD Language

AD Code: FL.Grade.Mod#.AD#

5 ▸ M5

EUREKA MATH2 Florida

FL.5.Mod5.AD13 Solve real-world problems involving volumes of right rectangular prisms.

Related Standard

RELATED B .E .S .T .

MA.5.GR.3.3 Solve real-world problems involving the volume of right rectangular prisms, including problems with an unknown edge length, with whole-number edge lengths using a visual model or a formula. Write an equation with a variable for the unknown to represent the problem.

Partially Proficient Solve real-world problems involving finding the volume of a right-rectangular prism with whole-number side lengths. Jada has a block of clay shaped like a right rectangular prism. It has a length of 6 inches, a width of 8 inches, and a height of 14 inches. What is the volume of Jada’s block of clay?

Proficient Solve real-world problems involving the volume of right rectangular prisms, including problems with an unknown edge length, with whole-number edge lengths by using a visual model or a formula, and write an equation with a variable for the unknown to represent the problem. Jada needs 192 cubic inches of clay for a project. She cuts a block of clay shaped like a right rectangular prism from a larger block of clay. The block Jada cuts out has a height of 6 inches and a width of 8 inches.

Highly Proficient Solve real-world problems involving volumes of multiple right rectangular prisms with whole-number edge lengths. A box measures 12 inches by 8 inches by 3 inches. Another box has a base measuring 6 inches by 6 inches. How many inches tall must the second box be to have the same volume as the first box?

AD Indicators

Part A Write an equation with a letter representing the unknown that can be used to find the length of the block that Jada cuts out. Part B What is the length of the block Jada cuts out?

591

EUREKA MATH2 Florida

5 ▸ M5

Achievement Descriptors: Proficiency Indicators FL.5.Mod5.AD1 Multiply mixed numbers and whole numbers, fractions, or mixed numbers. RELATED B.E.S.T.

MA.5.FR.2.2 Extend previous understanding of multiplication to multiply a fraction by a fraction, including mixed numbers and fractions greater than 1, with procedural reliability.

Partially Proficient Multiply a mixed number and a fraction or a whole number. Multiply.

2 _37 × __ 95

592

Proficient

Highly Proficient

Multiply two mixed numbers. Multiply. 7 4 __ × 3 __13 12

EUREKA MATH2 Florida 5 ▸ M5

FL.5.Mod5.AD2 Model products of fractions and mixed numbers by using rectangular areas. RELATED B.E.S.T.

MA.5.FR.2.2 Extend previous understanding of multiplication to multiply a fraction by a fraction, including mixed numbers and fractions greater than 1, with procedural reliability.

Partially Proficient

Proficient

Match area models with expressions for multiplication of fractions or mixed numbers and whole numbers.

Create an area model for multiplication of fractions or mixed numbers and whole numbers or for two mixed numbers.

Consider the area model shown.

Draw an area model to represent the product shown.

4 5

Highly Proficient

2 __61 × 4 __85

2 Select all expressions the area model represents. 4

A. 2 __ 5 × 3 4

B. 2 × 3 __ 5 4

C. 3 × 2 __ 5 4

D. 3 __ 5 × 2

593

EUREKA MATH2 Florida

5 ▸ M5

FL.5.Mod5.AD3 Solve real-world problems involving multiplication of mixed numbers. RELATED B.E.S.T.

MA.5.AR.1.2 Solve real-world problems involving the addition, subtraction or multiplication of fractions, including mixed numbers and fractions greater than 1.

Partially Proficient

Proficient

Identify a numerical expression that can be used to solve a real-world problem involving multiplication of mixed numbers. 2

Lisa has a rectangular garden. Her garden is 10 __ 3 feet 1

wide and 16 __ 2 feet long. How many square feet is Lisa’s garden?

Highly Proficient

Solve one-step real-world problems involving multiplication of mixed numbers. 2

Lisa has a rectangular garden. Her garden is 10 __ 3 feet 1

wide and 16 __ 2 feet long. How many square feet is Lisa’s garden?

Which expression can be used to solve the problem? 1

__ A. 16 __ 2 + 10 3 __ B. 16 __ 2 − 10 3 __ C. 16 __ 2 × 10 3 __ D. 16 __ 2 ÷ 10 3

594

EUREKA MATH2 Florida 5 ▸ M5

FL.5.Mod5.AD4 Classify triangles into categories based on attributes and explain the classification. RELATED B.E.S.T.

MA.5.GR.1.1 Classify triangles or quadrilaterals into different categories based on shared defining attributes. Explain why a triangle or quadrilateral would or would not belong to a category.

Partially Proficient

Proficient

Classify triangles into categories based on attributes.

Classify triangles into categories based on attributes and explain the classifications.

Consider the triangle shown.

Which names correctly describe the figure? Select all that apply.

Part A

Highly Proficient

Classify the triangle by its side lengths.

A. Acute B. Equilateral C. Isosceles

Part B Classify the triangle by its angle measures.

D. Obtuse

Part C

E. Right

Explain your answers for parts A and B.

F. Scalene

595

EUREKA MATH2 Florida

5 ▸ M5

FL.5.Mod5.AD5 Classify quadrilaterals into categories based on attributes and explain the classification. RELATED B.E.S.T.

MA.5.GR.1.1 Classify triangles or quadrilaterals into different categories based on shared defining attributes. Explain why a triangle or quadrilateral would or would not belong to a category.

Partially Proficient

Proficient

Classify quadrilaterals into categories based on attributes.

Classify quadrilaterals into categories based on attributes and explain the classifications.

Consider the figure shown.

Which names correctly describe the figure? Select all that apply. A. Parallelogram

Part A Classify the figure.

B. Quadrilateral

Part B

C. Rectangle

Classify the figure again in a different way.

D. Rhombus E. Square

Highly Proficient

Part C Explain your answers for parts A and B.

F. Trapezoid

596

EUREKA MATH2 Florida 5 ▸ M5

FL.5.Mod5.AD6 Classify three-dimensional figures into categories based on attributes. RELATED B.E.S.T.

MA.5.GR.1.2 Identify and classify three-dimensional figures into categories based on their defining attributes. Figures are limited to right pyramids, right prisms, right circular cylinders, right circular cones and spheres.

Partially Proficient

Proficient

Highly Proficient

Identify three-dimensional figures from a given image.

Classify three-dimensional figures into categories based on attributes.

Explain classifications of three-dimensional figures into categories based on attributes.

Consider the three-dimensional figure shown.

Select all three-dimensional figures that have a curved surface.

Consider the three-dimensional figure shown.

A. Cone B. Cylinder C. Right prism D. Right pyramid E. Sphere

Select the correct name for the three-dimensional figure.

Classify the three-dimensional figure. Explain how you know.

A. Cone B. Cylinder C. Right prism D. Right pyramid

597

EUREKA MATH2 Florida

5 ▸ M5

FL.5.Mod5.AD7 Find perimeters of rectangles with fraction or decimal side lengths. RELATED B.E.S.T.

MA.5.GR.2.1 Find the perimeter and area of a rectangle with fractional or decimal side lengths using visual models and formulas.

Partially Proficient

Proficient

Find perimeters of rectangles with fraction or decimal side lengths given a model.

Find perimeters of rectangles with fraction or decimal side lengths.

Find the perimeter.

A rectangle has a length of 3.3 feet and a width of 1.6 feet. Find the perimeter.

Highly Proficient

1 ft 3

2 ft 3

598

EUREKA MATH2 Florida 5 ▸ M5

FL.5.Mod5.AD8 Find and model the area of rectangles with fraction side lengths by tiling. RELATED B.E.S.T.

MA.5.GR.2.1 Find the perimeter and area of a rectangle with fractional or decimal side lengths using visual models and formulas.

Partially Proficient

Proficient

Find the area of a rectangle with fraction side lengths by tiling it by using a given model.

Find and model the area of a rectangle with fraction side lengths by tiling.

Use the model shown to find the area of the shaded

Find the area of a rectangle with side lengths of

5 1 rectangle with side lengths of __ 4 unit and __ 8 units.

5 1 __ 4 unit and __ 8 units. Sketch to show how you know.

Highly Proficient

599

EUREKA MATH2 Florida

5 ▸ M5

FL.5.Mod5.AD9 Find areas of rectangles with fraction or decimal side lengths. RELATED B.E.S.T.

MA.5.GR.2.1 Find the perimeter and area of a rectangle with fractional or decimal side lengths using visual models and formulas.

Partially Proficient

Proficient

Find areas of rectangles with fraction or decimal side lengths given a model.

Highly Proficient

Find areas of rectangles with fraction or decimal side lengths. A rectangle has a length of 3.4 meters and a width of

Find the area.

1.6 meters. Find the area.

1 ft 3

2 ft 3

600

EUREKA MATH2 Florida 5 ▸ M5

FL.5.Mod5.AD10 Recognize that volume can be measured by using unit cubes and that a solid packed without gaps

or overlaps by n unit cubes has a volume of n cubic units. RELATED B.E.S.T.

MA.5.GR.3.1 Explore volume as an attribute of three-dimensional figures by packing them with unit cubes without gaps. Find the volume of a right rectangular prism with whole-number side lengths by counting unit cubes.

Partially Proficient Identify volume as the most relevant attribute in a given scenario. Lacy mails a box to a friend and needs to know how much space the box takes up. What should Lacy calculate?

Proficient

Highly Proficient

Recognize that volume can be measured by using unit cubes and that a solid packed without gaps or overlaps by n unit cubes has a volume of n cubic units. Consider the right rectangular prism shown.

A. Area B. Length

Each

represents 1 cubic unit.

C. Perimeter D. Volume

Lacy says that the right rectangular prism shown has a volume of 10 cubic units. Is Lacy correct? Explain how you know.

601

EUREKA MATH2 Florida

5 ▸ M5

FL.5.Mod5.AD11 Measure volumes by counting unit cubes. RELATED B.E.S.T.

Partially Proficient

Proficient Measure volumes by counting unit cubes in right rectangular prisms.

Measure volumes in right rectangular prisms that are partially packed with unit cubes.

Determine the volume of the right rectangular prism shown.

Each

602

Highly Proficient

represents 1 cubic inch.

Each

represents 1 cubic inch.

EUREKA MATH2 Florida 5 ▸ M5

FL.5.Mod5.AD12 Find volumes of right rectangular prisms with whole-number edge lengths by using a formula. RELATED B.E.S.T.

MA.5.GR.3.2 Find the volume of a right rectangular prism with whole-number side lengths using a visual model and a formula.

Partially Proficient

Proficient

Identify expressions that can be used to determine the volume of a right rectangular prism with whole-number side lengths. Which expressions can be used to find the volume of the right rectangular prism shown?

Highly Proficient

Find volumes of right rectangular prisms with whole-number edge lengths by using a formula. Find the volume of the right rectangular prism shown.

Drag unit cubes into the larger prism to answer part A and part C.

1 cm 5 cm

3 cm

5 cm

Explain that finding the volume of a right rectangular prism with whole-number edge lengths by packing it with unit cubes yields the same result as multiplying edge lengths or multiplying the area of the base by the height.

2 cm 1 cm 1 cm 1 cm

2 cm

Select all that apply.

Part A

A. 2 × 3 × 5

Enter the value of each measurement for the larger prism.

B. 2 × 8 C. 3 × 5 × 2 D. 5 × 2 E. 5 × 5

Length: cm Width: cm Height: cm Part B Multiply your numbers from part A. Part C How many total unit cubes does it take to pack the larger prism? Part D Explain how the results of part B and part C relate.

603

EUREKA MATH2 Florida

5 ▸ M5

FL.5.Mod5.AD13 Solve real-world problems involving volumes of right rectangular prisms. RELATED B.E.S.T.

Part A Write an equation with a letter representing the unknown that can be used to find the length of the block that Jada cuts out. Part B What is the length of the block Jada cuts out?

604

EUREKA MATH2 Florida 5 ▸ M5

Module Assessment Sample Solutions The digital platform provides access to digital quizzes and assessments as well as three downloadable print versions. Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

A N SW E R K E Y ▸ M O DU LE AS S E S S M E N T 1 Quadrilateral Trapezoid

1. Circle all the names for each figure.

Parallelogram Rhombus Rectangle

Quadrilateral

Square

Trapezoid Parallelogram Rhombus

Quadrilateral

Rectangle

Trapezoid

Square

Parallelogram Rhombus Rectangle Quadrilateral

Square

Trapezoid Parallelogram Rhombus Rectangle Square

261

262

ANSWER KEY ▸ MODULE ASSESSMENT 1

605

EUREKA MATH2 Florida

5 ▸ M5

2. Multiply. 8

2 _9 × 1_2 =

3. Circle all the names for each figure.

1 _4 9

Acute triangle Equilateral triangle Isosceles triangle

1 _8 × 4_5 =

Obtuse triangle

1_ 10

Right triangle Scalene triangle

Acute triangle 3

5 _7 × 1 _7 =

Equilateral triangle

7_ 49

Isosceles triangle Obtuse triangle Right triangle Scalene triangle

5 2_5 × 2 1_6 =

Acute triangle

7 11 _ 10

Equilateral triangle Isosceles triangle Obtuse triangle Right triangle Scalene triangle

606

ANSWER KEY ▸ MODULE ASSESSMENT 1

263

264

ANSWER KEY ▸ MODULE ASSESSMENT 1

EUREKA MATH2 Florida 5 ▸ M5

1 1 4. Adesh measures a wall to find its area. The wall measures 9 _ feet tall and 12 _ feet wide. 3 4 Calculate the area of the wall in square feet.

The area of the wall is

1 114 _ 12

6. Consider the rectangle.

square feet. 3 meters 5

5. Riley uses 24 blocks to build a right rectangular prism. Each block represents 1 cubic unit. Part A Create a right rectangular prism with 24 blocks. Sample:

3 meters 4

Part A What size square tile can be used to tile the rectangle without any gaps or overlaps? Choose the side length of the square tile.

_1 meter 4 1 B. _5 meter 1 C. _ meter 9 A.

Part B

1 D. _ meter 20

Write an equation that can be used to find the volume of your prism. Use a letter to represent the unknown. Scoring Notes for Item 5 Part B Points Description 1 Student response includes the following component: •

Distractor rationale: A. Incorrect. The student may have used a unit fraction with the same 3_ denominator as . 4

B. Incorrect. The student may have used a unit fraction with the same 3_ denominator as . 5 C. Incorrect. The student may have used a unit fraction with a denominator equal to the sum of the given denominators. D. Correct.

Part B

Computation: 1 point The student provides an accurate equation involving a variable that can be used to find the volume.

What is the area of the given rectangle?

𝑉=2×3×4 Sample:

Student response is missing or incorrect. The area of the rectangle is

ANSWER KEY ▸ MODULE ASSESSMENT 1

265

266

9 _

square meters.

ANSWER KEY ▸ MODULE ASSESSMENT 1

607

EUREKA MATH2 Florida

5 ▸ M5

3 1 7. One batch of banana muffins uses 2 _ cups of flour. One batch of cinnamon muffins uses 1 _ cups 2 4 of flour. Kayla wants to make 3 batches of banana muffins and 4 batches of cinnamon muffins.

9. Eddie builds box B to have the same volume as box A. Both boxes are right rectangular prisms.

Box A

Box B

Which expression can be used to find the total cups of flour that Kayla needs?

3 1 A. 2 _ + 1 _ + 3 + 4 2 4 3 1 _ B. 2 × 3 + 1 _ × 4 2 4 3 1 C. 2 _ × 4 + 1 _ × 3 2 4 3 1 D. 2 _ × 1 _ + 3 × 4 2 4

Distractor rationale: A. Incorrect. The student may have added the four numbers. B. Correct.

6 cm

C. Incorrect. The student may have reversed the number of batches of each type of muffin.

12 cm

D. Incorrect. The student may have multiplied the amounts of flour for each type of muffin and the numbers of batches for each type of muffin and then added.

12 cm

8 cm

9 cm

What is the height of box B?

8. Determine the perimeter and area of the rectangle shown. Write one number or unit from the given answer choices in each blank.

3 1 inches 4

The perimeter of the rectangle is The area of the rectangle is

5 1_4

608

6 1_8

10 1_2

1 6_ 8

1 3 inches 2

inches

Answer Choices

10 1_2

square inches

12 1_4

. The height of box B is

ANSWER KEY ▸ MODULE ASSESSMENT 1

inches

cm.

square inches

267

268

ANSWER KEY ▸ MODULE ASSESSMENT 1

EUREKA MATH2 Florida 5 ▸ M5

10. Consider the right rectangular prism shown. Each cube represents 1 cubic unit.

Part A What is the volume of the prism? The volume of the prism is

cubic units.

Part B Explain how you determined the volume of the prism in part A. Scoring Notes for Item 10 Part B Points Description 1 Student response includes the following component: •

Reasoning: 1 point The student provides a valid explanation for determining the volume.

Sample: The front layer of the prism has 12 cubic units, and there are 7 layers. The volume is 84 cubic units because 12 × 7 = 84. Student response is missing or incorrect.

ANSWER KEY ▸ MODULE ASSESSMENT 1

269

609

EUREKA MATH2 Florida

5 ▸ M5

Terminology The following terms are critical to the work of grade 5 module 5. This resource groups terms into categories called New, Familiar, and Academic Verbs. The lessons in this module incorporate terminology with the expectation that students work toward applying it during discussions and in writing. Items in the New category are discipline-specific words that are introduced to students in this module. These items include the definition, description, or illustration as it is presented to students. At times, this resource also includes italicized language for teachers that expands on the wording used with students. Items in the Familiar category are discipline-specific words introduced in prior modules or in previous grade levels. Items in the Academic Verbs category are high-utility terms that are used across disciplines. These terms come from a list of academic verbs that the curriculum strategically introduces at this grade level.

New acute triangle An acute triangle is a triangle that contains all acute angles. (Lesson 3) base The base of a prism is one of the faces of the prism, often thought of as the surface on which the prism rests. (Lesson 16) composite figure A composite figure is a geometric figure composed of two or more smaller figures. (Lesson 14)

610

cone A cone is a solid with a flat, circular surface, called the base, and a curved surface that connects the base to a point called the apex. (Lesson 2) cubic centimeter Cubic centimeters are a unit for measuring volume. 1 cubic centimeter of volume is defined as the volume of a cube with side lengths of 1 centimeter. (Lesson 17) cubic inch Cubic inches are a unit for measuring volume. 1 cubic inch of volume is defined as the volume of a cube with side lengths of 1 inch. (Lesson 17) cubic unit Cubic units are a unit for measuring volume. 1 cubic unit of volume is defined as the volume of a cube with side lengths of 1 unit. (Lesson 17) cylinder A cylinder is a solid with two flat, parallel, circular surfaces that are identical, called bases, and a curved surface connecting the two bases. (Lesson 2) equilateral triangle An equilateral triangle is a triangle with all sides of equal length. (Lesson 3) face A face is a polygon that is a flat part of the surface of a solid. (Lesson 2) © Great Minds PBC

EUREKA MATH2 Florida 5 ▸ M5

isosceles triangle

right rectangular prism

An isosceles triangle is a triangle with at least two sides of equal length. Note that an equilateral triangle is also an isosceles triangle. (Lesson 3)

A right rectangular prism is a solid where all the faces are rectangles. (Lesson 16)

kite

right triangle A right triangle is a triangle that contains a right angle. (Lesson 3)

A kite is a quadrilateral with at least two pairs of adjacent sides that are the same length. (Lesson 6)

scalene triangle

midpoint

A scalene triangle is a triangle with no sides of equal length. (Lesson 3)

The midpoint of a line segment is the point on the line segment that divides it into two line segments of equal length. (Lesson 4)

unit cube

obtuse triangle

A unit cube is a cube with edges that each measure 1 unit. (Lesson 17)

An obtuse triangle is a triangle that contains an obtuse angle. (Lesson 3)

volume

plane

The volume of a solid is the amount of three-dimensional space it takes up. (Lesson 18)

A plane is a flat, two-dimensional surface that extends forever. (Lesson 1) property A property is a characteristic that is true of every item in a category. (Lesson 1) right prism A right prism is a solid with two identical polygonal faces, called bases, connected by rectangular faces. (Lesson 2)

Familiar acute angle attribute capacity cube diagonal

right pyramid

edge

A right pyramid is a solid with a polygonal face, called the base, connected to triangular faces that meet at a vertex, called the apex. (Lesson 2)

figure

formula height

611

EUREKA MATH2 Florida

5 ▸ M5

intersect

square

line of symmetry

square unit

liquid volume

straight angle

obtuse angle

three-dimensional figure

parallel

trapezoid

parallelogram

two-dimensional figure

perpendicular

unit square

quadrilateral

vertex

rectangle rhombus right angle

612

Academic Verbs Module 5 does not introduce any academic verbs from the grade 5 list.

EUREKA MATH2 Florida 5 ▸ M5

Math Past The Hidden Structure of Solids What is a Platonic solid? How can we look at familiar objects in new ways? What is an example of “old mathematics” being seen in a new way? At the beginning of this module, students spend time classifying different quadrilaterals by focusing on specific measurements, such as the measures of the angles at their vertices and the measures of their side lengths. Students also study the measurement of area and volume—a property of three-dimensional shapes, or solids, that is tied to measurement. Ask your students for real-world examples of solids. Can your students think of any examples of solids from the ancient world? If no one mentions the pyramids of ancient Egypt, bring them up. Humans have been studying solids for millennia. The ancient Greeks were the first to write extensively about the study of solids. One particular group of solids they studied in detail is the polyhedrons. A polyhedron is a solid made entirely out of flat, polygonal faces. Many of the solids students have seen in earlier grades, such as cubes, square pyramids, and different kinds of prisms, are polyhedrons. Solids with rounded parts, such as spheres and cylinders, are not. Because students have studied polygons before, they might notice the similarity between the two words polyhedron and polygon. Both start with the prefix poly-, which comes from © Great Minds PBC

A statue of Plato in Athens, Greece

the Greek word for “many.” The suffix -gon comes from the Greek word for “corner,” while -hedron comes from the Greek word for “seat.” So, a polyhedron is a solid that can be “seated” on any of its many faces. Of particular interest to the ancient Greeks were the regular polyhedrons. Today, they are often called Platonic solids, after the Greek philosopher Plato, who lived around the 4th century BCE. Platonic solids are the polyhedrons that have faces that are all identical regular polygons. For example, a cube, which has faces that are all equal-size squares, is a Platonic solid. Here are the five Platonic solids.

Tetrahedron Octahedron

Cube

4 equilateral 8 equilateral 6 squares triangles

triangles

Dodecahedron Icosahedron

12 regular pentagons

20 equilateral triangles

Show your students these solids and ask what they notice and wonder. They might notice that the solids are three-dimensional and that they are made of triangles, squares, and pentagons. Students might also notice that the Platonic solids have long, interesting-sounding names. Consider taking time to explore these names together as a class. Students should recognize the prefix octa- from their experience with octagons. Ask them why they 613

EUREKA MATH2 Florida

5 ▸ M5

think octagon and octahedron start the same way. Both derive from the Greek word for “eight.” The octagon has 8 sides, and the Platonic octahedron is made of 8 equilateral triangles. The other solids’ names also have prefixes that correspond to Greek words for numbers: tetra- for four, dodeca- for 12, and icosa- for 20. Ask students why they think these numbers belong with each solid. The only name that does not follow this pattern is cube, but cubes have another name in Greek. Ask your students whether they can guess what that name is. Ask them to recall that a six-sided polygon is called a hexagon. Reveal that a cube can also be called a hexahedron. These names shift our attention from measuring to counting. This shift can highlight different kinds of geometric structure. For example, a polygon always has the same number of vertices and sides. Ask your students why they think this is the case. Ask your students to recall the terminology used to talk about the characteristics of solids: vertices, edges, and faces. Students can count the vertices, edges, and faces of solids. They won’t find that the number of vertices and edges are equal, like with polygons, but there is a hidden pattern to discover for polyhedrons. A cube has 8 vertices, 6 faces, and 12 edges. Count these parts together as a class to confirm. (Use a cubical tissue box in place of a cube so that everyone can see, if needed.) Add the number of vertices and faces: 8 + 6 = 14. Then subtract the number of edges: 14 − 12 = 2.

8 vertices

614

6 faces

12 edges

Ask your students what numbers they think they will get if they try this addition and subtraction with other solids. You can choose from any of the Platonic solids or other common polyhedrons such as a square pyramid, a triangular prism, or a hexagonal prism. Try this with as many of these as you would like. Ask students what they notice. They should find that the answer is always 2! We can perform this calculation for any polyhedron. The number that we get is called the Euler number of that solid. The Euler number is named after the Swiss mathematician Leonhard Euler (1707–1783), A portrait of Euler on a pronounced “oiler,” who Swiss 10-franc note noticed this relationship and studied it extensively. Euler appears to have been the first person to consider that this way of looking at solids could lead to important new insights. The question to ask, then, is whether the Euler number of a polyhedron gives us any new information. Unlike area or volume, the Euler number does not describe anything about how much space the figure takes up, and the Euler number does not change with size. For example, even though the volume of a cube that has edges that are each 3 cm long and the volume of a cube that has edges that are each 10 cm long are very different (27 cubic centimeters vs. 1,000 cubic centimeters), they both have the same Euler number, 2. Inspired by Euler’s work, mathematicians began studying what they then called geometria situs (geometry of position), the branch of mathematics that explores characteristics of geometric objects that do not change with the objects’ size. Today, this branch of mathematics is called topology, not to be confused with the more familiar word topography, which has the same Greek root © Great Minds PBC

EUREKA MATH2 Florida 5 ▸ M5

word, topos, meaning “place.” Topology is one of the richest and most active areas of mathematical study today. Topologists have helped answer questions about biology, physics, computer science, and more. Are there any polyhedrons that have an Euler number other than 2? Imagine you have a sphere made out of clay. You could make this clay sphere look like any of the solids we have been discussing by flattening the sides. What we learn from topology is that if you can make a solid from a sphere in this way, then the Euler number of this solid will be 2. Show your students this donut-like solid and ask them how it is different from the other solids you have been discussing. They might notice that it has a hole in it. To make this shape from the clay sphere, you

need to tear a hole in the clay. Challenge students to find the Euler number of this solid. They should find that the Euler number of this solid is 0. If we need to tear a hole in the clay sphere to make a solid, then the Euler number of that shape can be a number other than 2! The work of Euler—and the mathematicians who were inspired by his work—shows that in mathematics, objects can always be looked at in new ways. Even if that object, like a Platonic solid, has been studied by mathematicians for thousands of years, a new perspective can bring about new questions and new problems to solve.

615

EUREKA MATH2 Florida

5 ▸ M5

Materials The following materials are needed to implement this module. The suggested quantities are based on a class of 24 students and one teacher. 25

4ʺ protractor (180°)

Patty paper, 4ʺ x 4ʺ, 500 sheets

130

Blank paper, sheets

Pencils

Centimeter cubes, set of 1,000

Personal whiteboards

Clear tape, roll

Personal whiteboard erasers

Dry-erase markers

Projection device

104

Envelopes

Rulers, plastic

Eureka Math2® Rectangular prisms set

Scissors

Glue sticks

Sticky notes

Highlighters

Teach book

135

Index cards

Teacher computer or device

960

Interlocking cubes, 1 cm

Toothpicks

Learn books

Please see lesson 10 for a list of organizational tools (cups, whiteboards, bags, etc.) suggested for the counting collection.

616

EUREKA MATH2 Florida 5 ▸ M5

Works Cited Boaler, Jo, Jen Munsen, and Cathy Williams. Mindset Mathematics: Visualizing and Investigating Big Ideas: Grade 3. San Francisco, CA: Jossey-Bass, 2018. Carpenter, Thomas P., Megan L. Franke, and Linda Levi. Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Portsmouth, NH: Heinemann, 2003. Carpenter, Thomas P., Megan L. Franke, Nicholas C. Johnson, Angela C. Turrou, and Anita A. Wager. Young Children’s Mathematics: Cognitively Guided Instruction in Early Childhood Education. Portsmouth, NH: Heinemann, 2017. CAST. Universal Design for Learning Guidelines version 2.2. Retrieved from http://udlguidelines.cast.org, 2018. (all) Clements, Douglas H. and Julie Sarama. Learning and Teaching Early Math: The Learning Trajectories Approach. New York: Routledge, 2014. Cooke, Roger L. The History of Mathematics: A Brief Course. Hoboken, NJ: Wiley, 2013. Danielson, Christopher. Which One Doesn’t Belong?: A Teacher’s Guide. Portland, ME: Stenhouse, 2016. Danielson, Christopher. Which One Doesn’t Belong?: Playing with Shapes. Watertown, MA: Charlesbridge, 2019. Empson, Susan B. and Linda Levi. Extending Children’s Mathematics: Fractions and Decimals. Portsmouth, NH: Heinemann, 2011. Florida Department of Education. Florida’s B.E.S.T. Standards: Mathematics. Tallahassee, FL: Florida Department of Education, 2019. © Great Minds PBC

Flynn, Mike. Beyond Answers: Exploring Mathematical Practices with Young Children. Portsmouth, NH: Stenhouse, 2017. Fosnot, Catherine Twomey, and Maarten Dolk. Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Portsmouth, NH: Heinemann, 2001. Franke, Megan L., Elham Kazemi, and Angela Chan Turrou. Choral Counting and Counting Collections: Transforming the PreK–5 Math Classroom. Portsmouth, NH: Stenhouse, 2018. Hattie, John, Douglas Fisher, and Nancy Frey. Visible Learning for Mathematics: What Works Best to Optimize Student Learning. Thousand Oaks, CA: Corwin Mathematics, 2017. Hemdal, Jay. “How Much Swimming Space Do Your Fish Need?” Practical Fishkeeping. July 2009. https://www .practicalfishkeeping.co.uk/features/how-much-swimming -space-do-your-fish-need/. Huinker, DeAnn and Victoria Bill. Taking Action: Implementing Effective Mathematics Teaching Practices. Kindergarten– Grade 5, edited by Margaret Smith. Reston, VA: National Council of Teachers of Mathematics, 2017. Kelemanik, Grace, Amy Lucenta, Susan Janssen Creighton, and Magdalene Lampert. Routines for Reasoning: Fostering the Mathematical Practices in All Students. Portsmouth, NH: Heinemann, 2016. Ma, Liping. Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. New York: Routledge, 2010.

617

5 ▸ M5

Malkevitch, Joe. “Euler’s Polyhedral Formula.” Feature Column. American Mathematical Society. Published December 2004. Accessed January 21, 2021. http://www.ams.org /publicoutreach/feature-column/fcarc-eulers-formula. National Council for Teachers of Mathematics, Developing an Essential Understanding of Multiplication and Division for Teaching Mathematics in Grades 3–5. Reston, VA: National Council for Teachers of Mathematics, 2011. Parker, Thomas and Scott Baldridge. Elementary Mathematics for Teachers. Okemos, MI: Sefton-Ash, 2004. “Plato’s Timaeus.” Stanford Encyclopedia of Philosophy. Updated December 18, 2017. Accessed January 21, 2021. https://plato.stanford.edu/entries/plato-timaeus/. Shumway, Jessica F. Number Sense Routines: Building Mathematical Understanding Every Day in Grades 3–5. Portland, ME: Stenhouse Publishing, 2018. Smith, Margaret S. and Mary K. Stein. 5 Practices for Orchestrating Productive Mathematics Discussions, 2nd ed. Reston, VA: National Council of Teachers of Mathematics, 2018.

618

EUREKA MATH2 Florida

Smith, Margaret S., Victoria Bill, and Miriam Gamoran Sherin. The 5 Practices in Practice: Successfully Orchestrating Mathematics Discussions in Your Elementary Classroom, 2nd ed. Thousand Oaks, CA: Corwin Mathematics; Reston, VA: National Council of Teachers of Mathematics, 2020. Suzuki, Jeff. Mathematics in Historical Context. Washington, DC: Mathematical Association of America, 2012. Van de Walle, John A. Elementary and Middle School Mathematics: Teaching Developmentally. New York: Pearson, 2004. Van de Walle, John A., Karen S. Karp, LouAnn H. Lovin, and Jennifer M. Bay-Williams. Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3–5, 3rd ed. New York: Pearson, 2018. Zwiers, Jeff, Jack Dieckmann, Sara Rutherford-Quach, Vinci Daro, Renae Skarin, Steven Weiss, and James Malamut. Principles for the Design of Mathematics Curricula: Promoting Language and Content Development. Retrieved from Stanford University, UL/SCALE website: https://ul.stanford. edu/resource/principles-design-mathematics-curriculaand-mlrs, 2017.

EUREKA MATH2 Florida 5 ▸ M5

Credits Great Minds® has made every effort to obtain permission for the reprinting of all copyrighted material. If any owner of copyrighted material is not acknowledged herein, please contact Great Minds for proper acknowledgment in all future editions and reprints of this module. For a complete list of credits, visit http://eurmath.link/media-credits. Cover, Adam Jones/Getty Images; page 48, (left) Mikael Damkier/ Shutterstock.com, (right) dvoevnore/Shutterstock.com, (bottom) Elnur Amikishiyev/Alamy Stock Photo; page 136, Vasilyev Alexandr/ Shutterstock.com; page 203, severija/Shutterstock.com; pages 345, 346, kovalto1/Shutterstock.com; page 355, Ase/ Shutterstock.com; page 386, (top) Ksenia_designer/Shutterstock .com, (bottom) HQ3DMOD/Shutterstock.com; page 433,

New Africa/Shutterstock.com; page 435, GO DESIGN/Shutterstock .com; page 451, Natalya Levish/Shutterstock.com; pages 462, 508, 509, 510, 511, Rvector/Shutterstock.com; page 529, ANI ROFIQAH/Shutterstock.com; page 546, Flas100/Shutterstock .com; pages 552, 557, (composite) Progdiz/Shutterstock.com, GraphicsRF.com/Shutterstock.com; pages 553, 558, (top) WichitS/ Shutterstock.com, (bottom) BNP Design Studio/Shutterstock.com; page 613, (left) Richard Panasevich/Shutterstock.com, (right) kovalto1/Shutterstock.com; page 614, Prachaya Roekdeethaweesab/Shutterstock.com; page 615, Public Domain via Wikimedia Commons; All other images are the property of Great Minds.

619

5 ▸ M5

EUREKA MATH2 Florida

Acknowledgments Kelly Alsup, Leslie S. Arceneaux, Lisa Babcock, Parisa Badizadegan, Adam Baker, Agnes P. Bannigan, Christine Bell, Reshma P Bell, Melissa Benner, Joseph T. Brennan, Dawn Burns, Amanda H. Carter, David Choukalas, Mary Christensen-Cooper, Julia Dantchev, Cheri DeBusk, Lauren DelFavero, Jill Diniz, Mary Drayer, Karen Eckberg, Melissa Elias, Danielle A Esposito, Janice Fan, Scott Farrar, Gail Fiddyment, Ryan Galloway, Anita Geevarghese, Krysta Gibbs, Danielle Goedel, January Gordon, Lisa Gradney, Torrie K. Guzzetta, Kimberly Hager, Jodi Hale, Karen Hall, Eddie Hampton, Andrea Hart, Stefanie Hassan, Tiffany Hill, Christine Hopkinson, Rachel Hylton, Travis Jones, Laura Khalil, Raena King, Jennifer Koepp Neeley, Emily Koesters, Liz Krisher, Chelsea LaFont, Leticia Lemus, Marie Libassi-Behr, Courtney Lowe, Whitney Lyle, Sonia Mabry, Bobbe Maier, Ben McCarty, Maureen McNamara Jones, Pat Mohr, Bruce Myers, Marya Myers, Amy Ng, Tara O’Hare, Kati O’Neill, Darion Pack, Geoff Patterson, Victoria Peacock, Maximilian Peiler-Burrows, Brian Petras, April Picard, Marlene Pineda, DesLey V. Plaisance, Lora Podgorny, Jay Powers, Janae Pritchett, Elizabeth Re, Meri Robie-Craven, Deborah Schluben, Michael Short, Erika Silva, Jessica Sims,

620

Danielle Stantoznik, Heidi Strate, Theresa Streeter, James Tanton, Cathy Terwilliger, Stella Tessarollo, Rafael Vélez, Jessica Vialva, Rachael Waltke, Allison Witcraft, Jackie Wolford, Caroline Yang, Rachel Zanders, Jill Zintsmaster Trevor Barnes, Brianna Bemel, Lisa Buckley, Adam Cardais, Christina Cooper, Natasha Curtis, Jessica Dahl, Brandon Dawley, Delsena Draper, Sandy Engelman, Tamara Estrada, Soudea Forbes, Jen Forbus, Reba Frederics, Liz Gabbard, Diana Ghazzawi, Lisa Giddens-White, Laurie Gonsoulin, Nathan Hall, Cassie Hart, Marcela Hernandez, Rachel Hirsh, Abbi Hoerst, Libby Howard, Amy Kanjuka, Ashley Kelley, Lisa King, Sarah Kopec, Drew Krepp, Crystal Love, Maya Márquez, Siena Mazero, Cindy Medici, Ivonne Mercado, Sandra Mercado, Brian Methe, Patricia Mickelberry, Mary-Lise Nazaire, Corinne Newbegin, Max Oosterbaan, Tamara Otto, Christine Palmtag, Andy Peterson, Lizette Porras, Karen Rollhauser, Neela Roy, Gina Schenck, Amy Schoon, Aaron Shields, Leigh Sterten, Mary Sudul, Lisa Sweeney, Samuel Weyand, Dave White, Charmaine Whitman, Nicole Williams, Glenda Wisenburn-Burke, Howard Yaffe

ON THE COVER Orange Grove Orlando, Florida Adam Jones/Getty Images

Grade 5 Modules Module 1

EXPONENTIALLY GREATER

Place Value Concepts for Multiplication and Division with Whole Numbers

Eureka Math² Florida is crafted—by teachers, for teachers—to bring Florida’s B.E.S.T. Standards to life. Grounded in research and shaped by real classroom experience, this program ensures college and workforce readiness. It equips educators and students in classrooms across the state to explore math with confidence, clarity, and joy.

Module 2

ACCESS & ENGAGEMENT Developing conceptual understanding begins with

Module 4

Addition and Subtraction with Fractions

Module 3 Multiplication and Division with Fractions

students accessing and engaging in learning. That’s why Universal Design for Learning principles are at the heart of the program. Lessons engage students through a variety of facilitation styles and hands-on learning activities. Embedded support helps all learners become active participants in every lesson.

Module 5

CONCEPTUAL UNDERSTANDING & FLUENCY Daily lessons guide students

Module 6

in growing through the three-stage hierarchy of mathematical fluency. Each lesson includes concrete, pictorial, and symbolic representations, opportunities to identify math relationships and patterns through simple-to-complex progressions and meaningful discourse, and distributed and interleaved practice through fluency and practice problems.

APPLICATION Real-world contexts and collaborative tasks consolidate student thinking through why and how questions. Students are invited to draw conclusions and make deep connections among math ideas.

Place Value Concepts for Decimal Operations Addition and Multiplication with Area and Volume Number Patterns, the Coordinate Plane, and Data ISBN 979-8-89417-722-9

798894 177229